arXiv:cond-mat/0508729v1 [cond-mat.supr-con] 30 Aug 2005 g aibeculn schemes Variable-coupling qubits (g) SCT of coupling Phase qubits charge of Coupling oscillators via qubits Coupling charge (f) of qubits coupling JJ JJ single (d) of coupling qubits Capacitive flux (c) of coupling Inductive (b) qubits two principles General for (a) schemes coupling Physical IX. detection Threshold oscillator (e) coupled SET via with Measurement qubit (d) charge of Measurement (c) measurement qubit how? Direct and (b) when why, quan- Readout: (a) of measurement information and tum read-out Qubit VIII. (PCQ) qubits qubit Potential current (e) persistent - SQUID 3-junction rf-SQUID qubits Flux (SCT) (d) Transistor pair Cooper (SCB) Single Box pair Cooper Single qubits Charge qubit (b) (JJ) junction Josephson (a) qubits Basic circuits VII. superconducting Quantum Box VI. pair Cooper Single (d) dc-SQUID (c) rf-SQUID junction (b) Josephson biased Current circuits (a) superconducting Classical systems V. qubit of oscillation Decoherence Rabi (g) and perturbation Harmonic (e) switching precession Adiabatic and (d) switching sphere sudden Bloch dc-pulses, the (c) on evolution State (b) state two-level The systems two-level (a) of Dynamics preparation IV. state and Readout (d) gates and Operations (c) entanglement and processing Qubits information (b) quantum for computation Conditions quantum (a) of Basics qubits III. and computers Nanotechnology, II. Introduction I. odcigeetia icisfrqatmifrainpr information quantum for circuits electrical conducting hspprgvsa nrdcint h hsc n principl and physics the to introduction an gives paper This uecnutn unu icis uisadComputing and Qubits Circuits, Quantum Superconducting al fcontents of Table eateto irtcnlg n aocec MC2, - Nanoscience and Microtechnology of Department hlesUiest fTechnology, of University Chalmers .Wni n ..Shumeiko V.S. and Wendin G. E426Gtebr,Sweden Gothenburg, SE-41296 Dtd eray2 2008) 2, February (Dated: References Glossary perspectives correction and error Conclusion quantum XV. and encoding transfer Qubit entanglement (d) and buses Qubit (c) multi- Teleportation (b) with measurements Bell engineering (a) systems state JJ qubit Quantum qubits phase XIV. JJ coupled Capacitively qubits (c) charge coupled coupled Inductively qubits (b) charge of coupled Capacitively (a) manipulation systems strip-line two-qubit approaches Experimental Yale a and Delft XIII. to the of coupled a Comparison (d) to qubit coupled charge-phase resonator qubit Yale flux (c) current oscillator persistent quantum Delft (b) quan- discussion General to (a) coupled qubits oscillators tum with Experiments XII. qubit Charge-phase (e) qubits qubit Flux junction (d) Josephson current-biased NIST procedures (c) measurement and Operation (b) detectors readout Readout (a) and qubits single decoherence devices with and noise Experiments environment: XI. the of Effects coupling (e) yy qubits, coupling Two xx (d) transverse qubits, Two point (c) degeneracy point the degeneracy at the Biasing from coupling away zz far transverse Biasing Ising-type qubits, Two (b) formulation N-qubit systems General (a) multi-qubit resonator of a Dynamics via X. coupled qubits Two (h) coupling capacitive Variable coupling phase Variable coupling Josephson Variable coupling inductive Variable ocessing. so prto fqatzdsuper- quantized of operation of es 2

I. INTRODUCTION tance of quantum computers on any time scale, but there is no doubt that the research will be a powerful driver of The first demonstration of oscillation of a supercon- the development of solid-state quantum state engineering ducting qubit by Nakamura et al. in 19991 can be and quantum technology, e.g. performing measurements said to represent the ”tip of the iceberg”: it rests on ”at the edge of the impossible”. a huge volume of advanced research on Josephson junc- This article aims at describing the inner workings of tions (JJ) and circuits developed during the last 25 superconducting Josephson junction (JJ) circuits, how years. Some of this work has concerned fundamen- these can form two-level systems acting as qubits, and tal research on Josephson junctions and superconduct- how they can be coupled together to multi-qubit net- ing quantum interferometers (SQUIDs) aimed at under- works. Since the field of experimental qubit applications standing macroscopic quantum coherence (MQC)2,3,4,5, is only five years old, it is not even clear if the field repre- providing the foundation of the persistent current flux sents an emerging technology for computers. Neverthe- qubit6,7,8. However, there has also been intense research less, the JJ-technology is presently the only example of a aimed at developing superconducting flux-based digital working solid state qubit with long coherence time, with electronics and computers9,10,11. Moreover, in the 1990’s demonstrated two-qubit gate operation and readout, and the single-Cooper-pair box/transistor (SCB, SCT)12, was with potential for scalability. This makes it worthwile to developed experimentally and used to demonstrate the describe this system in some detail. quantization of Cooper pairs on a small superconducting It needs to be said, however, that much of the basic island13, which is the foundation of the charge qubit1,14. theory for coupled JJ-qubits was worked out well ahead 14,41,42 Since then there has been a steady of experiment , defining and elaborating basic op- development15,16,17,18,19, with observation of microwave- eration and coupling schemes. We recommend the reader induced Rabi oscillation of the two-level populations to take a good look at the excellent research and re- 42 in charge21,22,23, and flux24,25,26,27 qubits and dc- view paper by Makhlin et al. which describes the basic pulse driven oscillation of charge qubits with rf-SET principles of a multi-JJ-qubit information processor, in- detection28. An important step is the development of the cluding essential schemes for qubit-qubit coupling. The charge-phase qubit, a hybrid version of the charge qubit ambition of the present article is to provide a both in- consisting of an SCT in a superconducting loop21,22, troductory and in-depth overview of essential Josephson demonstrating Rabi oscillations with very long coherence junction quantum circuits, discuss basic issues of readout time, of the order of 1 µs, allowing a large set of basic and measurement, and connect to the recent experimen- and advanced (”NMR-like”) one-qubit operations (gates) tal progress with JJ-based qubits for quantum informa- to be performed23. In addition, coherent oscillations tion processing (JJ-QIP). have been demonstrated in the ”simplest” JJ qubits of them all, namely a single Josephson junction30,31,32,33, or a two-JJ dc-SQUID34, where the qubit is formed by II. NANOTECHNOLOGY, COMPUTERS AND the two lowest states in the periodic potential of the JJ QUBITS itself. Although a powerful JJ-based quantum computer with The scaling down of microelectronics into the nanome- hundreds of qubits remains a distant goal, systems with ter range will inevitably make quantum effects like tun- 5-10 qubits will be built and tested by, say, 2010. Pair- neling and wave propagation important. This will even- wise coupling of qubits for two-qubit gate operations is tually impede the functioning of classical transistor com- then an essential task, and a few experiments with cou- ponents, but will also open up new opportunities for pled JJ-qubits with fixed capacitive or inductive cou- multi-terminal components and logic circuits built on e.g. plings have been reported35,36,37,38,39,40, in particular the resonant tunneling, ballistic transport, single electronics, first realization of a controlled-NOT gate with two cou- etc. pled SCBs36, used together with a one-qubit Hadamard There are two main branches of fundamentally differ- gate to generate an entangled two-qubit state. ent computer architectures, namely logically irreversible For scalability, and simple operation, the ability to con- and logically reversible. Ordinary computers are irre- trol qubit couplings, e.g. switching them on and off, will versible because they dissipate both energy and infor- be essential. So far, experiments on coupled JJ qubits mation. Even if CMOS circuits in principle only draw have been performed without direct physical control of current when switching, this is nevertheless the source the qubit coupling, but there are many proposed schemes of intense local heat generation, threatening to burn up for two(multi)-qubit gates based on fixed or controllable future processor chips. The energy dissipation can in physical qubit-qubit couplings or tunings of qubits and principle be reduced by going to single-electron devices, bus resonators. superconducting electronics and quantum devices, but All of the JJ-circuit devices introduced above are this does not alter the fact that the information process- based on nanoscale science and technology and represent ing is logically irreversible. In the simplest case of an emerging technologies for quantum engineering and, at AND gate, two incoming bit lines only result in a single best, information processing. One may debate the impor- bit output, which means that one bit is in practice erased 3

(initialized to zero) and the heat irreversibly dissipated to the environment. The use of quantum-effect devices does not change the fact that we are dealing with com- puters where each gate is logically irreversible and where discarded information constantly is erased and turned into heat. A computer with quantum device components therefore does not make a quantum computer. A quantum information processor has to be built on fundamentally reversible schemes with reversible gates where no information is discarded, and where all inter- nal processes in the components are elastic. This issue is connected with the problem of the minimum energy FIG. 1: 3-junction persistent current flux qubit (PCQ) (inner needed for performing a calculation43 (connected with loop) surrounded by a 2-junction SQUID. Courtesy of J.E. the entropy change created by erasing the final result, Mooij, TU Delft. i.e. reading the result and then clearing the register). A reversible information processor can in principle be built by classical means using adiabatically switched networks of different kinds. The principles were investigated in the 1980’s and form a background for much of the work on quantum computation44,45,46,47,48,49,50. Recently there has been some very interesting development of reversible computers (see51,52,53,54 and references therein). How- ever, a reversible computer still does not make a quan- tum computer. What is characteristic for a quantum computer is that it is reversible and quantum coherent, meaning that one can build entangled non-classical multi- FIG. 2: Single Cooper pair box (SCB) (right) coupled to a qubit states. single-electron transistor (SET) (left) for readout. Courtesy One can broadly distinguish between microscopic, of P. Delsing, Chalmers. mesoscopic and macroscopic qubits. Microscopic effec- tive two-level systems are localized systems confined by natural or artificial constraining potentials. Natural sys- EoH68. This is really an atomic-like microscopic qubit: tems then typically are atomic or molecular impurities a thin film of liquid He is made to cover a Si surface, utilizing electronic charge, or electronic or nuclear spins. and electrons are bound by the image force above the He These systems may be implanted55,56 or naturally oc- surface, forming an electronic two-level system. Qubits curring due to the material growth process57. A related are laterally defined by electrostatic gate patterns in the type of impurity qubit involves endohedral fullerenes, i.e. Si substrate, which also defines circuits for qubit control, atoms implanted into C60 or similar cages58, to be placed qubit coupling and readout. at specific positions on a surface prepared for control and Macroscopic superconducting qubits - the subject of readout. this article - are based on electrical circuits containing Mesoscopic qubit systems typically involve geomet- Josephson junctions (JJ). Looking at two extreme exam- rically defined confining potentials like quantum dots. ples, the principle is actually very simple. In one limit, Quantum dots (QD) for qubits59 are usually made in the qubit is simply represented by the two rotation direc- semiconductor materials. One type of QD is a small tions of the persistent supercurrent of Cooper pairs in a natural or artificial semiconductor grain with quantized superconducting ring containing Josephson tunnel junc- electronic levels. The electronic excitations may be ex- tions (rf-SQUID)(flux qubit)6,7,8, shown below in Fig. 1 citonic (excitons,60,61 or biexcitons,62), charge-like63, or for the Delft 3-junction flux qubit, spin-like (e.g. singlet-triplet)64,65. Another type of QD In another limit, the qubit is represented by the pres- is geometrically defined in semiconductor 2DEG by elec- ence or absence of a Cooper pair on a small supercon- trostatic split-gate arrangements. Although the host ma- ducting island (Single Cooper pair Box, SCB, or Transis- terials are epitaxially layered semiconducting materials tor, SCT) (charge qubit)1,13,28,69, as illustrated in Fig. 2. (e.g.GaAs/AlGaAs), the ungated 2DEG electronic sys- Hybrid circuits21,70,71 can in principle be tuned between tem is metallic. A split-gate arrangement can then define these limits by varing the relations between the electro- a voltage-controlled system of quantum dots coupled to static charging energy EC and the Josephson tunneling metallic reservoir electrodes, creating a system for elec- energy EJ . tron charge and spin transport through quantum point All of these solid-state qubits have advantages and dis- contacts and (effective) two-level quantum dots64,65,66,67. advantages, and only systematic research and develop- There is also a potential qubit based on liquid He ment of multi-qubit systems will show the practical and superfluid technology, namely ”Electrons on Helium”, ultimate limitations of various systems. The supercon- 4 ducting systems have presently the undisputable advan- tage of acutally existing, showing Rabi oscillations and responding to one- and two-qubit gate operations. In fact, even an elementary SCB two-qubit entangling gate creating Bell-type states has been demonstrated very recently36. All of the non-superconducting qubits are so far, promising but still potential qubits. Several of the impurity electron spin qubits show impressive relax- ation lifetimes in bulk measurements, but it remains to demonstrate how to read out individual qubit spins.

III. BASICS OF QUANTUM COMPUTATION

A. Conditions for quantum information processing

72 DiVincenzo has formulated a set of rules and con- FIG. 3: The Bloch sphere. Points on the sphere correspond ditions that need to be fulfilled in order for quantum to the quantum states |ψi; in particular, the north and south computing to be possible: poles correspond to the computational basis states |0i and 1. Register of 2-level systems (qubits), n = 2N states |1i; superposition cat-states |ψi = |0i + eiφ|1i are situated on 101..01 (N qubits) the equator. |2. Initializationi of the qubit register: e.g. setting it to 000..00 |3. Toolsi for manipulation: 1- and 2-qubit gates, e.g. can be characterised by a unit vector on the Bloch sphere: Hadamard (H) gates to flip the spin to the equator, U 0 = ( 0 + 1 )/2, and Controlled-NOT (CNOT ) The state vector can be represented as a unitary vector H | i | i | i gates to create entangled states, UCNOT UH 00 >= on the Bloch sphere, and general unitary (rotation) oper- ( 00 + 11 )/2 (Bell state) | ations make it possible to reach every point on the Bloch 4.| Read-outi | i of single qubits ψ = a 0 + beiφ 1 a, b sphere. The qubit is therefore an analogue object with (spin projection; phase φ of qubit| i lost)| i | i → a continuum of possible states. Only in the case of spin 5. Long decoherence times: > 104 2-qubit gate opera- 1/2 systems do we have a true two-level system. In the tions needed for error correction to maintain coherence general case, the qubit is represented by the lowest levels ”forever”. of a multi-level system, which means that the length of 6. Transport qubits and to transfer entanglement be- the state vector may not be conserved due to transitions tween different coherent systems (quantum-quantum in- to other levels. The first condition will therefore be to terfaces). operate the qubit so that it stays on the Bloch sphere 7. Create classical-quantum interfaces for control, read- (fidelity). Competing with normal operation, noise from out and information storage. the environment may cause fluctuation of both qubit am- plitude and phase, leading to relaxation and decoherence. It is a delicate matter to isolate the qubit from a perturb- B. Qubits and entanglement ing environment, and desirable operation and unwanted perturbation (noise) easily go hand in hand. It is a major A qubit is a two-level quantum system caracterized by issue to design qubit control and read-out such that the the state vector necessary communication lines can be blocked when not in use. θ θ ψ = cos 0 + sin eiφ 1 (3.1) The state of N independent qubits can be represented | i 2| i 2 | i as a product state, Expressing 0 and 1 in terms of the eigenvectors of the | i | i ψ = ψ1 ψ2 .... ψN = ψ1ψ2....ψN (3.4) Pauli matrix σz, | i | i| i | i | i

1 0 involving any one of all of the configurations 00...0 >, 0 = , 1 = . (3.2) 00...1 >, ...., 11...1 >. A general state of an| N-qubit | i 0 | i 1     |memory register| (i.e. a many-body system) can then this can be described as a rotation from the north pole be written as a time-dependent superposition of many- of the 0 state, particle configurations | i θ θ ψ(t) = c1(t) 0...00 + c2(t) 0...01 (3.5) 1 0 cos 2 sin 2 1 ψ = iφ θ − θ (3.3) | i | i | i | i 0 e sin cos 0 + c3(t) 0...10 + .... + cn(t) 1...11    2 2    | i | i 5 where the amplitudes ci(t) are complex, providing phase describing the time evolution of the entire N-particle information. This state represents a time-dependent su- state in the interval [t,t0]. If the total Hamiltonian com- perposition of 2N N-body configurations which in gen- mutes with itself at different times, the time ordering can eral cannot be written as a product of one-qubit states be omitted, and then represents an entangled (quantum correlated) t i ˆ ′ ′ h¯ H(t )dt many-body state. t0 U(t,t0)= e− . (3.17) In the case of two qubits, the maximally entangled R states are the so-called Bell states, This describes the time-evolution controlled by a ho- mogeneous time-dependent potential or electromagnetic ψ = ( 00 + 11 )/2 (3.6) field, e.g. dc or ac pulses with finite rise times but having | i | i | i ψ = ( 00 11 )/2 (3.7) no space-dependence. If the Hamiltonian is constant in | i | i − | i ψ = ( 01 + 10 )/2 (3.8) the interval [t0,t], then the evolution operator takes the | i | i | i simple form ψ = ( 01 10 )/2 (3.9) i | i | i − | i Hˆ (t t0) U(t,t )= e− h¯ − , (3.18) where the last one is the singlet state. In the case 0 of three qubits, the corresponding maximally entan- describing the time-evolution controlled by square dc gled (”cat”) states are the Greenberger-Horne-Zeilinger pulses. (GHZ) states73 The time-development will depend on how many terms are switched on in the Hamiltonian during this time in- ψ = ( 000 111 )/√8 (3.10) terval. In the ideal case, usually not realizable, all terms | i | i ± | i are switched off except for those selected for the specific Another interesting entangled three-qubit state appears computational step. A single qubit gate operation then in the teleportation process, involves turning on a particular term in the Hamiltonian for a specific qubit, while a two-qubit gate involves turn- ψ = [ 00 (a 0 + b 1 )+ 01 (a 0 b 1 ) (3.11) | i | i | i | i | i | i− | i ing on an interaction term between two specific qubits. + 10 (b 0 + a 1 )+ 11 (b 0 a 1 )]/√8 In principle one can perform an N-qubit gate operation | i | i | i | i | i− | i by turning on interactions for all N qubits. In practical cases, many terms in the Hamiltonian are turned on all C. Operations and gates the time, leading to a ”background” time development that has to be taken into account. Quantum computation basically means allowing the The basic model for a two-level qubit is the spin-1/2 in N-body state to develop in a fully coherent fashion a magnetic field. A system of interacting qubits can then through unitary transformations acting on all N qubits. be modelled by a collection of interacting spins, described The time evolution of the many-body system of N two- by the Heisenberg Hamiltonian level subsystems can be described by the Schr¨odinger 1 equation for the N-level state vector ψ(t) , Hˆ (t)= hi(t) Si + Jij (t) Si Sj (3.19) | i 2 X X i¯h∂ ψ = Hˆ ψ . (3.12) controlled by a time-dependent external magnetic field t| i | i hi(t) and by a time-dependent spin-spin coupling Jij (t). in terms of the time-evolution operator characterizing by Expressing the Hamiltonian in Cartesian components, ˆ the time-dependent many-body Hamiltonian H(t) of the hi(t) = (hx,hy,hz), Si(t) = (Sx,Sy,Sz) and introducing system determined by the external control operations and 1 the Pauli σ-matrices, (Sx,Sy,Sz) = 2 ¯h(σx, σy, σz) we the perturbing noise from the environment, obtain a general N-qubit Hamiltonian with general qubit- qubit coupling: ψ(t) = U(t,t0) ψ(t0) . (3.13) | i | i 1 Hˆ = (ǫiσzi + Re∆iσxi + Im∆iσyi) (3.20) The solution of Schr¨odinger equation for U(t,t0) −2 i X ˆ 1 i¯h∂tU(t,t0)= HU(t,t0) (3.14) + λ (t) σ σ 2 ν,ij νi νj ij;ν may be written as X + (fi(t)σzi + gxi(t)σxi + gyi(t)σyi) i t i U(t,t )=1 Hˆ (t′)U(t′,t )dt′ , (3.15) X 0 − ¯h 0 Zt0 We have here introduced time-independent components of the external field defining qubit energy level splittings and finally, in terms of the time-ordering operator T, as ǫi, Re∆i and Im∆i along the z and x,y axes, as well i t ′ ′ as time-dependent components f (t) and g (t) explicitly Hˆ (t )dt i i h¯ t U(t,t0)= T e− 0 , (3.16) describing qubit operation and readout signals and noise. R 6

Inserted into Eq.(3.17), this Hamiltonian determines to the operation fields. This also opens up the system the time evolution of the many-qubit state. In the ideal to a noisy environment, which puts great demands on case one can turn on and off each individual term of the signal-to-noise ratios. Hamiltonian, including the two-body interaction, giving The readout operation is a particularly critical step. complete control of the evolution of the state. The ultimate purpose is to perform a ”single-shot” quan- It has been shown that any unitary transformation can tum non-demolition (QND) measurement, determining be achieved through a quantum network of sequential the state ( 0 or 1 ) of the qubit in a single measure- application of one- and two-qubit gates. Moreover, the ment and then| i leaving| i the qubit in that very state. Dur- size of the coherent workspace of the multi-qubit memory ing the measurement time, the back-action noise from the can be varied (in principle) by switching on and off qubit- ”meter” will cause relaxation and mixing, changing the qubit interactions. qubit state. The ”meter” must then be sensitive enough In the common case of NMR applied to to detect the qubit state on a time scale shorter than the 74,75,76 molecules , one has no control of the fixed, induced relaxation time T1∗ in the presence of the dissipa- direct spin-spin coupling. With an external magnetic tive back-action of the ”meter” itself. Under these con- field one can control the qubit Zeeman level splittings ditions it is possible to detect the qubit projection in a (no control of individual qubits). Individual qubits can single measurement (single shot read-out). Performing a be addressed by external RF-fields since the qubits have single-shot projective measurement in the qubit eigenba- different resonance frequencies (due to different chemical sis then provides a QND measurement (note the ”trivial” environments in a molecule). Two-qubit coupling can fact that the phase is irreversibly lost in the measurement be induced by simultaneous resonant excitation of two process under any circumstances). qubits. The readout/measurement processes described above In the case of engineered solid state JJ-circuits, indi- can be related to the Stern-Gerlach (SG) experiment80. vidual qubits can be addressed by local gates, control- A SG spin filter acts as a beam splitter for flying qubits, ling the local electric or magnetic field. Extensive single- creating separate paths for spin-up and spin-down atoms, qubit operation has recently been demonstrated by Collin preparing for the measurement by making it possible in et al.23. Regarding two-qubit coupling, the field is just principle to distinguish spatially between the two states starting up, and different options are only beginning to of the qubit. The measurement is then performed by be tested. The most straightforward approaches involve particle counters: a click in, say, the spin-up path col- fixed capacitive35,36,40 or inductive37,38 coupling; such lapses the atom to the spin-up state in a single shot. systems could be operated in NMR style, by detuning There is essentially no decohering back-action from the specific qubit pairs into resonance. Moreover, there are detector until the atom is detected. However, after de- various solutions for controlling the direct physical qubit- tection of the spin-up atom the state is thoroughly de- qubit coupling strength, as will be described in Section stroyed: this qubit has not only decohered and relaxed, IX. but is removed from the system. However, if it was entan- gled with other qubits, these are left in a specific eigen- state. For example, if the qubit was part of the Bell pair D. Readout and state preparation ψ = ( 00 + 11 )/2, detection of spin up ( 0 ) selects |00i and| leavesi | thei other qubit in state 0 . If instead| i the |qubiti was part of the Bell pair ψ = ( |01i + 10 )/2, de- External perturbations, described by fi(t) and gi(t) in | i | i | i Eq.(3.20) can influence the two-level system in typically tection of spin up ( 0 ) selects 01 and leaves the other qubit in the spin-down| i state 1| .i Furthermore, for the two ways: (i) shifting the individual energy levels, which | i 3-qubit GHZ state ψ = ( 000 111 )/√8, detection may change the transition energy and the phase of the | i | i ± | i qubit; and (ii) inducing transitions between the levels, of the first qubit in the spin up ( 0 ) state leaves the remaining 2-qubit system in the 00| istate. changing the level populations. These effects arise both | i from desirable control operations and from unwanted Finally, in the case of the three-qubit entangled state noise (see42,77,78,79 for a discussion of superconducting in the teleportation process, Eq.(3.11), detection of the circuits, and Grangier et al.80 for a discussion of state first qubit in the spin up state 0 , leaves the remaining | i preparation and quantum non-demolition (QND) mea- 2-qubit system in the entangled state [ 0 (a 0 + b 1 )+ | i | i | i surements). 1 (a 0 b 1 )]/2. Moreover, detection of also the second qubit| i | ini− the| i spin-up state 0 will leave the third qubit To control a qubit register, the important thing is to | i control the decoherence during qubit operation and read- in the state (a 0 + b 1 )/√2. In this way, measurement | i | i out, and in the memory state. In the qubit memory state, can be used for state preparation. the qubit must be isolated from the environment. Op- Applying this discussion to the measurement and read- eration and read-out devices should be decoupled from out of JJ-qubit circuits, an obvious difference is that the the qubit and, ideally, not cause any dephasing or relax- JJ-qubits are not flying particles. The detection can ation. The resulting intrinsic qubit life times should be there not be turned on simply by the qubit flying into long compared to the duration of the calculation. In the the detector. Instead, the detector is part of the JJ-qubit operation and readout state, the qubit must be connected circuitry, and must be turned on to discriminate between 7 the two qubit states. These are not spatially separated and project onto the basis states and therefore sensitive to level mixing by detector noise with frequency around the qubit transition energy. The Hˆ m m ψ = E ψ (4.4) | ih | i | i sensitivity of the detector determines the time scale T m m X of the measurement, and the detector-on back-action de- obtaining the ususal matrix equation termines the time scale of qubit relaxation T1∗. Clearly T T ∗ is needed for single-shot discrimination of 0 m ≪ 1 | i k Hˆ m a = Ea (4.5) and 1 . This requires a detector signal-to-noise (S/N) h | | i m k | i m ratio 1, in which case one will have the possibility also X to turn≫ off the detector and leave the qubit in the deter- where ak = k ψ , and mined eigenstate, now relaxing on the much longer time h | i scale T1 of the ”isolated” qubit. This would then be the ultimate QND measurement. ǫ ∆ Hqp = k σz m k σx m (4.6) Note that in the above discussion, the qubit dephas- −2h | | i− 2 h | | i ing time (”isolated” qubit) does not enter because it is giving the Hamiltonian matrix assumed to be much longer than the measurement time, T T . This condition is obviously essential for uti- m ≪ φ lizing the measurement process for state preparation and 1 ǫ ∆ Hˆ = (4.7) error correction. −2 ∆ ǫ  −  The Schr¨odinger equation is then given by IV. DYNAMICS OF TWO-LEVEL SYSTEMS 1 ǫ +2E ∆ a (Hˆ E) ψ = 1 =0 − | i −2 ∆ ǫ +2E a2 − To perform computational tasks one must be able to     (4.8) put a qubit in an arbitrary state. This is usually done The eigenvalues are determined by in two steps. The first step, initialization, consists of relaxation of the initial qubit state to the equilibrium 1 det(Hˆ E)= E2 (ǫ2 + ∆2) = 0 (4.9) state due to interaction with environment. At low tem- − − 4 perature, this state is close to the ground state. During the next step, time dependent dc- or rf-pulses are ap- with the result plied to the controlling gates: electrostatic gate in the 1 E = ǫ2 + ∆2 (4.10) case of charge qubits, bias flux in the case of flux qubits, 1,2 ±2 and bias current in the case of the JJ qubit. Formally, p the pulses enter as time-dependent contributions to the The eigenvectors are given by: Hamiltonian and the state evolves under the action of the time-evolution operator. To study the dynamics of a sin- ∆ gle qubit two-level system we therefore first describe the a2 = a1 (4.11) two-level state, and then the evolution of this state under − ǫ +2E the influence of the control pulses (”perturbations”). After normalisation

