Physical implementations of quantum computing

Andrew Daley

Department of Physics and Astronomy University of Pittsburgh Overview (Review)

Introduction • DiVincenzo Criteria • Characterising coherence times

Survey of possible qubits and implementations • Neutral atoms • Trapped ions

• Colour centres (e.g., NV-centers in diamond) • Electron spins (e.g,. quantum dots) • Superconducting qubits (charge, phase, flux)

• NMR • Optical qubits • Topological qubits

Back to the DiVincenzo Criteria:

Requirements for the implementation of quantum computation 1. A scalable physical system with well characterized qubits

1 | i 0 | i

2. The ability to initialize the state of the qubits to a simple fiducial state, such as |000...⟩

1 | i 0 | i 3. Long relevant decoherence times, much longer than the gate operation time

4. A “universal” set of quantum gates control target (single qubit rotations + C-Not / C-Phase / .... )

U U

5. A qubit-specific measurement capability

D. P. DiVincenzo “The Physical Implementation of Quantum Computation”, Fortschritte der Physik 48, p. 771 (2000) arXiv:quant-ph/0002077

Neutral atoms

Advantages: • Production of large quantum registers • Massive parallelism in gate operations • Long coherence times (>20s)

Difficulties: • Gates typically slower than other implementations (~ms for collisional gates) (Rydberg gates can be somewhat faster) • Individual addressing (but recently achieved) Quantum Register with neutral atoms in an optical lattice

0 1 | |

Requirements: • Long lived storage of qubits • Addressing of individual qubits • Single and two-qubit gate operations

• Array of singly occupied sites • Qubits encoded in long-lived internal states (alkali atoms - electronic states, e.g., hyperfine) • Single-qubit via laser/RF field coupling • Entanglement via Rydberg gates or via controlled collisions in a spin-dependent lattice Rb: Group II Atoms

87Sr (I=9/2):

Extensively developed, 1 • P1 e.g., optical clocks 3 • Degenerate gases of Yb, Ca,... P2 3 • Stable lasers, especially for clock P1 transition frequency 3 P0

689nm

1 Key properties S0 • Metastable triplet states: 3 - P0 Lifetimes >150s (Fermions) 3 - P1 linewidth ~ kHz 3 - P2 lifetime >>150s • Many nuclear spin levels for fermionic isotopes • Nuclear spin states decoupled from electronic state on clock transition Quantum Computing with Alkaline Earth Atoms

87Sr (I=9/2):

1 Implementation of Quantum Computing: P1 • Nuclear spin states for qubit storage 3 P2 (insensitive to magnetic field fluctuations) 3 P1 D. Hayes, P. S. Julienne, and I. H. Deutsch, 3 PRL 98, 070501 (2007) P0 I. Reichenbach and I. H. Deutsch, PRL 99, 123001 (2007).

689nm • Electronic state for: - Access to qubits 1 S0 - Gate operations

HERE: Via state-dependent lattices

A. J. Daley, M. M. Boyd, J. Ye, and P. Zoller, Phys. Rev. Lett. 101, 170504 (2008) AC Polarisability (AC-Stark Shift per intensity) for 87Sr

Ye, Kimble, & Katori, Science 320, 1734 (2008).

