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- NASA/TP-2019-220319
- Quantum Supremacy Using a Programmable
- Superconducting Processor
- Eleanor G. Rieffel
- NASA Ames Research Center
- August 2019
- https://ntrs.nasa.gov/search.jsp?R=20190030475 2019-09-13T07:55:30+00:00Z
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- NASA/TP-2019-220319
- Quantum Supremacy Using a Programmable
- Superconducting Processor
- Eleanor G. Rieffel
- NASA Ames Research Center
- National Aeronautics and
- Space Administration
- Ames Research Center
- Moffett Field, California
- August 2019
- Acknowledgments
- This report is available in electronic form at
- http://www.sti.nasa.gov
- or http://ntrs.nasa.gov
- Quantum supremacy using a programmable superconducting processor
- Google AI Quantum and collaboratorsy
- The tantalizing promise of quantum computers is that certain computational tasks might be
- executed exponentially faster on a quantum processor than on a classical processor. A fundamen-
- tal challenge is to build a high-delity processor capable of running quantum algorithms in an
- exponentially large computational space. Here, we report using a processor with programmable
- superconducting qubits to create quantum states on 53 qubits, occupying a state space 253 ˘1016.
- Measurements from repeated experiments sample the corresponding probability distribution, which
- we verify using classical simulations. While our processor takes about 200 seconds to sample one
- instance of the quantum circuit 1 million times, a state-of-the-art supercomputer would require
- approximately 10,000 years to perform the equivalent task. This dramatic speedup relative to all
- known classical algorithms provides an experimental realization of quantum supremacy on a com-
- putational task and heralds the advent of a much-anticipated computing paradigm.
- In the early 1980s, Richard Feynman proposed that a
- quantum computer would be an eective tool to solve
- problems in physics and chemistry, as it is exponentially
- costly to simulate large quantum systems with classical
- computers [1]. Realizing Feynman’s vision poses signi-
- cant experimental and theoretical challenges. First, can
- a quantum system be engineered to perform a computa-
- tion in a large enough computational (Hilbert) space and
- with low enough errors to provide a quantum speedup?
- Second, can we formulate a problem that is hard for a
- classical computer but easy for a quantum computer? By
- computing a novel benchmark task on our superconduct-
- ing qubit processor[2{7], we tackle both questions. Our
- experiment marks a milestone towards full scale quantum
- computing: quantum supremacy[8].
- In reaching this milestone, we show that quantum
- speedup is achievable in a real-world system and is
- not precluded by any hidden physical laws. Quantum
- supremacy also heralds the era of Noisy Intermediate-
- Scale Quantum (NISQ) technologies. The benchmark
- task we demonstrate has an immediate application in
- generating certiable random numbers[9]; other initial
- uses for this new computational capability may include
- optimization optimization [10{12], machine learning[13{
- 15], materials science and chemistry [16{18]. However,
- realizing the full promise of quantum computing (e.g.
- Shor’s algorithm for factoring) still requires technical
- leaps to engineer fault-tolerant logical qubits[19{23].
- To achieve quantum supremacy, we made a number of
- technical advances which also pave the way towards er-
- ror correction. We developed fast, high-delity gates that
- can be executed simultaneously across a two-dimensional
- qubit array. We calibrated and benchmarked the pro-
- cessor at both the component and system level using a
- powerful new tool: cross-entropy benchmarking (XEB).
- Finally, we used component-level delities to accurately
- predict the performance of the whole system, further
- showing that quantum information behaves as expected
- when scaling to large systems.
- A COMPUTATIONAL TASK TO
- DEMONSTRATE QUANTUM SUPREMACY
- To demonstrate quantum supremacy, we compare our
- quantum processor against state-of-the-art classical com-
- puters in the task of sampling the output of a pseudo-
- random quantum circuit[24{26]. Random circuits are a
- suitable choice for benchmarking since they do not pos-
- sess structure and therefore allow for limited guarantees
- of computational hardness[24, 25, 27, 28]. We design the
- circuits to entangle a set of quantum bits (qubits) by re-
- peated application of single-qubit and two-qubit logical
- operations. Sampling the quantum circuit’s output pro-
- duces a set of bitstrings, e.g. f0000101, 1011100, ...g.