A. The two-level state ∆ 2 1 ǫ a1 =1/ 1+ = 1 (4.12) s ǫ +2E √2 ± 2E   r The general 1-qubit Hamiltonian has the form | |

1 2 1 ǫ Hˆ = (ǫ σz + ∆ σx) (4.1) a2 = 1 a = 1 (4.13) −2 ± − 1 ±√2 ∓ 2E q r | | The qubit eigenstates are to be found from the stationary We finally simplify the notation by fixing the signs of Schr¨odinger equation the amplitudes,

Hˆ ψ = E ψ (4.2) 1 ǫ 1 ǫ | i | i a1 = 1+ ; a2 = 1 ; (4.14) √2 2E √2 − 2E To solve the S-equation we expand the 1-qubit state in a r | | r | | complete basis, e.g. the basis states of the σz operator, and explicitly writing down all the energy eigenstates,

E1 = a1 0 + a2 1 (4.15) ψ = ak k = c0 0 + c1 1 (4.3) | i | i | i | i | i | i | i E2 = a2 0 a1 1 (4.16) Xk | i | i− | i 8 where giving a 3-vector representation for the Hamiltonian,

1 2 2 1 2 2 E = ǫ + ∆ ; E =+ ǫ + ∆ (4.17) H = (Hx, Hy, Hz). (4.25) 1 −2 1 1 2 2 1 1 q q shown in Fig. 4. The sign of E2 has been chosen to give the familiar expression for| thei superposition at the degeneracy point ǫ = 0 where a = a =1/√2, 0 1 | 1| | 2| H 1 E1 = ( 0 + 1 ) (4.18) | i √2 | i | i 1 E2 = ( 0 1 ) (4.19) | i √2 | i − | i ρ

B. The state evolution on the Bloch sphere

To study the time-evolution of a general state, a conve- nient way is to expand in the basis of energy eigenstates,

iE1t iE2t ψ(t) = c E e− + c E e− (4.20) | i 1| 1i 2| 2i If we know the coefficients at t = 0, then we know the time evolution. On the Bloch sphere this time evolution 1 is represented by rotation of the Bloch vector with con- stant angular speed (E1 E2)/¯h around the direction defined by the energy eigenbasis.− Indeed, by introduc- FIG. 4: The Bloch sphere: the Bloch vector ρ represents the ing parameterization, c = cos θ , c = sin θ eiφ , we see states of the two-level system (same as in Fig. 3). The poles 1 ′ 2 ′ ′ of the Bloch sphere correspond to the energy eigenstates; the that according to Eq. (4.20) the polar angle remains vector H represents the two-level Hamiltonian. constant, θ′ = const, while the azimuthal angle grows, φ′(t)φ′(0) + (E1 E2)t/¯h. The primed angles here refer to a new coordinate− system on the Bloch sphere related to The time evolution of the density matrix is given by the energy eigenbasis, which is obtained by rotation from the Liouville equation, the earlier introduced computational basis, Eq. (3.1). i¯h∂ ρˆ = [H,ˆ ρˆ]. (4.26) The dynamics on the Bloch sphere is conveniently de- t scribed in terms of the density matrix for a pure quantum 81 The vector form of the Liouville equation is readily de- state , rived by inserting Eqs.(4.24),(4.25) and using the com- ρˆ = ψ ψ . (4.21) mutation relations among the Pauli matrices, | ih | 1 This is a 2 2 Hermitian matrix whose diagonal ele- ∂ ρ = [ H ρ ]. (4.27) × t ¯h ments ρ1 and ρ2 define occupation probabilities of the × basis states, hence satisfying the normalization condition This equation coincides with the Bloch equation for a ρ1 +ρ2 = 1, while the off-diagonal elements give informa- magnetic moment evolving in a magnetic field, the role tion about the phase. The density matrix can be mapped of the magnetic moment being played by the Bloch vector on a real 3-vector by means of the standard expansion in ρ which rotates around the effective ”magnetic field” H terms of σ-matrices, associated with the Hamiltonian of the qubit (plus any 1 driving fields) (Fig. 4). ρˆ = (1 + ρ σ + ρ σ + ρ σ ). (4.22) 2 x x y y z z Direct calculation of the density matrix Eq.(4.21) using C. dc-pulses, sudden switching and free precession Eq.(3.1) and comparing with Eq.(4.22) shows that the vector ρ = (ρx,ρy,ρz) coincides with the Bloch vector, To control the dynamics of the qubit system, one method is to apply dc (square) pulses which suddenly ρ = (sin θ cos φ, sin θ sin φ, cos θ) (4.23) change the Hamiltonian and, consequently, the time- introduced in Fig. 3 and also shown in Fig. 4. In the evolution operator. Sudden pulse switching means that same σ-matrix basis, the general two-level Hamiltonian the time-dependent Hamiltonian is changed so fast on takes the form the time scale of the evolution of the state vector that the state vector can be treated as time-independent - Hˆ = (Hxσx + Hyσy + Hzσz). (4.24) frozen - during the switching time interval. This implies 9 that the system is excited by a Fourier spectrum with an upper cut-off given by the inverse of the switching time. H In the specific scheme of sudden switching of different terms in the Hamiltonian using dc-pulses, the initial state ρ is frozen during the switching event, and begins to evolve in time under the influence of the new H ρ 0 = ψ(0) = c E + c E (4.28) | i | i 1| 1i 2| 2i To find the coefficients we project onto the charge basis, a) b) k=0,1 H H 0 0 = c 0 E + c 0 E (4.29) h | i 1h | 1i 2h | 2i 1 0 = c 1 E + c 1 E (4.30) h | i 1h | 1i 2h | 2i and use the explict results for the energy eigenstates to calculate the matrix elements, obtaining ρ ρ

1= c1a1 + c2a2 (4.31)

0= c1a2 c2a1 (4.32) c) d) − As a result, FIG. 5: Qubit operations with dc-pulses: the vector H repre- sents the qubit Hamiltonian, and the vector ρ represents the 0 = a E + a E (4.33) | i 1| 1i 2| 2i qubit state. a) The qubit is initialized to the ground state; b) the Hamiltonian vector H is suddenly rotated towards x- This stationary state then develops in time governed by axis, and the qubit state vector ρ starts to precess around the constant Hamiltonian as H; c) when qubit vector reaches the south pole of the Bloch sphere, the Hamiltonian vector H is switched back to the iE1t iE2t ψ(t) = a1e− E1 + a2e− E2 (4.34) initial position; the vector ρ remains at the south pole, indi- | i | i | i cating complete inversion of the level population (π-pulse); d) Inserting the energy eigenstates we finally obtain the time if the Hamiltonian vector H is switched back when the qubit evolution in the charge basis, vector reaches the equator of the Bloch sphere (π/2-pulse), then the ρ vector remains precessing at the equator, repre- senting equal-weighted superposition of the qubit states (cat ψ(t) = iφ 2 iE1t 2 iE1t iE1t | iEi1t states) |ψi = |0i + e |1i; this operation is the basis for the 0 a e− + a e + 1 a a (e− e ) | i 1 2 | i 1 2 − Hadamard gate.    (4.35)

The probability amplitudes of finding the system in one Let us consider, for example, the diagonal qubit Hamil- of the two charge states is then tonian, Hˆ = (ǫ/2)σz, and apply a pulse δǫ during a

2 iE1t 2 iE1t time τ. This operation will shift phases of the qubit 0 ψ(t) = a e− + a e (4.36) h | i 1 2 eigenstates by, δǫτ/2¯h. If the applied pulse is such iE1t iE1t ± 1 ψ(t) = a1a2(e− e ) (4.37) that ǫ is switched off, and instead, the σx component, h | i − ∆, is switched on, Fig 5, then the state vector will ro-   If the system is driven to the degeneracy point ǫ1 = 0, tate around the x-axis, and after the time ∆τ/2¯h = π where a1 = a2 =1/√2, then (π-pulse) the ground state, + , will flip and become, | | | | + . This manipulation| correspondsi to the quan- 0 ψ(t) = cos E t (4.38) | i → |−i h | i 1 tum NOT operation. Furthermore, if the pulse duration 1 ψ(t) = sin E1t (4.39) is twice smaller (π/2-pulse), then the ground state vector h | i will approach the equator of the Bloch sphere and pre- In particular, the probability of finding the system in cess along it after the end of the operation. Such state is state 1 (level 2) oscillates like an equal-weighted superposition of the basis states (cat | i state). p (t)= 1 ψ(t) = sin2 E t 2 |h | i| 1 1 = [1 cos(E2 E1)t] (4.40) 2 − − D. Adiabatic switching with the frequency of the interlevel distance. On the Bloch sphere, this describes free precession around the Adiabatic switching represents the opposite limit to X-axis. sudden switching, namely that the state develops so fast 10 on the time scale of the Hamiltonian that this can be Thus, the dynamics of a driven qubit is characterized by regarded as ”frozen” , i.e. the time-dependence of the a linear combination of the two wave functions, Hamiltonian becomes parametric. This implies that en- ergy is conserved and no transitions are induced - the (1) 1 iλxt/2¯h iE1t/h¯ iE2t/h¯ ψ = e− e− E1 + e− E2 , system stays in the same energy level (although the state | i √2 | i | i changes).   (2) 1 iλxt/2¯h iE1t/h¯ iE2t/h¯ ψ = e e− E1 e− E2 . | i √2 | i− | i   E. Harmonic perturbation and Rabi oscillation (4.48) Let us assume that the qubit was initially in the ground A particularly interesting and practically important state, E1 , and that the perturbation was switched on case concerns harmonic perturbation with small ampli- instantly.| i Then the wave function of the driven qubit tude λ and resonant frequencyhω ¯ = E2 E1. Let us will take the form, consider the situation when the harmonic− perturbation is added to the z-component of the Hamiltonian corre- λxt iE1t/h¯ λxt iE2t/h¯ ψ = cos e− E + i sin e− E . sponding to a modulation of the qubit bias with a mi- | i 2¯h | 1i 2¯h | 2i crowave field. (4.49) In the eigenbasis of the non-perturbed qubit, E1 , Correspondingly, the probabilities of the level occupa- | i E2 , the Hamiltonian will take the form, tions will oscillate in time, | i ˆ H = E1σz + cos ωt (λz σz + λx σx) , (4.41) λ t 1 λ t P = cos2 x = 1+cos x , 1 2¯h 2 ¯h ǫ ∆   λ = λ , λ = λ . (4.42) 2 λxt 1 λxt z x P2 = sin = 1 cos , (4.50) E2 E2 2¯h 2 − ¯h   The first perturbative term determines small periodic os- with small frequency Ω = λ /¯h ω, Rabi oscillations, cillations of the qubit energy splitting, while the second R x ≪ term will induce interlevel transitions. Despite the am- illustrated in Fig. 6. plitude of the perturbation being small, λ/E2 1, the system will be driven far away from the initial≪ state be- cause of the resonance. Indeed, let us consider the wave 1 function of the driven qubit on the form

iE1t/h¯ iE2t/h¯ ψ = a(t)e− E + b(t)e− E . (4.43) | i | 1i | 2i Substituting this ansatz into the Schr¨odinger equation, i¯h ψ˙ = Hˆ (t) ψ , (4.44) | i | i we get the following equations for the coefficients, 0 i(E1 E2)t/h¯ t i¯ha˙ = λx cos ωte − b, T i(E2 E1)t/h¯ R i¯hb˙ = λx cos ωte − a. (4.45) (Here we have neglected a small diagonal perturbation, FIG. 6: Rabi oscillation of populations of lower level (full line) and upper level (dashed line) at exact resonance (zero λz.) detuning). TR = 2π/ΩR is the period of Rabi oscillations. Let us now focus on the slow evolution of the coeffi- cients on the time scale of qubit precession, and average Eqs.(4.45) over the period of the precession. This approx- imation is known in the theory of two-level systems as the ”rotating wave approximation (RWA)”. Then, taking F. Decoherence of qubit systems into account the resonance condition, we get the simple equations, Descriptions of the qubit dynamics in terms of the den- sity matrix and Liouville equation are more general than λx λx i¯ha˙ = b, i¯hb˙ a, (4.46) descriptions in terms of the wave function, allowing the 2 2 effects of dissipation to be included. The density ma- whose solutions read, trix defined in Section IV B for a pure quantum state 2 (1) (1) iλxt/2¯h possesses the projector operator property,ρ ˆ =ρ ˆ. This a (t) = b (t)= e− , assumption can be lifted, and then the density matrix a(2)(t) = b(2)(t)= eiλxt/2¯h. (4.47) describes a statistical mixture of pure states, say energy − 11 eigenstates, processes, relaxation and dephasing, are referred to as decoherence. ρˆ = ρ E E . (4.51) Coupling to the environment leads, in the simplest i| iih i| i case, to the following modification of the Liouville X equation83,84, The density matrix acquires off-diagonal elements when the basis rotates away from the energy eigenbasis. Such 1 (0) a mixed state cannot be represented by the vector on the ∂tρz = (ρz ρz ), (4.55) −T1 − Bloch sphere; however, its evolution is still described by the Louville equation (4.26), i 1 ˆ ∂tρ12 = Eρ12 ρ12 . (4.56) i¯h∂tρˆ = [H, ρˆ]. (4.52) ¯h − T2 The density matrix in Eq. (4.51) is the stationary solu- This equation is known as the Bloch-Redfield equa- tion of the Liouville equation. The evolution of an arbi- tion. The first equation describes relaxation of the (0) trary density matrix, off-diagonal in the energy eigenba- level population to the equilibrium form, ρz = sis, is given by the equations, (1/2) tanh(E/2kT ), T being the relaxation time. The − 1 i(E1 E2)t/h¯ second equation describes disappearance of the off- ρ1,ρ2 =const, ρ12 e − . (4.53) ∝ diagonal matrix element during characteristic time T2, Dissipation is included in the density matrix descrip- dephasing. tion by extending the qubit Hamiltonian and including The relaxation time is determined by the spectral den- interaction with an environment. The environment for sity of the environmental fluctuations at the qubit fre- macroscopic superconducting qubits basically consists of quency, various dissipative elements in external circuits which 1 λ2 provide bias, control, and measurement of the qubit. The = ⊥ Sφ(ω = E). (4.57) ”off-chip” parts of these circuits are usually kept at room T1 2 temperature and produce significant noise. Examples are The particular form of the spectral density depends on the fluctuations in the current source producing magnetic the properties of the environment, which are frequently field to bias flux qubits and, similarly, fluctuations of the expressed via the impedance (response function) of the voltage source to bias gate of the charge qubits. Elec- environment. The most common environment consists tromagnetic radiation from the qubit during operation is of a pure resistance, in this case, Sφ(ω) ω, at low another dissipative mechanisms. There are also intrinsic frequencies. ∝ microscopic mechanisms of decoherence, such as fluctuat- The dephasing time consists of two parts, ing trapped charges in the substrate of the charge qubits, and fluctuating trapped magnetic flux in the flux qubits, 1 1 1 = + . (4.58) believed to produce dangerous 1/f noise. Another intrin- T2 2T1 Tφ sic mechanism is possibly the losses in the tunnel junction dielectric layer. Various kinds of environment are com- The first part is generated by the relaxation process, monly modelled with an infinite set of linear oscillators while the second part results from the pure dephasing due in thermal equilibrium (thermal bath), linearly coupled to the longitudinal coupling to the environment. This to the qubit (Caldeira-Leggett model2,3). The extended pure dephasing part is proportional to the spectral den- qubit-plus-environment Hamiltonian has the form in the sity of the fluctuation at zero frequency. qubit energy eigenbasis82, 1 λ2 = z S (ω = 0). (4.59) 1 T 2 φ Hˆ = Eσz + (λiz σz + λi σ )Xi φ −2 ⊥ ⊥ i X There is already a vast recent literature on de- Pˆ2 mω2X2 coherence and noise in superconducting circuits, i + i i , (4.54) 2m 2 qubits and detectors, and how to engineer the i ! X qubits and environment to minimize decoherence and 20,42,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106 where E = E1 E2. The physical effects of the two relaxation coupling terms in− Eq. (4.54) are quite different. The Many of these issues will be at the focus of this article. ”transverse” coupling term proportional to λ induces interlevel transitions and eventually leads to⊥ the relax- ation. The ”longitudinal” coupling term proportional to V. CLASSICAL SUPERCONDUCTING CIRCUITS λz commutes with the qubit Hamiltonian and thus does not induce interlevel transitions. However, it randomly changes the level spacing, which eventually leads to the In this section we describe a number of elementary su- loss of phase coherence, dephasing. The effect of both perconducting circuits with tunnel Josephson junctions, 12

U(φ) current through the Josephson element has the form85,

IJ = Ic sin φ, (5.3)

where Ic is the critical Josephson current, i.e. I the maximum non-dissipative current that may flow J CR through the junction. The microscopic theory of superconductivity111,112,113 gives the following equation for the Josephson current, φ π∆ ∆ Ic = tanh , (5.4) FIG. 7: Current-biased Josephson junction (JJ) (left), equiv- 2eRN 2T alent circuit (center), and effective (washboard-like) potential where ∆ is the superconducting order parameter, and T (right). The superconducting leads are indicated with dark color, and the tunnel junction with light color. is the temperature. Using these relations and expressing voltage through the superconducting phase, we can write down Kirchhoff’s rule for the circuit, which are used as building blocks in qubit applications. ¯h ¯h These basic circuits are: single current biased Joseph- Cφ¨ + φ˙ + Ic sin φ = Ie, (5.5) 2e 2eR son junction; single Josephson junction (JJ) included in a superconducting loop (rf SQUID); two Josephson junc- where Ie is the bias current. This equation describes the tions included in a superconducting loop (dc SQUID); dynamics of the phase, and it has the form of a damped and an ultra-small superconducting island connected to non-linear oscillator. The role of the non-linear induc- a massive superconducting electrode via tunnel Joseph- tance is here played by the Josephson element. son junction (Single Cooper pair Box, SCB). The dissipation determines the qubit lifetime, and therefore circuits suitable for qubit applications must have extremely small dissipation. Let us assume zero A. Current biased Josephson junction level of the dissipation, dropping the resistive term in Eq. (5.5). Then the circuit dynamic equations, using the The simplest superconducting circuit, shown in Fig. 7, mechanical analogy, can be presented in the Lagrangian consists of a tunnel junction with superconducting elec- form, and, equivalently, in the Hamiltonian form. The trodes, a tunnel Josephson junction, connected to a cur- circuit Lagrangian consists of the difference between the rent source. An equivalent electrical circuit, which repre- kinetic and potential energies, the electrostatic energy of sents the junction consists of the three lumped elements the junction capacitors playing the role of kinetic energy, connected in parallel: the junction capacitance C, the while the energy of the Josephson current plays the role junction resistance R, which generally differs from the of potential energy. normal junction resistance RN and strongly depends on The kinetic energy corresponding to the first term in temperature and applied voltage, and the Josephson ele- the Kirchhoff equation (5.5) reads, ment associated with the tunneling through the junction. ¯h 2 Cφ˙2 K(φ˙)= . (5.6) The current-voltage relations for the junction ca- 2e 2   pacitance and resistance have standard forms, IC = This energy is equal to the electrostatic energy of the C (dV/dt), and IR = V/R. To write down a similar rela- 2 tion for the Josephson element, it is necessary to intro- junction capacitor, CV /2. It is convenient to introduce duce the superconducting phase difference φ(t) across the the charging energy of the junction capacitor charged junction, often simply referred to as the superconducting with one electron pair (Cooper pair), phase, which is related to the voltage drop across the (2e)2 junction, E = , (5.7) C 2C 2e φ(t)= V dt + φ, (5.1) in which case Eq. (5.6) takes the form ¯h Z ¯h2φ˙2 where φ is the time-independent part of the phase dif- K(φ˙)= . (5.8) ference. The phase difference can be also related to a 4EC magnetic flux, The potential energy corresponds to the last two terms in Eq. (5.5), and consists of the energy of the Josephson 2e Φ φ = Φ=2π , (5.2) current, and the magnetic energy of the bias current, ¯h Φ0 ¯h U(φ)= E (1 cos φ) I φ, (5.9) where Φ0 = h/2e is the magnetic flux quantum. The J − − 2e e 13 where EJ =h/ ¯ 2e Ic is the Josephson energy. This po- L tential energy has a form of a washboard (see Fig. 7). In the absence of bias current this potential corresponds to Φ a pendulum with the frequency of small-amplitude oscil- C lations given by J

2eIc R ωJ = . (5.10) r ¯hC This frequency is known as the plasma frequency of the FIG. 8: Superconducting quantum interference device - Josephson junction. When current bias is applied, the SQUID (left) consists of a superconducting loop (dark) in- pendulum potential becomes tilted, its minima becom- terrupted by a tunnel junction (light); magnetic flux Φ is sent through the loop. Right: equivalent circuit. ing more shallow, and finally disappearing when the bias current becomes equal to the critical current, Ie = IC . At this point, the plasma oscillations become unstable, U(φ) which physically corresponds to switching to the dissipa- tive regime and the voltage state. Now we are ready to write down the Lagrangian for the circuit, which is the difference between the kinetic and potential energies. Combining Eqs. (5.8) and (5.9), we get,

2 ˙2 ˙ ¯h φ ¯h L(φ, φ)= EJ (1 cos φ)+ Ieφ. (5.11) φ 4EC − − 2e 0

It is straightforward to check that the Kirchhoff equation FIG. 9: SQUID potential: the full (dark) curve corresponds (5.5) coincides with the dynamic equation following from to integer bias flux (in units of flux quanta), while the dashed the Lagrangian (5.11), using (light) curve corresponds to half-integer bias flux.