0.2 0.4 0.6 0.8 2 3 4 2

3 to states in the long-lived P2 manifold. As shown schematically in Fig. 1, we would transfer qubit states 3 |0" and |1" to the P0 level (which can be done state- selectively in a large magnetic field due to the differ- 1 3 ential Zeeman shift of 109Hz/G between S0 and P0), and2 then selectively transfer them to additional readout 3 87 Polarizability and State-dependent lattices: levels |0x" and |1x" in the P2 level (e.g., for Sr we 3 3 to states in the long-lived P2 manifold. As showncould choose |0x"≡| P2,F =13/2,mF = −13/2" and 3 schematically in Fig. 1, we would transfer qubit states|1x"≡| P2,F =13/2,mF = −11/2", where F is the 3 total angular momentum and mF the magnetic sublevel, |0" and |1" to the P0 level (which can be done state- 3 and connect these states to the P0 level via off-resonant selectively in a large magnetic field due to the differ- 3 1 3 Raman coupling to a S1 level). The individual qubit se- ential Zeeman shift of 109Hz/G between S0 and P0), 3 lectivity can be based on a gradient magnetic field, as P2 and then selectively transfer them to additional readout 3 1 3 87 is much more sensitive to magnetic fields P0 or S0.A levels |0x" and |1x" in the P level (e.g., for Sr we2 2 gradient field of 100 G/cm will provide an energy gradient could choose |0x"≡|3P ,F =13/2,m = −13/2" and 2 3 F 3 to states in the long-lived P2 manifold. As shownof 410 MHz/cm for |0x" or an energy difference of about FIG.|1 2:x"≡ (a) Energy| P2,F level=13 diagram/2,mF showing= −11 how/2", independent where F is the schematically3P in Fig. 1, we would transfer qubit states 0 3 1 3 15kHz between atoms in neighbouring sites. In the same optical lattices can be|0 produced" and |1" to for the theP0 levelS0 (whichand canP0 belevels done state- total angular momentum and mF the magnetic sublevel,field the 3P level states will be shifted by −m × 1Hz in by finding wavelengthsselectively where the in polarisability a large3 magnetic of field each due of to th thee differ- 0 I and connect these states to the P0 level via o1ff-resonant3 ential Zeeman shift of 109Hz/G between S0 and P0neighbouring), sites, which again indicates the advantage levels is zero and the other1 non-zero.3 (b) AC Polarisability 1 Raman3 couplingand toS then a0 S selectively1 level). transfer The them individual to additional qubitreadout se- of S0 and P0 levels near 627nm. (c) AC polarisability3 of87 of storing qubits on the nuclear spin states. This selec- 1 3 levels |0x" and |1x" in the P2 level (e.g., for Sr3 we lectivity can be based on a gradient3 magnetic field, as P2 S0 and P0 levels nearcould 689nm. choose (d)|0x"≡ AC| P polarisability2,F =13/2,mF of= − dif-13/2" andtively can be used to transfer atoms site-dependently to 3 3 3 1 ferentism muchF sublevels more of sensitive| the1x"≡P| 2P, to2F,F magnetic=13=13//22,m hyperfineF fields= −11/2" levelP, where0 or forFSis0.A thethe transport lattice, or to read out qubits by transfer- π−polarisedgradient light field at of 627nm 100total G/cm angular and 689.2nm. momentum will provide and m anF the energy magnetic gradient sublevel, 3 3 ring only the |0" state to P2, then making fluorescence and connect these states to the P0 level via off-resonant 3 of 410 MHz/cmRaman for |0 couplingx" or an to a energyS1 level). di Thefference individual of qubit about se-measurements (e.g., using the cycling transition between FIG. 2: (a) Energy level diagram showing how independent lectivity can be based on a gradient magnetic field, as 3P 3 3 1 3 15kHz between atoms in neighbouring sites. In the samethe2 P2,F =13/2 and D3 manifold). optical lattices can be produced for the S0 and P0 levels 3 1 field the 3P levelis much states more will sensitive be shifted to magnetic by fields−mP30×or1HzS0.A in by finding wavelengths where the polarisability of each of thestorage lattice for0 qubits,gradient which field of 100 will G/cm not will a provideffect thean energyI P0 gradient A necessary requirement here is that our states |0x" 3 states.neighbouring Similarly, thesites,of 410polarizability which MHz/cm again for |0x of" indicatesor anP0 energyat 689.2nm the difference advantage of about levels is zero and the other non-zero.FIG. (b) 2: (a) AC Energy Polarisability level diagram showing how independent and |1x" are trapped in the combination of the storage 1 3 1 3 15kHz between atoms in neighbouring1 sites. In the same 0 0 optical lattices can be producedis for∼ theof1550a.u,S0 storingand P0 levels whereas qubits on the the3 polarisability nuclear spin of states.S0 is zero. This selec-and transport lattices (these will both provide AC-Stark of S and P levels near 627nm. (c) AC polarisability of field the P0 level states will be shifted by −mI × 1Hz in 1 3 by finding wavelengths where the polarisability of each of the 3 S0 and P0 levels near 689nm. (d) AClevels polarisability is zero and the other of non-zero. dif-This (b)tively is AC largely Polarisability can because be usedneighbouring of to the transfer near-resonant sites, which atoms again site-dependently indicates coupling the of advantageshifts to for the P2 level). In Fig. 2d we plot the polaris- 3 1 3 1 3 ferent mF sublevels of the P2, F =13of /S20 and hyperfineP0 levels level near 627nm. for (c) AC polarisability of of storing qubits on the nuclear spin states. This selec- 3 1 3 S0 totheP transport1, which is lattice, made or possible to read without out qubits large by spon- transfer-ability of all of the magnetic sublevels of P2, F =13/2 S0 and P0 levels near 689nm. (d) AC polarisability of dif- tively can be used to transfer atoms site-dependently to π−polarised light at 627nm and 689.2nm. 3 3 ferent mF sublevels of the P2, Ftaneous=13/ring2 hyperfine emission only level the for rates|0"thestate transportdue to to lattice, theP2, narrow then or to read making linewidth out qubits fluorescence by of transfer-at our lattice wavelengths, and the large tensor shifts π−polarised light at 627nm and 689.2nm.3 3 P1.measurements This lattice can (e.g.,ring be only using used the | for0 the" statetransport cycling to P2, then transition, and making atoms fluorescence betweenmake certain mF levels suitable for trapping at the same 3 measurements3 (e.g., using the cycling transition between in itthe willP not,F be=13 affected/2 and3 byD themanifold). storage3 lattice. These locations as our qubit states. If the depths of the storage 2 the P2,F =133/2 and D3 manifold). 3 3 storage lattice for qubits, which willstorage notlattice affect for qubits, the whichP0lattices will notA can aff necessaryect be the madeP0 requirement toA have necessary the requirement same here is spatial here that is period thatour our states by states ||00xandx"" transport lattices are chosen to be equal, then both 3 states.3 Similarly, the polarizabilityusing of angledP0 at 689.2nm beams inand the|1x case" are of trapped the 627nm in the combination light, so of that the storagethe |0x" level and the |1x" will be trapped, in lattices states. Similarly, the polarizabilityis ∼ of1550a.u,P0 whereasat 689.2nm the polarisabilityand of|1Sx"isare zero. trapped in the combination of the storage 1 0 and transport lattices (these will both provide AC-Stark the lattice period is increased to3 match that formed by about 2/3 and 1/3 of the depth of the storage lattice is ∼ 1550a.u, whereas the polarisabilityThis is largely of becauseS0 is of zero. the near-resonantand transport coupling of latticesshifts for (these the P2 level). will both In Fig. provide 2d we plot AC-Stark the polaris- 1 3 3 S0 to P1, which is made possiblecounterpropagating without large spon- 3 beamsability of at all of 689.2nm, the magnetic and sublevels the of depthsP2, F =13/respectively.2 The timescale for all transfer processes be- This is largely because of the near-resonant coupling of shifts for the P2 level). In Fig. 2d we plot the polaris- taneous emission rates due to the narrow linewidth of at our lattice wavelengths, and the large tensor shifts 1 3 3 can be made equal by using light of intensity I03 for the tween lattices τtransfer is limited by the smallest trapping S0 to P1, which is made possibleP1 without. This lattice large can be spon- used for transportability, and of atoms all of themake magnetic certain m levels sublevels suitable for of trappingP2, F at the=13 same/2 storage and ∼ I /4 for the transportF lattice, facilitat- taneous emission rates due to thein narrow it will not linewidth be affected by of the storageat our lattice. lattice These0 wavelengths,locations as our qubit and states. the If large the depths tensor of the storage shiftsfrequency ωt (so that atoms are not coupled to excited lattices can be made to have the same spatial period by 3 ing transfer of atomsand between transport lattices the two are chosen lattices. to be equal, Gate then bothmotional states), and by the frequency shift ωe between P1. This lattice can be used for usingtransport angled beams, and in the atoms case of the 627nmmake light, certain so thatmF levels suitable for trapping at the same operations can then bethe performed|0x" level and thebetween|1x" will distant be trapped, sites in latticesneighbouring sites in the case of position-selective trans- in it will not be affected by the storagethe lattice lattice. period is increased These to matchlocations that formed as by ourabout qubit 2/3 states. and 1/3 If of the the depthdepths of the of storage the storage lattice counterpropagating beams at 689.2nm,by transferring and the depths atomsrespectively. state-selectively The timescale into for the all transfer transpor processest be-fer, as τ & max(2π/ω , 2π/ω ). lattices can be made to have the same spatial period by and transport lattices are chosen to be equal, then both transfer t e can be made equal by using lightlattice, of intensity andI0 movingfor the themtween lattices to theτtransfer appropriateis limited by distant the smallest site trapping Single-qubit gates can be performed either by transfer- using angled beams in the case of thestorage 627nm and ∼ light,I0/4 for so the that transport lattice, facilitat- the |0x" level andfrequency the ω|1t x(so" thatwill atoms be trapped, are not coupled in to lattices excited 3 ing transfer of atoms between(see the two below lattices. for Gatemore details). This is somewhat reminis- ring atoms to the P level and then rotating the qubit the lattice period is increased to match that formed by about 2/3 andmotional 1/3 of states), the depth and by the of frequency the storage shift ωe between lattice 0 operations can then be performedcent between of the distant use of sites spin-dependentneighbouring sites lattices in the case for of alkali position-selective atoms trans- counterpropagating beams at 689.2nm, and the depths states by directly applying Raman couplings, or alter- by transferring atoms state-selectively[2, 12,respectively. into 14], the transpor wheret The latticefer, timescale as lasersτtransfer & are formax(2 tuned allπ/ω transfert, 2 betweenπ/ωe). processes fine- be- lattice, and moving them to the appropriate distant site Single-qubit gates can be performed either by transfer-natively with single-qubit addressability. This would in- can be made equal by using light of intensity I0 for the tween lattices τtransfer is limited by the smallest trapping 3 (see below for more details). Thisstructure is somewhat states, reminis- whichring can atoms lead to the to3 largeP level heating and then rotating and de- the qubitvolve using the P level in an intermediate step to trans- storage and ∼ I /4 for the transport lattice, facilitat- 0 2 0 cent of the use of spin-dependentcoherence latticesfrequency for alkali from atomsω spontaneoust (sostates that by atoms emissions. directly applying are not Here, Raman coupled the couplings, lattices to excited or alter- 3 [2, 12, 14], where lattice lasers are tuned between fine- fer atoms position-selectively to the P0 level. Two-qubit ing transfer of atoms between the two lattices. Gate motional states),natively and bywith the single-qubit frequency addressability. shift ω Thise between would in- can be made completely independent3 by selection of the structure states, which can lead to large heating and de- volve using the P2 level in an intermediate step to trans-gates are then performed using the transfer lattice. In operations can then be performedcoherence between from distant spontaneous sites emissions.neighbouring Here, the lattices sites in the case of position-selective3 trans- correct wavelengths.fer Note atoms that position-selectively whilst we to illustrate the P0 level. our Two-qubitparticular, a phase gate between qubits in site i and j by transferring atoms state-selectivelycan be into made the completely transpor independentt by selection of the schemefer, in as oneτtransfer dimension&gatesmax(2 are here, thenπ/ theseω performedt, 2π ideas/ω usinge). are the generalis- transfer lattice. Incan be performed in a straight-forward manner by: (i) lattice, and moving them to the appropriatecorrect wavelengths. distant Note site that whilst we illustrate our particular, a phase gate between qubits in site i and j scheme in one dimension here,able these ideas toSingle-qubit storage are generalis- and gates transportcan be can performed be lattices performed in a in straight-forward 2D either and 3D. manner by transfer- by: (i)transferring atoms in |0" on site i (and j) to the trans- 3 (see below for more details). Thisable is somewhat to storage and reminis- transport latticesAnring in essential 2D atoms and 3D. ingredient to thetransferringP0 forlevel atoms general-purpose and in |0 then" on site rotatingi (and quantumj) to the the qubit trans-port lattice; (ii) moving the transport lattice relative to cent of the use of spin-dependent latticesAn essential for alkali ingredient atoms for general-purposestates by quantum directlyport applying lattice; (ii) moving Raman the transport couplings, lattice or relative alter- to information processing is the informationindividual addressing processingof the storage is the latticeindividual so that an atom addressing that was originallyof inthe storage lattice so that an atom that was originally in [2, 12, 14], where lattice lasers arequbits, tuned both between for readout fine- andqubits, gatenatively operations, both with which for readout single-qubitthe |0" state and on addressability. gatesite i would operations, now be present This which at would site j; (iii) in-the |0" state on site i would now be present at site j; (iii) can be achieved in this system by coupling selectively 3generating a phase φ for the state conditioned on whether structure states, which can lead to large heating and de-canvolve be achieved using the in thisP2 level system in an by intermediate coupling selectively step to trans-generating a phase φ for the state conditioned on whether 3 coherence from spontaneous emissions. Here, the lattices fer atoms position-selectively to the P0 level. Two-qubit can be made completely independent by selection of the gates are then performed using the transfer lattice. In correct wavelengths. Note that whilst we illustrate our particular, a phase gate between qubits in site i and j scheme in one dimension here, these ideas are generalis- can be performed in a straight-forward manner by: (i) able to storage and transport lattices in 2D and 3D. transferring atoms in |0" on site i (and j) to the trans- An essential ingredient for general-purpose quantum port lattice; (ii) moving the transport lattice relative to information processing is the individual addressing of the storage lattice so that an atom that was originally in qubits, both for readout and gate operations, which the |0" state on site i would now be present at site j; (iii) can be achieved in this system by coupling selectively generating a phase φ for the state conditioned on whether Quantum computing with Alkaline Earth Atoms

3 P 2 detection addressing addressing 3 P 2 Raman 3 transfer P 0 transport / operations 3 transport lattice P 0

1 S 0 storage lattice

|0 |1

Key ideas: 3 1 • Independent Lattices for P0 and S0 states: storage and transport • Qubits encoded on nuclear spin states, relatively insensitive to magnetic fields 3 • Local addressing via P2 level, which shifts in a gradient field (100 G/cm - 410 MHz/cm, 15 kHz shift between neighbouring sites) Collisional Gates (simple example): • Gate: controlled collisions D. Jaksch et al., PRL 82, 1975 (’99) • Operation performed in parallel for whole system • Simple preparation of a cluster state • Ideal setup for measurement-based quantum computing.

Atom 1 Atom 2

1 3 S0 P0 internal states

move

collision “by hand“ Extensions:

• Use additional nuclear spins: Quantum register encoded on a single atom

1 P1 3 P2 3 P1 3 P0

1 S0

• Flying qubits, e.g., coupling to an optical cavity Requirements: • Scalable physical system, well-defined qubits • Initialisable to a fiducial state such as |000...> • Long coherence times • Universal set of quantum gates • High efficiency, Qubit-specific measurements

Selected numbers: Gate/readout Timescales: • Lattice trapping frequency: 25 - 100 kHz / collisional gate limit Addressing 3 • P2 Shift mF=-13/2: 100 G/cm: 410 MHz / cm (ca. 15kHz per lattice site) 1 3 • S0/ P0 Shifts: 100 G/cm: ca. 1Hz / mI per lattice site 1 3 Spontaneous emission lifetime T1 (25kHz trap frequency lattices for S0 and P0 ): 1 • Storage S0: 20s 3 3 • Operations P0 / P2: 2s / 1s

Decoherence from Magnetic field fluctuations (T2) 1 • S0 shift: -185 Hz/G - Decoherence in mG fluctuations <<1 Hz, 3 • P0 shift: -195 Hz/G Lossy blockade gates: Realisations underway: Kyoto (in lattices) 3 3 • P2- P2 loss: 20 kHz Innsbruck, Houston (degenerate gases) Ion traps

Advantages: • Long coherence times (>20s for nuclear spins) • Basic gates somewhat faster than neutral atoms (~0.01 ms) • Individual addressing straightforward • High-precision experiments already commonplace (also optical clocks)

Difficulties: • Scaling to many qubits requires complicated traps • Slower gates than many solid state implementations Ion Trap Quantum Computer '95

