A NASA Perspective on Quantum Computing: Opportunities and Challenges
Rupak Biswas, Zhang Jiang, Kostya Kechezhi, Sergey Knysh, Salvatore Mandr?, Bryan O'Gorman, Alejandro Perdomo-Ortiz, Andre Petukhov, John Realpe-Gómez, Eleanor Rieffel, Davide Venturelli, Fedir Vasko, Zhihui Wang
AA NASA Perspective on Quantum Computing:Opportunities and Challenges
Rupak Biswas, Zhang Jiang, Kostya Kechezhi, Sergey Knysh, SalvatoreMandr`a, Bryan O’Gorman, Alejandro Perdomo-Ortiz, Andre Petukhov, JohnRealpe-G´omez, Eleanor Rieffel, Davide Venturelli, Fedir Vasko, Zhihui Wang
NASA Ames Research Center, Moffett Field, CA 94035
Abstract
In the last couple of decades, the world has seen several stunning instancesof quantum algorithms that provably outperform the best classical algorithms.For most problems, however, it is currently unknown whether quantum algo-rithms can provide an advantage, and if so by how much, or how to designquantum algorithms that realize such advantages. Many of the most challeng-ing computational problems arising in the practical world are tackled today byheuristic algorithms that have not been mathematically proven to outperformother approaches but have been shown to be effective empirically. While quan-tum heuristic algorithms have been proposed, empirical testing becomes possibleonly as quantum computation hardware is built. The next few years will be ex-citing as empirical testing of quantum heuristic algorithms becomes more andmore feasible. While large-scale universal quantum computers are likely decadesaway, special-purpose quantum computational hardware has begun to emergethat will become more powerful over time, as well as some small-scale universalquantum computers. ∗ Rupak Biswas
Email address: [email protected] (Rupak Biswas, Zhang Jiang, Kostya Kechezhi,Sergey Knysh, Salvatore Mandr`a, Bryan O’Gorman, Alejandro Perdomo-Ortiz, AndrePetukhov, John Realpe-G´omez, Eleanor Rieffel, Davide Venturelli, Fedir Vasko, ZhihuiWang )
Preprint submitted to Elsevier April 18, 2017 a r X i v : . [ qu a n t - ph ] A p r . Introduction In the last couple of decades, the world has seen several stunning instancesof quantum algorithms that provably outperform the best classical algorithms.For most problems, however, it is currently unknown whether quantum algo-rithms can provide an advantage, and if so by how much, or how to designquantum algorithms that realize such advantages. Many of the most challeng-ing computational problems arising in the practical world are tackled today byheuristic algorithms that have not been mathematically proven to outperformother approaches but have been shown to be effective empirically. While quan-tum heuristic algorithms have been proposed, empirical testing becomes possibleonly as quantum computation hardware is built. The next few years will be ex-citing as empirical testing of quantum heuristic algorithms becomes more andmore feasible. While large-scale universal quantum computers are likely decadesaway, special-purpose quantum computational hardware has begun to emergethat will become more powerful over time, as well as some small-scale universalquantum computers.Successful NASA missions require solution of many challenging computa-tional problems. The ambitiousness of such future missions depends on ourability to solve yet more challenging computational problems to support betterand greater autonomy, space vehicle design, rover coordination, air traffic man-agement, anomaly detection, large data analysis and data fusion, and advancedmission planning and logistics. To support NASA’s substantial computationalneeds, NASA Ames Research Center has a world-class supercomputing facilitywith one of the world’s most powerful supercomputers. In 2012, NASA es-tablished its Quantum Artificial Intelligence Laboratory (QuAIL) at Ames toexplore the potential of quantum computing for computational challenges aris-ing in future agency missions. The following year, through a collaboration withGoogle and USRA, NASA hosted one of the earliest quantum annealer proto-types, a 509-qubit D-Wave II machine, which last summer was upgraded to a1097-qubit D-Wave 2X system. 2ecause quantum annealers are the most advanced quantum computationalhardware to date, the main focus for the QuAIL team has been on both theoreti-cal and empirical investigations of quantum annealing, from deeper understand-ing of the computational role of certain quantum effects to empirical analyses ofquantum annealer performance on small problems from the domains of planningand scheduling, fault diagnosis, and machine learning. This paper will concen-trate on the team’s quantum annealing work, with only brief mention of researchrelated to capabilities of other near-term quantum computational hardware thatwill be able to run quantum heuristic algorithms beyond quantum annealing.For information on quantum computing more generally, and other algorithms,both heuristic and non, see quantum computing texts such as [1].The power of quantum computation comes from encoding information ina non-classical way, in qubits, that enable computations to take advantage ofpurely quantum effects, such as quantum tunneling, quantum interference, andquantum entanglement, that are not available classically. The beauty of quan-tum annealers is that users can program them without needing to know aboutthe underlying quantum mechanical effects. Knowledge of quantum mechanicsaids in more effective programming, just as an understanding of compilationprocedures can aid classical programming, but it is not necessary for a basicunderstanding.For this reason, the first three sections consist of an overview of quantumannealing (Sec. 2), a description of how to program a quantum annealer (Sec. 3),and a high-level review of our exploration of three potential application areasfor quantum annealing (Sec. 4). The quantum effects involved are only lightlymentioned, so these sections should be easily accessible to computer scientistswithout any knowledge of quantum mechanics or quantum computing. Sec. 5,which examines the role various physical processes play in quantum anneal-ing, requires more physics knowledge for a full understanding, as does Sec. 6that discusses hardware, though a classically-trained computer scientist with-out knowledge of quantum mechanics can get a high-level understanding. Weconclude with a brief section summarizing the outlook for the future.3 . Quantum annealing
Quantum annealing [2, 3] is a metaheuristic optimization algorithm thatmakes use of quantum effects such as quantum tunneling and interference. Itis one of the most accessible quantum algorithms to people versed in classicalcomputing because of its close ties to classical optimization algorithms suchas simulated annealing and because the most basic aspects of the algorithmcan be captured by a classical cost function and parameter setting. Quantumannealers are special-purpose quantum computational devices that can run onlythe quantum annealing metaheuristic. For readers not familiar with quantumannealing in physics, we refer to Sec. 5.1 for a general introduction.Quantum annealers are designed to minimize Quadratic Unconstrained Bi-nary Optimization (QUBO) problems; i.e., the cost function is of the form C ( x ) = (cid:88) i a i x i + (cid:88) i 3. Programming a quantum annealer This section discusses the two main steps in programming a quantum an-nealer: mapping the problems to QUBO; and embedding , which takes thesehardware-independent QUBOs to other QUBOs that match the specific quan-tum annealing hardware that will be used. For a cost function not natively in QUBO form, the typical procedure tomap the problem into QUBO is to properly choose binary variables, formulateconstraints, and embed the violation of constraints as energy penalties. Weillustrate this process with an example from Ref. [5]. Example: In a graph coloring problem, the task is to determine whethereach vertex of a graph G ( V, E ) can be colored from a set C so that no twovertices connected by an edge have the same color. The goal is to formulatea cost function such that the minimum is 0. One way to choose the binaryvariable is to use x v,c = 0 or 1 to express whether vertex v is assigned color7 . The ensuing constraints would be: (1) Each vertex needs to be assignedexactly one color that can be expressed in binary form as ( (cid:80) c x v,c − . (2)Connected vertices cannot share the same color; otherwise, the energy penaltyis raised, (cid:80) c (cid:80) v,v (cid:48) ∈ E x v,c x v (cid:48) ,c . The cost function expressed in QUBO is then H = (cid:80) v ( (cid:80) c x v,c − + (cid:80) c (cid:80) v,v (cid:48) ∈ E x v,c x v (cid:48) ,c . When no requirement is violated,the cost function has value 0, which is the ground state of H .In this example, the cost function H is naturally quadratic. More gener-ally, the cost functions of many optimization problems can be expressed ashigher-degree polynomials of the binary variables (PUBOs). Degree-reductiontechniques can then be applied to recast a PUBO as QUBO, usually at the priceof adding ancilla variables [6]. Because the physical hardware has limited connectivity, there usually doesnot exist a direct one-to-one mapping between the QUBO binary variables andthe physical qubits so that each binary term in the QUBO corresponds to apair of connected qubits. To obtain the needed connectivity in the embeddableQUBO, an additional step is required. Unlike the mapping step, the embeddingstep is hardware dependent. A cluster of qubits { y i,k } connected to each otherin the hardware graph will represent a single variable x i . For any term x i x j inthe mapped QUBO, there is a connection in the embeddable QUBO betweenone of the qubits in the cluster for x i and one qubit in the cluster for x j . Minorembedding is the process of determining a cluster for each binary variable inthe problem QUBO [7]. The problem of finding the optimal minor embeddingis itself NP-complete, but fortunately it is not necessary to find the optimalembedding. In general, for planar architectures, there are straightforward, fastalgorithms to embed an N -variable problem in hardware consisting of no morethan N physical qubits [7, 8, 9]. In the near term, while the hardware is soqubit constrained, heuristic algorithms [10] are used to try to minimize resourcesand maximize the size of the problems embeddable on the machine.To encourage the qubits in the cluster to all take the