d ∂L ∂L =0. (5.12) dt ∂φ˙ − ∂φ where Φe is the external magnetic flux threading the SQUID loop. The Kirchhoff rule for this circuit takes It is important to emphasize, that the resistance of the form the junction can only be neglected for low temperatures, ¯h ¨ ¯h ˙ ¯h and also only for slow time evolution of the phase; both Cφ + φ + Ic sin φ + (φ φe)=0. (5.14) 2e 2eR 2eL − the temperature and the characteristic frequency must be small compared to the magnitude of the energy gap in the While neglecting the Josephson tunneling (Ic = 0), this superconductor: T, ¯hω ∆. The physical reason behind equation describes a damped linear oscillator of a con- this constraint concerns≪ the amount of generated quasi- ventional LC-circuit. The resonant frequency is then, particle excitations in the system: if the constraint is 1 fulfilled, the amount of equilibrium and non-equilibrium ω = , (5.15) LC √ excitations will be exponentially small. Otherwise, the LC gap in the spectrum will not play any significant role, and the (weak) damping is γ =1/RC. dissipation becomes large, and the advantage of the su- In the absence of dissipation, it is straightforward to perconducting state compared to the normal conducting write down the Lagrangian of the rf-SQUID, state will be lost. 2 2 2 ¯h φ˙ (φ φe) L(φ, φ˙)= EJ (1 cos φ) EL − . (5.16) 4EC − − − 2 B. rf-SQUID The last term in this equation corresponds to the energy of the persistent current circulating in the loop, The rf-SQUID is the next important superconducting circuit. It consists of a tunnel Josephson junction in- Φ2 E = 0 . (5.17) serted in a superconducting loop, as illustrated in Fig. 8. L 4π2L This circuit realizes magnetic flux bias for the Josephson junction113. To describe this circuit, we introduce the The potential energy U(φ) corresponding to the last two current associated with the inductance L of the leads, terms in Eq. (5.16) is schetched in Fig. 9. For bias flux equal to integer number of flux quanta, ¯h 2e or φe2πn, the potential energy of the SQUID has one ab- I = (φ φ ), φ = Φ (5.13) L 2eL − e e ¯h e solute minimum at φ = φe. For half integer flux quanta 14

Cg

φ φ 1 Φ 2 I Vg

FIG. 11: Single Cooper pair box (SCB): a small supercon- FIG. 10: dc SQUID consists of two tunnel junctions included ducting island connected to a bulk superconductor via a tun- in a superconducting loop; arrows indicate the direction of nel junction; the island potential is controlled by the gate the positive Josephson current. voltage Vg. the potential energy has two degenerate minima, which and correspond to the two persistent current states circulat- I I φ ing in the SQUID loop in the opposite directions. This tan α = c1 − c2 tan e . (5.22) configuration of the potential energy provides the basis Ic1 + Ic2 2 for constructing a persistent-current flux qubit (PCQ). For a symmetric SQUID with Ic1 = Ic2, giving α = 0, Eq. (5.20) reduces to the form C. dc SQUID 2Ic cos(φe/2) sin φ = Ie. (5.23) − We now consider consider the circuit shown in Fig. 10 The potential energy generated by Eq. (5.20) has the consisting of two Josephson junctions coupled in paral- form, lel to a current source. The new physical feature here, compared to a single current-biased junction, is the de- φe ¯h U(φ)=2EJ cos (1 cos φ ) Ieφ , (5.24) pendence of the effective Josephson energy of the double 2 − − − 2e − junction on the magnetic flux threading the SQUID loop.   which indeed is similar to the potential energy of a sin- Let us evaluate the effective Josephson energy. Now gle current biased junction, Eq. (5.9) and Fig. 9, but the circuit has two dynamical variables, superconducting with flux-controlled critical current. This property of phases, φ1,2 across the two Josephson junctions. Defin- the SQUID is used in qubit applications for controlling ing phases as shown in the figure, and applying consider- the Josephson coupling, and also for measuring the qubit ations from Sections V A and V B we find for the static flux. Josephson currents, The kinetic energy of the SQUID can readily be writ- ten down noticing that it is associated with the charging Ic1 sin φ1 Ic2 sin φ2 = Ie, (5.18) energy of the two junction capacitances connected in par- − allel, where Ie is the biasing current. Let us further assume small inductance of the SQUID loop and neglect the mag- ¯h 2 φ˙2 K(φ˙)= (C + C ) . (5.25) netic energy of circulating currents. Then the total volt- 2e 1 2 2 age drop over the two junctions is zero, V1 + V2 = 0, and   therefore, φ1 + φ2 = φe, where φe is the biasing phase Thus the Lagrangian for the SQUID has a form similar 2 related to biasing magnetic flux. Introducing new vari- to Eq. (5.11) where EC = (2e) /2(C1 + C2), ables, 2 2 ¯h φ˙ φe ¯h φ1 φ2 L(φ, φ˙)= 2EJ cos (1 cos φ)+ Ieφ. φ = ± , (5.19) 4EC − 2 − 2e ± 2   (5.26) and taking into account that 2φ+ = φe, we rewrite equa- tion (5.18) on the form, D. Single Cooper Pair Box (SCB) Ic(φe) sin(φ + α)= Ie, (5.20) − There is a particularly important Josephson junction where circuit consisting of a small superconducting island con- nected via a Josephson tunnel junction to a large super- 2 2 Ic(φe)= Ic1 + Ic2 + Ic1Ic2 cos φe, (5.21) conducting reservoir (see Fig. 11). q 15

The effect of the gate electrode is only essential for the kinetic term, and repeating previous analysis we arrive at the following equation for the kinetic energy (assuming I for simplicity identical junctions), 2 CΣ ¯h Cg K(φ˙ )= φ˙ Vg , − 2 2e − − C  Σ  CΣ =2C + Cg. (5.29) Combining this kinetic energy with the potential en- ergy derived in previous subsection, we arrive at the La- Vg grangian of the SCB (cf. Eq. (5.28)) where both the charging energy and the Josephson energy can be con- trolled, FIG. 12: Single Cooper pair transistor (SCT): SCB with loop- shape bulk electrodes; charge fluctuations on the island pro- 2 CΣ ¯h ˙ Cg duces current fluctuation in the loop. L = φ Vg 2 2e − − C  Σ  φe +2EJ cos cos φ . (5.30) 2 − The island is capacitively coupled to another massive   electrode, which may act as an electrostatic gate. The voltage source Vg controls the gate potential. In the nor- mal state, such a circuit is named a Single Electron Box VI. QUANTUM SUPERCONDUCTING (SEB)114 for the following reason: if the junction resis- CIRCUITS tance exceeds the quantum resistance Rq 26 kΩ, and the temperature is small compared to the≈ charging en- One may look upon the Kirchhoff rules, as well as the ergy of the island, the system is in a Coulomb blockade circuit Lagrangians, as the equations describing the dy- regime1,115 where the electrons can only be transferred namics of electromagnetic field in the presence of the to the island one by one, the number of electrons on electric current. Generally, this electromagnetic field is a the island being controlled by the gate voltage. In the quantum object, and therefore there must be a quantum superconducting state, the same circuit is called a Sin- generalization of the equations in the previous section. gle Cooper pair Box (SCB)116,117; for a review see the At first glance, the idea of quantization of an equation de- book118. An experimental SCB device is shown in Fig. scribing a macroscopic circuit containing a huge amount 2. In this section we consider a classical Lagrangian for of electrons may seem absurd. To convince ourselves that the circuit. Since the structure now has two capacitances, the idea is reasonable, it is useful to recall an early ar- one from the tunnel junction, C, and another one from gument in favor of the quantization of electron dynamics the gate, Cg, the electrostatic term in the Hamiltonian in atoms, and to apply it to the simplest circuit, an rf- must be reconsidered. SQUID: When the current oscillations are excited in the Let us first evaluate the electrostatic energy of the SQUID, it works as an antenna radiating electromagnetic SCB. It has the form, waves. Since EM waves are quantized, the same should apply to the antenna dynamics. CV 2 C (V V )2 To quantize the circuit equation, we follow the conven- + g g − , (5.27) 2 2 tional way of canonical quantization: first we introduce the Hamiltonian and then change the classical momen- where V is the voltage over the tunnel junction. Then the tum to the momentum operator. The Hamiltonian is Lagrangian can be written (omitting the constant term), related to the Lagrangian as

2 ˙ C ¯h C H(p, φ)= pφ L, (6.1) L(φ, φ˙)= Σ φ˙ g V E (1 cos φ), (5.28) − 2 2e − C g − J −  Σ  where p is the canonical momentum conjugated to coor- dinate φ, where CΣ = C + Cg. The interferometer effect of two Josephson junctions ∂L p = . (6.2) connected in parallel (Fig. 12) can be used to control the ∂φ˙ Josephson energy of the single Cooper pair box. This setup can be viewed as a flux-biased dc SQUID where For the simplest case of a single junction, Eq. (5.6), the Josephson junctions have very small capacitances and are momentum reads, placed very close to each other so that the island confined 2 between them has large charging energy. The gate elec- ¯h p = Cφ.˙ (6.3) trode is connected to the island to control the charge. 2e   16

The so defined momentum has a simple interpretation: The commutation relation between the phase operator it is proportional to the charge q = CV on the junction and the pair number operator has a particularly simple capacitor, p = (¯h/2e)q, or the number n of electronic form, pairs on the junction capacitor, [φ, nˆ]= i. (6.13) p =¯hn. (6.4) The meaning of the quantization procedure is the fol- The Hamiltonian for the current-biased junction has lowing: the phase and charge dynamical variables can the form (omitting a constant), not be exactly determined by means of physical measure- ments; they are fundamentally random variables with the ¯h probability of realization of certain values given by the H = E n2 E cos φ I φ. (6.5) C − J − 2e e modulus square of the wave function of a particular state, Similarly, the Hamiltonian for the SQUID circuit has the φ = ψ∗(φ) φ ψ(φ) dφ, q = ψ∗(φ)ˆq ψ(φ) dφ. form h i h i Z Z (6.14) 2 (φ φe) The time evolution of the wave function is given by the H(n, φ)= E n2 E cos φ + E − . (6.6) C − J L 2 Schr¨odinger equation, The dc SQUID considered in the previous Section V C ∂ψ(φ, t) i¯h = Hψˆ (φ, t) (6.15) has two degrees of freedom. The Hamiltonian can be ∂t written by generalizing Eqs. (6.5), (6.6) for the phases φ . In the symmetric case we have, where H¯ is the circuit quantum Hamiltonian. The ex- ± plicit form of the quantum Hamiltonian for the circuits 2 2 H = EC n+ + EC n 2EJ cos φ+ cos φ considered above, is the following: − − − 2 (2φ+ φe) ¯h + EL − + Ieφ . (6.7) rf-SQUID: 2 2e − 2 2 (φ φe) For the SCB, Eq. (5.28), the conjugated momentum has Hˆ = EC nˆ EJ cos φ + EL − ; (6.16) the form, − 2 Current biased JJ: ¯hCΣ ¯h ˙ Cg p = φ Vg , (6.8) ¯h 2e 2e − CΣ Hˆ = E nˆ2 E cos φ + I φ; (6.17)   C − J 2e e and the Hamiltonian reads, dc-SQUID: 2 H = EC (n ng) EJ cos φ, (6.9) ˆ 2 2 − − H = EC nˆ+ + EC nˆ 2EJ cos φ+ cos φ − − − 2 2 where now EC = (2e) /2CΣ, and n = p/¯h has the mean- (2φ+ φe) ¯h + EL − + Ieφ . (6.18) ing of the number of electron pairs (Cooper pairs) on the 2 2e − island electrode. ng = CgVg/2e is the charge on the gate capacitor (in units of− Cooper pairs (2e)), which can and finally the single Cooper pair box (SCB): be tuned by the gate potential, and which therefore plays Hˆ = E (ˆn n )2 E cos φ, (6.19) the role of external controlling parameter. C − g − J The quantum Hamiltonian results from Eq. (6.2) by For junctions connecting macroscopically large elec- substituting the classical momentum p for the differential trodes, the charge on the junction capacitor is a con- operator, tinuous variable. This implies that no specific boundary ∂ conditions on the wave function are imposed. The sit- pˆ = i¯h . (6.10) uation is different for the SCB: in this case one of the − ∂φ electrodes, the island, is supposed to be small enough Similarly, one can define the charge operator, to show pronounced charging effects. If tunneling is for- bidden, electrons are trapped on the island, and their ∂ number is always integer (the charge quantization con- qˆ = 2ei , (6.11) − ∂φ dition). However, there is a difference between the ener- gies of even and odd numbers of electrons on the island: and the operator of the pair number, while an electron pair belongs to the superconducting condensate and has the additional energy EC , a single ∂ electron forms an excitation and thus its energy consists nˆ = i . (6.12) − ∂φ of the charging energy, EC /2 plus the excitation energy, 17

∆ (parity effect)116. To prevent the appearance of indi- resonance width, γ = 1/RC compared to the resonance vidual electrons on the island and to provide the SCB frequency, γ ω . It is intuitively clear that the quan- ≪ LC regime, the condition ∆ EC /2 must be fulfilled. Thus tization effect will be destroyed when the level broaden- when the tunneling is switched≫ on, only Josephson tun- ing exceeds the level spacing. For quantum behavior of neling is allowed since it transfers Cooper pairs and the the circuit, narrow resonances, γ ωLC, are therefore number of electrons on the island must change pairwise, essential. However, even in this case≪ it is hard to ob- n = integer. In order to provide such a constraint, peri- serve the quantum dynamics in linear circuits such as odic− boundary conditions on the SCB wave function are LC-resonators123 because the expectation values of the imposed, linear oscillator follow the classical time evolution. Thus the presence of non-linear circuit elements is essential. ψ(φ)= ψ(φ +2π). (6.20) The linear oscillator provides the simplest example of This implies that arbitrary state of the SCB is a super- a quantum energy level spectrum: it only consists of dis- position of the charge states with integer amount of the crete levels with equal distance between the levels. The Cooper pairs, level spectrum of the Josephson junction associated with a pendulum potential is more complicated. Firstly, be- inφ ψ(φ)= ane . (6.21) cause of the non-linearity (non-parabolic potential wells), n the energy spectrum is non-equidistant (anharmonic), X The uncertainty of the dynamical variables are not im- the high-energy levels being closer to each other than the portant as long as the relative mean deviations of dy- low-energy levels. Moreover, for energies larger than the namical variables are small, i.e. the amplitudes of the amplitude of the potential (top of the barrier), EJ , the quantum fluctuations are small. In this case, the particle spectrum is continuous. Secondly, one has to take into behaves as a classical particle. It is known from quan- account the possibility of a particle tunneling between tum mechanics, that this corresponds to large mass of neighboring potential wells: this will produce broadening the particle, in our case, to large junction capacitance. of the energy levels into energy bands. The level broad- Thus we conclude that the quantum effects in the circuit ening is determined by the overlap of the wave function dynamics are essential when the junction capacitances tails under the potential barriers, and it must be small are sufficiently small. The qualitative criterion is that for levels lying very close to the bottom of the potential the charging energy must be larger than, or comparable wells. Such a situation may only exist if the level spac- ing, given by the plasma frequency of the tunnel junction to, the Josephson energy of the junction, EC EJ . Typical Josephson energies of the tunnel junctions∼ in is much smaller than the Josephson energy,hω ¯ J EJ , i.e. when E E . This almost classical regime≪ with qubit circuits are of the order of a few degrees Kelvin C ≪ J or less. Bearing in mind that the insulating layers of Josephson tunneling dominating over charging effects, is the junctions have the thickness of few atomic distances, called the phase regime, because phase fluctuations are and modeling the junction as a planar capacitor, the esti- small and the superconducting phase is well defined. In the opposite case, E E , the lowest energy level lies mated junction area should be smaller than a few square C ≫ J micrometers to observe the circuit quantum dynamics. well above the potential barrier, and this situation corre- One of most important consequences of the quantum sponds to wide energy bands separated by small energy dynamics is quantization of the energy of the circuit. Let gaps. In this case, the wave function far from the gap us consider, for example, the LC circuit. In the classical edges can be well approximated with a plane wave, case, the amplitude of the plasma oscillations has con- iqφ tinuous values. In the quantum case the amplitude of ψ (φ) = exp . (6.23) q 2e oscillation can only have certain discrete values defined through the energy spectrum of the oscillator. The linear This wave function corresponds to an eigenstate of the oscillator is well studied in the quantum mechanics, and charge operator with well defined value of the charge, q. its energy spectrum is very well known, Such a regime with small charge fluctuations is called the charge regime. En =¯hωLC(n +1/2), n =0, 1, 2, ... (6.22) One may ask, why is quantum dynamics never ob- served in ordinary electrical circuits? After all, in high- VII. BASIC QUBITS frequency applications, frequencies up to THz are avail- able, which corresponds to a distance between the quan- The quantum superconducting circuits considered tized oscillator levels of order 10K, which can be observed above contain a large number of energy levels, while for at sufficiently low temperature. For an illuminative dis- qubit operation only two levels are required. Moreover, cussion on this issue see the paper by Martinis, Devoret these two qubit levels must be well decoupled from the and Clarke123. According to Ref. 2 it is the dissipation other levels in the sense that transitions between qubit that kills quantum fluctuations: as known from classi- levels and the environment must be much less probable cal mechanics, the dissipation (normal resistance) broad- than the transitions between the qubit levels itself. Typi- ens the resonance, and good resonators must have small cally that means that the qubit should involve a low-lying 18 pair of levels, well separated from the spectrum of higher levels, and not being close to resonance with any other transitions.

A. Single Josephson Junction (JJ) qubit

The simplest qubit realization is a current biased JJ with large Josephson energy compared to the charging energy. In the classical regime, the particle representing the phase either rests at the bottom of one of the wells of the ”washboard” potential (Fig. 7), or oscillates within the well. Due to the periodic motion, the average voltage FIG. 13: Quantized energy levels in the potential of a current biased Josephson Junction across the junction is zero, φ˙ = 0. Strongly excited states, where the particle may escape from the well, correspond 3/2 to the dissipative regime with non-zero average voltage where Umax =2√2(Φ0/2π)(1 Ie/Ic) is the height of across the junction, φ˙ = 0. the potential barrier at given− bias current. In the quantum regime6 described by the Hamiltonian (6.5), B. Charge qubits 2 ¯h Hˆ = EC nˆ EJ cos φ Ieφ, (7.1) − − 2e 1. Single Cooper pair Box - SCB particle confinement, rigorously speaking, is impossi- ble because of macroscopic quantum tunneling (MQT) An elementary charge qubit can be made with the SCB operating in the charge regime, E E . Neglecting through the potential barrier, see Fig. 13. However, C ≫ J the probability of MQT is small and the tunneling may the Josephson coupling implies the complete isolation of be neglected if the particle energy is close to the bot- the island of the SCB, with a specific number of Cooper pairs trapped on the island. Correspondingly, the eigen- tom of the local potential well, i.e. when E EJ . To find the conditions for such a regime, it is≪ convenient functions, to approximate the potential with a parabolic function, 2 2 EC (ˆn ng) n = En n , (7.4) U(φ) (1/2)EJ cos φ0 (φ φ0) , where φ0 corresponds to − | i | i ≈ − the potential minimum, EJ sin φ0 = (¯h/2e)Ie. Then the correspond to the charge states n =0, 1, 2..., with the en- lowest energy levels, Ek =hω ¯ p(k +1/2) are determined 2 ergy spectrum En = EC (n ng) , as shown in Fig. 14. by the plasma frequency, ω = 21/4ω (1 I /I )1/4. It − p J − e c The ground state energy oscillates with the gate voltage, then follows that the levels are close to the bottom of the and the number of Cooper pairs in the ground state in- potential if E E , i.e. when the Josephson junction C ≪ J creases. There are, however, specific values of the gate is in the phase regime, and moreover, if the bias current voltage, e.g. ng = 1/2 where the charge states 0 and is not too close to the critical value, Ie < Ic. 1 become degenerate. Switching on a small Josephson| i It is essential for qubit operation that the spectrum in |couplingi will then lift the degeneracy, forming a tight the well is not equidistant. Then the two lowest energy two-level system. levels, k =0, 1 can be employed for the qubit operation. The qubit Hamiltonian is derived by projecting the Truncating the full Hilbert space of the junction to the full Hamiltonian (6.19) on the two charge states, 0 , 1 , subspace spanned by these two states, 0 and 1 , we leading to | i | i may write the qubit Hamiltonian on the| form,i | i ˆ 1 1 HSCB = (ǫ σz + ∆ σx), (7.5) H = ǫσ , (7.2) −2 q −2 z where ǫ = E (1 2n ), and ∆ = E . The qubit level C − g J where ǫ = E1 E0. energies are then given by the equation The interlevel− distance is controlled by the bias cur- rent. When bias current approaches the critical cur- 1 2 2 2 E1,2 = E (1 2ng) + E , (7.6) rent, level broadening due to MQT starts to play a role, ∓2 C − J E E + iΓ /2. The MQT rate for the lowest level is q k k k the interlevel distance being controlled by the gate volt- given→ by82 age. At the degeneracy point, ng = 1/2, the diag- onal part of the qubit Hamiltonian vanishes, the lev- 52ω U 7.2U Γ = p max exp max , (7.3) els being separated by the Josephson energy, EJ , and MQT 2π ¯hω − ¯hω s p  p  the qubit eigenstates corresponding to the cat states, 19

current is distributed between the two junctions. The EC answer to this question is apparently equivalent to eval- uating the persistent current circulating in the SQUID loop. This current was neglected so far because the as- sociated induced flux φ˜ = 2φ φ was assumed to be + − e frozen, φ˜ = 0, in the limit of infinitely small SQUID in- ductance, L = 0. Let us now lift this assumption and allow fluctuation of the induced flux; the Hamiltonian 0 n will then take the form (6.7), in which a gate potential is g included in the charging term of the SCB, and the term n = −10 1 containing external current is dropped, ˆ 2 2 HSCT = EC (ˆn ng) + EC nˆ+ − − 2 (2φ+ φe) 2EJ cos φ+ cos φ + EL − . (7.7) − − 2 FIG. 14: Single Cooper pair box (SCB): charging energy (up- (ˆn = i∂/∂φ ). For small but non-zero inductance, per panel), and charge on the island (lower panel) vs gate + − + potential. Washed-out onsets of the charge steps indicate the amplitude of the induced phase is small, φ˜ =2φ+ − quantum fluctuations of the charge on the island. φe 1, and the cosine term containing φ+ can be ex- panded,≪ yielding the equation

HˆSCT = HˆSCB(φ )+ Hˆosc(φ˜)+ Hˆint . (7.8) E −

HˆSCB(φ ) is the SCB Hamiltonian (6.19) with the flux − ∆/2 dependent Josephson energy, EJ (φe)=2EJ cos(φe/2). Hˆosc(φ˜) describes the linear oscillator associated with the

variable φ˜, ∋ φ˜2 0 Hˆ (φ˜)=4E n˜ˆ2 + E , (7.9) osc C L 2 and the interaction term reads,

φe Hˆint = EJ sin cos(φ ) φ˜ . (7.10) 2 −   FIG. 15: Energy spectrum of the SCB (solid lines): it results Thus, the circuit consists of the non-linear oscillator of from hybridization of the charge states (dashed lines). the SCB linearly coupled to the linear oscillator of the SQUID loop. This coupling gives the possibility to mea- sure the charge state of the SCB by measuring the per- E , E = 0 1 . For theses states, the average 1 2 sistent currents and the induced flux. charge| i | oni the| islandi ∓ | isi zero, while it changes to 2e far Truncating Eq. (7.8) we finally arrive at the Hamil- from the degeneracy point, where the qubit eigenstates∓ tonian which is formally equivalent to the spin-oscillator approach pure charge states. Hamiltonian, The SCB was first experimentally realized by Lafarge 117 et al. , observing the Coulomb staircase with steps of 1 13 Hˆ = (ǫσ + ∆(φ )σ )+ λφσ˜ + H . (7.11) 2e and the superposition of the charge states, see also . SCT −2 z e x x osc Realization of the first charge qubit by manipulation of the SCB and observation of Rabi oscillations was done In this equation, ∆(φe) = 2EJ cos(φe/2), and λ = 1,119,120 by Nakamura et al. , and further investigated the- EJ sin(φe/2). oretically by Choi et al.121.