• Cold ions in a linear trap

Qubits: internal atomic states 1-qubit gates: addressing ions with a laser 2-qubit gates: entanglement via exchange of phonons of quantized collective mode • State vector

quantum register data bus

• QC as a time sequence of laser pulses • Read out by quantum jumps The 43Ca+ ion trap quantum computer

R. Blatt, Innsbruck D.Wineland R.Blatt NIST Innsbruck

What has been achieved in the laboratory

• 2 … 15 ions / qubits – high fidelity quantum gates – simple algorithms – teleportation (within a trap) – error correction – quantum simulation algorithms String of 40Ca+ Ions in a Linear Paul Trap

String of ions a quantum register: addressing & read out

70 µm R. Blatt@Innsbruck Experimental Achievements: Innsbruck & NIST Addressable Cirac-Zoller 2-ion Controlled-NOT

truth table CNOT:

Control Target

fidelity F=0.993

R. Blatt et al., Nature 2003; Nature Physics 2008 Experimental Achievements: Innsbruck & NIST Deterministic Teleportation

R. Blatt et al., Nature 2004 Experimental Achievements: Innsbruck & NIST

Deterministic• teleportation protocol Teleportation If Alice and Bob share a singlet (EPR) pair as a resource, we can teleport the unknown quantum state

Alice Bob Protocol: ü CNOT between A&C C ü measure A&C A B noise ü classical communication Alice to Bob ü rotate B

• Innsbruck ion trap experiment:

deterministic teleportation: ü no postselection ü complete Bell measurement ü on demand ü only 10 μm L EPR pair Quantum Byte

Reconstruction of quantum Fidelity: 0.76 state takes days on a classical computer

quantum byte

true 8 particle entanglement 656100 measurements, ~ 10 h measurement time total R. Blatt et al., Nature 2006 NATURE|Vol 453|19 June 2008 INSIGHT REVIEW

Prospects ~50 µs without loss of coherence; the excitation of the ion’s motion (in Although the basic elements of quantum computation have been its local well) was less than one quantum73. Multiple ions present in a demonstrated with atomic ions, operation errors must be significantly single zone can be separated46,73 by inserting an electric potential ‘wedge’ reduced and the number of ion qubits must be substantially increased if between the ions. In the tele portation experiment by the NIST group46, quantum computation is to be practical. Nevertheless, before fidelities two ions could be separated from a third in ~200 µs, with negligible and qubit numbers reach those required for a useful factoring machine, excitation of the motional mode used for subsequent entangling opera- worthwhile quantum simulations might be realized. tions between the two ions. This absence of motional excitation meant that an additional entangling-gate operation on the sepa rated ions could More ion qubits and better fidelity be implemented with reasonable fidelity. For algorithms that operate To create many-ion entangled states, there are two important goals: over long time periods, the ions’ motion will eventually become excited improving gate fidelity, and overcoming the additional problems that as a result of transportation and background noise from electric fields. are associated with large numbers of ions. For fault-tolerant operation, a To counteract this problem, additional laser-cooled ions could be used reasonable guideline is to assume that the probability of an error occur- to cool the qubits ‘sympathetically’ (Fig. 6a). These ‘refrigerator’ ions ring during a single gate operation should be of the order of 10−4 or could be identical to the qubit ions76, of a different isotope77 or of a dif- lower. An important benchmark is the fidelity of two-qubit gates. The ferent species60,78. They could also aid in detection and state preparation best error probability achieved so far is approximately 10−2, which was (described earlier). inferred from the fidelity of Bell-state generation63. In general, it seems For all multiqubit gates that have been implemented so far, the speeds that gate fidelities are compromised by limited control of classical com- are proportional to the frequen cies of the modes of the ions, which scale 2 ponents (such as "uctuations in the laser-beam intensity at the positions as 1/dqe, where dqe is the distance of the ion to the nearest electrode. of the ions) and by quantum limitations (such as decoherence caused Therefore, it would be valuable to make traps as small as possible. Many by spontaneous emission)64. These are daunting technical problems; groups have endeavoured to achieve this, but they have all observed however, eventually, with sufficient care and engineering expertise, significant heating of the ions, compromising gate fidelity. The heat- these factors are likely to be suppressed. ing is anomalously large compared with that expected to result from The multiqubit operations discussed in this review rely on the abil- thermal noise, which arises from resistance in, or coupled to, the trap 18,79–83 4 ity to isolate spectrally a single mode of the motion of an ion. Because electrodes . It scales approximately as 1/dqe (refs 18, 79–83), which there are 3N modes of motion for N trapped ions, as N becomes large, is consistent with the presence of independently "uctuating potentials the mode spectrum becomes so dense that the gate speeds must be on electrode patches, the extent of which is small compared with dqe significantly reduced to avoid off-resonance coupling to other modes. (ref. 79). The source of the heating has yet to be understood, but recent Several proposals have been put forward to circumvent this problem65,66. experiments80,82 indicate that it is thermally activated and can be signifi- Alternatively, a way to solve this problem with gates that have been cantly suppressed by operating at low temperature. demonstrated involves dis tributing the ions in an array of multiple trap For large trap arrays, a robust means of fabrication will be required, zones18,67–69 (Fig. 6a). In this architecture, multiqubit gate operations as well as means of independently controlling a very large number of could be carried out on a relatively small number of ions in mul tiple electrodes. Microelectromechanical systems (MEMS) fabrication tech- processing zones. Entanglement could be distributed between these nologies can be used for monolithic construction83,84, and trap struc- zones by physically moving the ions18,68,69 or by optical means25,67,70–72. tures can be further simplified by placing all electrodes in a plane84,85. For quantum communication over large distances, optical distribution To mitigate the problem of controlling many electrodes, it might be seems to be the only practical choice; for experiments in which local possible to incorporate ‘on-board’idea: electronics Wineland et close al. to individual trap entanglement is desirable, moving ions is also an option. zones86. Laser beams must also be applied in several locations simultane- Examples of traps that could beScalability: used for scaling Multizoneup the number Trapsof ions ously, because it will be essentialexp.: to carry Innsbruck, out parallelNIST operations when used in an algorithm are shown in Fig. 6b. Ions can be moved be tween implementing complex algorithms.Boulder, The Michigan, recycling Oxford,... of laser beams can be zones by applying appropriate control electric potentials to the various used86,87, but the overall laser power requirements will still increase. If electrode segments46,73–75. Individual ions have been moved ~1 mm in gates are implemented by using stimulated-Raman transitions, then a • implementation: physically sending the qubit a b Refrigerator ion trap quantumGate computer beam beam(s)

Qubit memory zone

To additional zones

Figure 6 | Multizone trap arrays. a, A schematic representation of a separated and usedR. for Slusher, algorithm Georgia demonstrations, Tech including teleportation46 multizone trap array is shown. Each control electrode is depicted as a (width of narrow slot (where the ions are located) = 200 µm). In the upper rectangle. Ions (blue circles) can be separated and moved to specific zones, right is a three-layer, two-dimensional multizone trap that can be used to including a memory zone, by applying appropriate electrical potentials. switch ion positions99 (width of slot = 200 µm). In the lower left is a single- Because the ions’ motion will become excited as a result of transport zone trap in which all of the electrodes lie in a single layer; this design (bidirectional arrow) and noisy ambient electric fields, refrigerator ions considerably simplifies fabrication85. In the lower right is a single-layer, (red; which are cooled by the red laser beam) are used to cool the ions linear multizone trap fabricated on silicon (width of open slot for loading before gate operations, which are implemented with the blue laser beam. ions Ӎ 95 µm), which can enable electronics to be fabricated on the same b, Examples of the electrode configurations of trap arrays are shown. In the substrate that contains the trap electrodes. (Image courtesy of R. Slusher, upper left is a two-layer, six-zone linear trap in which entangled ions can be Georgia Tech Research Institute, Atlanta).

1013 Nitrogen-Vacancy Centers

Advantages: • Combine advantages of atomic systems with solid state • Faster gate times (<µs) but faster decoherence (~2ms) • Room-temperature operation

Difficulties: • Lack of uniformity in qubit frequency • Coupling qubits is more difficult (e.g., optical processes)

Pitt: Experiments in group of G. Dutt NV-center in diamond System: NV color center in diamond

• Substitutional nitrogen atom replaces a single carbon atom in the lattice. " P1 centers (S=1/2)

• Vacancy (missing carbon in the lattice) becomes mobile at 450°C, but forms a stable NV center when pairing with N ! • Two flavors (NV0 and NV-) have different optical properties.

• NV-: Excite with 532 nm off-resonantly or resonantly with 637 nm, emission at 637 nm (ZPL) or 638 – 720 nm (PSB).

• Good single photon emitter.

Early work: S.Rand, N.Manson

GS Research talk, Mar 30th 2012! IsolatingIsolating a single Single spin by Spinslaser spectroscopy

GS Research talk, Mar 30th 2012! System: NV color center in diamond

• non-zero electronic spin (S=1) ground state With zero-field splitting Δ=2.87 GHz Early Pioneers: S.Rand, N.Manson

• Long T1 (~ 10ms – 4 sec) and T2 (~ 0.3 - 2 ms) times

• Spin-state dependent fluorescence allows for spin detection at room temp, also allows spin pumping. ~ 10 ns

• Proximal nuclear spins (T2 ~ 100s of ms) can be controlled and measured. ! F. Jelezko et al, PRL 2004 ! L. Childress et al, Science 2006 ! G. Dutt et al, Science 2007 ! Neumann et al, Science 2008 ! Balasubramanian et al, N. Mat. 2009

GS Research talk, Mar 30th 2012! Spin-Photon Entanglement E. Togan et al, Nature (2010)!

# Fidelity F=69 ± 7 %, Probability of entanglement = 99.7%

GS Research talk, Mar 30th 2012! Slide: G. Dutt Cavity-QED for Optical Interconnects

• Photons mediate entangled states of atoms (spins) at remote locations!

• Strong coupling results in deterministic interactions!

• Weak coupling allows for great improvement in probabilistic entanglement creation – loophole-free tests of Bell inequalities!