same value by the end8f the anneal so that the value of the variable they represent is unambiguous, theembeddable QUBO also includes constraint terms J F y i,p y i,q for any pair p, q ofqubits in the cluster that are connected to each other, where J F is the strength ofthe coupling. This is to ensure that in the most energy-favorable configuration,all qubits in the cluster take the same value. The Hamiltonian obtained fromthe embeddable QUBO shares the same ground state energy as the Hamiltonianfrom the mapped QUBO, but conforms to the hardware architecture. Thehigher energy spectrum may be considerably altered, so different embeddingscan significantly affect performance.The optimal strength of J F is a subject of extensive research [5, 11, 12]. Onemight think it should be as high as possible to force the qubits to all take thesame value at the end, but in practice there is a sweet spot. Coupling strengthsthat are too high degrade performance. Intuitively, a high coupling strengthmakes it harder to change the value of a variable in the cluster once they takeon a value that is not, ultimately, optimal, though the actual quantum dynamicsare more complicated than this simple explanation.The layout of the qubits and couplers of a D-Wave quantum annealer is a n × n lattice of unit cells called a Chimera graph. Each unit cell is composedof a bi-partite graph of 8 qubits. A schematic diagram of the graph formed by9 cells is shown in Fig. 2. The current D-Wave machine at NASA has 12 × N variables, N qubitsand couplers are needed in the worst case so that each binary variable canbe represented by N physical qubits and effectively couple to all other binaryvariables. As an illustration, Fig. 3 shows an example of embedding a triangleonto a bi-partite graph.When an Ising problem is programmed to the chip, errors due to noise ormanufacturing miscalibration associated with the bias fields ( h ’s) and couplers( J ’s) would affect the annealing performance. Simple offset errors can be cor-rected through software, but more complicated errors are harder to mitigate.9 igure 2: Nine unit cells in a Chimera graph. One strategy is to repeat the annealing with a gauge-transformed Hamiltonianin which the states used to represent 0 and 1 are swapped. The qubits areencoded into s (cid:48) j = g j s j where g j = ± 1, and the biases and couplers are ac-cordingly set as h (cid:48) j = g j h j and J (cid:48) i,j = g i g j J i,j . The resulting Hamiltonian H (cid:48) = (cid:80) j h (cid:48) j s (cid:48) j + (cid:80) i,j J (cid:48) i,j s (cid:48) i s (cid:48) j , which is equal to the original Hamiltonian, is sentto the annealer and the solution obtained is then decoded using s j = g j s (cid:48) j . Oneset of parameters { g j } is called a gauge. In the absence of errors, the annealingresults for H and H (cid:48) should be the same while the actual performance could begauge-dependent. Success probabilities averaged over a set of gauges are typi-cally used. Various error suppression and correction strategies exist, both fullyquantum [13], a mix of quantum and classical [14], and a more recent quan-tum approach [15]. Once the problem is programmed, the annealing is repeatedmultiple times (typically thousands to millions), and each time the final statemeasured in the computational basis is recorded.10 bc d1 23 Figure 3: Schematics of embedding the Hamiltonian H = J , s s + J , s s + J , s s on a graph. Left: Triangle graph to be embedded. Right: Graph after embedding on a bi-partite graph of size 4. The variable s is represented by two physical qubits s a and s b witha strong ferro-magnetic coupling J F < 0. The Hamiltonian after embedding is H embed = J F s a s b + J , s a s d + J , s b s c + J , s c s d . 4. Applications In this section, we give a high-level overview of our in-depth studies of threepotential applications areas: planning and scheduling, fault diagnosis, and ma-chine learning. Further technical details can be found in the publications refer-enced in each section. Automated planning and scheduling has many applications, from logistics,air traffic control, and industrial automation to conventional military missions,resource allocation, and assistance in disaster recovery. Many of the challengesin autonomous operations include significant planning and multi-agent coordi-nation tasks in which operational teams must generate courses of action priorto the event and adjust those plans as new information becomes available orunexpected events occur.Many planning and scheduling problems are very challenging to solve; as thenumber of events to plan or schedule grows, the number of possible solutionsgrows exponentially. These problems are often NP-hard or harder, and arecurrently tackled by classical heuristic algorithms. The emergence of quantumannealing hardware allows the exploration of quantum heuristic approaches tothese problems [3], with the objectives to search for significant improvements11ver existing techniques in the efficiency with which good plans can be found, orin finding better plans that satisfy more constraints, and/or in greater diversityin the plans found.Given the severe limitation in quantum memory of current quantum anneal-ers, in order to benchmark the machines, it is imperative to find prescriptions toidentify small problems that exhibit signature of hardness. Currently, the mostcommon approach to designing benchmark planning problems is to extract solv-able problems from real-world applications. This approach has the benefit oftuning algorithms toward the applications from which the benchmark problemsare obtained. A complementary approach is to design parametrized familiesthat capture aspects of practical planning problems and can be shown to beintrinsically hard. Such families support focused examination of these aspects,small problems that can be meaningfully considered to be hard, and scalinganalyses with respect to size. Families of small but hard problems are criti-cal for present research into quantum annealing because the current quantumannealers can handle only small problems. Families we have designed for thepurpose of assessing the performance of quantum annealers have proved usefulin distinguishing the strengths and weaknesses of state-of-the-art planners [16]. Classical planning problems are expressed in terms of binary state variables and actions . Examples of state variables in the domain of autonomous rovernavigation are “Rover R is in location X ” and “Rover R has a soil sample fromlocation X ,” which may be True or False. Actions consist of two lists, a set of preconditions and a set of effects (see Fig. 4). The effects of an action consistsof a subset of state variables with the values they take on if the action is carriedout. For example, the action “Rover R moves from location X to location Y ”has one precondition, “Rover R is in location X = True” and has two effects“Rover R is in location X = False” and “Rover R is in location Y = True.”A specific planning problem specifies an initial state , with values specifiedfor all state variables, and a goal , specified values for one or more state variables.12 igure 4: (a) Pictorial view of a planning problem. The initial state (e.g., Rover behindthe rocks without sample) is specified by assigning True (1) or False (0) to state variables(named A-J in this simplified example). The planning software navigates a tree, where apath represents a sequence (with possible repetitions) of actions selected from a pool (colors).Each action has preconditions on the state variables (e.g., moves can be done around the rocksbut not through them) that need to be satisfied for the actions to be executed (the circlesunder the state variables in the search tree need to be True) and has an effect on the state(colored variables in shaded regions of the new state have changed values). A valid search plan(multiple valid plans are possible) will reach the goal state (e.g., Rover in front of the rockswith a sample collected). (b) Direct time-indexed QUBO structure for a planning problemwith only positive preconditions and goals. Each node represents a state variable (left) or anaction (right) at any given time t . Time flows from top to bottom, and variables y ( t ) i for theactions at time t are shown between the state variables x ( t − i for one time step and the statevariables x ( t ) i for the next time step. The node grayscale intensity represents the magnitudeof local field (bias) h i applied to a given qubit i , and the double contour in a node indicatesa negative bias. As for preconditions, goals are conventionally positive, so the specified value forthe goal variables is True. Generally, the goal specifies values for only a smallsubset of the state variables. A plan is a sequence of actions. A valid plan, or asolution to the planning problem, is a sequence of actions A , ..., A L such thatthe state at time step t i − meets the preconditions for action A i , the effects ofaction A i are reflected in the state at time step t i , and the state at the end hasall of the goal variables set to True.Ref. [5] discusses a general QUBO formulation of planning problems (see13ig. 4(b)). If the original planning problem has N state variables and we arelooking for a plan of length L , then the QUBO problem will have N ( L + 1)binary variables x ( t ) i , where t ∈ { , . . . , L } is the time index, and i is the indexof the state variable in the original planning problem. In addition, if the originalplanning problem has M possible actions, we will have LM additional binaryvariables y ( t ) j which indicate whether the j th action is carried out at time step t or not. A QUBO can then be defined in terms of these variables, with termscapturing the goal, precondition, effect, single-action, and no-op (no variablechange without an action) constraints: H = H (cid:48) goal + H no − op + H (cid:48) precond + H effects + H single-action . (4)Ref. [5] describes a somewhat more general cost function that supports multipleactions per time step. Scheduling was recognized early on as one the most promising near-term tar-gets for quantum annealing due to its efficient quadratic time-indexed Mixed-Integer Linear Programming formulation. Furthermore, there is a rich litera-ture of complex pre-processing and hybrid classical techniques. Using this directquadratic formulation of scheduling instead of the most general planning formu-lation leads to very significant