C. Flux qubit 2. Single Cooper pair Transistor - SCT 1. Quantum rf-SQUID In the SCB, charge fluctuations on the island gener- ate fluctuating current between the island and large elec- An elementary flux qubit can be constructed from an trode. In the two-junction setup discused at the end of rf-SQUID operating in the phase regime, E E . Let J ≫ C Section V D, an interesting question concerns how the us consider the Hamiltonian (6.6) at φe = π, i.e. at half 20

E subspace spanned by these two levels. The starting point of the truncation procedure is to approximate the double well potential with Ul and Ur, as shown in Fig. 17, to confine the particle to the left or to the right well, respec- tively. The corresponding ground state wave functions l and r satisfy the stationary Schr¨odinger equation, | i

∆ | i 0 φ Hˆ l = E l , Hˆ r = E r . (7.14) l| i l| i r| i r| i

FIG. 16: Double-well potential of the rf-SQUID with de- The averaged induced flux for these states, φl and φr generate quantum levels in the wells (black). Macroscopic have opposite signs, manifesting opposite directions of quantum tunneling (MQT) through the potential barrier in- the circulating persistent currents. Let us allow the bias troduces a level splitting ∆, and the lowest level pair forms a flux to deviate slightly from the half integer value, φe = qubit. π + f, so that the ground state energies are not equal but still close to each other, E E . The tunneling will l ≈ r (φ) hybridize the levels, and we can approximate the true U eigenfunction, E , | i Hˆ E = E E , (7.15) Ur Ul | i | i with a superposition,

E = a l + b r . (7.16) | i | i | i 0 φ The qubit Hamiltonian is given by the matrix elements of the full Hamiltonian, Eq. (7.15), with respect to the FIG. 17: Flux qubit: truncation of the junction Hamiltonian; states l and r , dashed lines indicate potentials of the left and right wells with | i | i ground energy levels. H = E + l U U l , ll l h | − l| i H = E + r U U r , rr r h | − r| i integer bias magnetic flux. The potential, U(φ), shown Hrl = El r l + r U Ul l . (7.17) h | i h | − | i in Fig. 16 has two identical wells with equal energy levels when MQT between the wells is neglected (phase regime, In the diagonal matrix elements, the second terms are ωJ EJ ). These levels are connected with current fluc- small because the wave functions are exponentially small tuations≪ within each well around averaged values cor- in the region where the deviation of the approximated responding to clockwise and counterclockwise persistent potential from the true one is appreciable. The off di- currents circulating in the loop (the flux states). Let agonal matrix element is exponentially small because of us consider the lowest, doubly degenerate, energy level. small overlap of the ground state wave functions in the When the tunneling is switched on, the levels split, and left and right wells, and also here the main contribution a tight two-level system is formed with the level spacing comes from the first term. Since the wave functions can determined by the MQT rate, which is much smaller than be chosen real, the truncated Hamiltonian is symmet- ric, H = H . Then introducing ǫ = E E , and the level spacing in the well. In the case that the tunnel- lr rl r − l ing barrier is much smaller than the Josephson energy, ∆/2 = Hrl, we arrive at the Hamiltonian of the flux the potential in Eq. (5.9) can be approximated, qubit,

2 1 (φ φe) ˆ U(φ) = E (1 cos φ)+ E − H = (ǫσz + ∆σx). (7.18) J − L 2 −2 ˜2 φ 1+ ε 4 The Hamiltonian of the flux qubit is formally equiva- EL ε fφ˜ + φ˜ , (7.12) ≈ − 2 − 24 ! lent to that of the charge qubit, Eq. (7.5), but the phys- ical meaning of the terms is rather different. The flux where φ˜ = φ π, f = φe π, and where qubit Hamiltonian is written in the flux basis, i.e. the − − basis of the states with certain averaged induced flux, φl E ε = J 1 1. (7.13) and φr (rather than the charge basis of the charge qubit). EL − ≪ The energy spectrum of the flux qubit is obviously the same as that of the charge qubit, determines the height of the tunnel barrier. The qubit Hamiltonian is derived by projecting the 1 E = ǫ2 + ∆2, (7.19) whole Hilbert space of the full Hamiltonian (6.6) on the 1,2 ∓2 p 21

Φ

FIG. 18: Persistent current flux qubit (PCQ) with 3 junctions. The side junctions are identical, while the central junction has smaller area. as shown in Fig. 15. However, for the flux qubit the dashed lines indicate persistent current states in the ab- sence of macroscopic tunneling, and the current degener- acy point (ǫ = 0) corresponds to a half-integer bias flux. The energy levels are controlled by the bias magnetic flux (instead of the gate voltage for the charge qubit). At the flux degeneracy point, f = 0 (φe = π), the level spacing is determined by the small amplitude of FIG. 19: Potential energy landscape U(φ1,φ2) = U(φ+,φ−) tunneling through macroscopic potential barrier, and the of the Delft qubit as a function of the two independent phase wave functions correspond to the cat states, which are variables φ1 and φ2, or equivalently, φ+ (horizontal axis) and equally weighted superpositions of the flux states. Far φ− (vertical axis). Black represents the bottom of the poten- tial wells, and white the top of the potential barriers. The from the degeneracy point, the qubit states are almost qubit double-well potential is determined by the potential pure flux states. landscape centered around the origin in the horizontal direc- The possibility to achieve quantum coherence of tion, and typically has the shape shown in Fig. 16. Courtesy macroscopic current states in an rf-SQUID with a small of C.H. van der Wal. capacitance Josephson junction was first pointed out in 1984 by Leggett122. However, successful experimental observation of the effect was achieved only in 2000 by Friedman et al.8. To explain the idea, let us consider the potential energy. The three phases are not independent and satisfy the relation φ1 +φ2 +φ3 = φe. Let us suppose that the qubit 2. 3-junction SQUID - persistent current qubit (PCQ) is biased at half integer flux quantum, φe = π. Then introducing new variables, φ = (φ1 φ2)/2, we have ± ± The main drawback of the flux qubit with a single Josephson junction (rf-SQUID) described above concerns U(φ+, φ )= EJ [2 cos φ cos φ+ (1/2+ ε)cos2φ+]. the large inductance of the qubit loop, the energy of − − − − (7.21) which must be comparable to the Josephson energy to The potential landscape is shown in Fig. 19, and the form the required double-well potential profile. This im- qubit potential consists of the double well structure near plies large size of the qubit loop, which makes the qubit the points (φ+, φ ) = (0, 0). An approximate form of vulnerable to dephasing by magnetic fluctuations of the the potential energy− is given by environment. One way to overcome this difficulty was pointed out by Mooij et al.6, replacing the large loop in- φ4 ductance by the Josephson inductance of an additional U(φ , 0) E 2εφ2 + + . (7.22) + ≈ J − + 4 tunnel junction, as shown in Fig. 18, The design employs   three tunnel junctions connected in series in a supercon- ducting loop. The inductive energy of the loop is chosen to be much smaller than the Josephson energy of the Each well in this structure corresponds to clock- and junctions. The two junctions are supposed to be identi- counterclockwise currents circulating in the loop. The cal while the third junction is supposed to have smaller amplitude of the structure is given by the parameter area, and therefore smaller Josephson and larger charging ǫEJ , and for ǫ 1 the tunneling between these wells energy. The Hamiltonian has the form, dominates. Thus≪ this qubit is qualitatively similar to the single-junction qubit described above, but the quan- Hˆ = E [ˆn2 +ˆn2 +ˆn2/(1/2+ ε)] C 1 2 3 − titative parameters are different and can be significantly EJ [cos φ1 + cos φ2 + (1/2+ ε)cos φ3]. (7.20) optimized. 22

D. Potential qubits ∆ 1

The superconducting qubits that have been discussed in previous sections exploit the fundamental quantum un- certainty between electric charge and magnetic flux. This E 0 R ∆ uncertainty appears already in the dynamics of a single 2 Josephson junction (JJ), which is the basis for elementary JJ qubits. There are however other possibilities. One of them is to delocalize quantum information in a JJ net- work by choosing global quantum states of the network ∆ −1 as a computational basis. Recently, some rather com- 00 1φ 2π2 plicated JJ networks have been discussed, which have the unusual property of degenerate ground state, which might be employed for efficient qubit protection against FIG. 20: Energy spectrum of microscopic bound Andreev lev- decoherence124,125. els; the level splitting is determined by the conact reflectivity. An alternative possibility of further miniaturization of superconducting qubits could be to replace the stan- net Josephson current is negligibly small. dard Josephson junction by a quantum point contact In QPCs, the Josephson effect is associated with mi- (QPC), using the microscopic conducting modes in the croscopic Andreev levels, localized in the junction area, JJ QPC, the bound Andreev states, as a computational which transport Cooper pairs from one junction electrode basis, allowing control of intrinsic decoherence inside the 127,128 105,126 to the other . As shown in Fig. 20, the Andreev junction . levels, two levels per conducting mode, lie within the su- To explain the physics of this type of qubit, let us perconducting gap and have the phase-dependent energy consider an rf SQUID (see Fig. 8) with a junction that spectrum, has such a small cross section that the quantization of electronic modes in the (transverse) direction perpendic- E = ∆ cos2(φ/2)+ R sin2(φ/2), (7.23) ular to the current flow becomes pronounced. In such a a ± junction, quantum point contact (QPC), the Josephson q current is carried by a number of independent conduct- (here ∆ is the superconducting order parameter in the junction electrodes). For very small reflectivity, R 1, ing electronic modes, each of which can be considered an ≪ elementary microscopic Josephson junction characterized and phase close to π (half integer flux bias) the An- by its own transparency. The number of modes is propor- dreev two-level system is well isolated from the contin- tional to the ratio of the junction cross section and the uum states. The expectation value for the Josephson area of the atomic cell (determined by the Fermi wave- current carried by the level is determined by the Andreev length) of the junction material. In atomic-sized QPCs level spectrum, with only a few conducting modes, the Josephson current 2e dEa can be appreciable if the conducting modes are transpar- Ia = , (7.24) ¯h dφ ent (open modes). If the junction is fully transparent e (reflectivity R = 0) then current is a well defined quan- and it has different sign for the upper and lower level. tity. This will correspond to a persistent current with Since the state of the Andreev two-level system is deter- certain direction circulating in the qubit loop. On the mined by the phase difference and related to the Joseph- other hand, for a finite reflectivity (R = 0), the electronic son current, the state can be manipulated by driving 6 back scattering will induce hybridization of the persistent magnetic flux through the SQUID loop, and read out current states giving rise to strong quantum fluctuation by measuring circulating persistent current129,130. of the current. This microscopic physics underlines recent proposal for Such a quantum regime is distinctly different from the Andreev level qubit105,126. The qubit is similar to the macroscopic quantum coherent regime of the flux qubit macroscopic flux qubits with respect to how it is manip- described in Section VII, where the quantum hybridiza- ulated and measured, but the great difference is that the tion of the persistent current states is provided by charge quantum information is stored in the microscopic quan- fluctuations on the junction capacitor. Clearly, charging tum states. This difference is reflected in the more com- effects will not play any essential role in quantum point plex form of the qubit Hamiltonian, which consists of contacts, and the leading role belongs to the microscopic the two-level Hamiltonian of the Andreev levels strongly mechanism of electron back scattering. This mechanism coupled to the quantum oscillator describing phase fluc- is only pronounced in quantum point contacts: in classi- tuations, cal (large area) contacts, such as junctions of macroscopic qubits with areas of several square micrometers, the cur- iσx√R φ/2 φ φ Hˆ = ∆ e− cos σ + √R sin σ +Hˆ [φ], rent is carried by a large number (> 104) of statistically 2 z 2 y osc independent conducting modes, and fluctuation of the   (7.25) 23

2 Hˆosc[φ]= EC nˆ +(EL/2)(φ φe) . Comparing this equa- tion with e.g. the SCT Hamiltonian− (7.8), we find that the truncated Hamiltonian of the SCB is replaced here by the Andreev level Hamiltonian.

VIII. QUBIT READOUT AND MEASUREMENT OF QUANTUM INFORMATION

In this section we present a number of proposed, and realized, schemes for measuring quantum states of vari- ous superconducting qubits.

A. Readout: why, when and how? FIG. 21: Measurement of the phase qubit. Long-living levels form the qubit, while the dashed line indicates a leaky level As already mentioned in Section III D, the ultimate with large energy. objective of a qubit readout device is to distinguish the eigenstates of a qubit in a single measurement ”without destroying the qubit”, a so called ”single-shot” quantum eigenstate. Then the value is predetermined and the non-demolition (QND) projective measurement. This qubit left in the eigenstate (Stern-Gerlach-style). objective is essential for several reasons: state prepara- On the other hand, to extract the desired final result it tion for computation, readout for error correction during may be necessary to create an ensemble of calculations to the calculation, and readout of results at the end of the be able to perform a complete measurement to determine the expectation values of variables of interest, performing calculation. Strictly speaking, the QND property is only 131 needed if the qubit must be left in an eigenstate after quantum state tomography . the readout. In a broader sense, readout of a specific qubit must of course not destroy any other qubits in the system. B. Direct qubit measurement It must be carefully noted that one cannot ”read out the state of a qubit” in a single measurement - this is pro- Direct destructive measurement of the qubit can be il- hibited by quantum mechanics. It takes repeated mea- lustrated with the example of a single JJ (phase) qubit, surements on a large number of replicas of the quantum Section VII A. After the manipulation has been per- state to characterize the state of the qubit (Eq. (3.1)) - formed (e.g. Rabi oscillation), the qubit is left in a su- ”quantum tomography”131. perposition of the upper and lower energy states. To The measurement connects the qubit with the open determine the probability of the upper state, one slowly system of the detector, which collapses the combined sys- increases the bias current until it reaches such a value tem of qubit and measurement device to one of its com- that the upper energy level equals Or gets close to) the mon eigenstates. If the coupling between the qubit and top of the potential barrier, see Fig. 21. Then the junc- the detector is weak, the eigenstates are approximately tion, being at the upper energy level, will switch from the those of the qubit. In general however, one must con- Josephson branch to the dissipative branch, and this can sider the eigenstates of the total qubit-detector system be detected by measuring the finite average voltage ap- and manipulate gate voltages and fluxes such that the pearing across the junction (voltage state). If the qubit readout measurement is performed in a convenient en- is in the lower energy state the qubit will remain on the ergy eigenbasis (see e.g.42,79). Josephson branch and a finite voltage will not be de- Even under ideal conditions, a single-shot measure- tected (zero-voltage state). An alternative method to ment can only determine the population of an eigenstate activate switching30 is to apply an rf signal with reso- if the system is prepared in an eigenstate: then the an- nant frequency (instead of tilting the junction potential) swer will always be either ”0” or ”1”. If an ideal single- in order to excite the upper energy level and to induce the shot measurement is used to read out a qubit superpo- switching event, see Fig. 21 (also illustrating a standard sition state, e.g. during Rabi oscillation, then again the readout method in atomic physics). answer can only be ”0” or ”1”. To determine the qubit It is obvious that, in this example, the qubit upper 2 2 population (i.e. the a1 and a2 probabilities) requires energy state is always destroyed by the measurement. repetition of the measurement| | | to| obtain the expectation Single-shot measurement is possible provided the MQT value. During the intermediate stages of quantum com- rate for the lower energy level is sufficiently small to putation one must therefore not perform a measurement prevent the junction switching during the measurement on a qubit unless one knows, because of the design and time. It is also essential to keep a sufficiently small rate of timing of the algorithm, that this qubit is in an energy interlevel transitions induced by fluctuations of the bias 24

P

t << t ms t < t ms t = t ms SCB

0 t I(n 0 ) t I(n 1 ) m SET x x FIG. 23: Probability distributions P of counted electrons as functions of time after the turning on the measurement beam of electrons. Courtesy of G. Johansson, Chalmers. Vg capacitively coupled to a charge qubit has also been ex- tensively discussed in literature132,139,140,141,142,143,144. FIG. 22: Single Electron Transistor (SET) capacitively cou- The induced charge on the SET gate depends on the pled to an SCB. state of the qubit, affecting the SET working point and determining the conductivity and the average cur- rent. The development of the probability distributions current and by the current ramping. of counted electrons with time is shown in Fig. 23. A similar kind of direct destructive measurement was As the number of counted electrons grows, the distribu- performed by Nakamura et al.1 to detect the state of the tions separate and become distinguishable, the distance charge qubit. The qubit operation was performed at the between the peaks developing as N and the width √N. Detailed investigations144∼show that the two charge degeneracy point, ug = 1, where the level splitting ∼ is minimal. An applied gate voltage then shifted the SCB electron-number probability distributions correlate with working point (Fig. 15), inducing a large level splitting the probability of finding the qubit in either of two energy of the pure charge states 0 and 1 (the measurement levels. The long-time development depends on the inten- preparation stage). In this| processi | thei upper 1 charge sity and frequency distribution of the back-action noise state went above the threshold for Cooper pair| i decay, from the electron current. With very weak detector back action, the qubit can relax to 0 during the natural re- creating two quasi-particles which immediately tunnel | i out via the probe junction into the leads. These quasi- laxation time T1. With very strong back-action noise at particles were measured as a contribution to the classical the qubit frequency, the qubit may become saturated in charge current by repeating the experiment many times. a 50/50 mixed state. Obviously, this type of measurement is also destructive. D. Measurement via coupled oscillator C. Measurement of charge qubit with SET Another method of qubit read out that has attracted much attention concerns the measurement of the proper- Non-destructive measurement of the charge qubit has ties of a linear or non-linear oscillator coupled to a qubit. been implemented by connecting the qubit capacitively 69 This method is employed for the measurement of induced to a SET electrometer . The idea of this method is to magnetic flux and persistent current in the loop of flux use a qubit island as an additional SET gate (Fig. 22), qubits and charge-phase qubits, as well as for charge mea- controlling the dc current through the SET depending surement on charge qubits. With this method, the qubit on the state of the qubit. When the measurement is to affects the characteristics of the coupled oscillator, e.g. be performed, a driving voltage is applied to the SET, changes the shape of the oscillator potential, after which and the dc current is measured. Another version of the the oscillator can be probed to detect the changes. There measurement procedure is to apply rf bias to the SET 69,133,134,135 are two versions of the method: resonant spectroscopy of (rf-SET ) in Fig. 22, and to measure the dis- a linear tank circuit/cavity, and threshold detection us- sipative or inductive response. In both cases the trans- ing biased JJ or SQUID magnetometer. missivity will show two distinct values correlated with The first method uses the fact that the resonance fre- the two states of the qubit. Yet another version has re- 136 quency of a linear oscillator weakly coupled to the qubit cently been developed by the NEC group to perform undergoes a shift depending on the qubit state. The single-shot readout: the Cooper pair on the SCB island effect is most easily explained by considering the SCT then tunnels out onto a trap island (instead of the leads) Hamiltonian, Eq. (11.1), used as a gate to control the current through the SET. The physics of the SET-based readout has been exten- 1 42,137,138 Hˆ = (ǫσ + ∆(φ )σ )+ λ(φ )φσ˜ sively studied theoretically (see and references SCT −2 z e x e x therein). A similar idea of controlling the transmission 1 +4E n˜ˆ2 + E φ˜2. (8.1) of a quantum point contact (QPC) (instead of an SET) C 2 L 25

Let us proceed to the qubit energy basis, in which casee Vout (t) u(t) the qubit Hamiltonian takes the form (E/2)σ , E = − z √ǫ2 + ∆2. The interaction term in the qubit eigenbasis will consist of two parts, the longitudinal part, λzφσ˜ z, λz = (∆/E)λ, and the transverse part, λxφσ˜ x, λx = (ǫ/E)λ. In the limit of weak coupling the transverse part SCB of interaction is the most essential. In the absence of interaction (φe = 0), the energy spectrum of the qubit + oscillator system is E 1 En = +¯hω(n + ), (8.2) ∓ ∓ 2 2 where ¯hω = √8E E is the plasma frequency of the C L g oscillator. The effect of weak coupling is enhanced in V the vicinity of the resonance, when the oscillator plasma frequency is close to the qubit level spacing,hω ¯ E. Let ≈ us assume, however, that the coupling energy is smaller FIG. 24: SCT qubit coupled to a readout oscillator. The than the deviation from the resonance, λ ¯hω E . x ≪ | − | qubit is operated by input pulses u(t). The readout oscillator Then the spectrum of the interacting system in the lowest is controlled and driven by ac microwave pulses Vg(t). The perturbative order will acquire a shift, output signal will be ac voltage pulses Vout(t), the amplitude 2 or phase of which may discriminate between the qubit ”0” λx¯hω δEn = (n + 1) . (8.3) and ”1” states. ± ± E (¯hω E) L − This shift is proportional to the first power of the oscilla- tor quantum number n, which implies that the oscillator frequency acquires a shift (the frequency of the qubit is also shifted145,146,147,148,149). Since the sign of the oscil- lator frequency shift is different for the different qubit states, it is possible to distinguish the state of the qubit SCT Ib by probing this frequency shift. In the case of the SCT, the LC oscillator is a generic part of the circuit. It is equally possible to use an addi- Vg tional LC oscillator inductively coupled to a qubit. This type of device has been described by Zorin71 for SCT readout, and recently implemented for flux qubits by Il’ichev et al.27,39. FIG. 25: SCT qubit coupled to a JJ readout quantum oscil- Figure 24 illustrates another case, namely a charge lator. The JJ oscillator is controlled by dc/ac current pulses qubit capacitively coupled to an oscillator, again pro- Ib(t) adding to the circulating currents in the loop due to the viding energy resolution for discriminating the two qubit SCT qubit. The output will be dc/ac voltage pulses Vout(t) levels150. Analysis of this circuit is similar to the one dis- discriminating between the qubit ”0” and ”1” states. cussed below in the context of qubit coupling via oscilla- tors, Section IX. The resulting Hamiltonian is similar to Eq. (9.20), namely, inserted in the qubit loop, as shown in Fig. 25. When the measurement of the qubit state is to be per- ˆ ˆ ˆ H = HSCB + λσyφ + Hosc. (8.4) formed, a bias current is sent through the additional junc- In comparison with the case of the SCT, Eq. (8.4) has tion. This current is then added to the qubit-state depen- a different form of the coupling term, which does not dent persistent current circulating in the qubit loop. If change during rotation to the qubit eigenbasis. There- the qubit and readout currents flow in the same direction, fore the coupling constant λ directly enters Eq. (8.3). the critical current of the readout JJ is exceeded, which Recently, this type of read out has been implemented for induces the junction switching to the resistive branch, a charge qubit by capacitively coupling the SCB of the sending out a voltage pulse. This effect is used to distin- qubit to a superconducting strip resonator151,152,153. guish the qubit states. The method has been extensively used experimentally by Vion et al.21,22,23. To describe the circuit, we add the Lagrangian of a E. Threshold detection biased JJ, Eq. (5.11),

2 ˙2 To illustrate the threshold-detection method, let us ¯h φ m ¯h L = m + EJ cos φ + Ieφ, (8.5) consider an SCT qubit with a third Josephson junction 4EC 2e 26 to the SCT Lagrangian (generalized Eq. (5.30), cf. Eqs. U (6.7), (7.7)),

2 2 2 ¯h ˙ 2eCg ¯h ˙2 LSCT = φ Vg + φ+ 4E − − ¯hC 4E C  Σ  C 1 2 2EJ cos φ+ cos φ ELφ˜ (8.6) − − − 2 (here we have neglected a small contribution of the gate capacitance to the qubit charging energy). The phase quantization condition will now read: 2φ+ + φ = φe + φ˜. The measurement junction will be assumed in the phase m m θ regime, EJ EC , and moreover, the inductive energy will be the largest≫ energy in the circuit, E Em. The L ≫ J latter implies that the induced phase is negligibly small FIG. 26: Josephson potential energy of the measurement and can be dropped from the phase quantization condi- junction during the measurement: for the ”0” qubit eigen- tion. We also assume that φe = 0, thus 2φ+ + φ = 0. state there is a well (full line) confining a level, while for the Then, after having omitted the variable φ+, the kinetic ”1” qubit state there is no well (dashed line). energy term of the qubit can be combined with the much larger kinetic energy of the measurement junction lead- ing to insignificant renormalization of the measurement while for the excited state it does not. This implies that junction capacitance, Cm + C/4 Cm. As a result, the when the junction is in the ground state, no voltage will total Hamiltonian of the circuit will→ take the form, be generated. However, if the junction is in the excited state, it will switch to the resistive branch, generating a

2 φ voltage pulse that can be detected. Hˆ = EC (ˆn ng) 2EJ cos cos φ − − − 2 − With the discussed setup the direction of the persis-   tent current is measured. It is also possible to arrange m 2 m ¯h +EC nˆ EJ cos φ Ieφ. (8.7) the measurement of the flux by using a dc SQUID as a − − 2e threshold detector. Such a setup is suitable for the mea- Since the measurement junction is supposed to be almost surement of flux qubits. Let us consider, for example, the classical, its phase is fairly close to the minimum of the three junction flux qubit from Section VII C inductively junction potential. During qubit operation, the bias cur- coupled to a dc SQUID. Then, under certain assump- rent is zero; hence the phase of the measurement junction tions, the Hamiltonian of the system can be reduced to is zero. When the measurement is made, the current is the following form: ramped to a large value close to the critical current of 1 m ˆ the measurement junction, I = (2e/¯h)E δI, tilting H = (ǫσz + ∆σx) e J −2 the junction potential and shifting the minimum− towards s s ¯h π/2. Introducing a new variable φ = π + θ, we expand +EC nˆ2 (EJ + λσz)cos φ δI φ, (8.10) the potential with respect to small θ 1 and, truncating − − 2e ≪ s the qubit part, we obtain where EJ is an effective (bias flux dependent) Josephson energy of the SQUID, Eq. (5.24), and λ is an effective ǫ ∆ θ coupling constant proportional to the mutual inductance Hˆ = σz 1 σx −2 − 2 − 2 of the qubit and the SQUID loops.   θ3 ¯h +Em nˆ2 Em + δI θ, (8.8) C J 6 2e − IX. PHYSICAL COUPLING SCHEMES FOR TWO QUBITS where ∆ = 2√EJ . The ramping is supposed to be adi- abatic so that the phase remains at the minimum point. Let us analyze the behavior of the potential minimum A. General principles by omitting a small kinetic term and diagonalizing the Hamiltonian (8.8). The corresponding eigenenergies de- A generic scheme for coupling qubits is based on pend on θ, the physical interaction of linear and non-linear os- cillators constituting a superconducting circuit. The 3 2 E m θ ¯h ∆ Hamiltonians for the SCB, rf-SQUID, and plain JJ con- E (θ)= EJ + δI θ, (8.9) ± ∓ 2 − 6 2e ± 4E   tain quadratic terms representing the kinetic and poten- tial energies plus the non-linear Josephson energy term, as shown in Fig. 26. Then within the interval of the bias which is quadratic when expanded to lowest order, currents, δI (2e/¯h)(∆2/4E), the potential energy corresponding| |≤− to the ground state has a local minimum, Hˆ = E (ˆn n )2 + E (1 cos φ) (9.1) C − g J − 27

