Quantum Computing Joseph C. Bardin, Daniel Sank, Ofer Naaman, and Evan Jeffrey
©ISTOCKPHOTO.COM/SOLARSEVEN
uring the past decade, quantum com- underway at many companies, including IBM [2], Mi- puting has grown from a field known crosoft [3], Google [4], [5], Alibaba [6], and Intel [7], mostly for generating scientific papers to name a few. The European Union [8], Australia [9], to one that is poised to reshape comput- China [10], Japan [11], Canada [12], Russia [13], and the ing as we know it [1]. Major industrial United States [14] are each funding large national re- Dresearch efforts in quantum computing are currently search initiatives focused on the quantum information
Joseph C. Bardin ([email protected]) is with the University of Massachusetts Amherst and Google, Goleta, California. Daniel Sank ([email protected]), Ofer Naaman ([email protected]), and Evan Jeffrey ([email protected]) are with Google, Goleta, California.
Digital Object Identifier 10.1109/MMM.2020.2993475 Date of current version: 8 July 2020
24 1527-3342/20©2020IEEE August 2020
Authorized licensed use limited to: University of Massachusetts Amherst. Downloaded on October 01,2020 at 19:47:20 UTC from IEEE Xplore. Restrictions apply. sciences. And, recently, tens of start-up companies have Quantum computing has grown from emerged with goals ranging from the development of software for use on quantum computers [15] to the im- a field known mostly for generating plementation of full-fledged quantum computers (e.g., scientific papers to one that is Rigetti [16], ION-Q [17], Psi-Quantum [18], and so on). poised to reshape computing as However, despite this rapid growth, because quantum computing as a field brings together many different we know it. disciplines, there is currently a shortage of engineers
who understand both the engineering aspects (e.g., mi- a0 ||}HH==aa0101+ |,H (1) crowave design) and the quantum aspects required to ;a1E build a quantum computer [19]. The aim of this article is to introduce microwave engineers to quantum com- where |0 H and |1H are the zero and one computa- puting and demonstrate how the microwave commu- tional-basis vectors of the qubit, respectively, and the
nity’s expertise could contribute to that field. complex coefficients a0 and a1 are referred to as proba- To begin, we pose the question, What’s the big deal bility amplitudes. Physically, the probability amplitudes about quantum computing anyway? To put things in describe the likelihood that one would find the qubit perspective, let’s start by considering a regular digi- in the associated state were a measurement to be car- 2 2 tal computer—or, in the language of quantum engi- ried out: P ||0H = a0| and P |.1H =||a1 Note that in " , " , neers, a classical computer. The fundamental unit of (1) we have implicitly introduced Dirac notation, where information in a computer is the bit. In a conventional each of our computational-basis vectors is represented computer, one bit of memory is stored in the charge by a ket (i.e., |0 H and |)H .1 on a dynamic random-access memory (DRAM) cell. A quantum register comprises an ensemble of qubits, A memory register with N bits has 2N possible states and, just as with the digital register, an N-qubit regis- in the range {,00ff00 01,,01ff01ff,}11 . Thus, a ter has 2N possible states. However, since qubits can 300-b memory can store any of 2300 possible bit strings, exist in superposition states, the instantaneous state of a set so large that it is physically impossible to print the quantum register can be in any superposition of its N T it out since there is not enough matter in the universe 2 basis states; that is, |}aH ==00ff00aa01 f 11f 1 6 @ to make the ink, not to mention the paper (the infor- aa00ff00||00ff00HH++01 01 gf+ a11f 1|,111H where mation capacity of the observable universe, if every the probability amplitudes, ai, are constrained so that 2 2 2 degree of freedom of every particle was used to encode ||a00f 0 + ||aa00ff1 ++g ||111 = .1 Similar to the case a bit, is roughly 2b300 [20]). However, while a 300-b reg- of a single qubit, each probability amplitude describes ister can assume any of its 2300 possible states, we know the likelihood that the associated bit string would be that each bit is either zero or one, so at any given time measured the associated bit string if the state of the the register can store only one of the bit strings. quantum register were measured. In a classical computer, computations are carried out Computation on a quantum computer is typically by moving bits into a CPU, where logical operations are carried out as an in-memory operation. That is, rather implemented in physical circuitry. The essential func- than moving data from the memory to a CPU and back, tion of the CPU is to provide a prescribed output bit the quantum register is operated upon directly. An string for each possible input bit string so as to carry out operator that can be applied to a quantum register is a desired logical operation. Since this is digital circuitry, described by a unitary matrix t ,U which preserves the we know that for each input bit string the processor length of the unit state vector (UUtt@ = I). A universal will produce exactly one output bit string and that this quantum computer permits the application of any arbi- input-to-output mapping is defined in hardware, per- trary unitary operation to the quantum register; i.e., a haps with some subset of the input bits determining the universal quantum computer can perform any trans- logical operation applied to the remainder of the bits. formation of the form In a quantum computer, we still have physical objects representing bits, but the difference is that the ||}}lHH= t ,U (2) memory register exists as a quantum state. The basic building block in a quantum register is the quantum where Ut is an arbitrary unitary matrix. bit, or qubit. Just like a regular bit, a qubit has a zero Unitary operators should be nothing new to a micro- (ground) state and a one (excited) state. However, what wave engineer; the scattering matrix of a lossless passive distinguishes a qubit from a classical bit is that it can be network is unitary (SS@ = I). In fact, from a mathemati- in a superposition of its zero and one states. Mathemat- cal perspective, the way a quantum computation acts ically, the state of a qubit is described by a 2D complex upon the 2N states of a quantum register is identical to state vector of unit amplitude, the operation that occurs when N incident waves scatter
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Authorized licensed use limited to: University of Massachusetts Amherst. Downloaded on October 01,2020 at 19:47:20 UTC from IEEE Xplore. Restrictions apply. from a lossless N-port network (see Figure 1). Therefore, |.}HH=+05()||00 01HH++|| 10 11H , which can be real- we can see that familiar microwave concepts, such as ized by two isolated qubits, may be “factored” into superposition and interference, are also useful in pro- two single-qubit states |.}HH=+05()||01HH# (| 01+|)H . viding intuition with respect to quantum computing; On the other hand, the Bell state |(}HH=+||H)0011/ 2 complex microwave signal amplitudes are analogous cannot be factored (try it!), so it cannot be thought of as to complex quantum probability amplitudes, and, representing the information of two independent bits. whereas the signals in Figure 1(a) carry power, those in States that cannot be thought of as combinations of Figure 1(b) carry probabilities. single-qubit states are called entangled, and the genera- There is an important distinction between the tion of such states is essential in accessing the power of physical N-port model and the conceptual model of quantum computing. quantum computation: while the microwave N-port While universal quantum computing permits op has one physical port associated with each input/out- eration on all 2N computational states of a quantum put wave, the equivalent diagram describing quantum register, we cannot perform a measurement of a quan- computation has a port associated with each of its 2N tum superposition state. Instead, when we mea- computational states. Thus, the computational space of sure the state of the quantum register at the end of a universal quantum computer grows exponentially a computation, the computer must make a decision. with the number of qubits. This massive computational As dictated by the postulates of quantum mechan- capacity explains why quantum computing is so prom- ics, such a measurement returns a single classical bit ising: even at roughly 50 qubits, the computational of information per measured qubit, meaning that, if space is too large to simulate using the best supercom- one measures the state of an N-qubit quantum reg- puters, while an ideal quantum computer can easily ister, one will find the register in one of its 2N basis operate across the entire expanse [5]. states, with the likelihood of each outcome equal to Although superposition and interference make the modulus of the associated probability amplitude 2 2 qubits a little more complex than classical bits, those (i.e., P ||00f 0H = a00f 0|, P ||00f 1H = a00f 1|, and " , " , features alone do not provide the exponential com- so on). Additionally, the act of measuring the state putational space described previously. For instance, of a quantum register will cause any superposition a set of N isolated qubits is only about as complex as state to collapse, and the best one can do is that the a set of N classical bits because, when the qubits oper- postmeasurement state of the register agrees with the ate in isolation, only a small subset of possible super- measurement result. It is the job of the quantum algo- position states is accessed; practically speaking, we rithm designer to manipulate the quantum register really still have 2N degrees of freedom here. However, into a state where, if measured, a certain bit string that when the individual qubits are allowed to interact, the corresponds to the desired solution will be returned total register of qubits accesses the full set of available with high probability. superposition states. For example, the two-qubit state So, from a high level, that’s it. Quantum computing provides access to an exponential computational space, with the caveat that at the end of a computation you
" get only one answer. The remainder of this article is | ′〉 = U| 〉 a b = Sa | 1 00…00 00…00〉 aimed at digging deeper. The following questions are 1 1 addressed: What are the requirements of a technology | b1 00′ …00 00…00〉 to be used in a universal quantum computer? How can a | 〉 2 00…00 00…01 these requirements be met with practical components? 2 2