performance advantages in runs of the D-Wavemachines [5].Scheduling formalizes problems dealing with the optimal allocation of re-sources (machines, people) to tasks (jobs) over time, under various constraintsand figures of merit. In one direct QUBO formulation, a bit is associated tothe execution of a given job in a given machine (out of M possible) at a giventime (discretized in T slots), allowing for very efficient mappings on currentquantum annealers supporting two-body Ising-type interactions, using N M T qubits, where N is the number of jobs. While objective functions of the pri-ority maximization type are easily implementable as linear penalty functionsrequiring only local fields on the corresponding logical bits, objectives requiring14 akespan minimization require a more involved encoding with either T ancillaclock variables highly connected to the qubits relative to the jobs scheduled last,or by complementing the quantum solver with guidance from classical methods,such as binary search [12].Many planning and scheduling problems are of such scale and complexitythat they are by necessity solved in pieces, and so quantum hardware can benaturally integrated into the solution of such problems. Hybrid solvers employ-ing quantum annealing together with classical methods are particularly suitedto scheduling applications, because the state-of-the-art approaches for specificscheduling problems are typically combining different approaches in a mod-ular way, and decompositions can be employed to get around programmingbottlenecks such as high connectivity, precision requirements, continuous con-straints, or to employ quantum annealing as a heuristic module of a completesolver [17, 18]. As a heuristic module of a complete solver, quantum anneal-ing enables more directed search of the solution space. Building a completesolver out of a probabilistic quantum subroutine requires non-trivial classicalco-processing, but recent work has shown that it can be done successfully. Inparticular, partial solutions returned by a quantum solver can be used to derivebounds on the optimum value of the function to be optimized, and thereforefocus on the most promising or neglect the least promising parts of the solutionspace.Recent work on the application of quantum annealing to scheduling includesprogramming and benchmarking quantum annealers on small problems fromthe domains of graph coloring [5], job shop scheduling [12], Mars lander activityscheduling [17], air traffic runway landing [18], and alternative resource schedul-ing [18]. The question of speedup with respect to purely classical methodsare inconclusive due to the small size of the problems implementable on cur-rent quantum annealers and the inefficiency of embedding techniques [5]. Thisbody of work has identified precision and connectivity requirements that sug-gest future generations of annealers may be able to solve currently intractablescheduling problems within a decade. 15lanned technological advances in quantum annealing architectures will alsomake possible tighter integration of quantum and classical components in thehybrid approaches discussed above, both through more programmable devicesthat allow for greater flexibility as subroutines and through application-specificdevices that maximize the effectiveness of particular algorithms. In future, weexpect quantum hardware to be integrated into larger systems much as graphicalprocessing units are today [19]. Another application domain we have studied with quantum annealing de-vices is the diagnostics of electrical power-distribution systems (EPS); a collab-oration between QuAIL and the Discovery and System Health (DaSH) technicalarea at NASA Ames. Diagnosing the minimal number of faults capable of ex-plaining a set of given observations, e.g., from sensor readouts, is a hard combi-natorial optimization problem usually addressed with artificial intelligence tech-niques. In [20], we presented the first application of the Combinatorial Problem → QUBO Mapping → Direct Embedding process where we were able to em-bed instances with sizes comparable to those found in real-world problems. Wedemonstrated problem instances with over 100 electrical components (includ-ing circuit breakers and sensors) running on a quantum annealing device with509 quantum bits. In comparison, the number of components in the electricalcircuits used for diagnostics competitions from NASA’s Advanced Diagnosticsand Prognostics Testbed (ADAPT) ranges between 40 and 100 [21]. As shown in Fig. 5(a), there are two types of components. The first arecircuit breakers (CB), which in their healthy mode allow the flow of current, andare illustrated as the nodes of the quaternary tree. We denote them by the setof binary variables { x i } , with x i = 1 ( x i = 0) corresponding to CB i in a healthy(faulty) state. The other component type is the sensor or ammeter, which is notonly another electrical component that could potentially malfunction, but also16orms part of the observations from which one is asked to perform the diagnosisof the electrical network. Therefore, for each ammeter, we have an observationparameter and a status variable indicating its healthy or faulty state. Theobservations (or readouts) are part of the problem definition and provided asinput parameters. We denote this set of binary parameters { l i } , with l i = 1( l i = 0) if the i -th ammeter is showing a High (Low) readout. Similar to the { x i } variables for the CBs, the uncertainty in the ammeter readouts is introducedby assigning to them a set of binary variables, { y i } , with y i = 1 ( y i = 0)corresponding to ammeter i in a healthy (faulty) state.The goal is to find the minimum number of faults in the electrical compo-nents, either in the CBs and/or the ammeters, consistent with the circuit layoutand the readouts. We solve this as a minimization problem over the pseudo-Boolean function H problem ( { x i } , { y i } ; { l i } ), whose construction is explained be-low. After H problem is transformed into its QUBO form, we can subsequentlyuse the quantum annealer to find the assignment for each of the { x i } and { y i } .The construction of the pseudo-Boolean function contains two contributions: H problem = H numFaults + H consist . (5) H consist is constructed such that it is 0 whenever the prediction from the as-signment of all the { x i } and { y i } is consistent with the readouts { l i } from theammeters, and greater than 0 when the readouts and the prediction, given the { x i } and { y i } assignments, do not match. Consider the set P i as the set of CBindices in the path from the root node (CB 1) where power is input, all the wayto the CB connected to the i -th ammeter. Thus, for the network in Fig. 5(a), P = { , , } , P = { , , } , · · · , and P = { , , } . If we denote the numberof paths as n paths (equals the number of ammeters in this network), one canconstruct H consist as: H consist = λ path n paths (cid:88) i =1 y i g i , f i ( { x j } j ∈ P i ) = (cid:89) j ∈ P i x j , (6)with g i = l i + f i − f i l i , a binary function with g i = 0 when the prediction f i ,17 (b)$QUBO$form$(a)$Computa2onal$problem$ (c)$Hardware$embedding$ Obs.: Figure 5: General scheme of an experimental setup for the diagnosis of multiple faults witha quantum annealer. (a) A possible realization of the diagnosis of multiple faults in an EPSnetwork with one power source, 21 CBs and 16 sensors or ammeters. The orange crossesindicate faulty electrical components ( x i = 0). In this particular instance of 6 faults, aplausible explanation of the readouts places one of the faults on a CB and the remaining 5 onthe ammeters. However, this is only one of the 2 six-fault explanations that are equally likelyin this case. (b) QUBO form of the problem where coupling between two logical qubits isrepresented as edges. (c) The subsequent embedding into the Chimera graph usually requiresmore variables since some logical qubits are represented by several physical qubits (depictedhere as nodes in the graph) due to the sparse connectivity of the hardware graph. In thisproblem, 81 physical qubits are needed to implement the QUBO with 46 logical variables. P i , is consistent with the readouts l i ,and g i = 1 when the prediction and the readout are in disagreement. In otherwords, g i = xor ( f i , l i ). H numFaults is proportional to the number of faults (whenever x i = 0 or y i = 0) in the electrical network: H numFaults = λ CBfaults n CB (cid:88) i =1 (1 − x i ) + λ sensorfaults n sensor (cid:88) i =1 (1 − y i ) , (7)and when combined with H consist , as written in Eq. (5), defines the problem en-ergy function to be minimized by favoring the minimal set of faulty componentsthat are simultaneously consistent with the observations in the outermost sen-sors. A thorough discussion on setting the values of all the penalties is providedin [20].Notice the pseudo-Boolean H consist is a high-degree polynomial, and for thisparticular network, the order of the polynomial is related to the depth of thetree. We can reduce the degree of the polynomial to a quadratic expression, H QUBO , with the overhead of adding more binary variables, while conservingthe global minimum of the original function, H ( { x i } , { y i } ; { l i } ). Further detailson the techniques used for this reduction are provided in [20, 22].Assuming it requires n A ancilla variables { a i } to reduce the high-degreepolynomial to the quadratic expression, we can relabel the CB, sensor, andancilla variables, { x i } , { y i } , and { a i } , respectively, into a new set of binaryvariables { q i } for i = 1 , , · · · , n l , with n l = n CB + n sensor + n A as the totalnumber of logical qubits. The final quadratic cost function to be minimized canthen be written as H QUBO ( { q i } ) = E + (cid:88) i,j Q i,j q i q j = E + q T · Q · q . (8)As shown in Fig. 5, this expression can be represented as a graph with thenumber of vertices equal to the number of logical qubits n l corresponding to theset of variables { q i } . In this representation, Q i,i can be treated as the weights on19he vertices, while Q i,j are the weights for the edges representing the couplingsbetween variables i and j (see Fig. 5). Notice that since q i = q i , the expression q T · Q · q contains both linear terms Q i,i , and quadratic terms, Q i,j , when i (cid:54) = j . E corresponds to the constant independent term.Although the problems studied in [20] are