SCB SCB

C3 Vg Vg FIG. 27: Fixed inductive (flux) coupling of elementary flux qubit. The loops can be separate, or have a common leg like in the figure. FIG. 28: Fixed capacitive coupling of charge qubits 2 2 (φ φe) Hˆ = EC nˆ + EJ (1 cos φ)+ EL − (9.2) − 2 for each qubit. The last matrix element is exponentially small, while the first two ones are approximately equal to 2 ¯h the minimum points of the potential energy, φl and φr, Hˆ = EC nˆ + EJ (1 cos φ)+ Ieφ. (9.3) − 2e respectively. This implies that the truncated interaction In a multi-qubit system the induced gate charge in the basically has the zz-form, SCB, or the flux through the SQUID loop, or the phase Hˆint = λσz1σz2, in the Josephson energy, will be a sum of contributions 2 from several (in principle, all) qubits. The energy of the 1 ¯h 1 λ = (L− )12 (φl φr)1(φl φr)2. (9.7) system can therefore not be described as the sum of two 8 2e − −   independent qubits because of the quadratic dependence, and the cross terms represent interaction energies of dif- ferent kinds: capacitive, inductive and phase/current. C. Capacitive coupling of charge qubits Moreover, using JJ circuits as non-linear coupling ele- ments we have the advantage that the direct physical One of the simplest coupling schemes is the capacitive coupling strength may be controlled, e.g tuning the in- coupling of charge qubits. Such a coupling is realized by ductance via current biased JJs, or tuning the capaci- connecting the islands of two SCBs via a small capacitor, tance by a voltage biased SCB. as illustrated in Fig. 28. This will introduce an addi- tional term in the Lagrangian of the two non-interacting SCBs, Eq. (5.27), namely the charging energy of the B. Inductive coupling of flux qubits capacitor C3, C V 2 A common way of coupling flux qubits is the inductive δL = 3 3 . (9.8) coupling: magnetic flux induced by one qubit threads 2 the loop of another qubit, changing the effective external The voltage drop V3 over the capacitor is expressed via flux. This effect is taken into account by introducing the the phase differences across the qubit junctions, inductance matrix L , which connects flux in the i-th ik ¯h loop with the current circulating in the k-th loop, V = (φ˙ φ˙ ), (9.9) 3 2e 1 − 2 Φi = LikIk. (9.4) and thus the kinetic part of the Lagrangian (5.28) will Xk take the form The off-diagonal element of this matrix, L12, is the mu- 1 ¯h 2 tual inductance which is responsible for the interaction. K(φ˙1, φ˙2)= Cikφ˙iφ˙k By using the inductance matrix, the magnetic part of the 2 2e   Xi,k potential energy in Eq. (6.16) can be generalized to the 2 ¯h case of two coupled qubits, C V φ˙ , (9.10) −2e gi gi i 2 i 1 ¯h 1 X (L− )ik(φi φei)(φk φek). (9.5) 2 2e − − where the capacitance matrix elements are Cii = CΣi +   ik X C3, and C12 = C3. Then proceeding to the circuit quan- Then following the truncation procedure explained in tum Hamiltonian as described in Section VI, we find the Section VII C, we calculate the matrix elements, interaction term, 2 1 l φ˜ f l , r φ˜ f r , l φ˜ f r , (9.6) Hˆ =2e (C− ) nˆ nˆ . (9.11) h | − | i h | − | i h | − | i int 12 1 2 28

SCB SCB C3 JJ1 JJ2

JJ Vg Vg

FIG. 30: Capacitive coupling of single JJ qubits

FIG. 29: Fixed phase coupling of charge qubits

This interaction term is diagonal in the charge basis, and SCB SCB therefore leads to the zz-interaction after truncation,

2 e 1 Hˆ = λσ σ , λ = (C− ) . (9.12) int z1 z2 2 12 L C The qubit Hamiltonians are given by Eq. (7.5) with charging energies renormalized by the coupling capaci- Vg Vg tor.

FIG. 31: Two charge qubits coupled to a common LC- D. JJ phase coupling of charge qubits oscillator.

Instead of the capacitor, the charge qubits can be con- nected via a Josephson junction154, as illustrated in Fig. turns to σy, and the qubit-qubit interaction takes the 29, yy-form, In this case, the Josephson energy of the coupling junc- tion EJ3 cos(φ1 φ2) must be added to the Lagrangian 2 − ˆ 2 ¯h ωp1ωp2 1 in addition to the charging energy. This interaction term Hint = λσy1σy2, λ =2e (C− )12. (9.15) s EC1EC2 is apparently off-diagonal in the charge basis and, after truncation, gives rise to xx- and yy-couplings, E Hˆ = λ(σ σ + σ σ ), λ = J3 , (9.13) F. Coupling via oscillators int x1 x2 y1 y2 4 or equivalently, Besides the direct coupling schemes described above, several schemes of coupling qubits via auxiliary oscilla- Hˆint =2λ(σ+,1σ ,2 + σ ,1σ+,2). (9.14) 42 − − tors have been considered . Such schemes provide more flexibility, e.g. to control qubit interaction, to couple two remote qubits, and to connect several qubits. Moreover, E. Capacitive coupling of single JJs in many advanced qubits, the qubit variables are generi- cally connected to the outside world via an oscillator (e.g. Capacitive coupling of JJ qubits, illustrated in Fig. 30 the Delft and Saclay qubits). To explain the principles is described in a way similar to the charge qubit, in terms of such a coupling, we consider the coupling scheme for of the Lagrangian Eqs. (9.8), (9.9), and the resulting charge qubits suggested by Shnirman et al.14. interaction Hamiltonian has the form given in Eq. (9.11).

Generally, in the qubit eigenbasis, 0 and 1 , all ma- trix elements of the interaction Hamiltonian| i are| i non-zero. 1. Coupling of charge (SCB, SCT) qubits However, if we adopt a parabolic approximation for the Josephson potential, then the diagonal matrix elements In this circuit the island of each SCB is connected to turn to zero, n00 = n11 = 0, while the off-diagonal matrix ground via a common LC-oscillator, as illustrated in Fig. 1/4 elements remain finite, n01 = n10 = i(EJ /EC ) . 31. The kinetic energy (5.27) of a single qubit should Then, after truncation, the charge− number− operatorn ˆ now be modified taking into account the additional phase 29 difference φ across the oscillator, JJ 2 ˙ ˙ 1 ¯h ˙2 ˙ ˙ 2 K(φ ,i; φ)= 2Cφ ,i + Cg(Vgi φ φ ,i) . SCT − 2 2e − − − − SCT   h (9.16)i The cross term in this equation can be made to vanish by a change of qubit variable,

Cg φ ,i = φi aφ, a = . (9.17) − − CΣ Vg Vg The kinetic energy will then split into two independent parts, the kinetic energy of the qubit in Eq. (5.28), and an additional quadratic term, FIG. 32: Charge (charge-phase) qubits coupled via a com- mon Josephson junction providing phase coupling of the two 1 ¯h 2 CC circuits g φ˙2, (9.18) 2 2e C   Σ which should be combined with the kinetic energy of the the displacement does not change the oscillator ground oscillator, leading to renormalization of oscillator capac- state energy, which then drops out after the averaging, we finally arrive at the Hamiltonian of the direct effective itance. Expanding the Josephson energy, after the change of qubit coupling, variable, gives λ1λ2 Hˆint = σy1σy2. (9.23) E cos(φ aφ) E cos φ E aφ sin φ . (9.19) − EL Ji i − ≈ Ji i − Ji i provided the amplitude of the oscillations of φ is small. for the oscillator-coupled charge qubits in Fig. 31. The last term in this equation describes the linear cou- pling of the qubit to the LC-oscillator. Collecting all the terms in the Lagrangian and perform- 2. Current coupling of SCT qubits ing quantization and truncation procedures, we arrive at the following Hamiltonian of the qubits coupled to the Charge qubits based on SCTs can be coupled by con- oscillator (this is similar to Eq. (8.4) for the SCT), necting loops of neighboring qubits by a large Josephson junction in the common link155,156,157,158,159,160,161, as il- Hˆ = (HˆSCB,i + λiσyiφ)+ Hˆosc, (9.20) lustrated in Fig. 32, i=1,2 The idea is similar to the previous one: to couple qubit X variables to a new variable, the phase of the coupling ˆ (i) where HSCB is given by Eq. (7.5), and Josephson junction, then to arrange the phase regime for the junction with large plasma frequency (E Ccoupl ≪ EJiCg EJcoupl), and then to average out the additional phase. λi = , (9.21) CΣ Technically, the circuit is described using the SCT Hamil- tonian, Eqs. (7.7), (7.8), for each qubit, is the coupling strength, and ˆ 2 2 ˆ 2 2 HSCT = EC (ˆn ng) + EC nˆ+ Hosc = ECoscnˆ + ELφ /2, (9.22) − − 2 (2φ+ φe) 2EJ cos φ+ cos φ + EL − , (9.24) is the oscillator Hamiltonian where the term in Eq. (9.18) − − 2 has been included. The physics of the qubit coupling in this scheme is and adding the Hamiltonian of the coupling junction, the following: quantum fluctuation of the charge of one 2 qubit produces a displacement of the oscillator, which Hˆc = EC,cnˆ EJ,c cos φc. (9.25) c − perturbs the other qubit. If the plasma frequency of the LC oscillator is much larger than the frequencies of all The phase φc across the coupling junction must be added qubits, then virtual excitation of the oscillator will pro- to the flux quantization condition in each qubit loop; ˜ duce a direct effective qubit-qubit coupling, the oscillator e.g., for the first qubit 2φ+,1 + φc = φe,1 + φ1 (for the staying in the ground state during all qubit operations. second qubit the sign of φc will be minus). Assuming To provide a small amplitude of the zero-point fluctu- small inductive energy, EL EJ,c, we may neglect φ˜; ations, the oscillator plasma frequency should be small then assuming the flux regime≪ for the coupling Joseph- compared to the inductive energy, or ECosc EL. Then son junction we adopt a parabolic approximation for the ≪ 2 the fast fluctuations can be averaged out. Noticing that junction potential, EJ,cφc /2. 30

With these approximations, the Hamiltonian of the first qubit plus coupling junction will a take form sim- ilar to Eq. (9.24) where EJ,c will substitute for EL, and φ will substitute for2φ φ . Finally assuming the am- c + − e plitude of the φc-oscillations to be small, we proceed as in the previous subsection, i.e. expand the cosine term obtaining linear coupling between the SCB and the oscil- lator, truncate the full Hamiltonian, and average out the oscillator. This will yield the following interaction term,

λ1λ2 φi Hˆint = σx1σx2, λi = EJ sin (9.26) EJ,c 2

This coupling scheme also applies to flux qubits: in this FIG. 33: Flux transformer with variable coupling controlled case, the coupling will have the same form as in Eq. (9.7), by a SQUID. but the strength will be determined by the Josephson energy of the coupling junction, cf. Eq. (9.26), rather than by the mutual inductance. on the controlling flux is given by the equation167,

1 Φ E πΦ − G. Variable coupling schemes 2 = 1+ J cos cx , (9.27) Φ E Φ 1  L 0  Computing with quantum gate networks basically as- sumes that one-and two-qubit gates can be turned on where EJ is the Josephson energy of the SQUID junction, and off at will. This can be achieved by tuning qubits and EL is the inductive energy of the transformer. with fixed, finite coupling in and out of resonance, in NMR-style computing162. Here we shall discuss an alternative way, namely to vary the strength of the physical coupling between 2. Variable Josephson coupling nearest-neighbour qubits, as discussed in a number of re- cent papers155,156,158,159,160,163,164,165,166. A variable Josephson coupling is obtained when a sin- gle Josephson junction is substituted by a symmetric dc SQUID whose effective Josephson energy 2EJ cos(φe/2) 1. Variable inductive coupling depends on the magnetic flux threading the SQUID loop (see the discussion in Section V C). This property is To achieve variable inductive coupling of flux qubits commonly used to control level spacing in both flux and one has to be able to the control the mutual inductance charge qubits introduced in Section V D, and it can also of the qubit loops. This can be done by different kinds be used to switch on and off qubit-qubit couplings. For of controllable switches (SQUIDS, transistors)163 in the example, the coupling of the charge-phase qubits via circuit. In a recent experiment, a variable flux trans- Josephson junction in Fig. 32 can be made variable former was implemented as a coupling element (see Fig. by substituting the single coupling junction with a dc 33) by controlling the transforming ratio167. The flux SQUID155,156. transformer is a superconducting loop strongly induc- The coupling scheme discussed in Section IX F 1 is tively coupled to the qubit loops, which are distant from made controllable by using a dc SQUID design for the each other so that the direct mutual qubit inductance is SCB as explained in Section V D). Indeed, since the cou- negligibly small. Because of the effect of quantization of pling strength depends on the Josephson energy of the magnetic flux in the transformer loop112, the local vari- qubit junction, Eq. (9.21), this solution provides vari- ation of magnetic flux Φ1 induced by one qubit will af- able coupling of the qubits. Similarly, the coupling of fect a local magnetic flux Φ2 in the vicinity of the other the SCTs considered in Section IXF2 can be made con- qubit creating effective qubit-qubit coupling. When a dc trollable by employing a dc SQUID as a coupling ele- SQUID is inserted in the transformer loop, as shown in ment. A disadvantage of this solution is that the qubit Fig. 33, it will shortcircuit the transformer loop, and the parameters will vary simultaneously with varying of the transformer ratio Φ2/Φ1 will change. The effect depends coupling strength. A more general drawback of the dc on the current flowing through the SQUID, and is propor- SQUID-based controllable coupling is the necessity to tional to the critical current of the SQUID. The latter is apply magnetic field locally, which might be difficult to controlled by applying a magnetic flux Φcx to the SQUID achieve without disturbing other elements of the circuit. loop, as explained in Section V C and shown in Fig. 33. This is however an experimental question, and what are Quantitatively, the dependence of the transformer ratio practical solutions in the long run remains to be seen. 31

Ib

SCT SCT

Vg Vg FIG. 35: Variable capacitance tuned by a voltage-controlled SCB.

FIG. 34: Coupled charge qubits with current-controlled phase coupling: the arrow indicates the direction of the controlling nˆ2, and it also depends on the gate voltages of the qubits bias current. and the coupling junction. In contrast to the previous scheme, here the coupling junction is not supposed to be in the phase regime; however, it is still supposed to be 3. Variable phase coupling fast, EJ EJi. Then the energy gap in the spectrum of the coupling≫ junction is much bigger than the qubit An alternative solution for varying the coupling energy, and the junction will stay in the ground state is based on the idea of controlling the properties during qubit operations. Then after truncation, and av- of the Josephson junction by applying external dc eraging out the coupling junction, the Hamiltonian of the 158,159,160 current , as illustrated in Fig. 34. circuit will take the form, Let us consider the coupling scheme of Section IX F 2: the coupling strength here depends on the plasma fre- Hˆ = HˆSCB,i + ǫ0 (σz1 + σz2) , (9.30) quency of the coupling Josephson junction, which in turn i depends on the form of the local minimum of the junction X potential energy. This form can be changed by tilting the where the qubit Hamiltonian is given by Eq. (7.5), and junction potential by applying external bias current (Fig. the function ǫ0 is the ground state energy of the coupling 34), as discussed in Section V A. The role of the exter- junction. The latter can be generally presented as a linear nal phase bias, φe, will now be played by the minimum combination of terms proportional to σz1σz2 and σz1 + point φ0 of the tilted potential determined by the applied σz2, bias current, EJ,c sin φ0 = (¯h/2e)Ie. Then the interaction term will read, ǫ0 (σz1 + σz2)= α + νσz1σz2 + β (σz1 + σz2) . (9.31)

2 2 with coefficients depending on the gate potentials. The EJ sin (φ0/2) Hˆint = λσx1σx2, λ = , (9.28) second term in this expression gives the zz-coupling (in E cos φ J,c 0 the charge basis), and the coupling constant ν may, ac- 165 and local magnetic field biasing is not required. cording to the analysis of Ref. , take on both positive and negative values depending on the coupling junction gate voltage. In particular it may turn to zero, implying 4. Variable capacitive coupling qubit decoupling.

Variable capacitive coupling of charge qubits based on H. Two qubits coupled via a resonator a quite different physical mechanism of interacting SCB charges has been proposed in Ref.165. The SCBs are then connected via the circuit presented in Fig. 35. In the previous discussion, the coupling oscillator plays The Hamiltonian of this circuit, including the charge a passive role, being enslaved by the qubit dynamics. qubits, has the form However, if the oscillator is tuned into resonance with a qubit, then the oscillator dynamics will become essential, Hˆ = Hˆ +E (ˆn q(ˆn +ˆn ))2 E cos φ, (9.29) leading to qubit-oscillator entanglement. In this case, SCB,i C − 1 2 − J i the approximation of direct qubit-qubit coupling is not X appropriate; instead, manipulations explicitly involving where EC and EJ Ec are the charging and Josephson the oscillator must be considered. energies of the coupling∼ junction, andn ˆ and φ are the Let us consider, as an example, operations with two charge and the phase of the coupling junction. The func- charge qubits capacitively coupled to the oscillator. As- tion q is a linear function of the qubit charges,n ˆ1, and suming the qubits to be biased at the degeneracy point 32 and proceeding to the qubit eigenbasis (phase basis in The manipulation should not necessarily be step-like, it this case), we write the Hamiltonian on the form (cf. Eq. is sufficient to pass the resonance rapidly enough to pro- (7.10), vide the Landau-Zener transition, i.e. the speed of the frequency ramping should be comparable to the qubit ∆i level splittings. Hˆ = σzi λiσx φi + Hˆosc[φ]. (9.32) − 2 − References to recent work on the entanglement of i=1,2   X qubits and oscillators will be given in Section IV C. Let us consider the following manipulation involving the variation of the oscillator frequency (cf. Ref164): at time t = 0, the oscillator frequency is off-resonance with both X. DYNAMICS OF MULTI-QUBIT SYSTEMS qubits, A. General N-qubit formulation ¯hω(0) < ∆1 < ∆2. (9.33) Then the frequency is rapidly ramped so that the oscil- A general N-qubit Hamiltonian with general qubit- lator becomes resonant with the first qubit, qubit coupling can be written on the form

¯hω(t ) = ∆ , (9.34) ǫi ∆i 1 1 1 Hˆ = ( σ + σ ) + λ σ σ − 2 zi 2 xi 2 ν,ij νi νj i i,j;ν the frequency remaining constant for a while. Then the X X frequency is ramped again and brought into resonance (10.1) with the second qubit, To solve the Schr¨odinger equation ¯hω(t2) = ∆2. (9.35) Hˆ ψ = E ψ (10.2) Finally, after a certain time it is ramped further so that | i | i the oscillator gets out of resonance with both qubits at we expand the N-qubit state in a complete basis, e.g. the N N the end, 2 basis states of the (σz) operator,

¯hω(t>t3) > ∆2. (9.36) ψ = a q | i q| i When passing through the resonance, the oscillator is = a0 0..00 + a1 0..01 + a2 0..10 + ...a(2N X1) 1..11 | i | i | i − | i hybridized with the corresponding qubit, and after pass- (10.3) ing the resonance, the oscillator and qubit have become entangled. For example, let us prepare our system at and project onto the basis states t = 0 in the excited state ψ(0) = 100 = 1 0 0 , | i | i| i| i where the first number denotes the state of the oscilla- Hˆ p p ψ = E ψ (10.4) | ih | i | i tor (first excited level), and the last numbers denote the p (ground) states of the first and second qubits, respec- X tively. After the first operation, the oscillator will be obtaining the ususal matrix equation entangled with the first qubit, q Hˆ p ap = Eaq (10.5) iα h | | i ψ(t t3) = cos θ1 cos θ2 100 + sin θ2e 001 | i | i B. Two qubits, longitudinal (diagonal) coupling + sin θ eiα 010 . (9.38) 1 | i
To ensure that there are no more resonances during The first case is an Ising-type model Hamiltonian, rel- the described manipulations, it is sufficient to require evant for capacitively or inductively coupled flux qubits, ¯hω(0) > ∆2 ∆1. − ǫi ∆i If the controlling pulses are chosen so that θ2 = π/2, H = ( σ + σ ) + λ σ σ (10.7) − 2 zi 2 xi 12 z1 z2 then the initial excited state will be eliminated form the i=1,2 final superposition, and we’ll get entangled states of the X qubits, while the oscillator will return to the ground We expand a general 2-qubit state in the the σz basis state, q = kl , { } ψ(t>t ) = 0 cos θ eiβ 01 ) + sin θ eiα 10 . 3 | i 1 | i 1 | i (9.39) ψ = a1 00 + a2 01 + a3 10 + a4 11 (10.8)
| i | i | i | i | i 33

With q = kl and p = mn , we get dealing with pure charge states. The secular equation { } { } factorises according to ǫ ∆ H = kl i σ + i σ ) + λ σ σ mn det (Hˆ E)= qp −h | 2 zi 2 xi 12 z1 z2| i − i=1,2 2 2 2 2 X = [(λ E )+ ǫ(λ + E)][(λ E ) ǫ(λ + E)]=0 ǫ ∆ − − − = ( 1 k σ m + 1 k σ m ) l n (10.13) − 2 h | z1| i 2 h | x1| i h | i ǫ ∆ giving the eigenvalues k m ( 2 l σ n )+ 2 l σ n ) −h | i 2 h | z2| i 2 h | x2| i + λ k σ m l σ n E1 = ǫ + λ , E2,3 = λ , E4 = ǫ + λ (10.14) 12 h | z1| ih | z2| i − − (10.9) The corresponding eigenvectors and 2-qubit states are given by The longitudinal zz-coupling only connects basis states with the same indices (diagonal terms), kl kl . Eval- E = ǫ + λ : uation of the matrix elements results in| thei ↔ Hamiltonian | i 1 matrix (setting λ12 = λ, ǫ1 + ǫ2 =2ǫ, ǫ1 ǫ2 = 2∆ǫ) − a2 = a3 = a4 = 0 (10.15)

1 1 ǫ + λ 2 ∆2 2 ∆1 0 1 − − 1 ψ1 = 00 (10.16) ˆ 2 ∆2 ∆ǫ λ 0 2 ∆1 | i | i H =  − 1 − − 1  (10.10) 2 ∆1 0 ∆ǫ λ 2 ∆2 E = λ : − 1 − 1 − − 2,3  0 ∆1 ∆2 ǫ + λ  −  − 2 − 2 −    a = a = 0 (10.17) The one-body operators can only connect single- 1 4 particle states, and therefore the rest of the state must be unchanged (unit overlap). Zero overlap matrix elements ψ = 01 , (a =0); ψ = 10 , (a = 0) (10.18) | 2i | i 3 | 3i | i 2

k m = δ , l n = δ (10.11) h | i k,m h | i l,n E = ǫ + λ : therefore make some matrix elements zero. 4 − The two-body operators describing the qubit-qubit a = a = a = 0 (10.19) coupling connect two-particle states with up to two dif- 1 2 3 ferent indices (single-particle basis states). In the present case, the qubit coupling Hamiltonian H2 = λ σz1σz2 has ψ4 = 11 (10.20) the charge basis as eigenstates and cannot couple differ- | i | i ent charge states. It therefore only contributes to shifting the energy eigenvalues on the diagonal by λ. In con- 2. Biasing at the degeneracy point ± trast, qubit coupling Hamiltonians involving σx1σx2 or σy1σy2 couplings contribute to the off-diagonal matrix Parking both qubits at the degeneracy point, ǫ = 0, elements (in the charge basis), and specifically remove the secular equation factorises in a different way, the zero matrix elements above. In the examples below we will specifically consider the 2 2 1 2 det ( E) = 0 = [(λ E )+ (∆1 + ∆2) ] case when ǫ1 ǫ2 = 2∆ǫ = 0, which can be achieved H− − 4 by tuning the− charging energies via fabrication or gate 1 [(λ2 E2)+ (∆ ∆ )2] voltage bias. The eigenvalues of the matrix Schr¨odinger × − 4 1 − 2 equation with the Hamiltonian given by (10.10) are then (10.21) determined by again giving a set of exact energy eigenvalues 2 2 1 2 det (Hˆ E) = 0 = [(λ E )+ (∆1 + ∆2) ] − − 4 1 1 E = λ2 + (∆ + ∆ )2 (10.22) [(λ2 E2)+ (∆ ∆ )2] ǫ2 (λ + E)2 1,4 ± 4 1 2 × − 4 1 − 2 − r (10.12) 2 1 2 E2,3 = λ + (∆1 ∆2) (10.23) ±r 4 − 1. Biasing far away from the degeneracy point where E1 < E2 < E3 < E4. For E = E1,4, the corre- sponding eigenvectors and 2-qubit states are given by