" How is the state of a quantum processor controlled? b ′ |00…01〉 2 S 00…00 U How is it measured? What is the state of the art today, and how must this be improved in the future if the full aN | 〉 11…11 11…11 promise of quantum computing is to be realized? And, N N 2 finally, as a microwave engineer, why should you care? bN | 〉 11′ …11 11…11 Throughout the article, superconducting qubit technol- (a) (b) ogy is used to illustrate the key concepts. Items beyond the scope of the article include qu Figure 1. The similarity between (a) scattering from a antum algorithms and derivations of the quantum lossless microwave N-port and (b) quantum computation using an N-qubit quantum computer. While the mechanical properties of superconducting circuits. mathematical operations are identical, the number of For review articles describing these items in detail, ports for the microwave network is a physical number of the curious reader is referred to [21] and [22]. In addi- terminals, whereas the number of ports for the quantum tion, we refer the reader to [23] for a detailed review operator scale exponentially in the number of qubits. of superconducting qubits. A sidebar, “Key Concepts,”
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Authorized licensed use limited to: University of Massachusetts Amherst. Downloaded on October 01,2020 at 19:47:20 UTC from IEEE Xplore. Restrictions apply. Key Concepts probability distribution for a measurement of In quantum computing, we manipulate the state of a charge across the capacitor. quantum system through the Hamiltonian operator. But what exactly do we mean by “the state of a What Is a Quantum Operator? quantum system”? What is a quantum operator, and In quantum mechanics, all physical observables are what is the significance of the Hamiltonian operator? represented by linear transformations—or, as physicists Here, we answer each of these questions, using often call them, linear operators—which are applied the quantum mechanical LC circuit as an example. to the wave function. In fact, for every observable in Interested readers can find a more thorough classical mechanics, there is a corresponding operator background in one of the classic textbooks in the field, in quantum mechanics. For instance, one can write including [35], [36], and [46]. operators for the voltage, current, flux, charge, or other desired quantities. While, as with a wave function, the What Is a Quantum State? exact form of an operator depends on which basis In classical physics, we imagine an atom as a we write it in, it turns out that the eigenvalues of an planetary system, with the electron orbiting the operator are independent of basis. Moreover, in quantum nucleus, i.e., describing the position of the electron mechanics, eigenvalues of operators have special with a function r()t . Quantum mechanics replaces meaning: for a measurement outcome to be possible, it this picture of a solid point-like electron with an must be an eigenvalue of the associated operator. Moving “electron cloud” described by a complex-valued forward, we focus here on the total energy operator, function }(,rt) whose physical significance is that which has particular significance to quantum computing. ||}(,rt) 2 is the spatial probability density of the electron’s position at time t. The function }(,rt) is What Is the Importance of the called a wave function. Hamiltonian Operator? Similarly, in classical electrical engineering, we The linear operator corresponding to the system’s total can describe the state of a circuit—in this case, energy is called, for historical reasons, the Hamiltonian, the LC resonator—by its time-dependent flux U()t and it is denoted by H.t The Hamiltonian is of particular and charge Q(t), but quantum mechanics replaces importance to our study of quantum computing for that picture with the wave function }(,U t). two primary reasons. First, its eigenvalues are the 2 Analogous to the atomic system, ||}(,U t) is the discrete energies, En, that can be measured in a probability density for the flux U. What this means quantum system. Physicists refer to these energies as is that a measurement of the flux through the quantum levels or eigenenergies. For the undriven LC inductor at time t yields a random result, and the resonator of Figure S1(a), Enn =+'~0 12/, where ^h probability that the result is between Ua and Ub is ~0 = /1 LC is the resonant frequency of the circuit y Ub |(} UU,)td|.2 and n is an integer that is greater than or equal to zero. Ua A wave function provides a complete representation of When determining the state of a quantum processor, the state of a quantum system. While we have described we typically make an indirect measurement of the the wave function of the quantum LC resonator in terms qubits’ energies. In the case of transmon qubits, of flux, it is also possible to express the wave function we measure impedance, which depends on energy with respect to a different observable. For instance, one because of the nonlinearity of a transmon qubit (see could write the wave function in terms of charge, } Qt,, the “Readout” section). Since the energies are discrete, ^h where, interestingly, } U, t and } Qt, are related via a it is also helpful to expand the wave function in terms ^h ^h Fourier transform, of the eigenfunctions []}n U of the Hamiltonian ^h operator, which are referred to as energy eigenstates: 3 = 1 -jQU/' } U,,tQ# } te dQ, (S1) 3 ^^hh2r' -3 }aUU,,tt= n }n (S2) ^^hh/ ^ h where ' is the reduced Planck constant. Just n = 0 as the time and frequency domain pictures of where the energy eigenstates satisfy the typical a signal carry the same underlying information, eigenvalue equation, both } Qt, and } U, t are equally valid wave t ^h ^h H}}nnUU= E n . (S3) functions that fully describe the same underlying ^^hh quantum state. However, while } U, t describes Since (S3) is really a set of n independent equations, ^h the probability distribution for a measurement this choice of expansion leads us to a matrix of flux in the LC resonator, } Qt, describes the representation for the Hamiltonian operator and ^h
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Authorized licensed use limited to: University of Massachusetts Amherst. Downloaded on October 01,2020 at 19:47:20 UTC from IEEE Xplore. Restrictions apply. a vector representation for the wave function in the Note, that the left-hand side of (S5) represents the eigenenergy basis: state without giving its decomposition in any particular basis. This is common practice in the quantum physics RE0 0 0 gV S W community. For an n-level system, one typically writes S 0 E1 0 gW t = H , (S4) |,0H |,H f,1 | nH rather than, for instance, }0 U , S 0 0 E2 gW S W ^h S hhhjW }}1 UU,,f n , and, when we say in the main text T X ^^hh that we use the |0H and |1H states as computation- and basis states, it means that our computational bases
Ra0 t V are the }0 U and }1 U components, and we are S ^ hW ^h ^h Sa1 t W operating on a0 and a1. |.} t H = S ^ hW (S5) ^ h a2 t But what if we wanted to determine expressions S ^ hW S h W T X for the energy eigenstates in the flux basis? Each of the energy eigenstates can be found by solving the Note that |} t H is simply the wave function ^h eigenvalue (S3), plugging in the appropriate value expressed as a vector in the energy eigenstate basis. for En . Since the Hamiltonian operator in the flux There is something pretty deep here. Each entry in the basis is [22] Hdt =-'//22CL22dUU+ 2/, this Hamiltonian matrix corresponds to one of the possible ^^ h^ hh corresponds to solving a second-order differential measurement outcomes; that is, if a measurement equation of the form of the energy in the system were made, then the outcome must be one of the diagonal entries of (S4). '2 d2 U2 -+-=Enn} U 0. (S6) Since there is a 1:1 correspondence between the c 22CLdU2 m ^ h eigenvalues, En, and the energy eigenstates, }n U , ^h each of the energy eigenstates is associated with The first three energy eigenstates appear in a definite energy; in other words, one can find the Figure S1(b). Note that, if the wave function were 2 probability of measuring Ei as ||ai . Interestingly, this expressed with respect to a different basis, then the would not be true if we used a different basis for our general procedure would be exactly the same, but expansion of the wave function since the Hamiltonian the form of the Hamiltonian operator would change matrix would no longer be diagonal. in the new basis.
3
2.5 E 2 (Φ) + 2.5 2/ 0 C 2 D
1 (Φ) + 1.5 L C 1.5 vD(t ) L C E1/ 0 (a) 1 (Φ) + 0.5 0 (c)
0.5 E0/ 0 Offset Wave Function, Normalized Energy
0 –4 –2 024 Φ/ΦZPF (b) Figure S1. The quantum LC resonator. (a) An undriven LC resonator. (b) The first three wave functions (solid lines) and energy levels (dashed lines) for the undriven quantum LC resonator, plotted as a function of flux, normalized to UZPF = C '~0 /.2 The wave functions have been offset to align with the associated energy level. (c) A quantum LC resonator with a drive source. It is assumed that CCD % so that ~0 is not affected by this additional capacitance.
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Authorized licensed use limited to: University of Massachusetts Amherst. Downloaded on October 01,2020 at 19:47:20 UTC from IEEE Xplore. Restrictions apply. The second reason that the Hamiltonian operator can be accomplished by adding a drive source to the is significant to our current discussion is that the time circuit as in Figure S1(c). If we keep our basis states evolution of a quantum state is governed by the so- as the energy eigenstates of the undriven circuit [i.e., called Schrödinger equation: those of the circuit in Figure S1(a)], then the effect of the microwave drive is to add extra off-diagonal terms d j' ||}}ttHH= Ht . (S7) to the Hamiltonian: dt ^^hh R V In quantum computing, our goal is to control the time E0 -jvc D t 0 f S ^ h W S W evolution of a quantum system, so intuition into the jvc D t E1 -jv2 c D t f t = S ^ h ^ h W dynamics of the Schrödinger equation is essential. It H S W, (S10) 02jvc D t E2 f turns out that expanding the wave function in terms S ^ h W S hh h jW of the energy eigenstates as done earlier greatly T X simplifies the solution to the Schrödinger equation, which collapses to where c is a coupling parameter related to the microwave design. When the new Hamiltonian is dan t j plugged into the Schrödinger equation, the extra ^ h =- Etnna , (S8) dt ' ^ h off-diagonal terms cause mixing between adjacent levels: with the solution aanntj=-0 exp Etn /.' Thus, ^^hh" , one can write the full wave function in the energy da0 t j eigenstate basis as ^ h =- Et00ac- jv tta1D , (S11) dt ' ^^hh^^hh 3 -jt~n ||}ateHH= / n 0 n , (S9) and, for n $1, ^^hhn = 0
where ~~nn/ En//' =+0 12. So, the expansion dan t j ^h ^ h =- jncavtDn-1 + Etnna coefficients experience a phase progression that dt ' ^ ^ h ^ h varies with index n, but the overall energy in the -+jn1 cavtD n+1 t . (S12) ^^hhh system stays the same. That the energy does not change through time is exactly what one would expect Therefore, by engineering the Hamiltonian to add for an undriven lossless LC resonator. drive terms at appropriate frequencies, amplitudes, So how do we cause the circuit to switch between phases, and durations, we can control the evolution of states as we want to do when we control a qubit? This the circuit’s quantum state.