simpler than typical real-worldinstances, we believe that they still capture some non-trivial features, such asthe inclusion of uncertainty in the sensor readouts. Of course, aiming to embedall the details from realistic scenarios will require significantly more qubits andalso depend on the specific network/problem to be solved.As another realization of the fault detection application, the QuAIL teamis examining combinational digital circuits [23], a more realistic scenario usedto benchmark codes devoted to solving diagnostics related problems [21]. Pre-liminary results look very promising and harder than any other benchmarksreported in the literature and used to address the question of quantum speedupin quantum annealers. Sampling from high-dimensional probability distributions is at the core of awide spectrum of computational techniques with important applications acrossscience, engineering, and society. Examples include deep learning, probabilisticprogramming, and other machine learning and artificial intelligence applications.Much of the record-breaking performance of classical machine learning algo-rithms regularly reported in the literature pertains to task-specific supervisedlearning algorithms [24]. Unsupervised learning algorithms are more human-like, and in principle more general and powerful, but their development hasbeen lagging due to the intractability of traditional sampling techniques suchas Markov Chain Monte Carlo (MCMC). Indeed, as leading researchers in thefield have pointed out [24], future success of unsupervised learning algorithmsrequires breakthroughs in efficient sampling algorithms. Quantum annealingholds the potential to sample more efficiently and from more complex prob-abilistic models, which would significantly advance the field of unsupervised20earning. A computationally hard problem, key for some relevant machine learningtasks, is the estimation of averages over probabilistic models defined in termsof a Boltzmann distribution P B ( s ) = 1 Z exp (cid:88) i,j W ij s i s j + (cid:88) i b i s i , (9)where Z is the normalization constant or partition function, s = { s , . . . , s N } denotes a configuration of binary variables, and W ij and b i are the parametersspecifying the probability distribution.Sampling from generic probabilistic models, such as P B ( s ) in Eq. (9), ishard [25] in general. For this reason, algorithms relying heavily on samplingare expected to remain intractable no matter how large and powerful classicalcomputing resources become. Even though quantum annealers were designedfor challenging combinatorial optimization problems, it has been recently recog-nized as a potential candidate to speed up computations that rely on samplingby exploiting quantum effects, such as quantum tunneling [26, 27]. Indeed, some research groups have recently explored the use of quantumannealing hardware for the learning of Boltzmann machines and deep neuralnetworks (see [26, 28] and references therein). The standard approach to thelearning of Boltzmann machines relies on the computation of certain averagesthat can be estimated by standard sampling techniques, such as MCMC. An-other possibility is to rely on a physical process, like quantum annealing, thatnaturally generates samples from a Boltzmann distribution. In contrast to theiruse for optimization, when applying quantum annealing hardware to the learn-ing of Boltzmann machines, the control parameters (instead of the qubits’ states)are the relevant variables of the problem. The objective is to find the optimal21ontrol parameters that best represent the empirical distribution of a givendataset.These ideas are framed within a hybrid quantum-classical computing paradigm.Given a classical machine learning infrastructure, the idea is to replace the soft-ware module that generate samples, e.g., via MCMC, with a quantum annealingprocess. This quantum sampling module could be similarly employed in otherdomains where sampling is useful. Thus, demonstrating quantum speedup forsampling would have broad implications.In recent work [26], the QuAIL team has demonstrated how to properly usea quantum annealer by overcoming critical challenges such as the instances-dependent temperature estimation. In fact, while the probability distribution P B ( s ) in Eq. (9) is specified by parameters W ij and b i , the control parametersof a quantum annealer are instead J ij = T eff W ij and h i = T eff b i . Accordingto quantum dynamical arguments [27], T eff is an instance-dependent effectivetemperature, different from the physical temperature of the device. Unveilingthis unknown temperature is key to effectively using a quantum annealer forBoltzmann sampling. By introducing a simple effective temperature estimationalgorithm [26], it was possible to successfully use the D-Wave 2X system forthe learning of a special class of restricted Boltzmann machines that can serveas a building block for deep learning architectures. Experiments run using asynthetic dataset showed that the quantum-assisted algorithm outperformed interms of quality (i.e., the value of the likelihood reached) the standard classi-cal algorithm named CD-1 and approached the performance of CD-100, whichtakes about 100 times more computational effort than CD-1 (See [26] for details).Complementary work that appeared roughly simultaneously showed that quan-tum annealing can be used for supervised learning in classification tasks [28].These results are encouraging, but there remain numerous challenges beforethe full potential of quantum annealing hardware for sampling problems canbe harnessed. While each future generation will no doubt be an improvement,hardware advances alone will not suffice. The QuAIL team is therefore develop-ing algorithmic strategies to address these other problems, with promising initial22esults. For example, we recently experimentally demonstrated [29] the feasibil-ity of a fully unsupervised machine learning application by successfully trainingour quantum annealer, using up to 940 qubits, to generate, reconstruct, andclassify images that closely resemble (low resolution) handwritten digits, amongother synthetic datasets. We showed a Turing test (see Fig. 4 in [29]) to chal-lenge people to distinguish between handwritten digits and digits generated bythe quantum device; most people we informally showed this Turing test eitherfailed or found it difficult. To reach this milestone, we implemented denselyconnected hardware-embedded models that are more robust to noise and moreefficient to learn with state-of-the-art quantum annealers.The ultimate question that drives this endeavor is whether there is quantumspeedup in sampling applications. Current experience with the use of quantumannealers for combinatorial optimization suggest the answer is not straightfor-ward. This work is part of the emerging field of quantum machine learning [30],an essentially unexplored territory where quantum annealing might have a largeimpact in the near term. These explorations have spurred QuAIL to design advanced techniques toguide programming and improve performance. Software calibration methodsdevised by the team are described in [31]. In [5], we compare different mappingsand in [32], we present advanced techniques to intelligently select gauges basedon small numbers of trial runs that often improve performance by an order ofmagnitude. Compilation strategies for quantum annealers, including guidelinesfor optimally setting the strength of J F are discussed in [5, 11, 12]. Furthermore,we have identified certain common structures in the QUBO representations ofmany applications because different constraints often have similar forms [5]. 5. Physics of quantum annealing This section discusses results clarifying the role of various processes in quan-tum annealing that suggest where to look for potential quantum speedup and23here such an advantage would be unlikely. So far, we have been informal aboutwhat we mean by quantum speedup. However, knowing the different types ofquantum speedup is helpful in assessing results related to the computationalpower of quantum annealing. It is also necessary to improve our understandingof potential classes of problems for which such a quantum device can excel. The target of quantum annealing is to optimize a function of QUBO form, asin Eq. (1). The cost function has a physical realization in a system comprisingquantum bits (qubits) where each binary variable is encoded as a qubit. Thecoefficients ( a i ) translate into bias fields applied on the qubits and ( b i,j ) is rep-resented as the coupling strength between two qubits. The cost function thuscorresponds to a Hamiltonian , H , as in Eq. (2), which describes the energyof the system. The Hamiltonian bears strong similarity with the cost function.However, while in the classical cost function the binary variables can take valueeither 0 or 1, in a Hamiltonian the qubit is allowed to be (and in a physicalquantum system, can be) in a superposition of these two states α | (cid:105) + β | (cid:105) .The optimization problem translates into finding the ground state of the Hamil-tonian, i.e., the eigenstate of the lowest eigenvalue of H . In order to do so,quantum annealing introduces quantum fluctuation in the system, representedas a non-commuting term in the Hamiltonian, H . A typical H easy to preparephysically is H = (cid:80) j σ xj where each σ xj swaps states 0 and 1 on the j -th qubit.The weight of H with respect to H is the strength of the fluctuation. Theinitial state of the system is one with all possible classical configurations thatare equally likely. The system starts with a strong quantum fluctuation thatgradually quenches. The quantum fluctuation provided by H allows the dy-namics to explore a larger region of the search space and gradually concentrate(with large probability) at the global minimum. At the beginning of the search,the initial state is very far from the global minimum but a large fluctuationallows the system a better chance to accept a state that is energetically higher;thus allowing a more extensive search of the solution space. As the annealing24rogresses, the fluctuation is tuned down and the system spends more and moretime around the global minimum, eventually staying there once the fluctuationdisappears. This process resembles simulated annealing where the quantumfluctuation replaces the thermal fluctuations.Another perspective of the same process