Slide: G. Dutt GS Research talk, Mar 30th 2012! Electron Spins

Advantages: • Faster gate times (~ns) but faster decoherence (~30µs)

Difficulties: • Production of regular arrays of, e.g., quantum dots is non-trivial • Qubits are electron spins, e.g., in electrically gated quantum dots • Single qubits can be manipulated via electrode potentials, microwave fields

• Two-qubit gates based on spin-exchange interaction. Can be switched with electrical gates. H = JS~ .S~ 1 2 Superconducting qubits

Advantages: • Faster gate times (~ns) but faster decoherence (~0.5-9.6µs) • Many possibilities for coupling to AMO systems (microwave/optical photons, atoms/ions/molecules)

Difficulties: • Production of regular arrays of qubits is non-trivial

Pitt: Experiments J. Levy / S. Frolov NATURE|Vol 453|19 June 2008 INSIGHT REVIEW

Box 2 | Experimental issues with superconducting qubits a F Experiments on superconducting qubits are challenging. Most superconducting qubits are created by using electron-beam lithography, need millikelvin temperatures and an ultralow-noise environment to operate, and can be studied only by using very sensitive measurement techniques. Superconducting qubits generally require Josephson junctions with dimensions of the order of 0.1 × 0.1 µm2 — corresponding to a self-capacitance of about 1 fF — and are patterned by using shadow I 79 q NATURE|Vol 453|19 June 2008 evaporation and electron-beam lithography ; an exception is the phaseINSIGHT REVIEW qubit, which typically has aNATURE junction|Vol 453 of |119 × June 1 µ m20082 and can be patterned INSIGHT REVIEW photolithographically. The Josephson junctions are usually Al–AlxOy–Al

(where x ≤ 2 andINSIGHT y ≤ 3), and REVIEW Boxthe 2 oxidation | Experimental must issues be with controlled superconducting to yield qubits a NATURE|Vol 453|F19 June 2008 Box 2 | Experimental issues with superconducting qubits relatively precise valuesFlux of E Experimentsajqubit and Ec. Because on superconducting qubit frequencies qubitsF are challenging. are Most superconducting qubits are created by using electron-beam usually 5–10 GHz (which corresponds to 0.25–0.5 K), the circuits are 2 F0/2 Experiments on superconducting qubits are challenging. Most a lithography, need millikelvin temperatures and an ultralow-noiseof 2e on the island at zero voltage, (2e) /2CΣ, is much greater than the operated in dilution refrigerators, typically at temperatures of superconducting qubits are created by using electron-beam environment to operate, and can be studied only by using verythermal sensitive energy kBT (where CbΣ = Cg + Cj is the total capacitance). For 10–30 mK, to minimize thermalmeasurement population techniques. of the upper state. T = 1 K, this requires CΣ to be much less than 1 fF. TheU Cooper-pair box lithography, need millikelvin temperatures and an ultralow-noise Superconducting qubits generally require Josephson junctionsis connected to ground by a gate capacitance Cg in series with a potential Great efforts are made to attenuate external electrical and2 magnetic with dimensions of the order of 0.1 × 0.1 µm — correspondingVg and to a by a small Josephson junction with Ej << Ec. Given their weak environment to operate, and can be studied only by using very sensitive connection to the ‘outside world’, the number of Cooper pairs on the noise. The experiment is invariablyself-capacitance enclosed of about in 1 fFa —Faraday and are patterned cage — by either using shadow I measurement techniques. evaporation and electron-beam lithography79; an exceptionisland is the is phase a discrete variable n. The qubitq states correspond to adjacent a shielded room or the metal Dewar of the refrigerator with a2 contiguous ᎂ 〉 ᎂ Probability〉 qubit, which typically has a junction of 1 × 1 µm and can beCooper-pair patterned number states n and n + 1 . Superconducting qubits generally require Josephson junctionsmetal box on top. The electrical leads that are connected to the qubits To understand how to control a singleamplitudes Cooper pair, it is useful to first D 2 photolithographically. The Josephson junctions are usually Al–AlxOy–Al with dimensions of the order of 0.1 0.1 m — corresponding to a (where x 2 and y 3), and the oxidation must be controlledexamine to yield the electrostatic problem with an infinite junction resistance × µ and their read-out devices are heavily≤ filtered≤ or attenuated. For 2 2 relatively precise values of E and E . Because qubit frequencies(Ej = 0).are The total electrostatic energy of the circuit is Ech = (2e /Cg)(n − ng) , self-capacitance of about 1 fF — and are patterned by using shadow j c example, lines carrying quasistatic bias currents usually have multiplewhere ng = CgVg/2e (representing the gate voltage in terms of the gate usuallyI 5–10q GHz (which corresponds to 0.25–0.5 K), the circuits are F /2 evaporation and electron-beam lithography79; an exception is the phase charge, namely the polarization charge that the voltage induces0 on the low-pass filters at the variousoperated temperature in dilution refrigerators, stages of typically the refrigerator. at temperatures• Superconducting of b loop interrupted by Josephson junctions 2 gate capacitor). Although n is an integer, ng is a continuous variable. Ech qubit, which typically has a junction of 1 × 1 µm and can be patternedThese include both inductor–capacitor10–30 mK, to minimize and resistor–capacitor thermal population of the filters upper state. that U Great efforts are made to attenuate external electrical and Quantumversus magnetic ng is shown instates Fig. 3c for several of electronvalues of n. It should current be noted that in twoF directions • the curves for n and n + 1 cross at n = n + ½, the charge degeneracy point. photolithographically. The Josephson junctions are usually Al–AloperatexOy–Al up to a few hundrednoise. megahertz, The experiment as well is invariably as wires enclosed running in a Faraday through cage — either g At this point, the gate polarization corresponds to half a Cooper pair for a shielded room or the metal Dewar of the refrigerator with (with a5 contiguous different magneticProbability flux) constitute a qubit (where x ≤ 2 and y ≤ 3), and the oxidation must be controlled tocopper yield powder, which results in substantial loss at higher frequenciesboth. charge basis states. metal box on top. The electrical leads that are connected to the qubits amplitudes D By restoring the Josephson coupling to a small value, the behaviour relatively precise values of Ej and Ec. Because qubit frequencies Theare overall attenuation is typicallyand their read-out200 dB. devices Finally, are heavily the read-outfiltered or attenuated. process For 2 close to these crossing points is modified. The Josephson junction example, lines carrying quasistatic bias currents usually have multiple 0 usually 5–10 GHz (which corresponds to 0.25–0.5 K), the circuitsfor areprobing a quantum system is very delicate. F /2 allows Cooper pairs to tunnel onto the island one by one. The result- low-pass filters at the various temperature0 stages of the refrigerator. c

operated in dilution refrigerators, typically at temperatures of ant coupling between neighbouring charge states ᎂn〉 and ᎂn + 1〉 makes ᎂ 〉 Theseb include both inductor–capacitor and resistor–capacitor filters that the quantum superposition of charge eigenstates analogous to Fthe b operate up to a fewc hundred megahertz, as well as wires running through ᎂ 〉 10–30 mK, to minimize thermal population of the upper state. U superposition of flux states in equation (3) (identifying ᎂ 〉 = ᎂn〉 and copper powder, which results0 in substantial loss at higher frequencies5. 10 0 0.8 ᎂ 〉 = ᎂn + 1〉). The next excited charge state is higher in energy by E Great efforts are made to attenuate external electrical and magneticof Φ are shown in Fig. 1d. TheAt overallthe degeneracy attenuation is typically point, 200 dB.the Finally, probability the read-out processof c e and can safely be neglected. At the charge degeneracy point, where the for probing a quantum system is very delicate. 0

noise. The experiment is invariably enclosed in a Faraday cageobserving — either either state is ½. As Φe is reduced, the probability of observingJosephson coupling produces anc avoided crossing, the symmetrical and

ᎂ 〉 antisymmetrical superpositions areE split by an energy E . By contrast, a shielded room or the metal Dewar of the refrigerator with a contiguousᎂ 〉 increases while that of observingProbability ᎂ 〉 decreases. ᎂ 〉 j D n far from this point, Ec >> Ej, and the eigenstates are very close to being (GHz)

sw amplitudes I metal box on top. The electrical leads that are connected to the qubits of Φe are shown in Fig.m 1d. At the degeneracy point, the probability ofD The first observation of quantum superpositionf 5 in a flux qubitcharge was states. Again, the energy level structure is analogous to that of and their read-out devices are heavily filtered or attenuated. For observing either state is ½. As Φe is reduced, the probabilityflux of qubits, observing with ∆ replaced with Ej and ε with Ec × (ng − n − ½). Similarly,

made spectroscopically. Theᎂ 〉 state of the flux qubit ᎂis 〉 measured with E n

increases while that of observing decreases. 14the probabilities of measuring the ground state or excitedD state depend ᎂ 〉 example, lines carrying quasistatic bias currents usually have multiple on the gate voltage rather than the applied flux. a d.c. superconducting quantumThe first interference observation of quantum device superposition (SQUID) in a. flux This qubit was ᎂ 〉

low-pass filters at the various temperature stages of the refrigerator. made spectroscopically. The state of the flux qubit is measuredTo make with the qubit fully tunable, the Josephson junction is usually

device consists of two Josephson junctions, each with critical current I0,14 ᎂ 〉 a d.c. superconducting quantum interference device (SQUID)replaced by. This a d.c. SQUID with low inductance (βL << 1). Ej is then These include both inductor–capacitor and resistor–capacitor filters that 0 F adjusted by applying the appropriate magneticᎂ 〉 flux, which is kept con- e connected in parallel on0.498 adevice superconducting 0.500 consists 0.502 of two Josephson03 loop12 junctions, of inductance each with critical L. The current I0, stant throughout the subsequent measurements. e operate up to a few hundred megahertz, as well as wires running through connected/ in parallel on a superconducting∆ (10–3 ) loop of inductance L. The critical current of the SQUIDΦΦe I0c(Φs) is periodic in theΦΦ res externally0 appliedThe read-out of a charge qubit involves detecting the charge on the 5 Measurementscritical current of the SQUID viaIc(Φs) issensitive periodic in the externallymagnetic applied field detection (SQUIDs) copper powder, which results in substantial loss at higher frequencies . Figure 2 | Experimental• properties of flux qubits. ≡a, The configuration of the island to a much greater accuracy than 2e. This is accomplished by using magnetic flux Φs with periodmagnetic Φ0. In flux the Φ with limit period βL Φ . In 2 LIthe0 limit/Φ0 β<< ≡ 1 2LI in/ Φwhich << 1 in which 16 s 0 L 0 a0 single-electron transistor (SET), a sensitive electrometer . The SET The overall attenuation is typically 200 dB. Finally, the read-out process original three-junction• Controlthe Josephson flux qubit viainductance is shown. applied Arrows dominates indicate microwave the current geometrical flow inductance, fields the the Josephson inductancein the two qubit statesdominates (green denotes the ᎂ 〉, geometricaland yellow denotes ᎂ inductance, 〉). Scale bar, (Fig. the 3d), also based on a tiny island, is connected to two superconduct- 0 F0/2 for probing a quantum system is very delicate. 3 μm. (Image courtesycritical of current C. H. van for der Φ Wal,s = ( Rijksuniversiteitm + ½)Φ0 (m is Groningen, an integer) is reduceding leads to almost by two Josephson junctions. When the voltages across Fboth/2 critical current for Φs = (•m Coupling + ½)Φ0 (m is e.g., an integer) via magnetic is reduced to fields almost d 1 0

the Netherlands).zero, cb, Radiation and the offlux microwave dependence frequency of the f induces critical resonant current takes the approxi- d 1