- Due to quantum interference, the probability distribution
- of the bitstrings resembles a speckled intensity pattern
- produced by light interference in laser scatter, such that
- some bitstrings are much more likely to occur than oth-
- ers. Classically computing this probability distribution
- becomes exponentially more dicult as the number of
- qubits (width) and number of gate cycles (depth) grows.
- We verify that the quantum processor is working prop-
- erly using a method called cross-entropy benchmarking
- (XEB) [24, 26], which compares how often each bitstring
- is observed experimentally with its corresponding ideal
- probability computed via simulation on a classical com-
- puter. For a given circuit, we collect the measured bit-
- strings fx
- igand compute the linear XEB delity [24{
- 26, 29], which is the mean of the simulated probabilities
- of the bitstrings we measured:
- F
- XEB = 2
- nhP(x
- i)i
- i 1 (1)
- where nis the number of qubits, P(x
- i) is the probability
- of bitstring x
- i computed for the ideal quantum circuit,
- and the average is over the observed bitstrings. Intu-
- itively, F
- XEB is correlated with how often we sample high
- probability bitstrings. When there are no errors in the
- quantum circuit, sampling the probability distribution
- will produce F
- XEB = 1. On the other hand, sampling
- from the uniform distribution will give hP(x
- i)i
- i = 1=2n
- and produce F
- XEB = 0. Values of F
- XEB between 0 and
- 2
- Qubit Adjustable coupler
- a
- b
- 10 millimeters
- FIG. 1. The Sycamore processor. a, Layout of processor
- showing a rectangular array of 54 qubits (gray), each con-
- nected to its four nearest neighbors with couplers (blue). In-
- operable qubit is outlined. b, Optical image of the Sycamore
- chip.
- 1 correspond to the probability that no error has oc-
- curred while running the circuit. The probabilities P(x
- i)
- must be obtained from classically simulating the quan-
- tum circuit, and thus computing F
- XEB is intractable in
- the regime of quantum supremacy. However, with certain
- circuit simplications, we can obtain quantitative delity
- estimates of a fully operating processor running wide and
- deep quantum circuits.
- Our goal is to achieve a high enough F
- XEB for a circuit
- with sucient width and depth such that the classical
- computing cost is prohibitively large. This is a dicult
- task because our logic gates are imperfect and the quan-
- tum states we intend to create are sensitive to errors. A
- single bit or phase
- ip over the course of the algorithm
- will completely shue the speckle pattern and result in
- close to 0 delity [24, 29]. Therefore, in order to claim
- quantum supremacy we need a quantum processor that
- executes the program with suciently low error rates.
- BUILDING AND CHARACTERIZING A
- HIGH-FIDELITY PROCESSOR
- We designed a quantum processor named \Sycamore"
- which consists of a two-dimensional array of 54 trans-
- mon qubits, where each qubit is tunably coupled to four
- nearest-neighbors, in a rectangular lattice. The connec-
- tivity was chosen to be forward compatible with error-
- correction using the surface code [20]. A key systems-
- engineering advance of this device is achieving high-
- delity single- and two-qubit operations, not just in iso-
- lation but also while performing a realistic computation
- with simultaneous gate operations on many qubits. We
- discuss the highlights below; extended details can be
- found in the supplementary information.
- In a superconducting circuit, conduction electrons con-
- dense into a macroscopic quantum state, such that cur-
- rents and voltages behave quantum mechanically [2, 30].
- Our processor uses transmon qubits [6], which can be
- thought of as nonlinear superconducting resonators at 5
- to 7 GHz. The qubit is encoded as the two lowest quan-
- tum eigenstates of the resonant circuit. Each transmon
- has two controls: a microwave drive to excite the qubit,
- and a magnetic
- ux control to tune the frequency. Each
- qubit is connected to a linear resonator used to read out
- the qubit state [5]. As shown in Fig. 1, each qubit is
- also connected to its neighboring qubits using a new ad-
- justable coupler [31, 32]. Our coupler design allows us to
- quickly tune the qubit-qubit coupling from completely
- o to 40 MHz. Since one qubit did not function properly
- the device uses 53 qubits and 86 couplers.