is to view the total Hamiltonian asslow moving and time dependent. If the Hamiltonian is varying slowly enough,the system will follow its instantaneous eigen-state (this is known as the adi-abatic theorem). Since the initial state is actually the ground state of H , aslow tuning would eventually result in the ground state of the problem Hamil-tonian, H . A key question is: how slow is slow enough? During the evolutionwhen there is another energy level close to the ground state and if the changeof Hamiltonian is not slow enough, there is a risk the system would jump tothe higher level and never return, and the algorithm would fail. The closer thetwo energy levels are, the slower the Hamiltonian must vary in order to mitigatethis risk. The spectral gap (the minimal distance between the two energy levels)plays a crucial role in quantum annealing.Ref. [33] defines four classes of quantum speedups: • Provable quantum speedup: It is rigorously proven that no classical algo-rithm can scale better than a given quantum algorithm. • Strong quantum speedup: The quantum heuristic is faster than any knownclassical algorithm. This type of speedup has been established for dozensof special-purpose algorithms, with Shor’s polynomial-time algorithm forfactorization being the most prominent. The best classical algorithm maybe continually evolving, as is the case for most areas in which classicalheuristics prevail; the ICAPS (International Conference on AutomatedPlanning and Scheduling) planning competition and the SAT competitiongenerally see new algorithms every year. • Potential quantum speedup: The quantum speedup is in comparison to aspecific classical algorithm or a set of classical algorithms.25 Limited quantum speedup: There is a quantum speedup only if the quan-tum heuristic is compared to the closest classical counterpart.A finer-grained classification, which takes into account the type of classicalalgorithm used in the comparison, has been proposed in [34].To better understand where quantum annealing may confer an advantage,it is important to appreciate its major sources of error. The algorithm may failto find a solution due to escape from the ground state via either non-adiabatictransitions or decoherence processes. Yet another possibility is that the groundstate does not correspond to the optimal solution due to control noise. In thefollowing, we review some of the recent developments in assessing the impact ofthese error mechanisms. Some insight into the relative performance of quantum annealing can begained by studying random optimization problems using the tools of the thestatistical mechanics. Absent noise, non-adiabatic transitions can be preventedonly if the annealing proceeds slowly across points where the gap ∆ E that sepa-rates the instantaneous ground state from excited states becomes small (takingat least time t ∝ ¯ h/ ∆ E ). The most widely discussed bottleneck, where thegap reaches a local minimum, is the quantum phase transition. Some of thecomputationally hardest problems exhibit a discontinuous (first order) phasetransition, where the gap is exponentially small. In a common scenario, theground state wavefunction abruptly changes from being a superposition of alarge number of spin configurations to being nearly localized near a global min-imum. If the transverse field is lowered too fast, the algorithm performs nobetter than a random guess.Continuous (second order) phase transitions scale better, although strongfluctuations of disorder (randomness of the parameters of the problem) can stillmake the gap scale as a stretched exponential (exponential in some fractionalpower of problem size). This still leaves a large swath of problems — mostamenable to quantum annealing — where the disorder is irrelevant at the critical26oint (phase transition) so that the gap there is only polynomially small. Recentwork [35] addresses this practically relevant scenario and finds that after thephase transition bottleneck, the algorithm encounters further bottlenecks withgaps that scale as a stretched exponential.As annealing progresses, the number of spin configurations with significantamplitudes decreases until the wavefunction is completely localized. This isroughly equivalent to having a partial assignment of variables: An increasingfraction of binary variables have converged to a definite value, while the re-maining variables are in a superposition state. At times, a state with a differentassignment of already fixed variables becomes more energetically favorable, anda large number of variable have to flip simultaneously in a multi-qubit tunnel-ing, which is the source of ”hard” bottlenecks described above. This process isanalogous to ”backtracking” in classical search algorithms.The major finding is that the number of tunneling bottlenecks is proportionalto the logarithm of problem size. In practice, as the problem size increases, thetime complexity of quantum annealing will exhibit a crossover from polynomialscaling (when the phase transition bottleneck is dominant) to exponential (whenthe expected number of ”hard” bottlenecks exceeds one). This size thresholdis related to the ”density” of spin glass bottlenecks. Similar concept can beintroduced for other heurstic search algorithms, such as simulated annealing.The bottleneck density can thus be used as a metric of performance indicatingproblem sizes above which the time complexity increases exponentially.Interestingly, the minimum requirement for the annealing time is to avoidnon-adiabatic transition at the phase transition (polynomial scaling). As it turnsout, for fully coherent annealing, having one long annealing cycle versus choos-ing the best out of repeated short cycles results in identical time-to-solution (aslong as annealing time exceeds the aforementioned minimum). Shorter anneal-ing times minimize the effects of decoherence and have been favored in mostexperimental studies on the D-Wave hardware.Coupling to the environment affects these results in multiple ways. First, itchanges the universality class of the phase transition, worsening scaling of the27inimum annealing cycle [36]. Second, it suppresses multi-qubit tunneling sincein addition to flipping qubits, corresponding environmental degrees of freedomhave to adjust. If quantum-mechanical tunneling is strongly suppressed, equi-librium may be reached via thermal excitation due to finite temperature. In thisregime, performance would paradoxically improve with increasing temperatureas the system becomes more classical. Multi-qubit quantum co-tunneling is expected to be a key microscopic mech-anism responsible for quantum speedup in quantum annealing. In the following,we consider limited speedup; i.e., speedup compared to simulated annealing.Realistic hardware is subject to intrinsic noise that affects the quantum dy-namics of the system, and therefore needs to be considered when evaluatingthe efficiency of quantum annealing hardware. The effect of hardware noise istwofold: (1) Coupling to noise allows inelastic processes, prohibited by energyconservation in the closed system. Inelastic relaxation provides an efficient routeto a local minimum within a convex region of the potential energy landscape.(2) Dephasing noise leads to loss of coherence between the states on differentsides of the barrier, resulting in an incoherent tunneling regime, and, in thestrong coupling regime, causes renormalization of the tunneling rate.In the case of the flux qubits of the D-Wave system, the typical decoherencetime (a measure of how long quantum features of a single qubit can be main-tained, specifically the characteristic decay time of the off-diagonal elements ofthe qubits density matrix) is of the order of nanoseconds to tens of nanoseconds,which is shorter than the minimum run time of the annealing schedule, 5 mi-croseconds. Nevertheless, D-wave annealers demonstrate signatures of quantummany-body dynamics, particularly incoherent multi-qubit quantum tunnelingand evidence of 8-qubit tunneling has been reported [37]. In the course of quan-tum annealing, the dynamics of the device is limited to low-energy multi-qubitsuperposition states, which are more robust against the effects of noise and deco-herence than single qubit states. In this regime, single qubit excitations caused28y noise local to each qubit are strongly suppressed by the strong qubit-qubitcoupling energy. At the same time, slow fluctuations of local magnetic flux re-sult in a time-dependent spectrum of the multi-qubit low-energy states, whichintroduces decoherence of the multi-qubit dynamics.In the vicinity of the algorithm’s bottlenecks, quantum annealing hardwarerealizes incoherent tunneling [37]. Different tunneling regimes are determinedby comparing the quantum tunneling rate near the computational bottleneckto the characteristic dephasing rate. In a common regime, the tunneling ratenear the bottleneck is exponentially small, while the dephasing rate is at leastof order one. In this regime, quantum tunneling can be only incoherent innature [38]: an analog of the decay of a metastable state into a continuousspectrum encountered in nuclear physics and chemistry, as opposed to a coherentsuperposition of states on two sides of a potential barrier. The incoherent regimeis characterized by a quadratic slowdown of quantum tunneling. Nevertheless,there exist classes of problems where limited polynomial speedup is possible inthis regime, particularly in cases where the shape of the potential barrier favorsquantum tunneling over classical over-the-barrier escape, such as when barriersare tall and thin [39].An alternative [40], operational also in the case of thick barriers where theusual intuition would favor classical escape, is the class of problems character-ized by exponential degeneracy of the metastable state separated by a barrierfrom the global minima. The latter is typical for NP-hard problems; a commonfeature of classical mean-field spin glass models [41] is a polynomial numberof the global minima separated by large potential barriers from an exponentialnumber of metastable states. In such a landscape, simulated annealing slowsdown exponentially due to an additional entropic barrier associated with escap-ing the exponentially degenerate set of metastable states. In contrast, in thecourse of quantum annealing, the transverse field splits the degeneracy of theclassical problem and thereby avoids the additional entropic