explains some quantum mechanics fundamentals that 2) A mechanism for initializing each qubit into a known are relevant to the remainder of the article. It has been state: It is essential that we can reliably initialize provided for those who are new to the field, and it may the state of a quantum processor to some well- be helpful to read it before proceeding. known state. While the exact state (e.g., |000f H or |)111f H is not important, it is essential that any What Is Required to Build initialization procedure be robust for all possible a Quantum Computer? preinitialization states of the quantum processor. Just as the construction of a classical computer has well- 3) Fast gate operations with respect to the qubit coher- defined technological requirements, so does the design ence time: A qubit behaves as an analog object of a universal quantum computer. These requirements during computation, and its coherence time is can be summarized as five items known as the DiVin- a metric that quantifies the characteristic tim- cenzo criteria [24]: escale on which its state randomizes. Since a 1) A well-understood and scalable qubit technology: If quantum computation consists of a series of we are going to control a large array of qubits, gate operations, it is critical that the individual we will need a model that describes how the processes be carried out on a timescale orders qubits interact with external controls and with of magnitude shorter than the coherence time of one another. Beyond simply having a model, the individual qubits. the technology should also be amenable for 4) A universal set of quantum gates: A universal use in large arrays of qubits without a loss quantum computer allows for arbitrary unitary of performance. computations, which require a universal gate set.
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Authorized licensed use limited to: University of Massachusetts Amherst. Downloaded on October 01,2020 at 19:47:20 UTC from IEEE Xplore. Restrictions apply. Fortunately, this can be constructed from a set of called the transmon, which is a nonlinear microwave single qubit gates (i.e., operations applied to just resonator [30]. one qubit) and a suitable choice of a single two- To understand the transmon, we start by describ- qubit gate [25]. There are many different options ing the behavior in the quantum mechanical limit of for the constituents of a universal gate set. a simpler circuit, a parallel LC resonator [Figure 2(a)]
5) A method for measuring the state of individual qubits: characterized by a resonant frequency, ~0 = 1/,LC Finally, we must be able to determine the solution and a quality factor (Q-factor), QR= CL/. Normally, of a computation in a manner that is faithful to we assume that the amount of energy stored in an the probability amplitudes of the quantum state. LC resonator can adopt any positive value. However, What this really means is that we need a method if we were to measure the energy in a high Q-factor to measure the state of each qubit. resonator with very high sensitivity, we would find that the result is quantized to a set of allowed ener- How Can We Build a Quantum Computer? gies known as quantum energy levels, How can we actually go about building a quantum 1 computer that satisfies the requirements described Enn,LC =+'~0 , (3) 2 previously? Let us consider the constraints one by one. ` j where ' . 10-34 Js$ is the reduced Planck constant and A Well-Understood Qubit Technology the index n is a nonnegative integer. The corresponding First, we need a technology that provides scalable energy diagram appears in Figure 2(b). The equal spac- and well-understood qubits. A variety of candidate ing of the energy levels makes a lot of sense; the elec- qubit technologies exist, including trapped ions [26], tromagnetic quantum of energy is '~, and here we see
bulk and integrated photonics [27], cold atoms [28], that the step between allowed energies is equal to '~0. semiconductor spin qubits [29], and superconduct- Thus, moving between the various energy levels cor- ing circuits [23]. Here, we focus on superconducting responds to adding or removing an integer number of circuits, which are monolithically fabricated devices microwave quanta (photons) at the resonant frequency that are engineered, controlled, and measured using of the circuit. If we were to use this device as a qubit, familiar microwave circuit techniques. In particu- we could employ the |0 H and |1H states for our compu- lar, we discuss one type of superconducting qubit tational basis.
TA << 0/k
0 |3〉 0 C | 〉 D C LR 2 C L 0 |1〉 0 |0〉 (a) (b) (c)
T A << 01 /k |3〉 23 |2〉 C CL(Φ) 12 D J C Φ Q E Idc |1〉 01 Qubit vRF |0〉 (d) (e) (f)
Figure 2. From a linear resonator to the transmon. (a) The linear resonator circuit diagram. (b) The energy diagram for a high Q-factor LC linear resonator. (c) The drive circuit (left) and the linear resonator (right). (d) The fixed-frequency transmon (nonlinear resonator). The transmon is an LC resonator with a nonlinear inductance, realized with a Josephson junction (JJ) (represented by the X symbol). (e) The transmon energy diagram. The energy spacing between each level is unique due to the transmon nonlinearity. (f) The frequency-tunable transmon with RF and dc control ports: the XY drive (left), transmon qubit (nonlinear resonator) (center), and flux bias (Z drive) (right).
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Authorized licensed use limited to: University of Massachusetts Amherst. Downloaded on October 01,2020 at 19:47:20 UTC from IEEE Xplore. Restrictions apply. Why do we usually ignore these quantum energy in Figure 2(d), and its energy levels are illustrated in levels and just assume things are continuous? Micro- Figure 2(e). wave engineers rarely work in a situation where the Defining Josephson and capacitive energies, 2 electromagnetic quantum, '~, is anywhere near the EIJC= U0 / 2r and EqC = /,2CQ respectively, and ^h ^h energy scale associated with the thermal energy of in the usual case that EEJC& , the quantum energies of -23 the environment, kTA , where k . 13. 81# 0J/K is the transmon qubit are [30]
Boltzmann’s constant and TA is the ambient tem-
perature to which the resonator is thermalized. In 1 EC 2 Enn . '~0 +-66nn++3 , (4) the usual limit (e.g., at room temperature), Johnson 124444` 42413j 1244^44234444h noise from the resistor drives the resonator into a Linear resonator Anharmonicity energies random distribution across many energy levels. In this case, the energy levels are literally in the
noise. However, once we enter the regime where where ~00= 1/ LCJQ and h =-EC/.' For typical
TkA +.'~/(50mK # f/) 1 GHz , quantum mechani- transmon parameters (L 0J = 8nH and CQ = 80 fF [31]),
cal effects start to manifest. The thermal excitation ~r0/2 and hr=-EC/2 ' are roughly 6.2 GHz and
probability goes as exp -'~/;kTA so, to ensure that –240 MHz, respectively, (for these same parameters, " , thermal excitations of the resonator are lower than EJ/2r' is approximately 20 GHz, so our assumption
1%, we need to reduce TA to five times below '~/.k that EEJC& is valid). From (4), we can write the various A commercial dilution refrigerator can get down to transition frequencies 10 mK, so we expect to observe quantum mechanical - behavior as long as ~0 $ 1GHz. = EE10. + ~~01 ' 0 h, (5) So far, things look promising. We have a device that behaves quantum mechanically if cooled to low- = EE21- . + ~~12 ' 0 2h, (6) enough temperatures, and we understand its proper- - = EE32. + ties (e.g., ~0) well enough that we can engineer them ~~23 ' 0 3h, (7) through the choice of circuit parameters. However, if we are going to use it as a qubit, we will need a and so on. As depicted in Figure 2(e), the effect of the way to deterministically transition between the |0 H JJ nonlinearity is to create anharmonic energy levels and |1H states. To do this, a drive at ~0 can be capaci- ()~~~01!! 12 23 f . If we restrict the bandwidth of the tively coupled to the circuit, as in Figure 2(c). The microwave drive signal such that only ~01 is excited, drive causes the circuit to move between the vari- then it is possible to treat this device as if it were an ous energy levels, and it is the basis for controlling ideal two-level qubit. the quantum state of the resonator. However, there So how nonlinear is a transmon? From (4), we see is a problem. We want a system that behaves as if it that each time a photon is added to the resonator, the has only two levels (i.e., |0 H and |)H .1 But when we resonant frequency drops by ||h . Moreover, h is a excite the resonator at ~0, we end up driving all the parameter that can be engineered through the choice
transitions, which leads to a superposition state that of CQ. However, one cannot arbitrarily increase the includes the higher levels. Therefore, instead of the anharmonicity without paying a price in other per- two-level qubit described earlier, we have a system formance metrics, and a typical value for ||hr/2 is with many active levels. roughly 200 MHz. Nonetheless, this is a very strong Fortunately there is a fairly straightforward change nonlinearity: one photon of energy (e.g., 33. # 10-24 J at we can make to the LC resonator that enables us to 5 GHz) causes a 200-MHz shift in the transmon’s reso- independently address the ||01HH" transition: if we nant frequency. However, despite this, the transmon is add nonlinearity to the resonator, the energy level still referred to by quantum engineers as weakly anhar-
spacing becomes nonuniform, a property that quan- monic, since the relative difference between the ~01 and
tum engineers refer to as anharmonicity. In a transmon ~12 transitions is much smaller than ~01. This weak qubit, this nonlinearity is realized by replacing the anharmonicity means that the minimum duration of
inductor we had in the LC resonator by a Josephson the microwave pulses used to drive the ~01 transition junction (JJ), which is a superconducting tunnel junc- will be limited by the necessity to avoid exciting the
tion that behaves as a nonlinear inductor of inductance ~12 transition. 2 2 LLJJ=-0 / 1 IIJ /,C where IJ is the current through the In many architectures, one must be able to dynami- JJ, IC is the critical current of the JJ, LIJC00= U /2r is the cally tune the ||01HH" transition frequency ()~01 of a zero-flux inductance of the JJ, U0 = r'/.q . 207mVp$ s transmon. This is facilitated by replacing the JJ with a is the magnetic flux quantum, and q = 01. 6aC is the superconducting loop interrupted by two JJs. Such a charge of an electron. The transmon circuit is shown loop, known as a dc superconducting quantum interference
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Authorized licensed use limited to: University of Massachusetts Amherst. Downloaded on October 01,2020 at 19:47:20 UTC from IEEE Xplore. Restrictions apply. device (dc SQUID), behaves like a single JJ where LJ0 to the loaded Q-factor of the qubit ()QL . A qubit ini- depends on the magnetic flux bias through the loop. tially in its |1H state will randomly flip to the |0 H state The flux bias is typically provided by an on-chip bias due to electrical losses on a characteristic timescale of
coil near the SQUID. A complete schematic of a flux- T1L= Q /.~01 These losses can be resistances associated tunable transmon qubit with microwave (XY) and dc- with the qubit materials or losses due to the coupling SQUID flux (Z) drivelines is shown in Figure 2(f), and of the qubit to its control wiring. Achieving a typi-
photographs of a junction and complete qubit appear in cal value for T1 of 50n s for ~r01/2 = 5GHz requires 6 Figure 3(a)–(c). The reason for the terms XY and Z will QL . 16.;# 10 many years of materials research went be become apparent shortly. into achieving this specification.