14 m junctions are close to the degeneracy point (ng = n + ½), charges cross

ᎂ 〉 zero, and the fluxpeaks dependence and dips inmate the switching form of the Ic (currentΦ criticals) ≈ 2 IIsw0ᎂ withcos( current πrespectΦs/Φ 0to)ᎂ. the Thus, takes externally by biasingthe approxi- thethe SQUID junctions with to a produce a net current flow through the SET. Thus, the 14 applied magneticconstant flux Φe normalized magneticᎂ 〉 fluxto the near flux Φquantum0/2, and Φ measuring0. Frequencies the criticalcurrent current, near the the degeneracy points depends strongly on the gate volt- mate form Ic(Φs) ≈ 2I0ᎂcos(πΦs/Φ0)ᎂ. Thus, by biasing the SQUID with a range from 9.711changes GHz to 0.850 in flux GHz. produced Tick marks by on a nearby the y axis qubit show can steps be measuredage (Fig. with 3c). high Capacitively coupling the Cooper-pair-box island to the of 0.4 nA. (Panel reproduced, with permission, from ref. 15.) c, Microwave of Φe are shown in Fig. 1d. At the degeneracy point, the probabilityconstant magnetic of flux nearsensitivity. Φ0/2, and In most measuring experiments thewith qubits,critical a pulse current, of currentSET the island is applied makes a contribution0.5 to the SET gate voltage so that the SET frequency fm is plotted against half of the separation in magnetic flux,

to the SQUID, which either remains in the zero-voltagecurrent state or strongly makes depends on theProbability Cooper-pair-box state. This scheme observing either state is ½. As Φe is reduced, the probability ofchanges observing in flux produced∆Φres, between theby peak a nearby and the dip qubit at each frequency. can be The measured line is a linear with high fit through the dataa transition at high frequencies to the voltage and represents state, producing the classical a energy. voltage 2∆convertss/e. Because the measurement its of charge into a measurement of charge trans-

E 0.5 ᎂ 〉 increases while that of observing ᎂ 〉 decreases. sensitivity. In mostThe experiments inset is a magnifiedcurrent–voltage view with of the qubits, characteristic lower part a of pulse the is graph; hysteretic,D of the current curved the lineSQUID is applied portremainsn through at this a SET. In fact, for small Josephson junctions, this charge in the inset is a fitvoltage to equation until (4). the The current deviation bias of hasthe databeen points removed, from the allowing transport researchers is usually dissipative, because the phase coherence is destroyed to the SQUID, which either remains in the zero-voltage state or makes Probability The first observation of quantum superposition in a flux qubit was ᎂ 〉 ᎂ 〉 by environmental fluctuations. Thus,0 the read-out actually involves straight line demonstratesto determine quantum whether coherence the SQUIDof the has and switched. flux states. For sufficiently small Fe made spectroscopically. The state of the flux qubit is measureda transition with to the(Panel voltage reproduced, currentstate, with permission,pulses, producing the from probability ref. a 15.) voltage of the SQUID 2∆s/ eswitching. Because ismeasuring zero, its whereas the resistance of the SET, which depends on the state of the

14 Cooper-pair box. The preferred read-out device is a radio-frequency ᎂ 〉 current–voltage characteristicthe probability is hysteretic, is one for thesufficiently SQUID large remainspulses. The switchingat this17 event Figure 1 | The theory underlying flux qubits. a a d.c. superconducting quantum interference device (SQUID) . This SET , in which a SET is embedded in a resonant circuit. Thus, the, Flux Q qubits consist of a is a stochasticᎂ 〉process and needs to be repeated many times for the flux voltage until the versuscurrent applied bias flux are has shown been in Fig. removed, 2b for a succession allowing of microwave researchers fre- of the resonant circuitsuperconducting is determined loop by the interrupted resistance by of either the SET one and or three (shown) Josephson device consists of two Josephson junctions, each with critical current I0, in the SQUID to be measured accurately. 0 ᎂ 〉 quencies. As expected, the difference in the applied flux at which the peaks ultimately by the chargejunctions. on the The Cooper-pair two quantum box. statesA pulse are of magnetic microwaves flux Φ pointingF up to determine whether the SQUIDThe first step has in spectroscopicswitched. observation For sufficiently of quantume superpositionsmall is and Φ pointing down ᎂ 〉 or, equivalently, supercurrent I circulatinge in the connected in parallel on a superconducting loop of inductance L. The and dips appear, 2∆Φres, becomes greater as the microwave frequency slightly detuned from the resonant frequency is applied to the radio- q to determine the height of the current pulse at which the SQUID switches loop anticlockwise and I circulating clockwise. b, The double-well potential current pulses, theincreases. probability The microwave of the frequency SQUID versus switching ∆Φres is shown is in zero, Fig. 2c. whereas frequency SET, and the phase of the reflectedq signal enables the state of critical current of the SQUID Ic(Φs) is periodic in the externally applied (black) versus total flux Φ contained in a flux qubit is shown. The two wells The data have been— with, fitted for to example, equation a (4) probability with Iq = ( of½ )d½ν —/d Φase ain function the limit of Φthee over qubit a narrow to be determined. the probability is one for sufficiently large pulses. The switching event Figureare symmetrical1 | The theory when the underlying externally applied flux magnetic qubits. flux a Φ is (n + ½)Φ , ≡ ν >> ∆, using ∆range as a fitting (perhaps parameter. ± 5mΦ The0). Subsequently, data reveal the a pulseexistence of microwave of an fluxMany is appliedof the initial studies of superconducting qubits involved charge , eFlux qubits0 consist of a magnetic flux Φs with period Φ0. In the limit βL 2LI0/Φ0 << 1 in which where n is an integer (n = 0 in this case). The coloured curves are the is a stochastic processanticrossing and (that atneeds frequency is, an avoided to fbem, whichcrossing) repeated is of at sufficientΦe = many Φ0/2. amplitude times and for duration thequbits. flux to equalizeThat crossingsuperconducting is avoided at the degeneracy loop interrupted point was first by shown either one or three (shown) Josephson the Josephson inductance dominates the geometrical inductance, the the populations of the ground state and first excited state whenspectroscopically the energy- byeigenfunctions studying a charge (probability qubit9, and amplitudes) charge measurements for the ground state (symmetrical; in the SQUID to be measured accurately. junctions.red) and firstThe excited two statequantum (antisymmetrical; states are blue). magnetic c, The energy flux E of theΦ pointing up ᎂ 〉 Charge qubitslevel splitting difference ν = hfm. AssumingF0/2 that ᎂ 〉 is measured,revealed then, the oncontinuous, quantum-rounded form of the transition critical current for Φs = (m + ½)Φ0 (m is an integer) is reduced to almost two superpositions18 states in b versus the energy bias ε = 2I (Φ − Φ /2) is A charge qubit (alsod known1 as a Cooper-pair box) is shown in Fig. 3a, b. between quantum states . The coherent oscillations that occur with q e 0 The first step in spectroscopicresonance, observation there will be a of peak quantum in the switching superposition probability for isΦe < Φ0and/2 shown.Φ pointing The diagonal down dashed ᎂ black〉 or, li equivalently,nes show the classical supercurrent energies. The Iq circulating in the zero, and the flux dependence of the critical current takes the approxi- The key component is a tiny superconducting island that is small time at this11,15 avoided energy-level crossing were also first discovered by and a corresponding dip for Φe > Φ0/2. An example of these results is energy-level splitting is Δ at the degeneracy point, ε = 0, and is νb for ε ≠ 0. 14 to determine the heightenough that of the the electrostatic current charging pulse energy at which required the to place SQUID a charge switches studying a chargeloop qubit anticlockwise19. and Iq circulating clockwise. , The double-well potential mate form Ic(Φs) ≈ 2I0ᎂcos(πΦs/Φ0)ᎂ. Thus, by biasing the SQUID with a shown in Fig. 2. The configuration of the qubit and the SQUID is shown d, The probabilities of the qubit flux pointing up (green) or down (yellow) — with, for example,1034 a probabilityin Fig. 2a, of and ½ the — peaks as aand function dips in the ofamplitude Φe over of the a narrowswitching current (black) in the versus ground statetotal versus flux applied Φ contained flux are shown. in a flux qubit is shown. The two wells constant magnetic flux near Φ0/2, and measuring the critical current, the are symmetrical when the externally applied magnetic flux Φ is (n + ½)Φ , range (perhaps ± 5mΦ0). Subsequently, a pulse of microwave flux is applied 1033 e 0 changes in flux produced by a nearby qubit can be measured with high where n is an integer (n = 0 in this case). The coloured curves are the at frequency fm, which is of sufficient amplitude and duration to equalize sensitivity. In most experiments with qubits, a pulse of currentthe is populations applied of the ground0.5 state and first excited state when the energy- eigenfunctions (probability amplitudes) for the ground state (symmetrical; red) and first excited state (antisymmetrical; blue). c, The energy E of the

to the SQUID, which either remains in the zero-voltage state or makes Probability level splitting difference ν = hfm. Assuming that ᎂ 〉 is measured, then, on b a transition to the voltage state, producing a voltage 2∆ /e. Because its two superpositions states in versus the energy bias ε = 2Iq(Φe − Φ0/2) is s resonance, there will be a peak in the switching probability for Φe < Φ0/2 shown. The diagonal dashed black lines show the classical energies. The current–voltage characteristic is hysteretic, the SQUID remains at this 11,15 and a corresponding dip for Φe > Φ0/2. An example of these results is energy-level splitting is Δ at the degeneracy point, ε = 0, and is ν for ε ≠ 0. voltage until the current bias has been removed, allowing shownresearchers in Fig. 2. The configuration0 of the qubit and the SQUID is shown d, The probabilities of the qubit flux pointing up (green) or down (yellow) to determine whether the SQUID has switched. For sufficientlyin Fig. 2a,small and the peaks and dips in the amplitude of the switchingF currente in the ground state versus applied flux are shown. current pulses, the probability of the SQUID switching is zero, whereas 1033 the probability is one for sufficiently large pulses. The switching event Figure 1 | The theory underlying flux qubits. a, Flux qubits consist of a is a stochastic process and needs to be repeated many times for the flux superconducting loop interrupted by either one or three (shown) Josephson in the SQUID to be measured accurately. junctions. The two quantum states are magnetic flux Φ pointing up ᎂ 〉