barrier.To better understand multi-qubit tunneling processes, we developed an in-stantonic calculus for analytical treatment of the thermally-assisted tunneling29ecay rate of metastable states in fully-connected quantum spin models [42, 43].The tunneling decay problem can be mapped onto the Kramers escape problemof a classical random dynamical field. This dynamical field is simulated effi-ciently by path integral Quantum Monte Carlo (QMC). We show analyticallythat the exponential scaling with the number of spins of the thermally-assistedquantum tunneling rate and the escape rate of the QMC process are iden-tical [44]. This analytical result complements prior numerical work [45] andprovides an explanatory model. This effect is due to the existence of a dom-inant instantonic tunneling path. We solve exactly the nonlinear dynamicalmean-field theory equations for a single-site magnetization vector that describethis instanton trajectory. We also derive scaling relations for the “spiky” bar-rier shape when the spin tunneling and QMC rates scale polynomially with thenumber of spins while a classical over-the-barrier activation rate scales expo-nentially. Intrinsic noise cannot be eliminated from real quantum devices: manufac-turing imperfections, as well as thermal fluctuations, induce quantum dephasingand decoherence (see Section 6). Noise can sometimes be helpful (thermal fluc-tuations are responsible for the thermally-assisted annealing effects discussedearlier), but can cause quantum devices to work far from their ideal state, lim-iting the actual performance and hiding any potential quantum speedup.In addition, control noise can change the target Hamiltonian H with theconsequence that the target solution is no longer in the ground subspace of H . In this case, even a perfect quantum device, subject only to control noise,would find a “false” ground state, which could be far from any target solution.The maximum noise that can be added to H before the target solutions donot belong to the ground subspace of H is called resilience [46, 47]. In general,resilience can be increased by properly rescaling the energy of H . Real quantumdevices, however, have a limited range of energies so the resilience cannot becompletely neglected. Recent work shows that a low resilience could hide a30uantum speedup [46]. 6. Quantum annealing hardware To date, the most significant progress in quantum annealing hardware isbased on the engineering of quantum superconducting circuits with macroscopiccollective variables (e.g., electric charge and magnetic flux) exhibiting quantumcoherence. Here we review basic design and operational principles of such cir-cuits, focusing on different types of superconducting qubits, inter-qubit coupling,and decoherence processes caused by various sources of the environmental noise. Let us briefly describe quantization of zero-resistance superconducting cir-cuits, which is based on the lumped element method [48, 49, 50]. We canrepresent a circuit using two alternative sets of variables: current and voltage( I ( t ) and V ( t )) or charge and flux ( Q ( t ) and Φ( t )), connected with each othervia the relations I = dQ/dt and V = d Φ /dt . Let us start with the simplestcircuit such as an LC oscillator (see Fig. 6(a)), whose dynamics is governedby the Kirchhoff’s laws I L = I C ≡ I and V L + V C = 0. Using V L = LdI/dt and V C = Q C /C , one obtains the equation of motion ¨ I + ω LC I = 0, where ω LC = 1 / √ LC is the characteristic frequency for classical current (and voltage)oscillations. The magnetic flux Φ and charge Q are governed by similar equa-tions, e.g., ¨Φ + ω LC Φ = 0. Using variables ( Q, Φ) one can express the equationsof motion in the Hamiltonian form, ˙Φ = ∂H/∂Q and ˙ Q = − ∂H/∂ Φ, where theclassical Hamiltonian function is H = Q / C + Φ / L . Following the standardquantization procedure, we replace classical variables with corresponding oper-ators, introduce the commutator (cid:104) ˆΦ , ˆ Q (cid:105) = i ¯ h , and arrive at the Hamiltonian ofa quantum harmonic oscillator, ˆ H = ˆ Q / C + ˆΦ / L , describing the quantizedelectromagnetic modes of a macroscopic LC circuit with equidistant energy lev-els, E n = ¯ hω LC ( n + 1 / 2) with n = 1 , . . . . Clearly, this energy spectrum is notsuitable for an implementation of a two-level qubit.31 w (a) (b) (c) I V C V L Φ ext Φ ext - Figure 6: (a) Lumped element model for LC oscillator with current I and voltages V C = − V J .(b) Tunable SQUID loop biased by external flux Φ ext . (c) Effective circuit of a qubit. In order to separate two well-defined levels that can be used as logical states | (cid:105) and | (cid:105) , one should employ a non-harmonic circuit with almost negligiblecoupling of the qubit levels and the rest of the spectrum. A natural solutionis to introduce a Josephson junction as a nonlinear and non-dissipative elementof the circuit. Josephson junctions are formed by two superconductors weaklyconnected through a high barrier. Within the lumped element approach, theyare described by the current-voltage characteristics I J = I sin(2 π Φ / Φ ) where I is a critical current and Φ = π ¯ h/e is the flux quantum. Analysis of differentrealizations of a qubit is based on the Kirchhoff’s laws and on the descriptionof the junction’s contributions in terms of I J and V J (or Φ).A tunable qubit is realized if one replaces a single Josephson junction by aSQUID loop formed by two parallel junctions biased by an external flux, Φ ext (see Fig. 6(b)). The current passing through the SQUID is given by I J = I cos(2 π Φ ext / Φ ) sin(2 π Φ / Φ ) [51], which can be thought of as an effectivejunction with tunable critical current I eff = I cos(2 π Φ ext / Φ ) controlled byΦ ext . A typical tunable qubit can be represented as an effective junction shuntedby a linear circuit with admittance Y ω (see Fig. 6(c)). Below we consider twobasic types of such qubits shunted by either LC oscillator (flux qubit) or acapacitor (charge qubit). Effective Josephson junctions, inductance and capacitance, connected in par-allel, form a flux qubit (see Fig. 7(a)). The circuit is governed by the Kirchhoff’s32 b) I J V C I C V L I L V J I t V C V J (a) V f πΦ / Φ Figure 7: Effective circuits and potential energies vs. flux for: (a) Flux qubit (tunable Joseph-son junction shunted by LC oscillator); and (b) Junction shunted by capacitor only (chargequbit). laws for currents I J,L,C and voltages V J,L,C : I J = I C + I L , V J + V C = 0, and V J + V L = 0. Using these relations, we obtain the equation of motion for a fluxΦ threading through the device as: C ¨Φ + Φ /L + I eff sin (cid:16) π ΦΦ (cid:17) = 0, which leadsto the following Hamiltonian of a flux qubitˆ H = ˆ Q C + ( ˆΦ − Φ x ) L − Φ I eff π cos 2 π ˆΦΦ . (10)Here we assumed that the inductance loop L can be biased by an additionalexternal flux Φ x applied through inductive coupling. The first (capacitance)term ˆ Q / C in Eq. (10) can be interpreted as a kinetic energy while the secondand third terms describe a potential formed by inductance and Josephson terms,respectively.For further consideration, it is convenient to introduce dimensionless fluxˆ φ = 2 π ˆΦ / Φ + π and charge operators ˆ q = − id/d ˆ φ . Then the Hamiltonian inEq. (10) can be expressed asˆ H = 4 E C ˆ q + E L ( ˆ φ − φ x ) E eff J cos ˆ φ, (11)and it is different from the LC oscillator by adding the effective energy of Joseph-son junction, E eff J = Φ I eff / π . We also introduce here the capacitance and in-ductance energies, E C = e / C and E L = (Φ / π ) /L and φ x = 2 π Φ x / Φ + π .The Hamiltonian in Eq. (11) corresponds to a particle with kinetic energy pro-portional to E C and potential energy determined by the interplay between E L E eff J through the ratio β = E eff J /E L = 2 πI eff L/ Φ . If β < 1, Eq. (11)describes a single-well anharmonic oscillator, while for β > E eff J , and by tilting thetwo-well potential via the tilt flux φ x . Flux qubits described by Eq. (11) areimplemented in D-Wave quantum annealers [51].A typical charge qubit operates as an open circuit shown in Fig. 7(b). Toderive the Hamiltonian we must omit the inductance term in the equations ofmotion, which results in ˆ H = 4 E C ˆ q + E eff J cos φ, (12)and contains only the Josephson (periodic) part of the potential energy. Theeigenvalue problem is reduced to the Mathieu equation. Operational regimesof various qubits described by the generic Hamiltonian in Eq. (12) drasticallydepend on the ratio E J /E C .Several types of qubits have been realized during the last two decades. Thesimplest charge qubit, comprised of a voltage source in series with a Josephsonjunction ( the Cooper pair box ), had been implemented in [52]. Because of thelarge charging energy, E J /E C (cid:28) 1, the two charge states different by a singleCooper pair are the working states of this qubit. Unfortunately, the Cooper pairbox is highly sensitive to the charge noise. To overcome this difficulty, anotherqubit called the transmon was developed [53]. The transmon is derived fromthe Cooper pair box, but it operates in a different regime of E eff J /E C (cid:29) 1. Itbenefits from the fact that its charge dispersion and noise sensitivity decreasesexponentially with E J /E C . Tuning E eff J controls the amplitude of the potential,which forms a periodic array of minima and maxima shown as red and blueregions of a contour plot in Fig. 7(b). Since E J /E C (cid:29) 1, tunneling betweendifferent minima is greatly suppressed and the qubit is realized at an arbitraryminimum where the lower states are unevenly spaced due to the nonparabolicityof the cosine potential. Therefore, one can manipulate the lowest pair of levels34 passive ”— they seek to maintain coherence onlylong enough to entangle quantum bits or demon-strate some rudimentary capability before, inev-itably, decoherence sets in. The next stages ofQIP require one to realize an actual increase inthe coherence time via error correction, first onlyduring an idle “ memory ” state, but later also inthe midst of a functioning algorithm. This requiresbuilding new systems that are “ active, ” usingcontinuous measurements and real-time feedbackto preserve the quantum information through thestartling process of correcting qubit errors with-out actually learning what the computer is calcu-lating. Given the fragility of quantum information, itis commonly believed that the continual task oferror correction will occupy the