A critical property of a qubit is its coherence time: The second metric, T,2 is related to the frequency the timescale on which the state of the qubit is ran- jitter of the qubit. For example, fluctuations in the JJ domized by noise or loss. For a quantum technology critical current cause the qubit frequency to fluctuate, to be useful in a quantum computer, its coherence time leading to the accumulation of random phase. In tun- should be much greater than the timescale on which able qubit architectures, another important source of its quantum state can be manipulated. Quantum engi- noise is the flux-bias (frequency-control) line. Fixed- neers refer to two quantities to describe decoherence: frequency (i.e., nontunable) transmons can usually
T,1 which is the energy relaxation time, and T,2 which achieve a better T2 than tunable transmons. is the dephasing time (in the physics community, it Based on the preceding discussion, this tech-
is customary to use the notation TX to denote time nology appears to meet our first requirement; constant X related to decoherence; here, we maintain it is monolithic, so the qubits can be mass pro- this notation, with the caveat that we reserve the itali- duced, and we can engineer the properties of the
cized T to refer to noise temperatures.). T1 is related devices at the circuit level, so we have a pretty good
Readout AI Layer 1 AI Layer 2 Resonator
100 m
100 nm XY Control (a) SQUID
Oxide Layer AI Layer 2 JJs AI Layer 1 Substrate Z Control 100 nm
(b) (c)
Figure 3. A typical transmon qubit as seen through imaging. The (a) closeup view and (b) cross section of a JJ. (c) A complete qubit. The distributed qubit capacitance, XY coplanar waveguide (CPW) line, and Z CPW line are highlighted in blue, yellow, and green, respectively. A blow-up of the SQUID and flux-bias coupling is also shown. The JJs are typically patterned with electron beam lithography and made by a two-step aluminum (Al) deposition, interrupted by an oxidation step in a liftoff process, on sapphire or high-resistivity silicon wafers. Larger features in the circuit are typically patterned with photolithography and made with Al or niobium metallization in a liftoff or etching process. State-of-the-art JJ uniformity is typically in the 1% range, contributing a ~30-MHz variation in the qubit frequency. In most cases, superconducting qubits are made in low-volume research foundries on 100–200-mm tools, so, in practice, device parameter variation can significantly exceed 1%.
32 August 2020
Authorized licensed use limited to: University of Massachusetts Amherst. Downloaded on October 01,2020 at 19:47:20 UTC from IEEE Xplore. Restrictions apply. understanding of the quantum behavior. But what In quantum computing, where about the other criteria? individual bits can exist in a complex Initialization superposition state, there is a full Initialization is relatively easy. Qubits naturally relax continuum of gate operations that on the timescale T,1 so initialization into |0 H can be done simply by waiting. Of course, since we want our can be carried out.
qubit operations to be faster than 1,T we may need a
way to initialize faster than T.1 A variety of techniques Hamiltonian operator commutes with itself through- have been discovered and are in active use (e.g., [32] out the operation, and [33]). j t t ||}}t HH=-exp # Hdttll t0 . (10) ' ' t0 ^ h 1 ^ h 124444444444443 ^ h Fast and Universal Gates t U tt, 0 Our next two criteria—fast gates with respect to the ` j coherence time of our qubits and a universal gate Thus, we can control the state evolution of a quantum set—are related, so we treat them together. What we register if we can program its Hamiltonian; therefore, are really after here is to develop an understanding the name of the game in quantum control is to engi- of how the quantum state of a transmon-based pro- neer knobs that permit doing so deterministically. cessor is controlled via electrical signals. To develop These knobs may be microwave drive signals (super- this understanding, we begin by first considering how conducting [23] and semiconductor spin qubits [34]), quantum states can be controlled in an abstract sense. flux biases (superconducting qubits [23]), laser drive We will see that the total energy operator, known as signals (trapped ion qubits [26]), phase shifters (pho- the Hamiltonian operator, is the handle that enables con- tonic qubits [27]), and so on. Shortly, we will dive into trol of the quantum state. We then consider how quan- the details of how this is accomplished for transmon tum gates can be implemented, first in the abstract qubits. However, let us first consider what kind of gate sense, and then we apply this knowledge to see how operations we need to develop. single- and two-qubit gates can be carried out on For universal quantum computing, we need to transmon qubits. apply an arbitrary 22NN# unitary to a quantum regis- To begin, let’s consider how physics permits the con- ter. Producing a fully controllable 22NN# Hamiltonian trol of a quantum state in the first place. Recall that, would be intractable. Fortunately, any arbitrary quan- when we perform a quantum computation, our goal is tum operator Ut can be decomposed into a sequence of t to apply a unitary operation, U tt,,0 that takes a state elementary operations applied to just a single qubit or a ^h |} t0 H to a new state |} t H in time Tttt=-0, i.e., pair of qubits. These elementary operations are referred ^h ^h to as quantum gates, and the quantum algorithms used to t NN |,}}ttHH= U tt00|. (8) implement a desired 22# unitary operator are built ^^hh^ h up using sequences of them. Thus, we can focus our So what physics can we leverage to implement the attention on one- or two-qubit subsystems when consid- t unitary operator U tt,?0 The key here is understand- ering how to control a quantum computer. ^h ing how a quantum state evolves as a function of time and how to leverage this intuition to engineer Single-Qubit Gate Operations quantum gate operations. The time evolution of a First, we consider single-qubit gate operations in gen- quantum state is dictated by the time-dependent eral before seeing how one carries out such operations Schrödinger equation, on transmon qubits. In regular digital logic, there are really only two useful cells that operate on a single bit: d j |(}}t)|HH=- Ht tt, (9) an inverter and a buffer. However, in quantum com- dt ' ^^hh puting, where individual bits can exist in a complex which says that the time evolution of a quantum state is superposition state, there is a full continuum of gate entirely determined by the Hamiltonian (total energy) operations that can be carried out. Single-qubit states operator of the system, Ht t . Since the state of an are easy to understand using a visual representation of ^h N-qubit register is described by a state vector of length the qubit state. After removing an unobservable global N,2 the Hamiltonian operator for an N-qubit regis- phase term from (1), we can write the state in terms of ter is a 22NN# matrix. Hence, a single qubit has a two variables, i and z: 2 # 2 Hamiltonian, a pair of qubits share a 4 # 4
Hamiltonian, and so on. Solving (9), given an ini- iijz ||}HH=+coss0 e in |.1H (11) tial state at tt= 0 under the assumption that the ``2 jj2
August 2020 33
Authorized licensed use limited to: University of Massachusetts Amherst. Downloaded on October 01,2020 at 19:47:20 UTC from IEEE Xplore. Restrictions apply. The continuum of values that can be assigned to these from this graphical representation. On the other two variables maps to the surface of a sphere, which is hand, since any single-qubit gate can be viewed as referred to as the Bloch sphere. a rotation operation, its impact will depend on both The Bloch sphere representation of a qubit is i and z. shown in Figure 4(a). Any single-qubit state can be When we carry out single-qubit gate operations, we represented by a unit vector, known as a Bloch vector, are moving the Bloch vector of a qubit around the sur- on the surface of the Bloch sphere. The ground state face of the Bloch sphere, and the handle that enables |0 H is at the north pole, the excited state |1H is at the us to achieve this goal is the Hamiltonian operator south pole, and all other points on the surface of the of the qubit. But what exact Hamiltonian operator(s) sphere represent unique superposition states. Since would we like to engineer? That is, what form would all single-qubit states lie on the surface of the Bloch we like our single-qubit Hamiltonian to take if we wish sphere, any single-qubit gate operation corresponds to rotate the Bloch vector around an arbitrary axis? To to a rotation of the Bloch vector around some axis. answer this question, let’s consider an expression for a When a measurement of the qubit state is carried out, generic single-qubit Hamiltonian, the result depends solely on the elevation angle, i; that is, P |/02= cos2 i and P |/1 = 2 i 2 .sin So H H t '~R " , ^h " , ^h H =+wwXXvvvtt YY+ wZZt , (12) one can easily estimate the measurement probabilities 2 ^h
ZZZZ
θ
Y Y Y Y φ
X X X X " " "
" " " (a) (b) H = r σ (c) H = r σ (d) H = r σ 2 X 2 Y 2 Z 1 180 180 1 ( θ /2)
2 0.5 0 0 ( θ /2) φ (°) φ (°)
2 0.5 cos cos 0 –180 –180 0 12 12 rt/2π rt/2π " "
" " (e) H = r σ (f) H = r σ 2 Y 2 Z Z Z
" Y 0 –1 " Y U = 1 1 –1 Yπ UY = 1 0 π/2 2 1 1
X X
(g) Yπ Gate (h) Yπ /2 Gate
Figure 4. The Bloch sphere is a valuable tool to understand single-qubit states and control operations. (a) The Bloch sphere representation of a qubit state |(}iHH=+cose/)20|/xpjzi sin()21|.H The effect of (b) Pauli matrices vt X, (c) vt Y, and (d) vt Z " , is to rotate the qubit state around the x-, y-, and z-axes, respectively. The effect of (e) vt Y on an initial state ||}()HH= 00 and (f) vt Z on an initial state |(} 00)|HH=+()|12H /. The action of the (g) Yr and (h) Yr/2 gates. The Bloch sphere pictures show the effect of the gates on a qubit that is initially in the |0 H state, whereas the matrices describe the effect on an arbitrary state.