- The processor is fabricated using aluminum for metal-
- ization and Josephson junctions, and indium for bump-
- bonds between two silicon wafers. The chip is wire-
- bonded to a superconducting circuit board and cooled
- to below 20 mK in a dilution refrigerator to reduce am-
- bient thermal energy to well below the qubit energy.
- The processor is connected through lters and attenu-
- ators to room-temperature electronics, which synthesize
- the control signals. The state of all qubits can be read
- simultaneously by using a frequency-multiplexing tech-
- nique[33, 34]. We use two stages of cryogenic ampliers
- to boost the signal, which is digitized (8 bits at 1 GS/s)
- and demultiplexed digitally at room temperature. In to-
- tal, we orchestrate 277 digital-to-analog converters (14
- bits at 1 GS/s) for complete control of the quantum pro-
- cessor.
- We execute single-qubit gates by driving 25 ns mi-
- crowave pulses resonant with the qubit frequency while
- the qubit-qubit coupling is turned o. The pulses
- are shaped to minimize transitions to higher transmon
- states[35]. Gate performance varies strongly with fre-
- quency due to two-level-system (TLS) defects[36, 37],
- stray microwave modes, coupling to control lines and
- the readout resonator, residual stray coupling between
- qubits,
- ux noise, and pulse distortions. We therefore
- 3
- Pauli and measurement errors
- CDF am, E ted histogr Integra e
- 1
- e
- 2 e
- 2c e
- r
- a
- b
- Average error
- Single-qubit (e 1
- )
- Two-qubit (e 2
- )
- Two-qubit, cycle (e 2c )
- Readout (e r
- )
- Isolated
- 0.15%
- 0.36%
- 0.65%
- 3.1%
- Simultaneous
- 0.16%
- 0.62%
- 0.93%
- 3.8%
- Simultaneous
- Pauli error
- e
- 1
- , e 2
- 10 -2
- 10 -3
- Isolated
- FIG. 2. System-wide Pauli and measurement errors. a,
- Integrated histogram (empirical cumulative distribution func-
- tion, ECDF) of Pauli errors (black, green, blue) and readout
- errors (orange), measured on qubits in isolation (dotted lines)
- and when operating all qubits simultaneously (solid). The
- median of each distribution occurs at 0.50 on the vertical
- axis. Average (mean) values are shown below. b, Heatmap
- showing single- and two-qubit Pauli errors e
- 1 (crosses) and e
- 2
- (bars) positioned in the layout of the processor. Values shown
- for all qubits operating simultaneously.
- optimize the single-qubit operation frequencies to miti-
- gate these error mechanisms.
- We benchmark single-qubit gate performance by using
- the XEB protocol described above, reduced to the single-
- qubit level (n= 1), to measure the probability of an error
- occurring during a single-qubit gate. On each qubit, we
- apply a variable number mof randomly selected gates
- and measure F
- XEB averaged over many sequences; as m
- increases, errors accumulate and average F
- XEB decays.
- We model this decay by [1 e
- 1=(1 1=D2)]m where e
- 1 is
- the Pauli error probability. The state (Hilbert) space di-
- mension term, D= 2n = 2, corrects for the depolarizing
- model where states with errors partially overlap with the
- ideal state. This procedure is similar to the more typical
- technique of randomized benchmarking [21, 38, 39], but
- supports non-Cliord gatesets [40] and can separate out
- decoherence error from coherent control error. We then
- repeat the experiment with all qubits executing single-
- qubit gates simultaneously (Fig.2), which shows only a
- small increase in the error probabilities, demonstrating
- that our device has low microwave crosstalk.
- We perform two-qubit iSWAP-like entangling gates by
- bringing neighboring qubits on resonance and turning on
- a 20 MHz coupling for 12 ns, which allows the qubits to
- swap excitations. During this time, the qubits also ex-
- perience a controlled-phase (CZ) interaction, which orig-
- inates from the higher levels of the transmon. The two-
- qubit gate frequency trajectories of each pair of qubits are
- optimized to mitigate the same error mechanisms consid-
- ered in optimizing single-qubit operation frequencies.