vast majority of theeffort and the resources in any large quantumcomputer.Using the current approaches to error correc-tion, the next stages of development unfortunate-ly demand a substantial increase in complexity,requiring dozens or even thousands of physicalqubits per bit of usable quantum information, andchallenging our currently limited abilities to de-sign, fabricate, and control a complex Hamiltonian(second part of Table 1). Furthermore, all of theDiVincenzo engineering margins on each pieceof additional hardware still need to be maintainedor improved while scaling up. So is advancing tothe next stage just a straightforward engineeringexercise of mass-producing large numbers of ex- actly the same kinds of circuits and qubits thathave already been demonstrated? And will thismean the end of the scientific innovations thathave so far driven progress forward?We argue that the answers to both questionswill probably be “ No. ” The work by the com-munity during the past decade and a half, leadingup to the capabilities summarized in the first partof Table 1, may indeed constitute an existenceproof that building a large-scale quantum com-puter is not physically impossible. However, iden-tifying the best, most efficient, and most robustpath forward in a technology ’ s development is atask very different from merely satisfying oneselfthat it should be possible. So far, we have yet tosee a dramatic “ Moore ’ s law ” growth in the com-plexity of quantum hardware. What, then, are themain challenges to be overcome?Simply fabricating a wafer with a large num-ber of elements used today is probably not thehard part. After all, some of the biggest advan-tages of superconducting qubits are that they aremerely circuit elements, which are fabricated inclean rooms, interact with each other via con-nections that are wired up by their designer, andare controlled and measured from the outsidewith electronic signals. The current fabricationrequirements for superconducting qubits are notparticularly daunting, especially in comparison tomodern semiconductor integrated circuits (ICs).A typical qubit or resonant cavity is a few milli- meters in overall size, with features that aremostly a few micrometers (even the smallestJosephson junction sizes are typically 0.2 m m ona side in a qubit). There is successful experiencewith fabricating and operating superconductingICs with hundreds to thousands of elements ona chip, such as the transition-edge sensors withSQUID (superconducting quantum interferencedevice) readout amplifiers, each containing sev-eral Josephson junctions ( ), or microwave ki-netic inductance detectors composed of arrays ofhigh- Q (>10 ) linear resonators without Josephsonjunctions, which are being developed ( ) and usedto great benefit in the astrophysics community.Nonetheless, designing, building, and operat-ing a superconducting quantum computer presentssubstantial and distinct challenges relative to semi-conductor ICs or the other existing versions ofsuperconducting electronics. Conventional micro-processors use overdamped logic, which providesa sort of built-in error correction. They do notrequire high- Q resonances, and clocks or narrow-band filters are in fact off-chip and provided byspecial elements such as quartz crystals. There-fore, small interactions between circuit elementsmay cause heating or offsets but do not lead toactual bit errors or circuit failures. In contrast, anintegrated quantum computer will be essentiallya very large collection of very high- Q , phase-stableoscillators, which need to interact only in the wayswe program. It is no surprise that the leadingquantum information technology has been andtoday remains the trapped ions, which are thebest clocks ever built. In contrast with the ions,however, the artificially made qubits of a super-conducting quantum computer will never be per-fectly identical (see Table 1). Because operationson the qubits need to be controlled accurately toseveral significant digits, the properties of eachpart of the computer would first need to be char-acterized with some precision, have control sig-nals tailored to match, and remain stable whilethe rest of the system is tuned up and then op-erated. The need for high absolute accuracy mighttherefore be circumvented if we can obtain a veryhigh stability of qubit parameters (Table 1); recentresults ( ) are encouraging and exceed expecta-tions, but more information is needed. The powerof electronic control circuitry to tailor waveforms,such as composite pulse sequence techniques wellknown from nuclear magnetic resonance ( ), canremove first-order sensitivity to variations in qubitparameters or in control signals, at the expense ofsome increase in gate time and a requirement fora concomitant increase in coherence time.Even if the problem of stability is solved,unwanted interactions or cross-talk between theparts of these complex circuits will still causeproblems. In the future, we must know and controlthe Hamiltonian to several digits, and for manyqubits. This is beyond the current capability (~1 to10%; see Table 1). Moreover, the number ofmeasurements and the amount of data required C LL J e g E Φ Φ . . . . L J /L Phase qubitHybrid qubitFlux qubitFluxoniumQuantronium Cooper pair boxTransmon A f B C h C e L J Fig. 2. ( A ) Superconducting qubits consist of simple circuits that can be described as the parallel com-bination of a Josephson tunnel element (cross) with inductance L J , a capacitance C , and an inductance L . Theflux F threads the loop formed by both inductances. ( B ) Their quantum energy levels can be sharp and long-lived if the circuit is sufficiently decoupled from its environment. The shape of the potential seen by the flux F and the resulting level structure can be varied by changing the values of the electrical elements. This exampleshows the fluxonium parameters, with an imposed external flux of ¼ flux quantum. Only two of threecorrugations are shown fully. ( C ) A Mendeleev-like but continuous “ table ” of artificial atom types: Cooper pairbox ( ), flux qubit ( ), phase qubit ( ), quantronium ( ), transmon ( ), fluxonium ( ), and hybridqubit ( ). The horizontal and vertical coordinates correspond to fabrication parameters that determine theinverse of the number of corrugations in the potential and the number of levels per well, respectively. SCIENCE VOL 339 8 MARCH 2013 SPECIAL SECTION on M a r c h , . sc i en c e m ag . o r g D o w n l oaded f r o m E J /E C E L /E J FIG. 2 (color online). Different types of superconducting qubits. The basic types are the first three ones from above. The bottom three canbe thought of as improved versions, where additional components have been added. Xiang et al. : Hybrid quantum circuits: Superconducting ... Rev. Mod. Phys., Vol. 85, No. 2, April–June 2013 (a) (b) Figure 8: (a) Summary of the basic types of superconducting qubits [54]. (b) Ratios of energies E L /E J and E J /E C for different types of qubits (Mendeleev-like table) [55]. as in the case of a flux qubit. In Fig. 8, we present basic types of qubits [54]and show typical ratios E L /E J and E J /E C for these qubits (“Mendeleev-liketable”) [55]. Selection criteria among various qubits for particular applicationsare determined not only by internal device parameters but also by their couplingproperties and tolerance to the environmental noise. Controllable couplings between qubits is at the heart of any quantum com-puting application. The simplest and most commonly used couplers are basedon linear superconducting circuits; e.g., mutual inductances or capacitances,as shown in Figs. 9(a) or 9(b). A typical multi-qubit system is described byan anisotropic Heisenberg Hamiltonian: ˆ H = (cid:80) i,α B iα ˆ σ iα + (cid:80) i,j,α ( i (cid:54) = j ) J αij ˆ σ iα ˆ σ jα ,where ˆ σ iα are pseudo-spin Pauli matrices in a qubit 2 × B iα arethe components of local fields, and J αij are exchange coupling parameters. Mech-anism of inductive coupling between flux qubits i and j via mutual inductance M ij = M ji (Fig. 9(a)) is straightforward: if M ij (cid:54) = 0, the external flux fromqubit i threads through qubit j loop (or vice versa) and affects the energy lev-35 his system can be described by the Hamiltonian in Eq. (4),and the strength of the coupling between the charge qubit andthe SC resonator can in principle reach the ultrastrong-coupling regime (Devoret, Girvin, and Schoelkopf, 2007).A similar structure (see Fig. 4) and mechanism are also usedfor the electric coupling of phase qubits with SC resonators(Sillanpa¨a¨, Park, and Simmonds, 2007; Hofheinz et al. , 2008,2009), where the phase qubits are placed on the two sides ofthe transmission line and coupled to it via capacitors, sittingon two antinodes of the electric field. The photon in the CPWresonator acts as a quantum bus that transfers quantum statesbetween the two phase qubits.Flux qubits can also couple to CPW resonators via theinduced magnetic field (Yang, Chu, and Han, 2003, 2004;Niemczyk et al. , 2010; Peropadre et al. , 2010), as shown inFig. 