34 August 2020
Authorized licensed use limited to: University of Massachusetts Amherst. Downloaded on October 01,2020 at 19:47:20 UTC from IEEE Xplore. Restrictions apply. where ~R is an angular frequency of rotation; wX, What this means is that the state of a transmon qu 22 wY, and wZ are dimensionless weights (|wwXY||++| bit naturally rotates about the –z-axis of the Bloch 2 ||wZ = 1); and sphere at a frequency ~01, i.e., |/}it HH=+0 20|cos ^^hh sin iz00/|21exp jt- ~01 H, where i 0 and z0 are ^^hh" , 0 1 0 -j 1 0 the initial values of i and z, as defined in Fig- vvttXY==,,and vt Z = (13) ;1 0E =j 0 G ;0 -1E ure 4(a). For this reason, we typically refer to ~1 as the qubit frequency. are the Pauli spin matrices [37]. What we have done To understand the action of quantum gate opera- here is write the general Hamiltonian of our qubit in tions, it is helpful to first remove the natural rotation of
terms of three independent operators (,vt X vt Y, and vt Z). the qubit, analogous to working in the baseband enve- It turns out that breaking things up in this way is par- lope domain. After making this change of coordinates ticularly useful when trying to visualize the effect of to what we call the rotating frame, the isolated qubit t the Hamiltonian on the qubit dynamics. Hamiltonian simplifies as 0R, = 0H since there is no rela- As illustrated in Figure 4(b)–(d), the presence of tive motion in the new frame. Note that the subscript R each of the Pauli spin matrices in a Hamiltonian corre- has been added to signify that the expression is valid in sponds to rotation of the Bloch vector around one of the the rotating frame. Moving forward, we will work in the
major axes; i.e., vt X, vt Y, and vt Z describe rotation about rotating frame. the x-, y-, and z-axes, respectively. Example time evo- Now we are ready to talk about how one can apply
lutions for vt Y and vt Z operators appear in Figure 4(e) gates to the transmon. Let us assume that we want to and (f), respectively. Rotation around the y-axis causes apply a phase gate, which rotates the single qubit state the measurement probabilities to oscillate with time; around the z-axis by some desired angle. To see how these probability oscillations are referred to by the one might do that, we can rewrite (14) in the rotating
quantum community as Rabi oscillations. On the other frame, while allowing for a frequency detuning, T~01,
hand, rotation around the z-axis leaves the measure- from the idling value of ~01, ment probabilities unaffected and influences only the t 'T~01 qubit phase ()z . By controlling the duration through H00,R T~ 1 =- vt Z . (15) ^h 2 which the vt Y and vt Z operators are enabled, we can carry out deterministic rotations around the y- and So we can enable a rotation around the z-axis by con- z-axes, respectively. For instance, as shown in Fig- trolling the frequency of the qubit through the flux-
ure 4(g), if we enable vt Y for a duration t = r~/,R then tuning line, which, accordingly, is referred to as Z
we can execute a Yr gate, which initially flips a qubit control. This mechanism is similar to the way a phase- in state |0 H to state |1H and vice versa, analogous to a locked loop adjusts the phase of a voltage-controlled classical NOT gate. Similarly, as shown in Figure 4(h), oscillator (VCO) using the frequency-tuning voltage.
if we turn on vt Y for a duration t = r~/,2 R then we can What about rotations around the x- and y-axes? Since
execute a Yr/2-pulse, which creates a superposition these are transverse rotations (i.e., from |0 H to |1H and state from a qubit initialized to the |0 H or |1H states. If vice versa), a method to couple energy into the circuit we had control knobs that permitted us to enable and is required. This can be achieved through capacitive disable each of the Pauli operators at will, we would [as in Figure 2(e)] or inductive coupling of a microwave be able to perform any single-qubit gate operation we generator to the transmon, and it adds a drive term to
wanted. Now, let’s see how this is accomplished for the the Hamiltonian. If veRF t =+ttsin ~r01 - zD , ^^hh^ ^ hh transmon qubit. then the drive term corresponding to capacitive cou- pling is Control of Transmon Qubit
Let’s first write the Hamiltonian of the isolated qubit of t et CD ' HD,R . ^ h cosszvDXtt+ in zvDY, Figure 2(d) and consider the drive later. Throughout, we 22CCDQ+ ZQ ^ ^^hhh make the assumption that the control operations do not (16) take us out of the two-level computational space, and
we will deal with the constraints this imposes when where ZLQJ/ 0 /CQ (typically, ZQ is roughly 300 X tt we describe the practical requirements for single-qubit [31]). The overall Hamiltonian is now HH0R, + D,R. So control. The Hamiltonian of the isolated transmon can, the microwave drive induces a rotation of the Bloch then, be determined by considering the qubit’s two vector around an axis in the xy-plane whose angle is
lowest energies [see (4)]: determined by the carrier phase ()zD . The rate of this rotation is determined by the amplitude of the micro-
t '~01 wave signal. For a universal gate set, we really only H0 =- vt Z. (14) 2 require rotations around the x- and y-axes, so the key
August 2020 35
Authorized licensed use limited to: University of Massachusetts Amherst. Downloaded on October 01,2020 at 19:47:20 UTC from IEEE Xplore. Restrictions apply. requirements are that we can shift the carrier by 90° which can be employed to place a notch in the pulse’s
and that we have accurate control of the integrated spectral content at f12 . envelope amplitude, which sets the amount of rotation. How strong should these pulses be? If we want to Now, let’s consider some practical constraints. The carry out an XY gate of duration Tt that rotates the
required amplitude of the drive signal depends on CD, qubit state by an angle, Ti, while using a raised- which sets the drive-coupling strength. However, the cosine envelope, the required peak power available driveline presents a resistive loss channel to the qubit, lim- from the generator at the reference plane shown in -1 iting the qubit relaxation time to T1 # c~loading = QD/,10 Figure 2(e) is
where QD is the Q-factor imposed on the qubit due to 22 loading by the driveline. The loading is related to the PQAV = ' D TTi / t . (17) 2 ^^hh drive capacitance via QCDQ. //CZDQZ0 , where Z0 ^h is the generator impedance. We typically engineer QD It is important to note here that the rotation angle to be on the order of 51# 07 or so, which means that scales with amplitude as opposed to power since a
we need to keep our coupling very weak: CQ and CD qubit is a coherent device. Taking the value of QD given are usually on the order of 100 fF and 25 aF, respec- previously, we find that carrying out a r-pulse (cor- tively [31]. Referring to Figure 3(c), we see that the small responding to a 180º rotation) within 10 ns requires a drive capacitance has been realized as coupling from an peak available power of roughly -63 dBm, which is open-circuited coplanar waveguide (CPW) line to the only a tiny fraction of the energy makes it to the qubit. qubit across a region of the ground plane, which can The signal-to-noise ratio (SNR) of the XY control be designed using modern electromagnetic simula- signal is also an important consideration because tion tools. noise on the XY driveline causes decoherence by driv- Care also must be taken in coupling the dc flux bias ing small rotations about random axes on the Bloch to the qubit. While noise on the XY line drives XY sphere. The weak coupling of the XY line to the qubit rotations of random phase, noise on the Z driveline helps reject noise, but there’s still a limit to how much results in fluctuations of the qubit frequency, analo- noise can be tolerated. The characteristic timescale at gous to noise on the control line of a VCO. To keep which the qubit state randomizes due to noise on the these fluctuations in a range where they do not limit driveline is the coherence of the qubit, the magnetic coupling is -1 'QD very weak, with typical mutual inductances on the cnoise = , (18) Sa order of 2 pH. Such small inductances can be realized
by keeping both the self-inductance of the flux line where Sa is the spectral density of available noise
and the geometry of the SQUID small, as illustrated power at ~01. This formula makes sense because in Figure 3(c). more noise should cause more decoherence, and a
It is also important to consider which waveforms we higher QD means that the qubit is more weakly cou- want to apply to the microwave control port. We would pled to the XY driveline. How much attenuation do like to use microwave pulses to drive the qubit state we need between our pulse generator and the qubit around the surface of the Bloch sphere as quickly as to sufficiently suppress generator noise? Recall that
possible without introducing errors. But, since a trans- QD sets the decoherence rate due to resistive loading -1 mon qubit is a weakly anharmonic device, we need to from the XY line. If we require cnoise to be at least as -1 be especially careful that our microwave pulses do not large as cloading, then we need Sa 1 '~10. Rephrasing in
excite the ~12 transition. When engineering the qubit, terms of the effective noise temperature looking back
there is a tradeoff between the dephasing time and toward the generator from the drive port ()Tdrive , we
anharmonicity, with a larger anharmonicity (a smaller need Tkdrive 1 '~~10 /(= 48 mK # 10/)2r /GHz. So, for