- To characterize and benchmark the two-qubit gates,
- we run two-qubit circuits with mcycles, where each cy-
- cle contains a randomly chosen single-qubit gate on each
- of the two qubits followed by a xed two-qubit gate. We
- learn the parameters of the two-qubit unitary (e.g. the
- amount of iSWAP and CZ interaction) by using F
- XEB
- as a cost function. After this optimization, we extract
- the per-cycle error e
- 2c from the decay of F
- XEB with m,
- and isolate the two-qubit error e
- 2 by subtracting the two
- single-qubit errors e
- 1. We nd an average e
- 2 of 0:36%.
- Additionally, we repeat the same procedure while simul-
- taneously running two-qubit circuits for the entire array.
- After updating the unitary parameters to account for ef-
- fects such as dispersive shifts and crosstalk, we nd an
- average e
- 2 of 0.62%.
- For the full experiment, we generate quantum circuits
- using the two-qubit unitaries measured for each pair dur-
- ing simultaneous operation, rather than a standard gate
- for all pairs. The typical two-qubit gate is a full iSWAP
- with 1=6 of a full CZ. In principle, our architecture could
- generate unitaries with arbitrary iSWAP and CZ inter-
- actions, but reliably generating a target unitary remains
- an active area of research.
- Finally, we benchmark qubit readout using standard
- dispersive measurement [41]. Measurement errors aver-
- aged over the 0 and 1 states are shown in Fig 2a. We have
- also measured the error when operating all qubits simul-
- taneously, by randomly preparing each qubit in the 0 or
- 1 state and then measuring all qubits for the probability
- of the correct result. We nd that simultaneous readout
- incurs only a modest increase in per-qubit measurement
- errors.
- Having found the error rates of the individual gates and
- readout, we can model the delity of a quantum circuit
- as the product of the probabilities of error-free opera-
- 4
- single-qubit gate:
- 25 ns
- qubit
- XY control
- two-qubit gate:
- 12 ns
- qubit 1
- Z control
- qubit 2
- Z control
- coupler
- cycle: 1 2 3 4 5 6 m
- time
- column
- row
- 7 8
- A BC D
- A
- B
- D
- C
- a b
- FIG. 3. Control operations for the quantum supremacy circuits. a, Example quantum circuit instance used in our
- experiment. Every cycle includes a layer each of single- and two-qubit gates. The single-qubit gates are chosen randomly from
- f
- p
- X;
- p
- Y;
- p
- Wg. The sequence of two-qubit gates are chosen according to a tiling pattern, coupling each qubit sequentially to
- its four nearest-neighbor qubits. The couplers are divided into four subsets (ABCD), each of which is executed simultaneously
- across the entire array corresponding to shaded colors. Here we show an intractable sequence (repeat ABCDCDAB); we also
- use dierent coupler subsets along with a simpliable sequence (repeat EFGHEFGH, not shown) that can be simulated on a
- classical computer. b, Waveform of control signals for single- and two-qubit gates.
- tion of all gates and measurements. Our largest random
- quantum circuits have 53 qubits, 1113 single-qubit gates,
- 430 two-qubit gates, and a measurement on each qubit,
- for which we predict a total delity of 0:2%. This delity
- should be resolvable with a few million measurements,
- since the uncertainty on F
- XEB is 1=
- p
- N
- s, where N
- s is the
- number of samples. Our model assumes that entangling
- larger and larger systems does not introduce additional
- error sources beyond the errors we measure at the single-
- and two-qubit level | in the next section we will see how
- well this hypothesis holds.
- FIDELITY ESTIMATION IN THE SUPREMACY
- REGIME
- The gate sequence for our pseudo-random quantum
- circuit generation is shown in Fig.3. One cycle of the
- algorithm consists of applying single-qubit gates chosen
- randomly from f
- p
- X;
- p
- Y;
- p
- Wgon all qubits, followed
- by two-qubit gates on pairs of qubits. The sequences of
- gates which form the \supremacy circuits" are designed
- to minimize the circuit depth required to create a highly
- entangled state, which ensures computational complexity
- and classical hardness.