4. A flux qubit placed at or near an antinode of thestanding wave of the current on the SC wire can stronglycouple to the SC resonator via the mutual inductance. In such a SC circuit, the vacuum Rabi splitting in the trans-mission spectrum was observed, which means that strongcoupling was achieved. Furthermore, by placing an addi-tional Josephson junction at the central SC wire, where theflux qubit is fabricated, the inductive coupling between thequbit and the resonator can be enhanced and can bringthe system to the ultrastrong-coupling regime (Niemczyk et al. , 2010).The other type of resonators, LC resonators, can also beintegrated into SC circuits and can couple to charge and phasequbits via capacitors (electric field) or flux qubits via themutual inductance (magnetic field); see Fig. 4. For example,in flux qubits, the lowest two quantum states, which haveclockwise and anticlockwise supercurrents in the qubit loop,are used to denote the two basis states of the qubit. Byemploying the magnetic field produced by the current, theflux qubit can strongly couple to the LC circuit via a largemutual inductance between them. Such flux qubit-resonator ChargequbitFluxqubitPhasequbit LC oscillator Coplanar waveguide resonator (a)(b)(c) Electric fieldMagnetic field FIG. 4 (color online). Schematic diagrams of LC resonators (second column) and coplanar waveguide resonators (third column) coupled tothree types of superconducting qubits. Xiang et al. : Hybrid quantum circuits: Superconducting ... Rev. Mod. Phys., Vol. 85, No. 2, April–June 2013 M c C c (a) (d) (c) (b) (e) H = ! i =12 " Q i C i − E Ji cos ! i $ % + ! n U n ! n − ! nx $ $ where C i and E Ji = I ci " / represent the capacitance andJosephson energy of junction i , respectively, and & " ! i / , Q j ’ = i $% ij . The inductive terms originate from thetwo closed loops with n ! ( co,act ) , L act * L + L co / 4, and U n * " / $ / L n . The actuator and control loop phases aredefined as ! act * ! + ! $ / ! co * ! − ! , respectively.Hamiltonian $ can be reduced to an effective one-dimensional system if L act & L co because the plasma energyof the control loop will then be much higher than that of theactuator loop. Setting ! co = ! co x and combining the Josephsonterms, H + Q act2 C p + V ! act $ V ! act $ = U act , ! act − ! act x $ − ’ eff cos ! act − ! act0 $ - ’ eff * L act I c + " cos . ! co x / " I c − I c + tan . ! co x / % ! act0 * − arctan " I c − I c + tan ! co x % , $ where I c ( * I c ( I c and C p = C + C . Hamiltonian $ is ho-mologous to that of an rf-SQUID whose single junction pos-sesses a critical current that is a function of ! co x and whosephase has been shifted by ! act0 .Let the device described by Eq. $ be connected to twoqubits via mutual inductances M co,1 and M co,2 . The mutualinductance between the qubits will be M eff = M co,1 M co,2 ) $ , $ where ) $ * ! I act p / ! " act x represents the first-order linear $ sus-ceptibility of the coupler and the persistent current flowingabout the coupler actuator loop is I act p * ’ eff L act / " sin ! act − ! act0 $ . $ If V ! act $ is monostable and the first excited state can beneglected, then one can replace the operator ! act by the valuefor which V is a minimum dV / d ! act =0 $ ! act − ! act x + ’ eff sin ! act − ! act0 $ = 0, $ which can be solved for ! act given arbitrary ! act x , thus yield-ing I act p " act x , " co x $ . Differentiating Eqs. $ and $ with re-spect to " act x then yields ) $ ) $ * ! I act p ! " act x = 1 L act ’ eff cos ! act − ! act0 $ ’ eff cos ! act − ! act0 $ . $ Equation $ is similar to Eq. $ of Ref. 8, albeit ’ eff is afunction of ! co x and junction asymmetry results in a ! co x -dependent phase shift in the cosine terms.While rf-SQUID and CJJ rf-SQUID couplers possess similar expressions for ) $ , the latter holds two advantages:first, the CJJ coupler can be operated with " act x =0 and tunedvia " co x . If I c − / I c + * 1, then ! act0 * $ yields ! act + 0. Equation $ then predicts that I act p + 0. Thus the CJJcoupler need not invoke large persistent currents on the or-der of I c + $ when being tuned. Second, the CJJ coupler isusable over the range of " co x for which − min & ’ eff $’ +’ eff ! co x $ ,’ eff $ when " act x =0, where the lower boundhas been imposed by the condition that V ! act $ bemonostable. Thus the utility of the CJJ coupler is not com-promised if ’ eff $ - 1. As such, this device is robust againstfabrication variations.To test the CJJ rf-SQUID coupler, we fabricated a circuitcontaining 8 CJJ rf-SQUID flux qubits, each inductivelycoupled to its own hysteric dc-SQUID readout, and con-nected by a network of 16 CJJ rf-SQUID couplers. The chipwas fabricated from an oxidized Si wafer withNb / Al / Al O / Nb trilayer junctions, four Nb wiring layerscapped with SiN and separated by planarized plasma-enhanced chemical-vapor deposition SiO . The chip wasmounted to the mixing chamber of a dilution refrigeratorregulated at T =40 mK inside a Sn superconducting mag-netic shield with a residual field in the vicinity of the chip + f c + M co and M act , respectively. These give rise to the fluxes " co x and " act x .The qubits are controlled via fluxes " cjj . x and " q . x . =1,2 $ asdescribed in Ref. 13. The qubits interact with the coupler viamutual inductances M co, . . For brevity, we present resultsfrom a single coupler in this Brief Report and note that M eff " co x $ was identical to + % for all 16 couplers on thischip. For the particular coupler described herein, the relevantqubit critical currents were I q . c =3.25 ( / A and qubitinductances were L q $ =290 $ ( " co x =0.The flux wave forms used to obtain M act are depicted inFig. 3 a $ . In this case, " co x was held constant while the de-tector qubit . = d $ was annealed in the presence of a pulseon " act x t $ of amplitude " act i and a pulse on " q . x t $ of ampli-tude " qd . The sequence involves i $ initializing the qubit in amonostable potential with no net flux biases, ii $ setting " act x and " q . x , iii $ raising the detector qubit’s tunnel barrier to Qubit 1 M co Qubit 2Coupler F act F act x M act M co,1 M co,2 F co x F q F q x F q F q x F cjj1 x F cjj2 x FIG. 2. Color online $ Schematic of a CJJ rf-SQUID couplerinteracting with two CJJ rf-SQUID qubits.BRIEF REPORTS PHYSICAL REVIEW B , 052506 $ Figure 9: Effective circuits for different regimes of interqubit coupling: (a) between flux qubitsvia mutual inductance M c ≡ M , (b) through inductive loop controlled by SQUID, [56] (c)between transmons coupled via capacitance C c , and (d) tunable coupling between transmonscontrolled by Josephson junction with nonlinear inductance L c . [57] (e) Schematic of thecoplanar waveguide resonator (light blue), the transmon qubits and the first harmonic of thestanding wave electric field shown in red. [58, 54] els. Thus, the longitudinal coupling (proportional to ˆ σ z ˆ σ z ) can be expressedas J zij ∼ M ij I i I j . The direct inductive coupling is not tunable; however, atunable coupling strength can be realized if the inductance loop is driven by aSQUID. An example of such coupling, utilized in D-Wave quantum annealers,is schematically shown Fig. 9(b) [56]. It is based on bias currents that producecontrolled flux biases.A circuit diagram of two capacitively-coupled transmons is shown in Fig. 9(c),and can be analyzed using the lumped element method as above. As a result, theinteraction Hamiltonian for a pair of transmons can be expressed as ˆ q i ˆ q j C/C c .Calculating matrix elements of ˆ q i,j within the two-level approximation, we ob-tain the transverse coupling (proportional to ˆ σ ix ˆ σ jx ) with the coupling param-eter J xij ∼ (cid:112) ∆ E i ∆ E j ( C/C c ), where ∆ E i is level splitting of i -th transmon.The purely capacitive coupling is not tunable, but the coupling strength canbe controlled using a non-linear coupler with Josephson junction (a tunable in-ductor). This circuit is depicted in Fig. 9(d), where arrows indicate the flowof current for an excitation in the left qubit [57]. It is important that the cou-36ling be tunable with nanosecond resolution, making this circuit suitable forvarious applications ranging from quantum logic gates to quantum simulations.Similar circuits are employed for readout of a flux qubit state in D-Wave quan-tum annealers, where each qubit is connected inductively with a quantum fluxparametron ( rf -SQUID with a small inductance, a large capacitance and a verylarge critical current) [51, 59]. Another approach is to couple all qubits to ashared passive element (quantum bus) such as a cavity or a coplanar waveguideresonator (CPW) [54]. Superconducting qubits are macroscopic quantum objects whose genericquantum properties, such as superposition of states and entanglement, inher-ently suffer from detrimental effects caused by a macroscopic, noisy environ-ment [60]. To describe environmental noise phenomenologically, one shouldtake into account random charge, flux, and Josephson junction noise sourcesthat modulate lumped elements of the equivalent circuit in the qubit Hamilto-nians in Eqs. (11) or (12).After tracing over the environmental variables, the qubit dynamics is gov-erned by the Bloch equation with two transition rates Γ and Γ (or times T and T ) describing qubit relaxation and decoherence , respectively. The tworates are related: Γ = Γ / d , where Γ d describes dephasing due to the lowfrequency noise. The flux qubits (e.g., D-Wave qubits) studied to date sufferfrom a low-frequency flux noise due to environmental spins [61, 62]. This leadsto a substantial dephasing rate Γ d and, in turn, to a large difference betweenthe relaxation and decoherence rates, Γ ∼ Γ d (cid:29) Γ . In transmon qubits, theflux noise is absent and the low-frequency charge noise is suppressed; i.e., thedecoherence rate is low and Γ and Γ are close to each other.A particular choice of a qubit depends on its suitability for a given appli-cation. For instance, quantum annealing requires strong coupling between thequbits. Therefore, in this case the flux qubit is a preferred choice because atypical value of the coupling parameter for D-Wave flux qubits is several GHz.37n the other hand, the coupling between transmon qubits is much weaker (onthe order of 10 MHz). Thus, the coupling and connectivity requirements of thequantum annealing outweigh the disadvantages caused by the higher decoher-ence rate of the flux qubits. 7. Conclusions The emergence of quantum annealers in the past few years has enabled theexplorations described in this paper. The next few years promise to be yet moreexciting as more sophisticated quantum annealers become available and one seesthe advent of the first universal quantum computers able to run other quantumheuristic algorithms. The NASA QuAIL team is excited to be at the forefrontof these developments, and looks forward to working with quantum hardwareand algorithms teams from around the world to explore quantum heuristics andthereby broaden the areas in which quantum computation has clear applications. 8. Acknowledgements The authors would like to acknowledge support from the NASA AdvancedExploration Systems (AES) program and NASA Ames Research Center. Thiswork was supported in part by the AFRL Information Directorate under grantF4HBKC4162G001, the Office of the Director of National Intelligence (ODNI),and the Intelligence Advanced Research Projects Activity (IARPA), via IAA145483. The views and conclusions contained herein are those of the authorsand should not be interpreted as necessarily representing the official policies orendorsements, either expressed or implied, of ODNI, IARPA, AFRL, or the U.S.Government. The U.S. Government is authorized to reproduce and distributereprints for Governmental purpose notwithstanding any copyright annotationthereon. 38 eferencesReferenceseferencesReferences