CQ) corresponding to more dephasing [30]. In practice, a qubit frequency of 5 GHz, we need Tdrive 1 240 mK, a typical value of - hr/2 is roughly 200 MHz. So car- which is roughly 30 dB lower than room tempera- rying out fast XY pulses is quite challenging, since ture. A typical conservative approach is to attenuate
the relative energy at this offset frequency must be the drive signal by a ratio of TTGENB/ ASE , where TGEN
suppressed by more than 40 dB to prevent errors due and TBASE are the temperatures to which the genera- to |2H population from being the dominant source of tor and quantum integrated circuit are thermalized, gate error. On the other hand, gates with a duration of respectively, with the attenuation distributed along approximately 10 ns are necessary to limit errors due to the thermal gradient to balance the thermal noise of decoherence. To balance this tradeoff, XY pulses typi- the attenuators with heating due to dissipation. This cally employ Gaussian or raised-cosine envelopes in approach assumes that the noise floor of the genera- conjunction with spectral-shaping techniques, such as tor is thermally limited, and additional attenuation Derivative Removal by Adiabatic Gate (DRAG) [38], [39], may be required if the pulse generator has a higher
36 August 2020
Authorized licensed use limited to: University of Massachusetts Amherst. Downloaded on October 01,2020 at 19:47:20 UTC from IEEE Xplore. Restrictions apply. equivalent output noise floor, as is typically the case. Assuming a 30-mK base temperature and a thermally
limited pulse generator at room temperature, this ap CC proach would require 40 dB of attenuation, meaning that pulses with a peak power of roughly −23 dBm must be generated at room temperature. This number grows Qubit 1 Qubit 2 proportionally as the generator noise floor increases. Figure 5. Capacitively coupled qubits. Two-Qubit Gates In addition to controlling the state of individual qubits, if the two qubits are in the same initial state. The |01H we also need to interact pairs of qubits in a well-con- and |10H amplitudes also experience a −90° phase shift, trolled manner to implement two-qubit gates. There which creates entanglement. It can be shown that this are many possible two-qubit gate operations; to under- gate satisfies the requirements for universal quantum stand how one can be implemented, we consider the computing [40]. circuit shown in Figure 5, which consists of a pair of While just one two-qubit gate is required for univer- frequency-tunable qubits coupled through a capaci- sal quantum computing, it is desirable to minimize the tance. Imagine that the qubits are tuned to be at the steps in an algorithm. In this respect, it is often advan- same frequency and that both are initially in the same tageous to employ other gates in the family described state (e.g., ||}HH= 00 or ||}HH= 11 ). This is analogous to by (19) as well as controlled Z [41] and controlled NOT a common mode in classical electronics, so no energy [42] gates, which can be realized through flux tuning is exchanged between the qubits. On the other hand, and a cross-resonant microwave drive, respectively. if only one of the qubits is initially in the excited state, energy will periodically swap back and forth between Putting It All Together the qubits. The rate of the energy exchange will depend Now that we understand how to implement the gates in on the coupling strength, which is set by the relative a quantum algorithm, we can consider the simple gate
size of CC with respect to the qubit capacitances. This sequence shown in Figure 6, which is used to generate interaction can be turned on and off by tuning the rela- a maximally entangled two-qubit state. To begin, the tive qubit frequencies using the Z control line so that qubits are initialized to the |00H state and detuned in the qubits are on- and off-resonance, respectively. Thus, frequency. Next the XY line of each qubit is excited by a we can perform deterministic operations by enabling r/2-pulse (90º rotation), with qubit 1 rotating about the the interaction for a controlled duration. Gates leverag- –y-axis and qubit 2 rotating about the +y-axis, result- ing this form of interaction are referred to as iSWAP ing in a superposition state |.}HH=-05||00j 01H + ^ gates and are characterized by a time evolution opera- j||10HH+ 11 . This state is the equivalent of uniform h tor (recall, a two-qubit gate operates on a state vector of noise in that, if a measurement were carried out, any T the form |)}aH = 00 aaa01 10 11 , of the four possible outcomes would be equally likely. 6 @ Next, the qubits are momentarily brought on- R1 0 0 0V resonance to perform an iSWAP r gate, resulting S W ^h 0 cos ~s Tt/2 -jtsin ~s T /2 0 in state 0.|5000HH-+||1101HH+|.1 At this point, t S W ^h UiSWAP ~s Tt = S ^ h ^ h W, ^ h 0 -jtsin ~s T /2 cos ~s Tt/2 0 if we were to measure qubit 1, the state of qubit 2 S ^^hhW S0 0 0 1W would depend upon the measurement outcome. T X (19) That is, if we found qubit 1 in the |0 H state, qubit 2 would collapse to 12/ ||01HH+ , whereas, if we ^ h^ h where the parameter ~s describes the rate at which found qubit 1 in the |1H state, qubit 2 would collapse energy moves back and forth between the two qubits. to --12/ ||01HH. Therefore, the iSWAP gate has ^ h^ h If we enable the interaction for exactly half of the entangled the two qubits.
exchange cycle (i.e., ~rs Tt = ), the time evolution oper- Finally, we can apply a r/2-pulse about the x-axis to ator becomes qubit 1 to interfere with the |01H and |10H states, result- ing in the final state ||}HH=+12/ 00|. 11H This is ^ h^ h R1 0 0 0V a pretty interesting state. If we were to measure either S W t S0 0 -j 0W qubit 1 or 2, we would find |0 H with a 50% probability UiSWAP r = . (20) ^h S0 -j 0 0W and |1H with a 50% probability. However, once we mea- S W S0 0 0 1W sured the first qubit, we would then know the state of T X the other qubit with a 100% probability. This entangle- This swaps the |10H and |01H amplitudes while leaving ment, or correlation, is unique to quantum systems, |00H and |11 H unchanged, as swapping does nothing and it is the essential feature that enables one to build
August 2020 37
Authorized licensed use limited to: University of Massachusetts Amherst. Downloaded on October 01,2020 at 19:47:20 UTC from IEEE Xplore. Restrictions apply. up computational spaces that are exponential with respect is the qubit’s energy, since the states |0 H and |1H are to the number of qubits. precisely those where the qubit has definite and differ- ent energies. However, this amounts to detecting the Readout presence (if the qubit is in |)1H or absence (if the qubit Our fifth and final requirement is that we must be is in |)0H of a single ~5-GHz microwave photon carry- able to perform measurements of individual qubits. As ing an energy of about 20n eV, and conventional pho- mentioned, such measurements are nominally projec- todetection technology is far too insensitive. There is tive, meaning that at the end of a one, the state of a qubit another parameter that does differ between the two qubit that was initially ab||HH+ 10 becomes |0 H with prob- states. Because the transmon is a nonlinear resonator ability ||a 2 and |1H with probability ||b 2, and the evalu- its admittance depends on its oscillation amplitude. So ation tells us which of these outcomes has occurred. one can determine the state of a qubit by measuring its For practical quantum algorithms, we would like our reflection coefficient. readout technique to provide error rates of 1% or lower. A generic approach is shown in Figure 7(a). A non- How can we do this? linear resonator (the qubit) whose admittance is state- Let’s consider the requirements. First, we need a dependent is connected via a passive network, Y()~ , to physical parameter of the qubit whose value in the |0 H a load that represents the measuring device. The goal and |1H states differs enough that measuring that prop- is to design Y()~ such that a probe signal scattering erty distinguishes the states with a high probability. In off the network can robustly distinguish between the conventional DRAM, this parameter is the charge of a two qubit states while preserving the qubit’s internal
storage capacitance. Similarly, each superconducting Q-factor at ~01. Naively, one could imagine connect-
qubit type has different electric and magnetic proper- ing the qubit through a capacitor Cl to a transmis- ties available for measurement. Second, and critically, sion line and using the qubit to scatter a microwave we must be able to selectively activate the measurement: probe pulse, as illustrated in Figure 7(b). Note that a qubits do their thing (superposition, entanglement, and circulator has been employed to separate the forward
so on) only in private. Therefore, our measurement ()a1 and reflected ()b1 waves. The admittance differ- apparatus must not be extracting information, either ence between the two quantum states could, then, intentionally or through parasitic losses, while the qubits be detected as a phase shift in the reflected signal are being controlled for logic gates. (while we demonstrate the essential concepts here What characteristic of the qubit |0 H or |1H states can using a reflection measurement setup, in many cases be used to distinguish them during measurement? For state readout can also be performed with a transmis- the transmon qubit, one choice of a physical parameter sion measurement).