Far away from the degeneracy points in the limit ǫ ∆1 + ∆2 ≫ a1 = a4 , a2 = a3; , a2 = a1 (10.24) ∆1, ∆2, the Hamiltonian matrix is diagonal and we are − λ +2E 34

E1 = a1( 00 + 11 )+ a2( 01 + 10 ) (10.25) Inserting the energy eigenstates from expressions | i | i | i | i | i (10.25),(10.26),(10.29),(10.30) we obtain the time evolu- E = a ( 00 + 11 ) a ( 01 + 10 ) (10.26) tion in the charge basis, | 4i 2 | i | i − 1 | i | i After normalisation ψ(t) = 2 iE1t 2 iE1t 2 iE2t | 2 iEi2t 00 a e− + a e + b e− + b e | i 1 2 1 2 1 λ 1 λ iE1t iE1t iE2t iE2t a1 = 1+ ; a2 = 1 (10.27) + 01 a1a2(e− + e )+ b1b2(e− + e ) 2 2E1 2 − 2E1 | i s s iE1t iE1t iE2t iE2t | | | | + 10 a a (e− + e ) b b (e− + e ) | i  1 2 − 1 2  For E = E2,3, the corresponding eigenvectors and 2- 2 iE1t 2 iE1t 2 iE2t 2 iE2t + 11 a e− + a e b e− b e qubit states are given by | i 1 2 − 1 − 2  (10.36) ∆ ∆   b = b , b = b ; , b = b 1 − 2 (10.28) 1 − 4 2 − 3 2 − 1 λ +2E This state could be the input state for another gate oper- ation, where some other part of the Hamiltonian is sud- E = b ( 00 11 )+ b ( 01 10 ) (10.29) denly varied. It might also be that this is the state to | 2i 1 | i − | i 2 | i − | i be measured on. For that purpose one needs in principle E = b ( 00 11 ) b ( 01 10 ) (10.30) | 3i 2 | i − | i − 1 | i − | i the probabilities After normalisation 2 p1(t)= 00 ψ(t) (10.37) |h | i|2 1 λ 1 λ p2(t)= 01 ψ(t) (10.38) b1 = 1+ ; b2 = 1 (10.31) |h | i|2 2 2E 2 − 2E p3(t)= 10 ψ(t) (10.39) s | 2| s | 2| |h | i| p (t)= 11 ψ(t) 2 (10.40) 4 |h | i| 3. Two-qubit dynamics under dc-pulse excitation As one example, we will calculate p3(t)+ p4(t), which represents the probability of finding the qubit 1 in the upper state 1 , independently of the state of qubit 1 We now investigate the dynamics of the two-qubit state | i in the specific case that the dc-pulse is applied equally (the states of which are summed over). to both gates, ǫ1(t) = ǫ2(t), where ǫ1 = ǫ1(ng1) = 2 2 2 2 p3(t)= (a1 + a2)cos E1t (b1 + b2)cos E2t EC (1 2ng1), ǫ2 = ǫ2(ng2) = EC (1 2ng2), taking the | − − − i (a2 a2) sin E t (b2 b2) sin E t 2 (10.41) system from the the pure charge-state region to the co- − 1 − 2 1 − 1 − 2 2 |  2 degeneracy point and back. This is the precise analogy p4(t)= [2ia1a2 sin E1t 2ib1b2 sin E2t] (10.42) of the single-qubit dc-pulse scheme discussed in Section | − | IVC. Adding p3(t) and p4(t) we obtain Since the charge state 00 of the starting point does not have time to evolve during| i the steep rise of the dc- p3(t)+ p4(t)= pulse, it remains frozen and forms the initial state 00 1 1 1 = cos E1t cos E2t + χ sin E1t sin E2t at time t = 0 at the co-degeneracy point, where it can| bei 2 − 4 4 expanded in the energy eigenbasis, 1 1 = (1 + χ)cos E+t + (1 χ)cos E t (10.43) 2 − 4 − − 00 = ψ(0) = c1 E1 + c2 E2 + c3 E3 + c4 E4 | i | i | i | i | i | i where E+ = E2 + E1, E = E2 E1, and where χ = (10.32) 2 λ − − (1 2 E1E2 ). To find the coefficients we project onto the charge basis, − | | kl=0,1 C. Two qubits, transverse x-x coupling kl 00 = c kl E + c kl E + c kl E + c kl E h | i 1h | 1i 2h | 2i 3h | 3i 4h | 4i (10.33) This model is relevant for charge qubits in current- and use the explict results for the energy eigenstates to coupled loops. The Hamiltonian is now given by calculate the matrix elements, obtaining ǫ ∆ Hˆ = ( i σ + i σ + λ σ σ (10.44) − 2 zi 2 xi x1 x2 00 = a E + b E b E a E (10.34) i=1,2 | i 1| 1i 1| 2i− 2| 3i− 2| 4i X For t > 0 this stationary state then develops in time Proceeding as before we have governed by the constant Hamiltonian as

iE1t iE2t ψ(t) = a1e− E1 + b1e− E2 ǫi ∆i | i | i | i− Hqp = kl σzi σxi) + λ σz1σz2 mn +iE2t +iE1t h | 2 − 2 | i b2e E3 a2e E4 (10.35) i=1,2 − | i− | i X 35

ǫ ∆ = ( 1 k σ(1) m 1 k σ m ) l n 2 h | z | i− 2 h | x1| i h | i ǫ ∆ + k m ( 2 l σ n ) 2 l σ n ) h | i 2 h | z2| i 2 h | x2| i + λ k σ m l σ n h | x1| ih | x2| i (10.45)

This coupling only connects basis states which differ in both indices (”anti-diagonal” terms), i.e. 00 11 and 01 10 , meaning that interaction will| i only ↔ | appeari FIG. 36: Control pulse sequences involved in quantum state |oni the ↔ anti-diagonal. | i Evaluation of the matrix elements manipulations and measurement. Top: microwave voltage pulses u(t) are applied to the control gate for state manipula- result in the Hamiltonian matrix (again setting λ = 12 tion. Middle: a readout dc ac pulse (DCP) or ac pulse (ACP) λ, ǫ1 + ǫ2 =2ǫ, ǫ1 ǫ2 = 2∆ǫ) − Ib(t) is applied to the threshold detector/discriminator a time 1 1 td after the last microwave pulse. Bottom: output signal V(t) ǫ 2 ∆2 2 ∆1 λ from the detector. The occurence of a output pulse depends 1 − − 1 ˆ 2 ∆2 ∆ǫ1 λ 2 ∆1 on the occupation probabilities of the energy eigenstates. A H =  − 1 − 1  , (10.46) 2 ∆1 λ ∆ǫ 2 ∆2 discriminator with threshold Vth converts V(t) into a boolean − 1 −1 −  λ ∆1 ∆2 ǫ  0/1 output for statistical analysis.  − 2 − 2 −   

XI. EXPERIMENTS WITH SINGLE QUBITS In Fig. 36 the readout control pulse can be a dc pulse AND READOUT DEVICES (DCP) or ac pulse (ACP). A DCP readout most often leads to an output voltage pulse, which may be quite In this section we shall describe a few experiments destructive for the quantum system. An ACP readout with single-qubits that represent the current state-of- presents a much weaker perturbation by probing the the-art and quite likely will be central components in ac-response of an oscillator coupled to the qubit, creating the developmement of multi-qubit systems during the much less back action, at best representing QND readout. next five to ten years. The first experiment presents Rabi oscillations induced and observed in the elemen- tary phase qubit and readout oscillator formed by a single JJ-junction30,31,32,33,34. The next example de- 24 scribes a series of recent experiments with a flux qubit 1. Spectroscopic detection of Rabi oscillation coupled to different kinds of SQUID oscillator read- out devices25,26,168. A further example will discuss the charge-phase qubit coupled to a JJ-junction oscillator21 In the simplest use of the classical oscillator, it does and the recent demonstration of extensive NMR-style op- not discriminate between the two different qubit states, eration of this qubit23. The last example will present the but only between energies of radiation emitted by a lossy case of a charge qubit (single Cooper pair box, SCB) resonator coupled to the qubit. In this way it is possible coupled to a microwave stripline oscillator147,148,151,152, to detect the ”low-frequency” Rabi oscillation of a qubit representing a solid-state analogue of ”cavity QED”. driven by continuous (i.e. not pulsed) high-frequency Before describing experiments and results, however, we radiation tuned in the vicinity of the qubit transition en- will discuss in some detail the measurement procedures ergy. If the oscillator is tunable, the resonance window that give information about resonance line profiles, Rabi can be swept past the Rabi line. Alternatively, the Rabi oscillations, and relaxation and decohrence times. The frequency can be tuned and swept past the oscillator win- 27 illustrations will be chosen from Vion et al.21 and related dow by changing the qubit pumping power . work for the case of the charge-phase qubit, but the ex- amples are relevant for all types of qubits, representing fundmental procedures for studying quantum systems. 2. Charge qubit energy level occupation from counting electrons: rf-SET A. Readout detectors In this case, the charge qubit is interacting with a beam Before discussing some of the actual experiments, it of electrons passing through a single-electron transistor is convenient to describe some of basic readout-detector (SET) coupled to a charge qubit (e.g. the rf-SET,69), as principles which more or less the same for the SET, rf- discussed in Section VIII and illustrated in Fig. 22. In SET, JJ and SQUID devices. A typical pulse scheme for these cases the transmissivity of the electrons will show exciting a qubit and reading out the response is shown two distinct values correlated with the two states of the in Fig. 36: qubit. 36

I

Ic

2∆ eV FIG. 38: Schematics of readout dc SQUID coupled to flux FIG. 37: Current-voltage characteristic of tunnel junction qubit; left - inductive coupling, right - direct phase coupling (solid line) consists of the Josephson branch - vertical line at V = 0, and the dissipative branch - curve at eV ≥ 2∆. When the current is ramped, the junction stays at the Josephson branch and when the current approaches the critical value Ic, the junction switches to the dissipative branch (dashed line). pacitance Csh and lead inductance L is used to ”tune” ωp). Thus, the plasma frequency takes different values (0) (1) ωp or ωp depending on the state of qubit, representing 3. Coupled qubit-classical-oscillator system: switching two different shapes of the SQUID oscillator potential. detectors with dc-pulse (DCP) output In the dc-pulse-triggered switching SQUID7,24,25, a dc-current readout-pulse is applied after the operation In Section VIII we analyzed the case of an SCT qubit pulse(s) (Fig. 36), setting a switching threshold for the current-coupled to a JJ-oscillator (Fig. 25) and discussed critical current. The circulating qubit current for one the Hamiltonian of the coupled qubit-JJ-oscillator sys- qubit state will then add to the critical current and make tem. The effect of the qubit was to deform the oscillator the SQUID switch to the voltage state, while the other potential in different ways depending on the state of the qubit state will reduce the current and leave the SQUID qubit. The effect can then be probed in a number of in the zero-voltage state. ways, by input and output dc and ac voltage and current In an application of ac-pulse-triggered switching pulses, to determine the occupation of the qubit energy SQUID26, readout relies on resonant activation by a mi- levels. crowave pulse at a frequency close to ωp, adjusting the Using non-linear oscillators like single JJs or SQUIDS power so that the SQUID switches to the finite voltage one can achieve threshold and switching behaviour where state by resonant activation if the qubit is in state 0 , the JJ/SQUID switches out of the zero-voltage state, re- whereas it stays in the zero-voltage state if it is in state| i sulting in an output dc-voltage pulse. 1 . The resonant activation scheme is similar to the read- Switching JJ: The method is based on the dependence |outi scheme used by Martinis et al.30,31,32,33. (see Section of the critical current of the JJ on the state of the qubit, XI C). and consists of applying a short current DCP to the JJ at a value Ib during a time ∆t, so that the JJ will switch out of its zero-voltage state with a probability Psw(Ib). For well-chosen parameters, the detection efficiency can approach unity. The switching probability then directly 4. Coupled qubit-classical-oscillator system: ac-pulse measures the qubit’s energy level population. (ACP) non-switching detectors Switching SQUID: In the experiments on flux qubits by the Delft group, two kinds of physical coupling of This implementation of ACP readout uses the qubit- the SQUID to the qubit have been implemented, namely 7 inductive coupling (Fig. 38 (left))7,168 and direct cou- SQUID combination shown in Fig. 38 (left), but with pling (Fig. 38 (right)):24,25,26 The critical current of ACP instead of DCP readout, implementing a nonde- structive dispersive method for the readout of the flux the SQUID depends on the flux threading the loop, and 168 therefore is different for different qubit states. The prob- qubit . The detection is based on the measurement lem is to detect a two percent variation in the SQUID of the Josephson inductance of a dc-SQUID inductively coupled to the qubit. Using this method, Lupascu et critical current associated with a transition between the 168 qubit states in a time shorter than the qubit energy relax- al. measured the spectrum of the qubit resonance line and obtained relaxation times around 80 µs, much longer ation time T1. The SQUID behaves as an oscillator with a 1/2 than observed with DCP. characteristic plasma frequency ωp = [(L + LJ )Csh]− . This frequency depends on the bias current Ib and on A related readout scheme was recently implemented by 169 the critical current IC via the Josephson inductance Siddiqi et al. using two different oscillation states of L =Φ /2πI 1 I2/I2 (the shunt capacitor with ca- the non-linear JJ in the zero-voltage state. J 0 C − b c p 37

FIG. 39: Left: qubit energy level scheme. The vertical dashed line marks the qubit working point and transition energy. The FIG. 40: Decay of the switching probability of the charge- arrow marks the detuned microwave excitation. Right: pop- qubit readout junction as a function of the delay time td ulation of the upper level as a function of the detuning; the between the excitation and readout pulses. Courtesy of D. inverse of the half width (FWHM) of the resonance line gives Esteve, CEA-Saclay. the total decoherence time T2. Courtesy of D. Esteve, CEA- Saclay. .

B. Operation and measurement procedures

A number of operation and readout pulses can be ap- plied to a qubit circuit in order to measure various prop- erties. The number of applied microwave pulses can vary depending on what quantities are to be measured: reso- nance line profile, relaxation time, Rabi oscillation, Ram- sey interference or Spin Echo, as discussed below.

1. Resonance line profiles and T2 decoherence times

To study the resonance line profile, one applies a sin- FIG. 41: Left: Rabi oscillations of the switching probability gle long weak microwave pulse with given frequency, fol- measured just after a resonant microwave pulse of duration. lowed by a readout pulse. The procedure is then repeated Right: Measured Rabi frequency (dots) varies linearly with for a spectrum of frequencies. The Rabi oscillation am- microwave amplitude (voltage) as expected. Courtesy of D. plitude, the upper state population, and the detector Esteve, CEA-Saclay. switching probability p(t) will depend on the detuning and will grow towards resonance. The linewidth gives directly the total inverse decoherence lifetime 1/T2 = 3. Rabi oscillations and T2,Rabi decoherence time 1/2T1 +1/Tφ. The decoherence-time contributions from relaxation (1/T1) and dephasing (1/Tφ) can be (approx- imately) separately measured, as discussed below. To study Rabi oscillations (frequency Ω u, the am- plitude of driving field) one turns on a∼ resonant mi- crowave pulse for a given time tµw and measures the up- per 1 state population (probability) p1(t) after a given 2. T1 relaxation times | i (short) delay time td. If the systems is perfectly coher- ent, the state vector will develop as cos Ωt 0 +sinΩt 1 , | i | i To determine the T1 relaxation time one measures the and the population of the upper state will then oscil- decacy of the population of the upper 1 state after a late as sin2 Ωt between 0 and 1. In the presence of de- | i long microwave pulse saturating the transition, varying coherence, the amplitude of the oscillation of p1(t) will the delay time td of the detector readout pulse. The decay on a time scale TRabi towards the average value measured T1 =1.8 microseconds is so far the best value p1(t = )=0.5. This corresponds to incoherent satura- for the Quantronium charge-phase qubit. tion of∞ the 0 to 1 transition. 38

FIG. 42: Ramsey fringes of the switching probability af- ter two phase-coherent microwave π/2 pulses separated by the time delay t. The continuous line represents a fit by FIG. 43: Spin-echo experiment. The left part shows the exponentially damped cosine function with time constant ∗ basic Ramsey oscillation. The right part shows the echo signal T2 = Tφ = 0.5µs. The oscillation period coincides with the appearing in the time window around twice the time delay inverse of the detuning frequency δ (here δ = ν − ν01 = 20.6 between the first π/2-pulse and the π-pulse. Courtesy of D. MHz). Courtesy of D. Esteve, CEA-Saclay. Esteve, CEA-Saclay.

5. Spin-echo 4. Ramsey interference, dephasing and T2,Ramsey decoherence time The spin-echo and Ramsey pulse sequences differ in that a π-pulse around the x-axis is added in between the two π/2-pulses in the spin-echo experiment, as shown in The Ramsey interference experiment measures the de- Fig.43. As in the Ramsey experiment, the first π/2-pulse coherence time of the non-driven, freely precessing, qubit. makes the Bloch vector start rotating in the equatorial In this experiment a π/2 microwave pulse around the x- x-y plane with frequency E/¯h = ν . The effect of the axis induces Rabi oscillation that tips the spin from the 01 π-pulse is now to flip the entire x-y plane with the rotat- north pole down to the equator. The spin vector rotates ing Bloch vector around the x-axis, reflecting the Bloch in the x-y plane, and after a given time ∆t, another π/2 vector in the x-z plane. The Bloch vector then continues microwave pulse is applied, immediately followed by a to rotate in the x-y plane in the same direction. Finally readout pulse. a second π/2-pulse is applied to project the state on the Since the π/2 pulses are detuned by δ from the qubit z-axis. 0 1 transition frequency, the qubit will precess If two Bloch vectors with slightly different frequency |withi → frequency | i δ relative to the rotating frame of the start rotating at the same time in the x-y plane, they will driving field. Since the second microwave pulse will be move with different angular speeds. The effect of the π- applied in the plane of the rotating frame, it will have pulse at time ∆t will be to permute the Bloch vectors, a projection cos δt on the qubit vector and will drive and then let the motion continue in the same direction. the qubit towards the north or south poles, resulting This is similar to reversing the motion and letting the in a specific time-independent final superposition state Bloch vectors back-trace. The net result is that the two cos δt 0 + sin δt 1 of the qubit at the end of the last Bloch vectors re-align after time 2∆t. π/2 pulse.| i The readout| i pulse then catches the qubit In NMR experiments, the different Bloch vectors cor- in this superposition state and forces it to decay if the respond to different spins in the ensemble. In the case qubit is in the upper 1 state. The probability will of a single qubit, the implication is that in a series of re- oscillate with the detuning| i frequency, and a single-shot peated experiments, the result will be insensitive to small experiment will then detect the upper state with this variations δE of the qubit energy between measurements, probability. Repeating the experiment many times for as long as the energy (rotation frequency) is constant different pi/2 pulse separation ∆t will then give 0 or during one and the same measurement. If fluctuations 1 with probabilities cos2 δt and sin2 δt. Taking| i the occur during one measurement, then this cannot be cor- average,| i and then varying the pulse separation, will rected for. The spin-echo procedure can therefore remove trace out the Ramsey interference oscillatory signal. the measurement-related line-broadening associated with Dephasing will make the signa decay on the timescale Tφ. slow fluctuations of the qubit precession, and allow ob- servation of the intrinsic coherence time of the qubit. 39

C. NIST Current-biased Josephson Junction Qubit

Several experimental groups have realized the Joseph- son Junction (JJ) qubit29,30,31,32,33,34. Here we describe the experiment performed at NIST32,33. In this experi- ment the junction parameters and bias current were cho- sen such that a small number of well defined levels were formed in the potential well (Fig. 21), with the inter- level frequencies, ω01/2π =6.9GHz and ω12/2π6.28GHz, the quality factor was Q=380. The experiment was per- formed at very low temperature, T= 25 mK.

The qubit was driven from the ground state, 0 , to FIG. 44: Upper panel: Scanning electron micrograph of a the upper state, 1 , by the resonance rf pulse with| i fre- | i small-loop flux-qubit with three Josephson junctions of crit- quency ω01, and then the occupation of the upper qubit ical current ∼ 0.5 mA, and an attached large-loop SQUID level was measured. The measurement was performed by with two big Josephson junctions of critical current ∼ 2.2 mA. exciting qubit further from the upper level to the auxil- Arrows indicate the two directions of the persistent current iary level with higher escape rate by applying the second in the qubit. Lower panels: Schematic of the on-chip cir- resonance pulse with frequency ω12. During the whole cuit; crosses represent the Josephson junctions. The SQUID operation, across the junction only oscillating voltage de- is shunted by two capacitors to reduce the SQUID plasma fre- velops with zero average value over the period. When the quency and biased through a small resistor to avoid parasitic tunneling event occurred, the junction switched to the resonances in the leads. Symmetry of the circuit is introduced dissipative regime, and finite dc voltage appeared across to suppress excitation of the SQUID from the qubit-control pulses. The MW line provides microwave current bursts in- the junction, which was detected. Alternatively, post- ducing oscillating magnetic fields in the qubit loop. The cur- measurement classical states ”0” and ”1” differ in flux rent line provides the measuring pulse Ib and the voltage line by Φ0, which is readily measured by a readout SQUID. allows the readout of the switching pulse V out. Adapted from Chiorescu et al.24

The relaxation rate was measured in the standard way by applying a Rabi pumping pulse followed by a mea- D. Flux qubits suring pulse with a certain delay. Non-exponential relax- ation was observed, first rapid, 20 nsec, and then more slow, 300 nsec. By reducing the∼ length of the pumping 1. Delft 3-junction persistent current qubit with dc-pulse (DCP) readout pulse∼ down to 25 nsec, i.e. below the relaxation time, Rabi oscillations were observed. In this experiment the amplitude of the pumping pulse rather than duration was The original design of the 3-junction qubit (Fig. 1 was varied, which affected the Rabi frequency and allowed the published by Mooij et al. in 19996, and the first spec- observation of oscillation of the level population for fixed troscopic measurements by van der Wal et al. in 20007 duration of the pumping pulse. (and simultaneously by Friedman et al.8 for a single- junction rf-SQUID). Recently the Delft group has also investigated designs where the 3-junction qubit is shar- The Grenoble group34 has observed coherent oscil- ing a loop with the measurement SQUID to increase the lations in a multi-level quantum system, formed by a coupling strength24, as shown in Fig. 44. current-biased dc SQUID. These oscillations have been To observe and study Rabi oscillations, the qubit was induced by applying resonant microwave pulses of flux. biased at the degeneracy point and the qubit 0 1 Quantum measurement is performed by a nanosecond transition excited by a pulse of 5.71 GHz microwave| i → ra-| i flux pulse that projects the final state onto one of two diation of variable length, followed by a bias-current (Ib) different voltage states of the dc SQUID, which can be readout pulse applied to the SQUID (Fig. 44, lower left read out. The number of quantum states involved in the panel). The first (high) part of the readout pulse (about coherent oscillations increases with increasing microwave 10 ns) has two functions: It displaces the qubit away from power. The dependence of the oscillation frequency on the degeneracy point so that the qubit eigenstates carry microwave power deviates strongly from the linear regime finite current, and it tilts the SQUID potential so that expected for a two-level system and can be very well ex- the SQUID can escape to the voltage state if the qubit is plained by a theoretical model taking into account the in its upper state. The purpose of the long lower plateau anharmonicity of the multi-level system. of the the readout pulse is to prevent the SQUID from 40

FIG. 45: AFM picture of the charge-phase qubit (”quantro- nium”) corresponding to the circuit scheme in Fig. 25. The working point is controlled by two external ”knobs”, a voltage FIG. 46: Charge-phase qubit energy surface. (Ng = ng ; gate controlling the induced charge (ng) on the SCT island, δ/2π = φe.) Courtesy of D. Esteve, CEA-Saclay. and a current gate controlling the total phase across the SCT via the external flux (φe) threading the loop. The large read- out JJ is seen in the left part of the figure. Courtesy of D. Esteve, CEA-Saclay. ∆(φe)=2EJ cos(φe/2). The qubit energy levels 1 E = ǫ(n )2 + ∆(φ )2 (11.2) 1,2 ∓2 g e returning (”retrapping”) to the zero-voltage state. With q these operation and readout techniques, Rabi (driven) then form a 2-dimensional landscape as functions of the oscillations, Ramsey (free) oscillations and spin-echos of gate charge ng(V ) and phase φe(Φ), which are functions 24 the qubit were observed , giving a Ramsey free oscil- of the gate voltage V = Cg2eng and gate magnetic flux lation dephasing time of 20 ns and spin echo dephasing Φ = (¯h/2e)φe. The energy level surfaces are therefore time of 30 ns. functions of two parameters, gate voltage and flux, giv- As mentioned above, relaxation times around 80 µs ing us two independent knobs for controlling the working have been measured with ACP readout168, demonstrat- point of the (charge-phase) qubit. Expanding the energy ing that the properties of readout devices are criti- in Taylor series around some bias working point (Vb, Φb), cally important for observing intrinsic qubit decoherence one obtains times. Recent further improvements have resulted in δE(V, Φ) = E(V, Φ) E(Vb, Φb) Ramsey decoherence times T2,Ramsey of 200 ns, Rabi 2 − 2 (driven) decoherence times T2,Rabi of 5 µs, and relax- δE δE 1 δ E 2 1 δ E 2 = δV + δΦ+ 2 δV + 2 δΦ (11.3) ation times T1 of more than 100 µs. δV δΦ 2 δV 2 δΦ The derivatives are response functions for charge, cur- rent, capacitance and inductance (omitting the cross E. Charge-phase qubit term) (i =1, 2),

2 2 1. General considerations δEi(V, Φ) = Qi δV + Ii δΦ+ Ci δV + Li δΦ (11.4) or, equivalently, As described in Section VII, the charge-phase qubit 2 2 circuit consists of a single-Cooper-pair transistor (SCT) δEi(ng, φe)= Q˜i δng + I˜i δφe + C˜i δng + L˜i δφe in a superconducting loop, The Hamiltonian for the SCT (11.5) part of this circuit is given by On the energy level surfaces (Fig. 46), the special point 1 (n , φ ) = (0.5, 0) is an extreme point with zero first Hˆ = [ ǫ(n ) σ + ∆(φ ) σ ] (11.1) g e SCT −2 g z e x derivative. This means that the energies of the 0 and 1 states will be invariant to first order to small| varia-i where the charging and tunneling parameters are them- tions| i of charge and phase, which will minimize the qubit selves functions of external control parameters, gate sensitivity to fluctuations of the working point caused by charge n and loop flux φ , ǫ(n ) = E (1 2n ) and noise. g e g C − g 41

FIG. 48: Free-evolution decoherence times for the quantron- 170 FIG. 47: The charge-phase qubit frequency surface. (Ng = ium charge-phase SCT qubit. (Ng = ng ; δ/2π = φe.) Full ng ; δ/2π = φe.) Courtesy of D. Esteve, CEA-Saclay. and dashed curves represent results of theoretical modeling. Courtesy of D. Esteve, CEA-Saclay.