The first step in spectroscopic observation of quantum superposition is and Φ pointing down ᎂ 〉 or, equivalently, supercurrent Iq circulating in the to determine the height of the current pulse at which the SQUID switches loop anticlockwise and Iq circulating clockwise. b, The double-well potential — with, for example, a probability of ½ — as a function of Φe over a narrow (black) versus total flux Φ contained in a flux qubit is shown. The two wells range (perhaps ± 5mΦ0). Subsequently, a pulse of microwave flux is applied are symmetrical when the externally applied magnetic flux Φe is (n + ½)Φ0, where n is an integer (n = 0 in this case). The coloured curves are the at frequency fm, which is of sufficient amplitude and duration to equalize the populations of the ground state and first excited state when the energy- eigenfunctions (probability amplitudes) for the ground state (symmetrical; red) and first excited state (antisymmetrical; blue). c, The energy E of the level splitting difference ν = hfm. Assuming that ᎂ 〉 is measured, then, on two superpositions states in b versus the energy bias ε = 2Iq(Φe − Φ0/2) is resonance, there will be a peak in the switching probability for Φe < Φ0/2 11,15 shown. The diagonal dashed black lines show the classical energies. The and a corresponding dip for Φe > Φ0/2. An example of these results is energy-level splitting is Δ at the degeneracy point, ε = 0, and is ν for ε ≠ 0. shown in Fig. 2. The configuration of the qubit and the SQUID is shown d, The probabilities of the qubit flux pointing up (green) or down (yellow) in Fig. 2a, and the peaks and dips in the amplitude of the switching current in the ground state versus applied flux are shown. 1033 NATURE|Vol 453|19 June 2008 INSIGHT REVIEW

Cooper-pair boxes are particularly sensitive to low-frequency noise Although this switching read-out scheme is efficient, it has two from electrons moving among defects (see the section ‘Decoherence’) major drawbacks. First, the resultant high level of dissipation destroys and can show sudden large jumps in ng. The development of more the quantum state of the qubit, making sequential measurements of advanced charge qubits such as the transmon20 and quantronium12 has the state impossible. Second, the temperature of the read-out junction greatly ameliorated this problem. The transmon is a small Cooper- and substrate increase because of the energy that is deposited while the pair box that is made relatively insensitive to charge by shunting the SQUID is in the voltage state — typically for 1 µs — and the equilibrium Josephson junction with a large external capacitor to increase Ec and by is not restored for ~1 ms. This limits the rate at which measurements can NATURE|Vol 453|19 June 2008 increasing the gate capacitor to the same size.INSIGHT Consequently, REVIEW the energy be made to ~1 kHz, resulting in long data-acquisition times. bands of the type shown in Fig. 3c are almost flat, and the eigenstates are These drawbacks have been overcome by the introduction of the a combination of many Cooper-pair-box charge states. For reasons that Josephson bifurcation amplifier (JBA)22, a particularly powerful read- will be discussed later (see the section ‘Decoherence’), the transmon is out device in which there is no dissipation because the junction remains Cooper-pair boxes are particularly sensitive to low-frequency noise Althoughthus insensitive this switching to low-frequency read-out scheme charge noise is efficient, at all operating it has twopoints. in the zero-voltage state (Fig. 4c). The JBA exploits the nonlinearity of from electrons moving among defects (see the section ‘Decoherence’) major drawbacks.At the same First, time, thethe resultantlarge gate highcapacitor level provides of dissipation strong destroys coupling to the Josephson junction when a capacitor is connected across it, resulting and can show sudden large jumps in ng. The development of more the quantumexternal state microwaves of the qubit, even at making the level sequential of a single photon, measurements greatly increas- of in the formation of a resonant (or tank) circuit. When small-amplitude advanced charge qubits such as the transmon20 and quantronium12 has the stateing impossible. the coupling Second, for circuit the temperaturequantum electrodynamics of the read-out (QED) junction (see the microwave pulses are applied to the resonant circuit, the amplitude and greatly ameliorated this problem. The transmon is a small Cooper- and substratesection increase ‘Quantum because optics ofon the a chip’). energy that is deposited while the phase of the reflected signal are detected, with the signal strength boosted pair box that is made relatively insensitive to charge by shunting the SQUID is Thein the principle voltage bystate which — typically quantronium for 1 µs operates — and theis shown equilibrium in Fig. 4a, by a cryogenic amplifier. From this measurement, the resonant frequency and an actual circuit is shown in Fig. 4b. The Cooper-pair box involves of the tank circuit can be determined, then the inductance of the junction Josephson junction with a large external capacitor to increase Ec and by is not restored for ~1 ms. This limits the rate at which measurements can increasing the gate capacitor to the same size. Consequently, the energy be madetwo to Josephson~1 kHz, resulting junctions, in withlong a data-acquisition capacitance Cg connected times. to the island — which depends on the current flowing through it — and, finally, the separating them. The two junctions are connected across a third, larger, state of the quantronium. For larger-amplitude microwaves, however, the bands of the type shown in Fig. 3c are almost flat, and the eigenstates are These drawbacks have been overcome by the introduction of the junction, with a higher critical current, to form a closed superconduct- behaviour of the circuit is strongly nonlinear, with the resonance frequency a combination of many Cooper-pair-box charge states. For reasons that Josephson bifurcation amplifier (JBA)22, a particularly powerful read- ing circuit to which a magnetic flux Φe is applied. The key to eliminating decreasing as the amplitude increases. In particular, strong driving at fre- will be discussed later (see the section ‘Decoherence’), the transmon is out devicethe in effects which of there low-frequency is no dissipation charge because and flux the noise junction is to maintain remains the quencies slightly below the plasma frequency leads to a bistability: a weak, thus insensitive to low-frequency charge noise at all operating points. in the zero-voltagequbit at the double state (Fig. degeneracy 4c). The point JBA atexploits which the twononlinearity qubit states of are off-resonance lower branch during which the particle does not explore the At the same time, the large gate capacitor provides strong coupling to the Josephson(to first junction order) insensitive when a capacitor to these is noise connected sources. across To achieve it, resulting insensi- nonlinearity, and a high-amplitude response at which frequency matches external microwaves even at the level of a single photon, greatly increas- in the formationtivity to charge of a resonant noise, the (or qubit tank) is operatedcircuit. When at ng = small-amplitude ½, where the energy the driving frequency (Fig. 4d). The two qubit states can be distinguished ing the coupling for circuit quantum electrodynamics (QED) (see the microwavelevels pulses have are zero applied slope toand the the resonant energy-level circuit, splitting the amplitude is Ej (Fig. and 3c). by choosing driving frequencies and currents that cause the JBA to switch section ‘Quantum optics on a chip’). phase ofInsensitivity the reflected to signal flux noiseare detected, is achieved with by the applying signal strength an integer boosted number to one response or the other, depending on the qubit state. This technique The principle by which quantronium operates is shown in Fig. 4a, by a cryogenicof half-flux amplifier. quanta From to the this loop. measurement, The success the of thisresonant optimum frequency working is extremely fast and, even though it is based on a switching process, it and an actual circuit is shown in Fig. 4b. The Cooper-pair box involves of the tankpoint circuit has been can be elegantly determined, shown then experimentally the inductance21. The of the insensitivity junction to never drives the junction into the voltage state. Furthermore, the JBA two Josephson junctions, with a capacitance Cg connected to the island — whichboth depends flux and on charge the current implies, flowing however, through that the it —two and, states finally, of the the qubit remains in the same state after the measurement has been made. cannot be distinguished at the double degeneracy point. To measure The JBA has been shown to approach the quantum non-demolition separating them. The two junctions are connected across a third, larger, state of the quantronium. For larger-amplitude microwaves, however, the 22 junction, with a higher critical current, to form a closed superconduct- behaviourthe of qubit the circuit state, a is current strongly pulse nonlinear, that moves with thethe qubitresonance away frequencyfrom the flux (QND) limit . This limit is reached when the perturbation of the quan- degeneracy point is applied to the loop, and this produces a clockwise tum state during the measurement does not go beyond that required by ing circuit to which a magnetic flux Φe is applied. The key to eliminating decreasing as the amplitude increases. In particular, strong driving at fre- the effects of low-frequency charge and flux noise is to maintain the quenciesor slightly anticlockwise below thecurrent plasma in the frequency loop, depending leads to aon bistability: the state of a theweak, qubit. the measurement postulate of quantum mechanics, so repeated meas- The direction of the current is determined by the third (read-out) junc- urements of the same eigenstate lead to the same outcome23. Reaching qubit at the double degeneracy point at which the two qubit states are off-resonance lower branch during which the particle does not explore the tion: the circulating current either adds to or subtracts from the applied the QND limit is highly desirable for quantum computing. (to first order) insensitive to these noise sources. To achieve insensi- nonlinearity,current and pulse, a high-amplitude so the read-out response junction at switcheswhich frequency out of the matches zero-volt- A similar scheme that approaches the QND limit has been imp- tivity to charge noise, the qubit is operated at ng = ½, where the energy the drivingage frequencystate at a slightly (Fig. 4d). lower The or two slightly qubit higher states value can be of distinguished the bias current, lemented for the flux qubit, with the single Josephson junction replaced 24 levels have zero slope and the energy-level splitting is Ej (Fig. 3c). by choosingrespectively. driving frequencies Thus, the state and ofcurrents the qubit that can cause be inferredthe JBA to by switch measur- by a read-out SQUID . Dispersive read-out for a flux qubit has also Insensitivity to flux noise is achieved by applying an integer number to one responseing the switching or the other, currents. depending With on the the advent qubit state.of quantronium, This technique much been achieved by inductively coupling a flux qubit to the inductor of a of half-flux quanta to the loop. The success of this optimum working is extremelylonger fast relaxation and, even and though decoherence it is based times on can a switching be achieved process, than with it a resonant circuit and then measuring the flux state from the shift in the point has been elegantly shown experimentally21. The insensitivity to never drivesconventional the junction Cooper-pair into the box. voltage state. Furthermore, the JBA resonance frequency 25. both flux and charge implies, however, that the two states of the qubit remains in the same state after the measurement has been made. cannot be distinguished at the double degeneracy point. ToCharge measure qubit The JBA has been shown to approach the quantum non-demolition –V /2 the qubit state, a current pulse that moves the qubit away from the flux (QND) alimit22. This limitb is reached when the perturbation of the quan-cdtr degeneracy point is applied to the loop, and this produces a clockwise tum state during the measurement does not go beyond that required by SET island SCB island or anticlockwise current in the loop, depending on the state of the qubit. the measurementE postulateC of quantum mechanics, so repeated meas- SET SET j j 23 E Φe Cg Vg j Cint The direction of the current is determined by the third (read-out) junc- urements of then same eigenstate lead to the same outcomeSCB. Reaching gate I Cj tion: the circulating current either adds to or subtracts from the applied the QND limit is highly desirable for quantum computing. Energy n Cg current pulse, so the read-out junction switches out of the zero-volt- A similar scheme that approaches the QND limit has been imp- Cg age state at a slightly lower or slightly higher value of the bias current, lemented forVg the flux qubit, with the single Josephson junction replaced V respectively. Thus, the state of the qubit can be inferred by measur- by a read-out SQUID24. Dispersive read-out for a flux qubit has also g 0.5 1.5 ing the switching currents. With the advent of quantronium, much been achieved by inductively coupling a flux qubit to the inductor of a Vtr/2 ng longer relaxation and decoherence times can be achieved than• withSuperconductor a resonant connected circuit and to then a Cooper measuring pair thebox flux state from the shift in the 25 conventional Cooper-pair box. • Qubit stateresonance is determinedFigure frequency 3 | Charge by the .qubits. (quantised) a, A single number Cooper-pair-box of cooper (SCB) pairs circuit in the is box lines indicate the contribution of the charging energy Ech(n, ng) alone. The • Control via electricalshown. gating The superconducting island is depicted in brown and the junction energy-level splitting at ng = ½ is Ej. Red curves show the current I through in blue. Ej and Cj are the Josephson coupling energy and self-capacitance, the SET as a function of ng. Transport is possible at the charge degeneracy –V /2 a b cdrespectively, and n is the number of Cooper pairs on the island, whichtr points, where the gate strongly modulates the current. (Panel reproduced, is coupled to a voltage source with• Readoutvoltage V byvia way a single of a capacitor electron with transistor with permission, from ref. 28.) d, A charge qubit with two junctions (left) • Also couplingg to microwave fields SET island SCB island capacitance Cg. (Panel reproduced, with permission, from ref. 28.) coupled to a SET biased to a transport voltage Vtr (right) is shown. The b (cavity QED) critical current of the junctions coupled to the island is adjusted by means E C , A micrograph of a Cooper-pair box coupled to a single-electronSET SET j j E Φe Cg Vg transistor (SET) is shown. Scale bar,j 1 μm. (Panel reproduced,Cint with of an externally applied magnetic flux Фe. The gate of the SET is coupled to n SCB gate I permission, from ref. 78.) c, Black curvesCj show the energy of the Cooper- an externally controlled charge induced on the capacitor with capacitance