1 2 3 4
|0〉 Y– /2 X /2 iSWAP ( ) Initialize |0〉 Y /2 1 |00〉 Create Superposition 1 2 (|00〉 + j |01〉 – j |10〉 + |11〉) 2
(a.u.) Entangle
XY Control 1 3 (|00〉 – |01〉 + |10〉 + |11〉) 2
Interfere 1 4 (|00〉 + |11〉) 2 Control Z (Frequency) Time (a) (b)
Figure 6. An example quantum algorithm used to generate a maximally entangled pair. (a) The quantum circuit diagram and associated waveforms. (b) The effect of each step of the quantum algorithm. a.u.: arbitrary units.
38 August 2020
Authorized licensed use limited to: University of Massachusetts Amherst. Downloaded on October 01,2020 at 19:47:20 UTC from IEEE Xplore. Restrictions apply. a1 b1 a1 Qubit C Z b Z0 κ 0 Z 1 0 q Y ( ) C YIN( ) YQ( ) Q LJ
Qubit
(a) (b)
a b b 180 1 1 a1 1 |1〉 90 (°) 2 Z0 Z0 11 0 Z0 Z0 S
∠ r Cκ –90 |0〉 Purcell C κ Filter Cr –180 p Cr Frequency Readout Readout (d) Resonator Resonator r Q Cg |1〉 r Cg CQ q
I CQ Qubit q |0〉 Qubit
(c) (e) (f)
Figure 7. Qubit readout as a microwave design problem. (a) A qubit is generically coupled to a load via an admittance Y()~ . (b) Directly coupled measurement in reflection. (c) Dispersive readout using a detuned resonator. (d) Example phase shifts associated with the circuit in (c). (e) Typical experimentally measured constellations. Each data point was acquired by initializing the qubit to the |0 H (blue) or |1 H state (red) and then performing a state measurement. (f) A typical readout system, including a Purcell filter.
However, the circuit shown in Figure 7(b) has a To circumvent this problem, a typical readout sys- fundamental limitation. The time it takes for the probe tem is configured as in Figure 7(c) and contains an pulse to interact with the qubit and acquire an ampli- additional linear readout resonator that is coupled
tude and phase shift is characterized by a ring-up rate, to the readout line through capacitor Cl. The read- 2 l~= qm/,Q where QCmQ= //CZl Q Z0 is the con- out resonator has resonance frequency ~~r = 01 ! T, ^^hh tribution to the loaded Q-factor of the qubit associated where T/2r is typically between 500 MHz and with its coupling to the transmission line. To make the 1.5 GHz. The qubit and the readout resonator inter- measurement fast, we want a high l, corresponding to act dispersively, meaning that the qubit pulls the a strong coupling between the qubit and transmission readout resonator frequency. Because of the qubit
line, i.e., a low Qm. However, in this circuit, the qubit’s nonlinearity, this frequency pull is different in the
energy can leak through Cl into the transmission line; two states by an amount called the dispersive shift, in fact, it does so with precisely the same timescale, |,2 where -1 so 1 . l .T In other words, the measurement ring- 2 up time is also the ring-down time of the qubit itself. g h ~ | = r (21) Therefore, in this circuit, the qubit lifetime and mea- TTc + h m` ~10 j surement time are necessarily equal, which precludes
any possibility of a high-accuracy readout because the and gC. gr~~01 / 2 CCQr is the strength of the ^^hh qubit has a high probability to decay while we’re mea- coupling between the qubit and the resonator through
suring it. Cg [30], [43], [44].
August 2020 39
Authorized licensed use limited to: University of Massachusetts Amherst. Downloaded on October 01,2020 at 19:47:20 UTC from IEEE Xplore. Restrictions apply. The readout resonator can now be coupled rela- measurement time, which is sufficient for research
tively strongly to the measurement line through Cl for experiments, but we will still have a + %1 readout fast ring-up and measurement times. Since the qubit error due to the qubit relaxing during measurement.
frequency is detuned from the readout resonator by Additionally, the limit of T1 due to loading by the many linewidths, the readout resonator effectively readout circuit is shorter than the state-of-the-art lim- shorts the resistive load presented by the readout line, its due to material losses, so the setup in Figure 7(c)
and T1 can remain much larger than T.m In this cir- leaves performance on the table.
cuit, the ring-up rate, l, and the qubit lifetime, T,1 are Dispersive measurement with the architecture of Fig-
constrained by lT1 K T/|. In addition to speed, the ure 7(c) was standard for transmon qubits between 2005 measurement result also depends on the phase con- and 2010. However, as superconducting qubit coherence trast between the |0 H and |1H states. If we choose the times increased and gate error rates decreased, the read- dispersive shift to be roughly equal to the readout res- out accuracy had to keep up. To decrease the number of onator linewidth, 2|l. [30], and probe between the errors stemming from qubit relaxation due to loading |0 H and |1H state readout resonator frequencies, a phase by the readout circuit and enable the readout of mul- contrast as high as 180º can be realized, as shown in tiple qubits using frequency domain multiplexing, it is Figure 7(d) and (e), and the state can be determined now common to further shape the admittance function using demodulation techniques similar to those used Y()~ in Figure 7(a). This can be done by adding addi- for binary phase-shift keying. tional transmission zeros at the qubit frequency [33], For an optimal design achieving a 180º phase con- incorporating secondary readout resonators [47], or 2 trast, l 1 K T,T detuning the readout resonator from including a bandpass filter near the readout resonator
the qubit eases the tradeoff between T1 and l [45]. To frequency, with strong rejection at the qubit frequency, get a feel for the numbers, suppose we want to mea- as in Figure 7(f) [48], [49]. Such a filter is referred to as sure the transmon in 500 ns. Assuming that we need a Purcell filter in the superconducting qubit literature. the ring-up time to be, at most, half the total measure- Figure 8 shows a four-transmon device [48], with each
ment time, the readout circuit would limit the qubit’s transmon (one of whose capacitance, Cq , is highlighted 2 lifetime to T1 K (1 GHzn)(# 250ss).= 62 5 n . So we in red) connected to a CPW readout resonator (blue)
have made the qubit lifetime 125 times larger than the via capacitance Cg (yellow). The four readout resona- tors are coupled to a common quarter-wave resonator that serves as a Purcell filter (green). In this example, the readout probe tone is injected via a small capacitor at the left of the photograph and transmitted to a bond b 1 pad at the right. a1 Cκ Having explained how the state of a qubit can be read out through a dispersive measurement of an r ancillary linear resonator, we ask the question, Just C g how easy or hard is it to actually do this? Similar to the CQ design of a communication system, we need to under- Figure 8. A four-transmon device with individual readout stand how the error probability depends on design resonators (blue) coupled to a common Purcell filter (green). parameters. Assume that the reflected signal we are (Source: [48]; used with permission.) In this case, the trying to measure has amplitude AS and root-mean-
readout is accomplished via a transmission measurement. square noise voltage vn within some measurement The input probe signal ()a1 is injected on the left, and the bandwidth and that this signal takes on two distinct output signal ()b1 is taken from the right. phases, ! i S/,2 corresponding to the |0 H and |1H states. The measurement will consist of distinguishing between two points in the in-phase/quadrature (IQ) Q plane, as shown in Figure 9. The error probability 2 given this geometry is [50] |1〉 N∼(AD, n )
AD 1 AD 1 i Threshold Perror ==erfc erfc sin SNR, (22) 2 c 2 vn m 2 ``2 jj I S | 〉 2 0 N∼(–AD, n ) where SNR is the RF SNR in the measurement band- width.