The point (ng, φe) = (0.5, 0) is often referred to as the ”degeneracy” point because the charging energy is zero, ǫ(ng) = EC (1 2ng) = 0. The level splitting at this point is determined− by the Josephson tunneling in- teraction EJ , and is a function of the external bias φe) 22 ∆(φe)=2EJ cos(φe/2). The spin-echo results in Ref. gave a lifetime as long as Figure 47 shows the frequency surface ν01 = ∆E/h, T2 = 1µs. ∆E = E E . During operation, the qubit is preferably 2 − 1 parked at the degeneracy point (ng, φe) =(0.5, 0) where Q = I = 0, in order to minimize the decohering influence In a recent systematic experimental and theoretical in- of noise. In order to induce a qubit response, for gate vestigation of a specific charge-phase device, investigat- operation or readout, one must therefore either (i) move ing the effects of relaxation and dephasing on Rabi os- 170 the bias point away from point (0.5, 0) to have finite first- cillation, Ramsey fringes and spin-echos , one obtains order response with Q =0or I = 0, or (ii) stay at (0.5, 0) the following picture of different coherence times for the and apply a perturbing6 ac-field6 that makes the second- quantronium charge-phase qubit: order response significant.

Fluctuations of charge δng and flux δφe will shift energy levels and make the qubit transition energy 2. The CEA-Saclay ”quantronium” charge-phase qubit 2 2 ǫ(ng + δng) + ∆(φe + δφe) fluctuate. However, the charge degeneracy point is a saddle point, which means The ”quantronium” charge-phase circuit is given by pthat at that working point the qubit transition frequency Fig. 45 adding a large readout Josephson junction in is insensitive to low-frequency noise to first order, allow- the loop (cf. Fig. 25). The minimum linewidth (Fig. 39) ing long coherence times. As can be seen in Fig. 48, corresponds to a Q-value of 20000 and a decoherence time the coherence is sharply peaked ound the ”magic” de- T2 = 0.5 µs. With the measured relaxation time T1=1.8 generacy point. The spin-echo experiment indicates the µs (Fig. 40), the dephasing time can be estimated to presence of slow charge fluctuations from perturbing im- Tφ=0.8 µs. purity two-level systems (TLS) (1/f noise) and that long A set of results for Rabi oscillations of the Saclay coherence time can be recovered by spin-echo techniques quantronium qubit is shown in Fig. 41. The decoher- until the decoherence becomes too rapid at significant ence time represents decoherence under driving condi- distances from the magic point along the charge axis. In tions. Another, more fundamental, measure of decoher- contrast, in the experiments with this device the decoher- ence is the free precession dephasing time T2 when the ence due to phase fluctuations could not be significantly qubit is left to itself. This is measured in in the Ramsey compensated for by spin-echo techniques, indicating that two-pulse experiment, as already described above in Fig. the phase noise (current and flux noise) in this device is 21,22 42 showing the CEA-Saclay data , giving T2 = 0.5 µs. high-frequency noise. 42

XII. EXPERIMENTS WITH QUBITS COUPLED TO QUANTUM OSCILLATORS

A. General discussion

The present development of quantum information pro- cessing with Josephson Junctions (JJ-QIP) goes in the direction of coupling qubits with quantum oscillators, for operation, readout and memory. In Section VIII we dis- cussed the SCT, i.e. a single Cooper-pair transistor in a superconducting loop71, providing one typical form of the Hamiltonian for a qubit-oscillator coupled system. FIG. 49: Qubit-oscillator level structure. The notation for We also showed with perturbation theory how the Hamil- the states is: |qubit; oscillatori = |0/1; n = 0, 1, ..i; (a) E ≈ tonian gave rise to qubit-dependent deformed oscillator 2¯hω: very large ”detuning”, weak coupling. (b) E ≈ ¯hω: potential and shifted oscillator frequency. In VIII, in resonance, strong coupling and hybridization, level (”vacuum Rabi”) splitting. addition we discussed the SCT charge-coupled to an LC- oscillator, which is also representative for a flux qubit coupled to a SQUID oscillator, and which describes a charge qubit in a microwave cavity. Figure 49 shows the basic level structure of the qubit- To connect to the language of quantum optics and cav- oscillator system in the cases of (a) weak and (b) strong ity QED and the current work on solid-state applications, coupling. we now explicitly introduce quantization of the oscillator. What is weak or strong coupling is determined by the Quantizing the oscillator, φ˜ (a+ + a), a representative strength of the level hybridization, which in the end de- form of the qubit-oscillator Hamiltonian∼ reads: pends on qubit-oscillator detuning and level degeneracies. The perturbative Hamiltonian in Eq. (12.7) is valid for Hˆ = Hˆ + Hˆ + Hˆ (12.1) large detuning (g δqr) and allows us to approach the q int osc ≪ 1 case of strong coupling between the qubit and the oscil- Hˆ = E σ (12.2) q −2 z lator. Close to resonance the degenerate states have to be treated non-perturbatively. Hˆ = g σ (a+ + a) (12.3) int x Figure 49(a) corresponds to the weak-coupling (non- + 1 Hˆosc =¯hω (a a + ) (12.4) resonant) case where the levels of the two subsystems are 2 only weakly perturbed by the coupling, adding red and blue sideband transitions 01 10 and 00 11 to Introducing the step operators σ = σx i σy, the inter- ± ± the main zero-photon transition| i → |00 i 10| . i → | i action term can be written as | i → | i Figure 49(b) corresponds to the resonant strong- + + Hˆint = g (σ+ a + σ a )+ g (σ+ a + σ a) (12.5) coupling case when the 01 > and 10 > states are de- − − generate, in which case the| qubit-oscillator| coupling pro- The first term duces two ”bonding-antibonding” states in the usual way, and the main line becomes split into two lines (”vacuum + Hˆint = g (σ+ a + σ a ) (12.6) − Rabi splitting”. Of major importance are the linewidths of the qubit and the oscillator relative to the splittings represents the resonant (co-rotating) part of the interac- caused by the interaction. To discriminate between the tion, while the second term represents the non-resonant two qubit states, the oscillator shift must be larger than counter-rotating part. In the rotating-wave approxima- the average level width. tion (RWA) one only keeps the first term, which gives the Since the qubit and the oscillator form a coherent Jaynes-Cummings model171,172,173,174. Diagonalizing the multi-level system, as described before (Section X B 3), Jaynes-Cummings Hamiltonian to second order by a uni- the time evolution can be written as c (t) 00 +c (t) 01 + tary transformation gives 1 2 c (t) 10 +c (t) 11 , which in general does| noti reduce| toi a 3 | i 4 | i 2 2 product of qubit and oscillator states, and therefore rep- 1 g g + H = (E + ) σz + (¯hω + σz) a a (12.7) resents (time-dependent) entanglement. This provides −2 δ δ opportunities for implementing quantum gate operation where δ = E ¯hω is the so called detuning. The re- involving qubits and oscillators. sult illustrates− what we have already discussed in detail, Generation and control of entangled states can be namely that (i) the qubit transition energy E is shifted achieved by using microwave pulses to induce Rabi oscil- (renormalized) by the coupling to the oscillator, and (ii) lation between specific transitions of the coupled qubit- the oscillator energyhω ¯ is shifted by the qubit in differ- oscillator system, and the result can be studied by spec- ent directions depending on the state of the qubit, which troscopy on suitable transitions or by time-resolved de- allows discriminating the two qubit states. tection of suitable Rabi oscillations. 43

FIG. 50: Resonant frequencies indicated by peaks in the SQUID switching probability when a long (300 ns) microwave pulse excites (saturates) the system before the readout pulse. Data are represented as a function of the external flux Φext through the qubit area away from the qubit symmetry point. The blue |00i → |11i and red |01i → |10i sidebands are shown by down- and up-triangles, respectively; continuous lines are obtained by adding 2.96 GHz and -2.90 GHz, respectively, to the central continuous line (numerical fit). These values are close to the oscillator resonance frequency νp at 2.91 GHz (solid circles). Courtesy of J.E. Mooij, TU Delft. FIG. 51: Generation and control of entangled states via π and 2π Rabi pulses on the qubit transition |00i → |10i, followed by Rabi driving of blue sideband transitions |00i→.A π pulse B. Delft persistent current flux qubit coupled to a excites the system from |00i to |10i, which suppresses the blue quantum oscillator sideband transitions |00i → |11i (second curve from the top). On the other hand, with a 2π pulse the system returns to |00i, which allows Rabi oscillation on the blue sideband. Adapted 25 The Delft experiment demonstrates entanglement from Chiorescu et al.25 between a superconducting flux qubit and a SQUID quantum oscillator. The SQUID provides the measure- ment system for detecting the quantum states (threshold switching detector, Fig. 44). It is also presents an effec- dynamics in microwave multi-pulse experiments: by co- tive inductance that, in parallel with an external shunt herently (de)populating selected levels via proper timing capacitance, acts as a low-frequency harmonic oscillator; of Rabi oscillations induced by a first pulse, a second the qubit and oscillator frequencies are approximately pulse can induce Rabi oscillations on another transition connected to the levels controlled by the first pulse. Al- hνq = hν01 6 GHz and hνr 3 GHz, corresponding to Fig. 49(a).≈ ≈ ternatively, such Rabi oscillations can instead be blocked In the Delft experiment25, performing spectroscopy on by the first excitation. the coupled qubit-oscillator multi-level system reveals the This is shown experimentally in Fig. ??: by preparing variation of the main and sideband transitions with flux the initial state with initial π and 2π pulses, the side- bias Φext, as shown in Fig. 50. The presence of visible band Rabi oscillations could be turned off and on again. sideband transitions demonstrates the qubit-oscillator in- The corresponding Rabi oscillations are shown in the left teraction and (weak) level hybridization). part of Fig. ??(b), demonstrating rapid decay due to the strong damping of the SQUID oscillator (lifetime 3 ns; Short microwave pulses can now be used to induce Rabi ∼ oscillations between the various qubit-oscillator transi- Q=100-150). tions. In the Delft experiment25, microwave pulses with In the previous example, the control pulse (first pulse) frequency νq = ν01 ∆/h 5.9 GHz (qubit symme- was applied to the main 00 10 transition, control- try point) were first≈ used to≈ induce Rabi oscillations ling the populations of the| i ”0” → | andi ”1” levels. In the with 25 ns decay time at the main qubit transition next example, microwave control pulse is applied to the 00 ∼ 10 (and 01 11 ) for different values of 00 01 transition, inducing Rabi oscillations which the| i microwave → | i power.| i The → | Rabii frequency as function |populatei → | thei first excited state of oscillator. A second of the amplitude of the microwave voltage demonstrated microwave pulse (in principle, with different frequency) qubit-oscillator hybridization (avoided crossings) at the can then induce Rabi oscillations on the 01 10 | i →25 | i oscillator νp (νr) and Larmor ∆/h (!) frequencies. transition (red-sideband). The experimental result is The dynamics of the coupled qubit-oscillator system shown Fig. 53, Clearly, for sufficiently long qubit and os- can be studied by inducing microwave-driven Rabi oscil- cillator lifetimes, one can prepare entangled Bell states, 1 1 lation between the blue 00 11 and red 01 10 2 ( 00 01 ) and 2 ( 01 10 ) by applying π/2 mi- sidebands. In particular| onei → can | studyi the| conditionali → | i crowave| i ± pulses | i to the |00i ± | 11i and 01 10 tran- | i → | i | i → | i 44

FIG. 54: Yale SCB charge-phase qubit coupled to an oscil- lator in the form of a superconducting microwave stripline resonator. Adapted from Wallraff et al.151

C. Yale charge-phase qubit coupled to a strip-line resonator

In the Yale experiments151,152 a coherent qubit- quantum oscillator system is realized in the form of a Cooper pair box capacitively coupled to a superconduct- FIG. 52: Generation and control of entangled states via π and ing stripline resonator (Fig. 54) forming one section of a 2π Rabi pulses on the qubit transition |00i → |10i, followed microwave transmission line. The stripline resonator, a by Rabi driving of red sideband transitions |10i → |01i.A π finite length (24 mm) of planar wave guide, is a solid-state pulse excites the system from |00i to |10i, which In the right analogue of the cavity electromagnetic resonator used in panel, a π pulse excites the system from |00i to |10i, which quantum optics to study strong atom-photon interaction populates the |10i state and allows Rabi oscillation on the red sideband transitions |01i → |10i. Adapted from Chiorescu et and entanglement. In the Yale experiments, the qubit is al.25 placed in the transverse field in the gap between the res- onator strip lines, i.e. inside the microwave cavity. The EC and EJ parameters are such that the SCT is effec- tively in the charge-phase region, and it is operated by controlling both the charge and phase (flux) ports. The large effective electric dipole moment d of Cooper pair box and the large vacuum electric field E0 in the transmission line cavity lead to a large vacuum Rabi fre- quency νRabi =2dE0/h, which allows reaching the strong coupling limit of cavity QED in the circuit. In the Yale experiment, spectroscopic measurements are performed by driving the qubit with resonant mi- crowave pulses and simultaneously detecting the fre- quency and intensity-dependent amplitude and phase of probe pulses of microwave radiation sent through a trans- mission line coupled to the stripline resonator via in- put/out capacitors (Fig. 54). The oscillator frequency is fixed at hνr 6 GHz. The qubit level separation, on the other hand,≈ is tunable over a wide frequency range FIG. 53: Generation and control of entangled states and around 6 GHz in two independent ways: (i) by vary- study of decay and lifetimes. Here a Rabi π pulse on the os- ing the magnetic flux φe through the loop, forming the cillator transition |00i → |01i populates the state |01i, which phase (flux) gate V D, or (ii) by varying the charge ng via then allows Rabi oscillation on the red sideband transition the voltage gate, primarily around the charge degeneracy |01i → |10i. The decay of both the Rabi oscillation and the point, ng=1/2, to minimize the effect of charge fluctua- average probability gives evidence for short oscillator life time tions. (∼ 3 ns; Q=100-150). Adapted from Chiorescu et al.25 The flux bias is used to tune the qubit transition fre- quency at ng=1/2 to values larger or smaller than the res- onator frequency. The tuning the qubit frequency with the charge gate will provide two distinct cases: the qubit always detuned from the oscillator, and the qubit being sitions. degenerate with the resonator at certain values of ng. 45

A central result151 is the evidence for the qubit- oscillator hybridization and splitting of the degenerate 01 and 10 states, ν01 ν , as illustrated in Fig. 49| i (right).| Thisi splitting is→ often± called ”vacuum Rabi” splitting because the ocillator is in its ground (vacuum) state. The 01 , 10 level splitting is observed through mi- crowave| excitationi | i of the 0 1 resonance transition and observing the splitting of→ the resonance line as the qubit and the oscillator are tuned into resonance (by tun- ing the qubit frequency to zero qubit-oscillator detuning, δ = ν ν = 0). q − r Another central result152 is the evidence for a long qubit dephasing time of T2 > 200 ns under optimal con- ditions: qubit parked at the charge degeneracy point, and weak driving field (low photon occupation number in the resonator cavity). In the experiment151 the qubit- oscillator detuning is large, the qubit resonance transi- tion νq = ν01 is scanned by microwave radiation, and the dispersive shift of the resonator frequency νr seen by a microwave probe beam is used to determine the occupa- FIG. 55: The NEC 2-qubit system: two capacitively cou- tion of the qubit levels. Scanning the qubit 00 10 pled charge qubits. The left qubit is a single Cooper pair resonance line profile allowed to determine the| lineshapei → | i box (SCB) and the right qubit is an SCT with flux-tunable Josephson energy. Courtesy of J.S. Tsai, NEC, Tsukuba, and linewith as a function of microwave power, giving the Japan. best value of T2 > 200 ns. Moreover, observation of the postion of the resonance as a function of microwave power allowed the determination of the ac-Stark shift, i.e. the XIII. EXPERIMENTAL MANIPULATION OF Rabi frequency as a function of the photon occupation COUPLED TWO-QUBIT SYSTEMS (electric field) of the cavity. The measurement induces an ac-Stark shift of 0.6 MHz per photon in the qubit level A. Capacitively coupled charge qubits separation. Fluctuations in the photon number (shot noise) induce level fluctuations in the qubit leading to An AFM picture of the NEC SCB 2-qubit system36 is dephasing which is the characteristic back-action of the shown in Fig. 55 and the corresponding circuit JJ circuit measurement. A cross-over from Lorentzian to Gaus- in Fig. 28 (note that in the NEC circuit the left SCT is sian resonance line shape with increasing measurement replaced by a simple SCB). power is observed and explained by dependence of the Two coupled qubits constitute a 4-level system. The resonance linewidth on the cavity occupation number, Hamiltonian for the NEC system of two capacitively cou- exceeding the linewidth due to intrinsic decoherence at pled charge qubits (SCBs) was analyzed in the Appendix. high rf-power. The four energy eigenvalues E1,2,3,4(ng1,ng2) are plot- ted in Fig. 56 as a functions of the gate charges. At each point in the parameter space, an arbitrary two-qubit state can be written as a superposition of the four two- qubit energy eigenstates. D. Comparison of the Delft and Yale approaches As described in Sections IV and X, the general proce- dure for operating with dc-pulses is to initialize the sys- tem to its ground state 00 at the chosen starting point 25 | i In comparison, the Delft experiment corresponds to (ng10,ng20) and then suddenly change the Hamiltonian the case of very large detuning (νq = 6 GHz, νr = 3 GHz; at time t = 0 to the gate bias (ng1,ng2). If the change is δ = νq νr νr; Fig. 49 (left)), so that one will observe strictly sudden, then the state at (ng1,ng2) at time t =0 a main− resonance∼ line 00 10 and two sidebands, is ψ(t = 0) = 00 , which can be expanded in the energy 00 11 (blue) and |01 i →10 | i(red). Decreasing the basis| of thei Hamiltonian,| i |detuningi → | δi, the blue and| i red → sidebands | i will move away ψ(0) = 00 = c E + c E + c E + c E (13.1) to higher resp. lower frequencies, and one will arrive at | i | i 1| 1i 2| 2i 3| 3i 4| 4i the case of zero detuning and qubit-oscillator degeneracy. This stationary state then develops in time governed by If the qubit-oscillator coupling is larger than the average the constant Hamiltonian as linewidth, one will then observe a splitting of the main iE1t iE2t line (Fig. 49 (right)). ψ(t) = c e− E + c e− E + | i 1 | 1i 2 | 2i 46

FIG. 57: The NEC 2-qubit system: pulse sequences for a CNOT gate (actually, the gate is a NOT − CNOT gate). Courtesy of J.S. Tsai, NEC, Tsukuba, Japan.

tional gate operation due to the different time evolution FIG. 56: The NEC 2-qubit system: Energy-level structure of the states departing from A or B. as a function of the gate charges ng1 and ng2, independently controlled by the gate voltages Vg1 and Vg2. Courtesy of J.S. Specifically, an entangling gate of controlled-NOT Tsai, NEC, Tsukuba, Japan. (CNOT) type was demonstrated by Yamamoto et al.36 using the pulse sequences shown in Fig. 57: First one applies a dc-pulse to gate 1 of the control qubit, mov- iE3t iE4t +c e− E + c e− E (13.2) 3 | 3i 4 | 4i ing out (down left) left from 00 (point A in Fig. 56) in the n direction to the single-qubit| i degeneracy point If one re-expands this state in the charge basis of the g1 (point E), staying for a certain time, and then moving starting point A (Fig. 56), then one obtains a system back (turning off the pulse), putting the system in a su- with periodic coefficients a (t), i perposition α 00 + β 10 (points A and B). Next one applies a dc-pulse| i to gate| i 2 of the target qubit, moving ψ(t) = a1(t) 00 + a2(t) 01 + a3(t) 10 + a4(t) 11 | i | i | i | i |(13.3)i out (right) in the ng2 direction and back. The pulse is developing in time through all of the charge states. designed to reach the first degeneracy point (point G), To perform a two-qubit conditional gate operation, one allowing the state to develop into a superposition of 00 performs a series of pulses moving the system around (point A) and 01 (point C) if the control state was 00. If in parameter space. Specifically, the NEC scheme is to instead the control state was 10 (point B), the dc-pulse apply sequential dc-pulses to the charge pulse gates of on gate 2 will not reach the two-qubit degeneracy point each of the two qubits in Fig. 55. Two cases have been (point H) and the development will be roughly adiabatic, studied so far: taking the state back to 10 (point B), never reaching 11 (point D). (1) Vg1(t) = Vg2(t): This is the case with common 35 control dc-pulses for studied by Pashkin et al. . Since With timing such (π/2-pulse) that 00 01 (A G ǫ1(t) = ǫ2(t), the plane of operation is at 45 degrees to → → → 35 C), the control gate leads to 00+10, and the target gate the axes in Fig. 56. As a result, Pashkin et al. observed only modifies the first component (00 01), resulting in interference effects and beating oscillations between the 01+10, i.e. one of the Bell states. This→ is shown in the two qubits, as described by Eq. (10.43). top panel of Fig. 57. (2) Vg1(t) = 0; Vg2(t), Vg1(t); Vg2(t) = 0: This is the case with separate sequential dc pulses on separate gates If instead one first applies a π/2-pulse in the ng2 di- recently studied by Yamamoto et al.36. The scheme is rection to the target qubit, inducing 00 01 (C), and → illustrated in Fig. 56: Starting at point A, first the sys- then applies a π/2-pulse in the ng1 direction to this state, tem is pulsed in the ng1 direction, putting the system reaching 01+11, and then again applies a π/2-pulse to the in a superposition of the states at points A and B; then target qubit 2, resulting 00+11, one obtains the other the system is pulsed in the ng2 direction, allowing condi- Bell state. This is shown in the bottom panel of Fig. 57. 47

FIG. 59: Circuit scheme for two capacitively coupled current (flux) biased JJ qubits with rf and dc control/readout lines. FIG. 58: Scanning microscope image of two inductively cou- Adapted from McDermott et al.178 pled qubits surrounded by a DC-SQUID. Courtesy of J. Ma- jer, TU Delft.