Energy n SET SET C pair box as a function of the scaled gate voltage ng = CgVg/2e for different Cg by the voltage Vg , as well as to the qubit charge by way of the g C numbers (n) of excess Cooper pairs on gthe island. The parabola on the interaction capacitance Cint. (Panel reproduced, with permission, from Vg far left corresponds to n = 0 and the central parabola to n = 1. Dashed ref. 28.) Vg 1035 0.5 1.5 Vtr/2 ng

Figure 3 | Charge qubits. a, A single Cooper-pair-box (SCB) circuit is lines indicate the contribution of the charging energy Ech(n, ng) alone. The shown. The superconducting island is depicted in brown and the junction energy-level splitting at ng = ½ is Ej. Red curves show the current I through in blue. Ej and Cj are the Josephson coupling energy and self-capacitance, the SET as a function of ng. Transport is possible at the charge degeneracy respectively, and n is the number of Cooper pairs on the island, which points, where the gate strongly modulates the current. (Panel reproduced, is coupled to a voltage source with voltage Vg by way of a capacitor with with permission, from ref. 28.) d, A charge qubit with two junctions (left) capacitance Cg. (Panel reproduced, with permission, from ref. 28.) coupled to a SET biased to a transport voltage Vtr (right) is shown. The b, A micrograph of a Cooper-pair box coupled to a single-electron critical current of the junctions coupled to the island is adjusted by means transistor (SET) is shown. Scale bar, 1 μm. (Panel reproduced, with of an externally applied magnetic flux Фe. The gate of the SET is coupled to permission, from ref. 78.) c, Black curves show the energy of the Cooper- an externally controlled charge induced on the capacitor with capacitance SET SET pair box as a function of the scaled gate voltage ng = CgVg/2e for different Cg by the voltage Vg , as well as to the qubit charge by way of the numbers (n) of excess Cooper pairs on the island. The parabola on the interaction capacitance Cint. (Panel reproduced, with permission, from far left corresponds to n = 0 and the central parabola to n = 1. Dashed ref. 28.) 1035 CoherenceINSIGHT REVIEW measurements NATURE|Vol 453|19 June 2008

ᎂ0〉 Figure 6 | Qubit manipulation in the time domain. a, The Bloch sphere is depicted, a b 100 with an applied static magnetic field B0 and B0 a radio-frequency magnetic field Brf. Any given superposition of the six states shown is represented by a unique point on the surface ᎂ0〉 + i ᎂ1〉 80 of the sphere. b, Rabi oscillations in a flux

ᎂ0〉 – ᎂ1〉 (%) qubit are shown. The probability psw that the sw

p detector (SQUID) switches to the normal state Brf ᎂ0〉 + ᎂ1〉 60 versus pulse length is shown, and the inset is a ᎂ0〉 – i ᎂ1〉 magnification of the boxed region, showing that the dense traces are sinusoidal oscillations. As 40 expected, the excited-state population oscillates 0 1 2 under resonant driving. (Panel reproduced, with c Pulse length (µs) permission, from ref. 40.) , Ramsey fringes in ᎂ1〉 a phase qubit are shown. Coherent oscillations

of the switching probability p1 between two detuned π/2 pulses is shown as a function of c 1 d 1 pulse separation. (Panel reproduced, with permission, from ref. 31.) d, The charge echo in a Cooper-pair box is shown as a function of the

time difference δt = t1 − t2, where t1 is the time

1 0 p 0.5 between the initial π/2 pulse and the π pulse,

and t2 is the time between the π pulse and the second π/2 pulse. The echo peaks at δt = 0. (Panel (arbitrary units) units) (arbitrary I reproduced, with permission, from ref. 39.) –1 NATURE0 |Vol 453|19 June 2008 INSIGHT REVIEW 0 100 200 –600 –300 0 300 600 Time (ns) δt (ps)