We can now express Perror in terms of physical Figure 9. The geometry for readout SNR calculation. parameters. Assuming an ideal measurement, the RF
40 August 2020
Authorized licensed use limited to: University of Massachusetts Amherst. Downloaded on October 01,2020 at 19:47:20 UTC from IEEE Xplore. Restrictions apply. signal power referenced to the readout resonator port Superconductive processors have can be related to the energy stored in the resonator as
PERF = l/,2 so PnRF = r'~lp /,2 where ~p is the mea- advanced tremendously, culminating surement frequency and nr is the average number of in the recent demonstration of photons populating the readout resonator. In prac- quantum supremacy. tice, this is a dynamic measurement that is carried out before the resonator has a chance to settle. Addition-
ally, photons are lost since the measurement has finite Taking e = 02., lr= 23# MHz, nr = 8 photons, i S = 180o
quantum efficiency. To compensate for these facts, we and ~p = 5 GHz, we require a receiver noise tempera-
can introduce a loss factor, e, giving PnRF = er'~lp /.2 ture of roughly 0.56 K, which is approximately twice the Typically, e is in the range of 10–20%. The system noise quantum limit. Needless to say, achieving this level of
is PknoiseS==TTYS //x~' pR21x + XQ/,T where TRX performance is a very challenging task. ^^hh is the input-referred noise temperature of the measure- State-of-the art cryogenic semiconductor amplifiers
ment receiver and TkQp= '~ /2 is the quantum limit still have approximately four times too much noise for for noise added by a phase-preserving amplifier. The this application [52]. Moreover, even if the noise of semi- error probability is thus conductor low-noise amplifiers could be brought down
to TQ , the devices still dissipate far too much power to
1 i S elnr x be heatsunk to the 10-mK stage of a dilution refrigera- Perror = erfc sin . (23) 2 c ` 21j + TTRX/ Q m tion system. Therefore, we need to use something else if we are going to achieve the system sensitivity required As such, error rates improve if we use more pho- to perform one-shot dispersive readout of transmon tons, integrate longer, improve the phase contrast, or qubits. In fact, it turns out that the performance we
decrease TRX. However, in optimizing the measure- want for transmon qubit readout became possible only ment, the noise temperature is constrained by the quan- during the past 10–15 years through the development
tum limit TTRX $ Q , and the integration time must be of near-quantum-limited Josephson parametric ampli- kept low enough such that |1H-state readout errors due fiers [53]–[56], which are currently an essential element to relaxation are insignificant (/Perror,relaxation . x 2T1). that enables high-speed, one-shot dispersive readout of An obvious approach to improving the readout transmon qubits. For more details related to parametric SNR is to drive the readout resonator harder. How- amplifiers, we refer the reader to a review article on ever, the interaction goes both ways, and the average this topic, appearing elsewhere in this focus issue [57]. number of microwave photons nr “stored” in the res- Of course, semiconductor low-noise amplifiers are also onator affects the qubit frequency as well. When the required in the readout chain as second-stage ampli- nonlinear frequency pull on the qubit due to the field fiers, since nearly 60 dB of cryogenic gain is required to intensity in the resonator (also known as the ac Stark overcome practical room-temperature noise floors and
shift T~|01 = 2 nr r) becomes comparable to the qubit’s parametric amplifiers with the requisite linearity are nonlinearity ()h , we start breaking down the distinct not available. energy-level structure that defines the qubit, and the qubit becomes essentially useless. For typical trans- The Road to Practical Applications mon parameters, this can already happen when the To review, a quantum processor can be built using non- resonator stores just nr + 25 photons. Even worse, at linear microwave resonators whose quantum state is lower powers, the drive can already cause transitions controlled using a combination of microwave pulses between the qubit states [51]. For this reason, the read- and time-varying bias signals and measured using out resonator drive power is typically limited such that microwave reflectometry. During the past decade, the resonator is populated with only a handful of pho- these superconductive processors have advanced tre- tons (nr K 10). The power associated with populating mendously, culminating in the recent demonstration of
the resonator with nr photons is Pn= '~lp r /;2 for typi- quantum supremacy: the ability to perform a computa- cal design parameters, this is on the order of just 30 aW tion that is impractical even for today’s most powerful per photon. Since the energy in the resonator can be, at of supercomputers [5]. However, today’s quantum pro- most, roughly 10 photons, our peak drive power can be cessors are still limited. no more than approximately −125 dBm, referenced to With a regular digital computer, error rates are so the resonator drive port. low that we can write software while assuming that
Our final degree of freedom is TRX , so let’s try to under- the bits never suffer physical errors. Conversely, super- stand the requirements to achieve error rates of 1%. Let’s conducting qubits suffer relatively high error rates, assume that we want to limit x to 300 ns to avoid exces- meaning that they can perform only a finite number of sive errors due to relaxation during the measurement. logic operations before the quantum state randomizes.
August 2020 41
Authorized licensed use limited to: University of Massachusetts Amherst. Downloaded on October 01,2020 at 19:47:20 UTC from IEEE Xplore. Restrictions apply. For example, a transmon with an energy decay con- faults. According to current estimates based on today’s
stant of T1 = 50n s and a two-qubit gate time of 20 ns best physical qubit systems, a single logical qubit with can undergo only 2,500 gates before it has a high low-enough error rates to be used in a fault-tolerant probability of suffering an error. With such high error computer would need roughly 1,000 constituent physi- rates, how can a quantum computer actually complete a cal qubits, and a fault-tolerant computer would need full algorithm? There are currently two main approaches. approximately 1,000 logical qubits, necessitating In the near term, it is assumed that errors are un roughly 1 million physical qubits [64]. avoidable, and researchers are currently considering Let’s take stock of what that means in terms of control how to conduct useful quantum computation with a electronics, wiring, and packaging. In today’s state-of- modest number of imperfect qubits. The key thing here the-art technology [5], each qubit is driven by dedicated is that the algorithms must be short compared to the on-chip lines that carry microwave signals for XY rota- qubits’ coherence times. This is tricky, though: if the tions and 500-MHz dc signals for Z rotations. The XY quantum algorithm is too short, it can be simulated on pulses are generated by a pair of high-speed (1 gigas- a conventional computer, obviating the need for the ample per second) digital-to-analog converters (DACs) quantum computer. The most promising uses seem driving an IQ mixer, while the Z pulses are generated to be in simulating chemistry [58]–[62]. Applications by another high-speed DAC, so we need three DACs in this regime of so-called noisy intermediate-scale per qubit. The signals are carried from the pulse gen- quantum (NISQ) computation [63] constitute an area of erators to individual qubits through dedicated coaxial active research. cables and CPW interconnects. The cables interface to However, the long-term goal of the community is the package via blind-mate connectors, and the package to develop fault-tolerant quantum computers. Much in connects to the chip via wire bonds (see Figure 10). Can the same way that redundancy is used in conventional these technologies be scaled to a 1-million-qubit com- error-correcting code memory to lower error rates, it puter? For the control system, this would mean at least is theoretically possible to combine many imperfect 3 million DACs and millions of coaxial cables running physical qubits into a single logical qubit whose error into a cryostat holding a chip package with millions of rate is lowered exponentially in the number of constitu- blind-mate connectors. This is where microwave engi- ent physical qubits by using quantum error correction. neering really comes in: the cost, size, thermal, assem- Quantum error correction is somewhat more subtle bly, reliability, and signal integrity concerns arising in than conventional error correction because quantum a system with that much microwave hardware demand states cannot be copied (this fact is called the no-cloning innovations in the microwave domain. theorem). This limitation makes the subject of quan- Let us consider the optimization of the microwave tum error-correcting codes somewhat more delicate interface required to control a million-qubit system. If and richer than conventional error-correction codes. A the control system used in the recent quantum suprem- computer operating with logical qubits is called fault- acy experiment at Google [5] were scaled to control a tolerant because it continues to work without logical 1-million-qubit system, it would occupy three football errors, even when individual physical qubits suffer fields of floor space and consume roughly 40 MW of dc power, not to mention the significant power that would be required to compensate for losses associated with signal distribution. Development of integrated con- trol systems specifically tailored toward this applica- tion [65]–[68] is an important area for future research that could impact quantum computing in the way that integrated beamformers have enabled more scalable phased arrays [69]. Innovation is also required in the interconnections used in quantum computing systems. Control sig- nals are currently brought from the electronics into the cryostat and down to the quantum chip through coaxial cable, which raises several important ques- tions. How do we connect the cables to the package? Figure 10. The package used in a recent quantum computing experiment at Google. (Source: [5]; used The connection has to have a low loss to avoid ther- with permission.) Vertical launch connectors around the mally loading the 20-mK stage of the cryostat. Dilu- perimeter connect to coaxial cabling and a printed circuit tion refrigerators can cool to approximately 10 nW at board (PCB). The PCB is covered by a magnetic shield (the 20 mK, so, assuming that we have one dilution cooler square box in the center). per 65,000 qubits and that we need to source roughly
42 August 2020
Authorized licensed use limited to: University of Massachusetts Amherst. Downloaded on October 01,2020 at 19:47:20 UTC from IEEE Xplore. Restrictions apply. 300A n of dc current per qubit, the connector at the [5] F. Arute et al., “Quantum supremacy using a programmable super- 20-mK stage must have a contact resistance of K 2mX conducting processor,” Nature, vol. 574, pp. 505–510, Oct. 2019. doi: 10.1038/s41586-019-1666-5. per line. The microwave connector also must have low [6] Y. Sun, “Why Alibaba is betting big on AI chips and quantum com- crosstalk despite its high density. Developing a cost- puting,” MIT Technology Review, Cambridge, MA, Sept. 25, 2018. effective connector with a low-enough profile while [Online]. Available: https://www.technologyreview.com/s/612190/ maintaining acceptable channel-to-channel crosstalk why-alibaba-is-investing-in-ai-chips-and-quantum-computing/ [7] S. K. Moore and A. 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