B. Inductively coupled flux qubits

Figure 58 shows a circuit with two inductively coupled flux qubits forming a four-level quantum system, excited by a single microwave line and surrounded by a single measurement SQUID37,38. With this circuit Majer et al.37,38 have spectroscopically mapped large portions of FIG. 60: Potential energies and quantized energy levels the the level structure and determined the device parameters Josephson phase qubit: left, during qubit operation; right, during state measurement, in which case the qubit well is entering in the Hamiltonian matrix Eq. (10.10), finding much shallower and state |1i rapidly tunnels to the right hand good agreement with the design parameters. Majer et al. well. Adapted from McDermott et al.178 found clear manifestations of qubit-qubit interaction and hybridization in the level structure as a function of bias 175 flux. Presently, ter Haar et al. are investigating a more become hybridized and split by the interaction. A mi- strongly coupled system with a JJ in the common leg (cf. crowave π-pulse tuned to the 0 1 transition of one of Fig. 32), inducing Rabi oscillations and performing con- the isolated qubits will ”suddenly”| i → | inducei a 00 10 ditional spectroscopy along the lines described in Section | i → | i transition, populating the E+ = 10 + 01 and E = XII in connection with the coupled qubit-oscillator sys- 10 01 states with equal| i weights.| i | Thisi will| leave−i tem. |thei system − | i oscillating between the two qubits, between iδEt A similar device with two flux qubits inside a cou- the 10 and 01 states, ψ(t) = E+ + e− E = pling/measurement SQUID has recently been investi- cos(δEt/| i 2) 10| +i sin(δEt/|2) 01i where| iδE = E | −Ei . 176,177 + gated by et al. , so far demonstrating Rabi oscilla- This means| thati the two-qubit| systemi oscillates− between− 39 tion of individual qubits. Finally, Izmalkov et al. have non-entangled and entangled states. With one ideal de- spectroscopically demonstrated effects of qubit-qubit in- tector for each qubit we can measure the state of each teraction and hybridization for two inductively coupled qubit with any prescribed time delay between measure- flux qubits inside a low-frequency tank circuit. ments. With simultaneous measurements we can deter- 2 mine all the probabilities pij = ij ψ(t) , ideally giv- 1 |h 1| i| ing p10 = 2 (1 + cos(δEt)), p01 = 2 (1 cos(δEt)), and C. Two capacitively coupled JJ qubits − p00 = p11 = 0 in the absence of relaxation, decoherence and perturbations caused by the detectors. Capacitive coupling of two JJ qubits (Section IX Figure 61 shows the actual experimental results of 178 E, Fig. 30) has recently been investigated by several Ref. for the probabilities pij (t): The p10 and p01 prob- groups40,178, showing indirect40 and direct178 evidence abilities oscillate out-of-phase, as expected. In addition, for qubit entanglement. Figure 59 shows the circuit used the average probabilities and oscillation amplitudes all by McDermott et al.178, The potential-well and level decay with time. The results are compatible178 with the structure of each JJ qubit under operation and measure- the finite rise time of the initial π-pulse (5 ns), the single- ment conditions are shown in Fig. 60, qubit readout fidelity (70 percent) and the single-qubit The 2-qubit circuit behaves as two dipole-coupled relaxation time T1 (25 ns), and the limited two-qubit pseudo-2-level systems, illustrated in Fig.49. With ”iden- readout fidelity, as used in the numerical simulations. Of tical” qubits, the 01 and 10 states are degenerate and particular interest is that for simultaneous (within 2 ns) | i | i 48

FIG. 62: EPR anticorrelation in the singlet state

A. Bell measurements

The first essential step is to study the quantum dy- namics of a two-qubit circuit and to perform a test of FIG. 61: Interaction of the coupled qubits in the time do- Bell’s inequalities by creating entangled two-qubit Bell main. The qubits are tuned into resonance and the system states (Section III, Ref.49) and performing simultaneous is suddenly prepared in state (10) by a microwave π-pulse projective measurements on the two qubits, similarly to applied to qubit 1. Simultaneous single-shot measurement 181,184 of each of the qubits 1 and 2 reads out the probabilities the ion trap experiments . Clearly the experiment p00,p10,p01,p11 for finding the 1+2 system in states 00, 10, will be a test of the quantum properties of the circuit 01, and 11, respectively. The full lines represent results of and the measurement process rather than a test of a Bell numerical simulations. Adapted from McDermott et al.178 inequality. The general principle is to (a) entangle the two qubits, (b) measure the projection along different detector axes (”polarization directions”), (c) perform four independent readout of the qubits, the experiments show that readout measurements, and finally (d) analyze the correlations. If of one qubit only leads to small perturbation of the other the detector axes are fixed, as is the case for the JJ read- qubit178, which is promising for multi-qubit applications. out (measuring charge or flux), one can instead rotate each of the qubits. Figure 62 shows the quantum network for creating a Bell state ψ , perform single qubit rotations R (φ ), | i x 1 Rx(φ2), and finally perform a projective measurement on each of the qubits along the same common fixed quan- XIV. QUANTUM STATE ENGINEERING WITH tization axis. For each setting of the angles, (φ1, φ2), MULTI-QUBIT JJ SYSTEMS on then performs a series of measurements obtaining the probabilities P = ij ψ 2 = ψ ij ij ψ . These can ij |h | i| h | ih | i Due to the recent experimental progress, protocols be combined into the results for finding the two qubits and algorithms for multi-qubit JJ systems can soon in the same state, Psame = P00 + P11, or in different begin to be implemented in order to test the perfor- states, Pdiff = P01 + P10, and finally into the difference q(φ , φ )= P P . This quantity q(φ , φ ) is eval- mance of JJ qubit and readout circuits and to study 1 2 same− diff 1 2 the full dynamics of the quantum systems. The gen- uated in four experiments for two independent settings eral principles are well known (see e.g.49,50) and have of the two angles, φ1 = (α, δ), φ2 = (β,γ), constructing very recently been successfully applied in several other the function systems to achieve interesting and significant results B(α, δ; β,γ)= q(α, β)+ q(δ, β) (14.1) in well-controlled quantum systems: ion-trap technol- | | ogy has been used to entangle 4 qubits180, implement + q(α, γ) q(δ, γ) | − | 2-qubit gates and test Bell’s inequalities181,184, per- form the Deutsch-Josza algorithm182, achieve telepor- The Clauser-Horne-Shimony-Holt (CSCH) tation (within the system)185,186,187, and perform er- condition192 for violation of classical physics is then ror correction188; quantum optics has recently demon- B > 2 (maximum value 2√2). strated long-distance teleportation190 as well as 4-photon The application to JJ charge-phase qubits has been entanglement190,191. There are presently a considerable discussed by Refs.193,194. Experimentally, a first step theoretical literature on how to implement these and sim- in this direction has been taken by Martinis et al.178 ilar protocols and algorithms in JJ circits. Below we will who directly detected the anticorrelation in the oscillat- describe a few examples to illustrate what may need to ing superposition of Bell states ψ(t) = cos(δEt/2) 10 + be done experimentally. sin(δEt/2) 01 . | i | i | i 49

FIG. 64: Teleportation in a 3-qubit system without measure- ment and classical transmission. The unknown qubit (1) to be FIG. 63: Teleportation in a 3-qubit system. The unknown transferred is again |ψi = a|0i+b|1i. The state of qubits 1 and qubit (1) to be teleported is |ψi = a|0i + b|1i. The result 2 are now used to control CNOT gates to restore the original of the measurement is sent by Alice via classical channels to single-qubit state on qubit 3. As a result of the teleportation, Bob who performs the appropriate unitary transformations to a specific but unknown state has been transferred from qubit restore the original single-qubit state. As a result of the tele- 1 to qubit 3, leaving qubits 1 and 2 in superposition states. portation, a specific but unknown state has been transferred from one qubit to another. unitary transformation back to the original state:

B. Teleportation I(a 0 + b 1 )= ψ (14.5) | i | i | i σ (b 0 + a 1 )= ψ (14.6) x | i | i | i In the simplest form of teleportation an unknown σ (a 0 b 1 )= ψ (14.7) z | i− | i | i single-qubit quantum state is transferred from one part σ σ ( b 0 + a 1 )= ψ (14.8) of the system to another, i.e. from one qubit to another, z x − | i | i | i as illustrated in Fig. 63. In Fig. 63, the initial state is corresponding to resp. no change, bit flip, phase flip given by and bit-plus-phase flip of the original unknown state ψ . Alice’s measurement causes instantaneous collapse, after| i (a 0 + b 1 )( 00 ) (14.2) | i | i | i which Alice by classical means (e.g. e-mail!) can tell Bob which unitary transformation to apply to recover Next, applying CNOT and Hadamard gates entangles the original single-qubit state. qubits 2 and 3 into a Bell state, An alternative approach, without measurement and classical transmission, is shown in Fig. 64, In this case, (a 0 + b 1 )(1/ 2)( 00 + 11 ) (14.3) | i | i ∗ | i | i the final 3-qubit state is still a disentangled product state with the correct state ψ = a 0 + b 1 on qubit 3, which is the resource needed for teleportation (in quan- | i | i | i tum optics this corresponds to the entangled photon pair 1 1 ( 0 + 1 ) ( 0 + 1 )(a 0 + b 1 ) (14.9) shared between Alice and Bob). One member of the Bell √2 | i | i √2 | i | i | i | i pair (qubit 2) is now ”sent” to Alice and entangled with the unknown qubit to be teleported, creating a 3-qubit while qubits 1 and 2 are now in superposition states. If entangled state, desired, these states can be rotated back to 00 by single qubit gates. | i 00 (a 0 + b 1 ) (14.4) The teleportation protocol has been implemented in | i | i | i ion trap experiments186,187 (and of course in quantum + 01 (b 0 + a 1 ) | i | i | i optics189). A proposal for a setup for implementing a + 10 (a 0 b 1 ) | i | i− | i teleportation scheme in JJ circuitry is shown in Fig. 65, + 11 ( b 0 + a 1 ) describing a 3-qubit chain of charge-phase qubits coupled | i − | i | i by current-biased large JJ oscillators, in principle allow- At this point, qubit 3 is sent to Bob, meaning that a 3- ing controllable nearest-neighbour qubit couplings158,159: qubit entangled state is shared between Alice and Bob. In the absence of bias currents, to first order the qubits Moreover, at this point, in each component of this 3-qubit are non-interacting and isolated from each other. The entangled state in Eq. (14.4), the two upper qubits are in basic two-qubit gates are achieved by switching on the eigenstates. This means that a projective measurement bias-current-controlled qubit-qubit interaction for a given of the these two qubits by Alice will collapse the total time. CNOT gates can be achieved in combination with state to one of the four components. If the result of single-qubit Hadamard and phase gates158,159,160. Byap- Alice’s measurement is (ij), the first two qubits are in plying the sequence of gates shown in Fig. 64 (or an the state ij and the 3-qubit state is known, given by equivalent sequence), the unknown state will be physi- the corresponding| i component in the previous equation. cally moved from the left end to the right end of the chain. Any of these 3-qubit products can be transformed by a Moreover, by applying coupling pulses simultaneously to 50

Qubit i−1 Qubit i Qubit i+1 φb φb φb i−2 i−1 i φ φ φφ φ φ 1(i−1) 2(i−1) 1i 2i 1(i+1) 2(i+1)

Ib Cgi Ib i−1 i Vgi

FIG. 65: Network of loop-shaped SCT charge qubits, coupled by large Josephson junctions. The interaction of the qubits FIG. 66: Teleportation with error correction in a 5-qubit (i) and (i + 1) is controlled by the current bias Ibi Individ- system without measurement. ual qubits are controlled by voltage gates, Vgi. Single-qubit readout is performed by applying an ac current pulse to a par- ticular JJ readout junction (not shown), or using an RF-SET capacitively coupled to the island [27]. several qubits, one can in principle create multi-qubit entangled states in fewer operations than with sequential two-qubit gates195,196 as well as perform operations in parallel on different parts of the system. Extending the system to a five-qubit chain one can in FIG. 67: Coding, decoding and correcting a bit flip error in a similar way transfer an entangled two-qubit state from a 3-qubit logic qubit. The first gate on the left represents two the left to the right end of the chain. sequential CNOT gates. The last gate befor the garbage can is a control-control-NOT (Toffoli) gate.

C. Qubit buses and entanglement transfer oretical studies213,214,215,216,217,218,219. With entangled ”flying qubits” like photons, quan- tum correlations can be shared in a spatially highly ex- tended states. However, with solid-state circuitry the D. Qubit encoding and quantum error correction issue becomes how to transfer entanglement among spa- tially fixed qubits. Quantum error correction (QEC) (see e.g. The standard answer is to apply entangling two-qubit Refs.49,220,221,222,223,224,225,226) will most certainly gates between distant qubits. The qubit-qubit interac- be necessary for successful operation of solid state quan- tion can be direct, e.g. dipole-dipole-type interaction be- tum information devices in order to fight decoherence. tween distant qubits, or mediated by excitations in the The algorithmic approach to QEC follows the principles system. The transfer can be mediated via virtual or real of classical error correction, by encoding bits to create excitations in a passive polarizable medium, a ”system redundancy, and by devising procedures for identifying bus”, or via a protocol for coupling qubits and bus oscil- and correcting the errors based on specific models for lators. the errors. A classic example of protocol-controlled oscillator- The quantum circuit in Fig. 66 illustrates QEC in 197 mediated coupling is the Cirac-Zoller gate between terms of five-qubit teleportation with bit errors, illus- two ions sequentially entangled via exchange coupling trating restoration of the state including error correction with the lowest vibrational mode of the ions in the trap. by measurement-controlled unitary transformation. 198,199,200 Another example is the Molmer-Sorensen gate The quantum circuit in Fig. 67 demonstrates some which creates qubit-qubit interaction via virtual ex- essential steps of QEC in the case of one logical qubit citation of ion-trap modes. Similar concepts for encoded in three physical qubits: The first step is to controlling entanglement have recently been theoreti- encode the physical qubit in logical qubit basis states cally investigated in applications to JJ-qubit-oscillator 000 and 111 by applying two CNOT gates to create a 164,201,202,203,204,205,206,207,208 circuits |3-qubiti entangled| i state: An different approach is to allow the bus (”spin-chain”) states to develop in time governed by the fixed bus Hamil- (a 0 + b 1 ) 00 (a 000 + b 111 ) (14.10) tonian and to tailor the interactions and the initital con- | i | i | i → | i | i ditions such that useful dynamics and information trans- Next, there may occur a bit-flip error in one of the qubits: fer is achieved209,210,211,212. Also these concepts have been applied to JJ-circuits in a number of recent the- a 000 + b 111 (no bit flip) (14.11) | i | i 51

qubits 2 and 3 without inducing new errors. Recently an error correcting procedure with cooling of the system without measurement has been proposed227. Phase flips (sign changes), e.g. a 0 +b 1 a 0 b 1 can be handled by similar 3-qubit encoding,| i | i decoding → | i− and| i correction. Combining these two approaches gives the 9- qubit Shor code220 for correcting also combined bit and phase flips, e.g. a 0 + b 1 a 1 b 0 . The minimum code to achieve the| i same| i thing → | isi− a 5-qubit| i code221 and there are more efficient codes with 7 qubits223,224,225. FIG. 68: Coding, decoding and correcting a bit flip error in A related approach to fighting decoherence is to en- 3-qubit logic qubit. code the quantum information in noiseless subsystems, so called Decoherence Free Subspaces (DFS)124,125,228,229 A different approach is the so called ”bang-bang” a 100 + b 011 (bit flip in qubit 1) (14.12) 230,231,232,233,234,235 | i | i dynamic correction method , basically a 010 + b 101 (bit flip in qubit 2) (14.13) corresponding to very frequent application of the spin- | i | i a 001 + b 110 (bit flip in qubit 3) (14.14) echo technique. This is related to the quantum Zeno | i | i effect (see e.g. Ref.236, describing situations where the The next step applies a number of disentangling CNOT quantum system via very strong interaction with the en- gates to check for the type of error. For the four possible vironment is forced into a subspace from where it cannot states above we obtain: decay. In this Section we have restricted the discussion to a 000 + b 111 (a 0 + b 1 ) 00 (14.15) a few different protocols for quantum state engineering, | i | i → | i | i | i a 100 + b 011 (a 1 + b 0 ) 11 (14.16) representing basic steps in algorithms for solving specific | i | i → | i | i | i a 000 + b 111 (a 0 + b 1 ) 10 (14.17) computational problems. For a discussion of quantum | i | i → | i | i | i algorithms and computational complexity we refer to the a 000 + b 111 (a 0 + b 1 ) 01 (14.18) 49 50 | i | i → | i | i | i books by Nielsen and Chuang and Gruska , and to At this stage, qubits 2 and 3 have become independent the original papers. We would however like to mention a eigenstates, just as in teleportation. The correspond- few papers discussing how to implement a few well-know algorithms in JJ circuitry. The basics of quantum gates ing eigenvalues are called syndrome, and indicate which 42,237,238 corrective operations should be implemented. Remark- in JJ-circuits can be found in e.g. Refs. . The Deutsch-Josza (DJ) and related algorithms have been ably, in two of the above bit-flipped cases, with error 238 syndromes ((01) and (10), qubit 1 is now correct, the er- discussed by Siewert and Fazio describing a 3-qubit ror residing in qubits 2 or 3 in the workspace (”ancillas”). DJ implementation with three charge-phase qubits con- The only case where a transformation is needed is when nected in a ring via phase-coupling JJs with variable Josephson coupling energy (SQUIDS), and by Schuch the error syndrom is (11), which corresponds to a bit 239 flip in qubit 1, to be corrected with a CCNOT (control- and Siewert analyzing a 4-qubit implementation . For control-NOT, or Toffoli, gate), controlled by the truth the Grover search algorithm there seems to be nothing table of an AND gate. In Fig. 67 we have used a com- published on the implementation in JJ circuitry. Con- pact single-gate notation for CCNOT, while in reality it cerning Shor’s factorization algorithm there is a recent paper by Vartiainen et al.242. There is also recent work must be implemented through a sequence of eight two- 243 49 on optimization of two-qubit gates , and relations be- qubit gates (there is no direct three-qubit interaction 208 in the Hamiltonian). At this stage, the 3-qubit state is tween error correction and entanglement . completely disentangled into a product state. Finally there are a number of papers connecting JJ- The final step consists in re-encoding the physical qubit ciruits wih geometric phases and holononomic quantum computing246,247,248,249 and on adiabatic computation252 1 to the logical qubit a 000 + b 111 . However, although 250,251 the physical qubit 1 is| alwaysi correct| i at this stage, the and Cooper pair pumps . For a recent paper dis- cussing the universality of adiabatic quantum computing, workspace is not. This can be handled in a number of 253 ways. Figure 67 dumps the ”hot” qubits in the garbage see Aharonov et al. . can (i.e. leaves them, or forces them, to relax), and re-encodes with fresh qubits from a larger workspace. Alternatively, Fig. 68 describes a variation where the old XV. CONCLUSION AND PERSPECTIVES qubits 2 and 3 are re-initialized by entanglement with a measurement device which then dissipates the heat from Within 5 years, engineered JJ quantum systems with the bitflip and the error correction procedure. 5-10 qubits will most likely begin seriously to test the In both of these cases, as described, we need a total scalability of solid state QI processors. workspace with five qubits. In principle, however, one For this to happen, a few decisive initial steps and can do with three qubits if we can rapidly re-initialize breakthroughs are needed and expected: The first essen- 52 tial step is to develop JJ-hardware with long coherence Anticrossing - Lifting of degeneracy (level crossing) of time to study the quantum dynamics of a two-qubit cir- quantum levels during variation of the system parameters cuit and to perform a ”test” of Bell’s inequalities (or when an interaction is switched on rather the JJ-ciruitry) by creating entangled two-qubit Average measurement - Measurement of an expecta- Bell states and performing simultaneous projective mea- tion value of a dynamic variable in a certain state surements on the two qubits. Bloch sphere - Geometrical representation of the man- A first breakthrough would be to perform a significant ifold of quantum states of a two-level system as points number of single- and two-qubit gates on a 3-qubit clus- on the unit sphere. ter to entangle three qubits. Combined with simultane- Bloch vector - Normalized state vector of a two-level ous projective readout of individual qubits, not disturb- system represented by a radial unit vector of the Bloch ing unmeasured qubits, this would form a basis for the sphere. first solid-state experiments with teleportation, quantum Charging energy - Electrostatic energy of a capacitor error correction (QEC), and elementary quantum algo- charged with a single electron (e) (or a single Cooper pair rithms. This will provide a platform for scaling up the (2e)). system to 10 qubits. Charge qubit - Superconducting qubit based on a a This may not look very impressive but nevertheless single Cooper pair box (SCB), whose computational basis would be an achievement far beyond expectations only consists of the two charge states of the superconducting a decade back. The NMR successes, e.g. running Shor- island. type algorithms using a molecule with 7 qubits76, are Charge-phase qubit - Superconducting qubit based on based on technologies developed during 50 years using a single Cooper pair box (SCB) whose charging energy is natural systems with naturally long coherence times. of the order of the Josephson energy of tunnel junctions Similarly, semiconductor technologies have developed for CNOT gate - Controlled-NOT gate: two-qubit gate 50 years to reach today’s scale and performance of clas- which changes or does not change the state of a target sical computers. It is therefore to be expected that QI qubit depending on the state of a controlling qubit. technologies will need several decades to develop truly Cooper pair - Bound state of two electrons (2e), the significant potential. Moreover, in the same way as for elementary charge carrier in superconducting equilibrium the classical technologies, QI technologies will most prob- state. ably develop slowly step by step, ”qubit by qubit”, which Coulomb blockade - Suppression of current through in itself will be an exponential development. a tunnel junction or small metallic island due to large Moreover, in future scalable information processors, charging energy associated with a passage of a single elec- different physical realizations and technologies might be tron. combined into hybrid systems to achieve fast processing CPHASE gate - Controlled-phase gate: two-qubit gate in one system and long coherence and long-time informa- which changes or does not change the phase of a target tion storage in another system. In this way, solid state qubit depending on the state of a controlling qubit. technologies might be combined with ion trap physics to 254 Decoherence - Evolution of a quantum system, inter- build large microtrap systems , which in turn might be acting with its environment; cannot be described with coupled to superconducting Josephson junctions proces- 255 a unitary operator; consists of decay of phase coherence sors via microwave transmission lines . (dephasing) and/or changing of level population (relax- ation). Density matrix - Characteristics of a quantum system, Acknowledgments which contains full statistical information about the state of the system. This work has been supported by European Commis- Dephasing - Decay of phase coherence of a superpo- sion through the IST-SQUBIT and SQUBIT-2 projects, sition state, represented by decreasing off-diagonal ele- by the Swedish Research Council, the Swedish Founda- ments of the density matrix. tion for Strategic Research and the Royal Academy of Entanglement - Specific non-local coupling of quantum Sciences. systems when the wave function of whole system cannot be presented as a product of partial wave functions Flux qubit - Superconducting qubit based on a SQUID, GLOSSARY whose computational basis consists of the two states of the SQUID having opposite directions of the induced Adiabatic evolution - Development of a quantum sys- flux. tem without transitions among the quantum levels. Hadamard gate - Transformation of computational ba- Algorithm - Finite sequence of logical operations, which sis states of a single qubit to equally weighted superpo- produces a solution for a given problem. sitions of the basis states (cat states). Level crossing - Degeneracy of quantum levels appear- Holonomic quantum computation - Using the geo- ing at a certain value of a controlling system parameter metric phases when a quantum system is taken around a (e.g. gate voltage, bias flux, etc.). closed circuit in the space of control parameters. 53

Gate operation - Controlled transformation of the state resonant driving perturbation, consists of periodic oscil- of one or several qubits; a basic element of an algorithm. lation of the level populations with the frequency propor- Josephson effect - Non-dissipative current flow be- tional to the amplitude of the perturbation. tween two superconductors separated by a non- Readout - Measurement of a qubit state. superconducting material (insulator, normal metal, etc.). Relaxation - Change of population of the energy eigen- Josephson junction - Junction of two superconductors, states resulting in approaching the equilibrium popula- which exhibits the Josephson effect. tion. Josephson critical current - Maximal value of the SCB - Single Cooper pair Box: superconducting analog Josephson current maintained by a particular junction. of SEB, where it is energetically favorable to have only Josephson energy - Inductive energy of a Josephson paired electrons on the island. junction proportional to the critical Josephson current. SCT - Single Cooper pair Transistor: a superconducting π pulse - High frequency control pulse with a specific device containing a small island whose charging energy is duration applied to a qubit, producing inversion of the controlled by an electrostatic gate electrode to increase qubit level populations (π rotation; qubit flip) or decrease current flowing through the island from one π/2 pulse - High frequency control pulse with a specific large electrode (source) to another (drain). duration applied to a qubit, typically tipping the Bloch SEB - Single Electron Box: small metallic island con- vector from a pole to the equator, or from the equator to nected to a large electrode via resistive tunnel junction, a pole, on the Bloch sphere. whose charging energy hence amount of trapped electrons Phase gate - Single qubit gate, transforms a superpo- is controlled by an electrostatic gate. sition of two quantum states into another superposition SET - Single Electron Transistor: a device containing a with different relative phase of the states small island whose charging energy is controlled by an Precession - Dynamic evolution of a two-level system in electrostatic gate electrode to increase or decrease cur- a superposition state, i.e. linear combination of energy rent flowing through the island from one large electrode eigenstates. (source) to another (drain). QND measurement - Quantum Non-Demolition Mea- rf-SET - SET driven by an rf signal, is used as an ul- surement: measurement of a state of a quantum system, tra sensitive electrometer by monitoring a linear response which does not destroy the quantum state and makes function of the SET, which is sensitive to the electrostatic possible repeated measurements of the same state. gate potential. QPC - Quantum Point Contact: a constriction in a con- SQUID - Superconducting Quantum Interferometer De- ductor with ballistic transport through a small number vice: a device consisting of a one or more Josephson junc- of conduction channels. tions included in a superconducting loop. Qubit - Quantum two-level system; basic element of a dc-SQUID - SQUID containing two Josephson junc- quantum processor. tions. PCQ - Persistent Current Qubit: synonymous with flux rf-SQUID - SQUID containing one Josephson junction. qubit. Single-shot measurement - A measurement which Rabi oscillation - Dynamics of two-level system under gives an ”up/down” answer in one single detection event.

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