- While we cannot compute F
- XEB in the supremacy
- regime, we can estimate it using three variations to re-
- duce the complexity of the circuits. In \patch circuits",
- we remove a slice of two-qubit gates (a small fraction
- of the total number of two-qubit gates), splitting the cir-
- cuit into two spatially isolated, non-interacting patches of
- qubits. We then compute the total delity as the product
- of the patch delities, each of which can be easily calcu-
- lated. In \elided circuits", we remove only a fraction of
- the initial two-qubit gates along the slice, allowing for
- entanglement between patches, which more closely mim-
- ics the full experiment while still maintaining simulation
- feasibility. Finally, we can also run full \verication cir-
- cuits" with the same gate counts as our supremacy cir-
- cuits, but with a dierent pattern for the sequence of two-
- qubit gates which is much easier to simulate classically
- [29]. Comparison between these variations allows track-
- ing of the system delity as we approach the supremacy
- regime.
- We rst check that the patch and elided versions of the
- verication circuits produce the same delity as the full
- verication circuits up to 53 qubits, as shown in Fig.4a.
- For each data point, we typically collect N
- s = 5 106
- total samples over ten circuit instances, where instances
- dier only in the choices of single-qubit gates in each
- cycle. We also show predicted F
- XEB values computed
- by multiplying the no-error probabilities of single- and
- two-qubit gates and measurement [29]. Patch, elided,
- and predicted delities all show good agreement with
- the delities of the corresponding full circuits, despite
- the vast dierences in computational complexity and en-
- tanglement. This gives us condence that elided circuits
- can be used to accurately estimate the delity of more
- complex circuits.
- We proceed now to benchmark our most computa-
- tionally dicult circuits. In Fig.4b, we show the mea-
- sured F
- XEB for 53-qubit patch and elided versions of the
- full supremacy circuits with increasing depth. For the
- largest circuit with 53 qubits and 20 cycles, we collected
- N
- s = 30 106 samples over 10 circuit instances, obtaining
- F
- XEB = (2:24 0:21) 10 3 for the elided circuits. With
- 5˙condence, we assert that the average delity of run-
- ning these circuits on the quantum processor is greater
- than at least 0.1%. The full data for Fig.4b should have
- similar delities, but are only archived since the simula-
- tion times (red numbers) take too long. It is thus in the
- quantum supremacy regime.
- 5
- number of qubits, n number of cycles, m
- n = 53 qubits
- a Classically veriable b Supremacy regime
- idelity, XEB F
- XEB
- m = 14 cycles
- Prediction from gate and measurement errors
- Full circuit Elided circuit Patch circuit
- Prediction
- Patch
- E F G H A B C D C D A B
- Elided (±5 error bars)
- 10 millennia
- 100 years
- 600 years
- 4 years
- 4 years
- 2 weeks
- 1 week
- 2 hour sC la ic mp ng @ Sycamore
- 5 hours
- Classical verication
- Sycamore sampling (N s
- = 1M): 200 seconds
- 10 15 20 25 30 35 40 45 50 55 12 14 16 18 20
- 10 -3
- 10 -2
- 10 -1
- 10 0
- FIG. 4. Demonstrating quantum supremacy. a, Verication of benchmarking methods. F
- XEB values for patch, elided,
- and full verication circuits are calculated from measured bitstrings and the corresponding probabilities predicted by classical
- simulation. Here, the two-qubit gates are applied in a simpliable tiling and sequence such that the full circuits can be simulated
- out to n= 53;m= 14 in a reasonable amount of time. Each data point is an average over 10 distinct quantum circuit instances
- that dier in their single-qubit gates (for n= 39;42;43 only 2 instances were simulated). For each n, each instance is sampled
- with N
- s between 0:5M and 2:5M. The black line shows predicted F
- XEB based on single- and two-qubit gate and measurement
- errors. The close correspondence between all four curves, despite their vast dierences in complexity, justies the use of elided
- circuits to estimate delity in the supremacy regime. b, Estimating F
- XEB in the quantum supremacy regime. Here, the
- two-qubit gates are applied in a non-simpliable tiling and sequence for which it is much harder to simulate. For the largest
- elided data (n= 53, m= 20, total N
- s = 30M), we nd an average F
- XEB >0.1% with 5˙condence, where ˙includes both
- systematic and statistical uncertainties. The corresponding full circuit data, not simulated but archived, is expected to show
- similarly signicant delity. For m= 20, obtaining 1M samples on the quantum processor takes 200 seconds, while an equal
- delity classical sampling would take 10,000 years on 1M cores, and verifying the delity would take millions of years.