Measuring the times T1, T2 and T2* provides an important initial 0–10 GHz. It is particularly difficultstrength to avoid can resonances be increased over such a broadby having the two qubits use a common line. characTable terization 1 | Highest of qubit reported coherence. values However, of T 1other, T2* andfactors T2 such as range of frequencies. Clever engineering has greatly reduced this source of Even stronger coupling can be achieved by including a Josephson junc- pulseQubit in accuracy,T1 relaxation(μs) T 2during* (μs) measurementT2 (μs) Source and more complex decoherence, but it would be optimistic toFor consider a review, that this problem see has decoherence effects result in measurement errors. A more complete been completely solved. Flux 4.6 1.2 9.6 Y. Nakamura, personal communication tionJ. Clarkein this line and to F.increase K. Wilhelm, the line’s self-inductance (equation (6), Box 1). measure of a qubit is fidelity, a single number that represents the dif- The main intrinsic limitationIn onthe the case coherence of charge of superconducting and phase qubits, nearest-neighbour interactions ferenceCharge between 2.0 the ideal and 2.0 the actual 2.0 outcome ref. of the 77 experiment. qubits results from low-frequencyNature noise, notably 453, ‘1/ f1031 noise’ (in (2008) which Determining the fidelity involves quantum-process tomography the spectral density of the noiseare at mediatedlow frequency by f scales capacitors as 1/f α, where rather than inductors. Fixed interaction has (aPhase repeated set of 0.5 state tomographies), 0.3 which 0.5 characterizes J. Martinis, a quantum- personalα communication is of the order of unity). In thebeen solid implemented state, many 1/f noise for sources flux, arecharge and phase qubits42–45. These experi- mechanical process for all possible initial states. In a Ramsey-fringe well described by the Dutta–Horn model as arising from a uniform dis- tomography experiment, Matthias Steffen et al.31 found a fidelity of tribution of two-state defects32ments. Each defect show produces the energy random telegraph levels that are expected for the superposition of ~80%, where 10% of the loss was attributed to read-out errors and noise, and a superposition of twosuch uncorrelated pseudospin processes states: leads namely, to a 1/f a ground state and three excited states; another 10% to pulse-timing uncertainty. power spectrum. There are three recognized sources of 1/f noise. The In general, all three low-frequency processes leadfirst to is decoherence. critical-current fluctuations, the first which and arise second from fluctuations excited in statesthe may be degenerate. The entanglement DecoherenceThey do not contribute to relaxation because this processtransparency requires of the junction an causedof these by the states trapping for and two untrapping phase qubits of has been shown explicitly by means of Superconductingexchange of energy qubits are with macroscopic, the environment so — along the at linesthe energy-levelof electrons in the splitting tunnel barrier quantum-state33. All superconducting tomography qubits are subject 46. The most general description (including Schrödinger’s cat — they could be expected to be very sensitive to to dephasing by this mechanism. The slow fluctuations modulate energy- defrequencycoherence. In of fact, the given qubit, the unique which properties is typically of the superconducting in the gigahertz level splitting, range. even How- at the degeneracyall imperfections) point, so each measurementof the qubit is state based on the four basis states of the state,ever, careful there engineering is strong has evidence led to remarkable that chargeincreases influctuations decoherence aremade associated on a qubit with with a slightly coupled different frequency. qubits is The a four-by-four resultant phase array known as a density matrix. Steffen timesthe high-frequency compared with those resonatorsof early devices. that have been observed,errors in leadparticular, to decoherence. in et al.46 carried out a measurement of the density matrix; they prepared a Ideally, each type37 of qubit is described by a single degree of freedom. The second source of 1/f noise is charge fluctuations, which arise Thephase central qubits challenge. isImprovements to eliminate all other indegrees the of quality freedom. ofIn broadthe oxidefrom layersthe hopping that of electronsare system between traps in a on particular the surface of entangled the super- state and showed that only the correct terms,used there in the are twojunctions classes of decohering and capacitors element: extrinsic have resultedand intrinsic. in largeconducting reductions film or the in surface four of the matrix substrate. elements This motion were induces non-zero — and that their magnitude was in Obvious extrinsic sources include electromagnetic signals from radio and charges38 onto the surface of nearby superconductors. This decoherence televisionthe concentration transmitters; these of can these generally high-frequency be eliminated by using resonators careful mechanism. is particularly problematicgood agreement for charge qubits, with except theory. at the This experiment is a proof-of-principle shieldingThe andstrategy enough ofbroadband operating filters. a A qubit more challenging at the optimum extrinsic degeneracypoint, which point, wherewas the qubitsdemonstration are (to first order) of insensitive. a basic functionIf the required for a quantum computer. 47,48 sourcefirst carriedto exclude isout the withlocal electromagnetic quantronium environment: but is nowfor exam- appliedvalue to of Eallc/Ej increases,types of however, Simple the energy quantum bands (Fig. gates 3c) become have flat- also been demonstrated . ple, contributions from the leads that are coupled to read-out devices or ter, and the qubit is correspondingly less sensitive to charge noise away aresuper used conducting to apply flux or chargequbit biases. (except These for leads phase allow great qubits), flexibility has beenfrom the successful degeneracy point. at This meTwochanism flux underlies qubits the can substantially be coupled by flux transformers — in essence 20 inincreasing control of the phase-coherence system at the expense of times considerable by large coupling factors. to the Furtherincreased substantial values of T2 in the transmona closed. loop of superconductor surrounding the qubits — enabling environ ment. This issue was recognized in the first proposals of macro- The third source of 1/f noise is magnetic-flux fluctuations. Although improvements have resulted from the use of charge- or flux-echo tech- their interaction to be mediated34 over longer distances. Because the scopic quantum39,40 coherence and largely motivated the Caldeira–Leggett such fluctuations were first characterized more than 20 years ago , the theoryniques of quantum. In NMR, dissipation the6. Thisspin-echo theory maps technique any linear dissipationremoves themechanism inhomogeneous by which these occursuperconducting remained obscure untilloop recently. conserves It magnetic flux, a change in the state ontobroadening a bath of harmonic that is oscillators. associated The effectswith, of for these example, oscillators canvariations is now thought in magnetic that flux noiseof arises one from qubit the fluctuations induces aof circulatingunpaired current in the loop and hence a flux be calculated from the Johnson–Nyquist noise that is generated by the electron spins on the surface of the superconductor or substrate35,36, but complexfield, and impedance hence of inthe theenvironment. NMR frequency,In the weak-damping over regime,the sample. the details In the of the case mechanism of in remain the othercontroversial. qubit. Flux Flux noise transformers causes that contain Josephson junctions bothqubits, T1 and the τϕ can variation be computed is indirectly the fromqubit the energy-level power spectrum ofsplitting this decoherence frequency in flux from qubits, exceptenable at the the degeneracy interaction point, as of well qubits as in to be turned on and off in situ. One such noise, and then the impedance can be engineered to minimize decoher- phase qubits, which have no degeneracy point. The increased value of 49 measurement28,29 to measurement. For some qubits, using a combination device consists of a d.c. SQUID surrounding two flux qubits (Fig. 7a). ence . The experimental difficulty is to ensure that the complex imp- T2 in quantronium results from its insensitivity to both flux noise and edancesof echo ‘seen’ techniques by the qubit are and high optimum over a broad bandwidth, point operation for example, has charge eliminated noise at the pure double degeneracyThe inductance point. between the two qubits has two components: that of the 1038dephasing, so decoherence is limited by energy relaxation (T2* = 2T1). In direct coupling between the qubits, and that of the coupling through general, however, the mechanisms that limit T1 are unknown, although the SQUID. For certain values of applied bias current (below the critical resonators that are associated with defects may be responsible36,41. The current) and flux, the self-inductance of the SQUID becomes nega- highest reported values of T1, T2* and T2 are listed in Table 1. tive, so the sign of its coupling to the two qubits opposes that of the direct coupling. By choosing parameters appropriately, the inductance Coupled qubits of the coupled qubits can be designed to be zero or even have its sign An exceedingly attractive and unique feature of solid-state qubits in reversed. This scheme has been implemented by establishing the val- general and superconducting qubits in particular is that schemes can ues of SQUID flux and bias current and then using microwave manip- be implemented that both couple them strongly to each other and ulation and measuring the energy-level splitting of the first and second turn off their interaction in situ by purely electronic means. Because excited states50 (Fig. 7b). A related design — tunable flux–flux coupling the coupling of qubits is central to the architecture of quantum compu- mediated by an off-resonant qubit — has been demonstrated51, and ters, this subject has attracted much attention, in terms of both theory tunable capacitors have been proposed for charge qubits52. and experiment. In this section, we illustrate the principles of coupled Another approach to variable coupling is to fix the coupling strength qubits in terms of flux qubits and refer to analogous schemes for other geometrically and tune it by frequency selection. As an example, we superconducting qubits. consider two magnetically coupled flux qubits biased at their degeneracy Because the flux qubit is a magnetic dipole, two neighbouring flux points. If each qubit is in a superposition of eigenstates, then its magnetic qubits are coupled by magnetic dipole–dipole interactions. The coupling flux oscillates and the coupling averages to zero — unless both qubits oscillate at the same frequency, in which case the qubits are coupled. This I phenomenon is analogous to the case of two pendulums coupled by a abpb weak spring. Even if the coupling is extremely weak, the pendulums will 140 be coupled if they oscillate in antiphase at exactly the same frequency. Flux 120 Implementing this scheme is particularly straightforward for two qubit 100 phase qubits because their frequencies can readily be brought in and 37 80 out of resonance by adjusting the bias currents . For other types of qubit, the frequency at the degeneracy point is set by the as-fabricated param- 60 d.c. eters, so it is inevitable that there will be variability between qubits. As SQUID 40 a result, if the frequency difference is larger than the coupling strength, 20 Energy-level splitting (MHz) Energy-level the qubit–qubit interaction cancels out at the degeneracy point. Several 53–55 0 pulse sequences have been proposed to overcome this limitation , 0 0.2 0.4 0.6 0.8 none of which has been convincingly demonstrated as yet. The two- I ( A) pb µ qubit gate demonstrations were all carried out away from the optimum point, where the frequencies can readily be matched. Figure 7 | Controllably coupled flux qubits. a , Two flux qubits are shown On the basis of these coupling schemes, several architectures have surrounded by a d.c. SQUID. The qubit coupling strength is controlled been proposed for scaling up from two qubits to a quantum computer. by the pulsed bias current Ipb that is applied to the d.c. SQUID before measuring the energy-level splitting between the states ᎂ1〉 and ᎂ2〉. b, The The central idea of most proposals is to couple all qubits to a long central coupling element, a ‘quantum bus’56,57 (Fig. 8), and to use frequency selec- filled circles show the measured energy-level splitting of the two coupled 56–60 flux qubits plotted against Ipb. The solid line is the theoretical prediction, tion to determine which qubits can be coupled . This scheme has been fitted for Ipb; there are no fitted parameters for the energy-level splitting. experimentally demonstrated. As couplers become longer, they become Error bars, ±1σ. (Panels reproduced, with permission, from ref. 50.) transmission lines that have electromagnetic modes. For example, two 1039 Other systems / hybrid systems NMR: • First system with gate operations on multiple qubits • Demonstrated factorization of 15 with Shor’s algorithm • Difficulties in scaling for liquid phase (limited by molecule size) • Somewhat replaced by NMR in solid state qubits

• See Nielsen & Chuang for a detailed summary

Optical qubits • Polarisation or two-rail encoding, single qubit gates by linear elements • Probabalistic quantum gates, measurement-based entanglement • Optical C-Not gate (O’Brien et al, Nature 2003) HybridHybrid systems quantum systems Combine advantages of different physical quantum systems, e.g. fast (but decohering) qubits with slow (but protected) qubits; or matter qubits (robust, strongly interacting) with flying qubits (fragile, weakly interacting) !! Neutral atoms in lattices and optical cavities! Electron and nuclear spins in semiconductors!

Superconducting qubits with B. E. Kane, Nature (1998)! microwave cavity photons! Diamond color centers with microwave cavity photons!

Topological qubits GS Research talk, Mar 30th 2012! • Protected quantum memories based on non-trivial state topology (solid state, e.g., groups of J. Levy/S. Frolov at Pitt)