- DETERMINING THE CLASSICAL
- COMPUTATIONAL COST
- We simulate the quantum circuits used in the exper-
- iment on classical computers for two purposes: verify-
- ing our quantum processor and benchmarking methods
- by computing F
- XEB where possible using simpliable
- circuits (Fig.4a), and estimating F
- XEB as well as the
- classical cost of sampling our hardest circuits (Fig.4b).
- Up to 43 qubits, we use a Schrodinger algorithm (SA)
- which simulates the evolution of the full quantum state;
- the Julich supercomputer(100k cores, 250TB) runs the
- largest cases. Above this size, there is not enough RAM
- to store the quantum state [42]. For larger qubit num-
- bers, we use a hybrid Schrodinger-Feynman algorithm
- (SFA)[43] running on Google data centers to compute
- the amplitudes of individual bitstrings. This algorithm
- breaks the circuit up into two patches of qubits and e-
- ciently simulates each patch using a Schrodinger method,
- before connecting them using an approach reminiscent of
- the Feynman path-integral. While it is more memory-
- ecient, SFA becomes exponentially more computation-
- ally expensive with increasing circuit depth due to the
- exponential growth of paths with the number of gates
- connecting the patches.
- To estimate the classical computational cost of the
- supremacy circuits (gray numbers, Fig.4b), we ran por-
- tions of the quantum circuit simulation on both the Sum-
- mit supercomputer as well as on Google clusters and ex-
- trapolated to the full cost. In this extrapolation, we
- account for the computational cost scaling with F
- XEB,
- e.g. the 0.1% delity decreases the cost by 1000[43, 44].
- On the Summit supercomputer, which is currently the
- most powerful in the world, we used a method inspired
- by Feynman path-integrals that is most ecient at low
- depth[44{47]. At m= 20 the tensors do not reasonably
- t in node memory, so we can only measure runtimes
- up to m= 14, for which we estimate that sampling 3M
- bitstrings with 1% delity would require 1 year.
- 6
- On Google Cloud servers, we estimate that perform-
- ing the same task for m= 20 with 0:1% delity using
- the SFA algorithm would cost 50 trillion core-hours and
- consume 1 petawatt hour of energy. To put this in per-
- spective, it took 600 seconds to sample the circuit on
- the quantum processor 3 million times, where sampling
- time is limited by control hardware communications; in
- fact, the net quantum processor time is only about 30
- seconds. The bitstring samples from this largest circuit
- are archived online.
- One may wonder to what extent algorithmic innova-
- tion can enhance classical simulations. Our assumption,
- based on insights from complexity theory, is that the cost
- of this algorithmic task is exponential in nas well as m.
- Indeed, simulation methods have improved steadily over
- the past few years[42{50]. We expect that lower simula-
- tion costs than reported here will eventually be achieved,
- but we also expect they will be consistently outpaced by
- hardware improvements on larger quantum processors.
- VERIFYING THE DIGITAL ERROR MODEL
- A key assumption underlying the theory of quantum
- error correction is that quantum state errors may be con-
- sidered digitized and localized [38, 51]. Under such a dig-
- ital model, all errors in the evolving quantum state may
- be characterized by a set of localized Pauli errors (bit-
- and/or phase-
- ips) interspersed into the circuit. Since
- continuous amplitudes are fundamental to quantum me-
- chanics, it needs to be tested whether errors in a quantum
- system could be treated as discrete and probabilistic. In-
- deed, our experimental observations support the validity
- of this model for our processor. Our system delity is
- well predicted by a simple model in which the individ-
- ually characterized delities of each gate are multiplied
- together (Fig 4).
- To be successfully described by a digitized error model,
- a system should be low in correlated errors. We achieve
- this in our experiment by choosing circuits that ran-
- domize and decorrelate errors, by optimizing control to
- minimize systematic errors and leakage, and by design-
- ing gates that operate much faster than correlated noise
- sources, such as 1=f
- ux noise [37]. Demonstrating a pre-
- dictive uncorrelated error model up to a Hilbert space of
- size 253 shows that we can build a system where quantum
- resources, such as entanglement, are not prohibitively
- fragile.
- WHAT DOES THE FUTURE HOLD?
- Quantum processors based on superconducting qubits
- can now perform computations in a Hilbert space of di-
- mension 253 ˇ9 1015, beyond the reach of the fastest
- classical supercomputers available today. To our knowl-
- edge, this experiment marks the rst computation that
- can only be performed on a quantum processor. Quan-
- tum processors have thus reached the regime of quantum
- supremacy. We expect their computational power will
- continue to grow at a double exponential rate: the clas-
- sical cost of simulating a quantum circuit increases expo-
- nentially with computational volume, and hardware im-
- provements will likely follow a quantum-processor equiv-
- alent of Moore’s law [52, 53], doubling this computational
- volume every few years. To sustain the double exponen-
- tial growth rate and to eventually oer the computational
- volume needed to run well-known quantum algorithms,
- such as the Shor or Grover algorithms [19, 54], the engi-
- neering of quantum error correction will have to become
- a focus of attention.
- The \Extended Church-Turing Thesis" formulated by
- Bernstein and Vazirani [55] asserts that any \reasonable"
- model of computation can be eciently simulated by a
- Turing machine. Our experiment suggests that a model
- of computation may now be available that violates this
- assertion. We have performed random quantum circuit
- sampling in polynomial time with a physically realized
- quantum processor (with suciently low error rates), yet
- no ecient method is known to exist for classical comput-
- ing machinery. As a result of these developments, quan-
- tum computing is transitioning from a research topic to a
- technology that unlocks new computational capabilities.
- We are only one creative algorithm away from valuable
- near-term applications.
- Acknowledgments We are grateful to Eric Schmidt,
- Sergey Brin, Je Dean, and Jay Yagnik for their executive
- sponsorship of the Google AI Quantum team, and for their
- continued engagement and support. We thank Peter Norvig
- for reviewing a draft of the manuscript, and Sergey Knysh
- for useful discussions. We thank Kevin Kissel, Joey Raso,
- Davinci Yonge-Mallo, Orion Martin, and Niranjan Sridhar
- for their help with simulations. We thank Gina Bortoli and
- Lily Laws for keeping our team organized. This research used
- resources from the Oak Ridge Leadership Computing Facility,
- which is a DOE Oce of Science User Facility supported un-
- der Contract DE-AC05-00OR22725. A portion of this work
- was performed in the UCSB Nanofabrication Facility, an open
- access laboratory.
- Author contributions The Google AI Quantum team
- conceived of the experiment. The applications and algorithms
- team provided the theoretical foundation and the specics of
- the algorithm. The hardware team carried out the experiment
- and collected the data. The data analysis was done jointly
- with outside collaborators. All authors wrote and revised the
- manuscript and supplement.
- Competing Interests The authors declare that they have
- no competing nancial interests.
- Correspondence and requests for materials should
- be addressed to John M. Martinis (jmartinis@google.com).
- 7
- y Frank Arute1, Kunal Arya1, Ryan Babbush1, Dave
- Bacon 1, Joseph C. Bardin ;2, Rami Barends , Ru-
- pak Biswas3, Sergio Boixo1, Fernando G.S.L. Brandao1;4,
- David Buell 1, Brian Burkett , Yu Chen , Zijun Chen1,
- Ben Chiaro5, Roberto Collins 1, William Courtney , An-
- drew Dunsworth 1, Edward Farhi , Brooks Foxen5, Austin
- Fowler 1, Craig Gidney , Marissa Giustina1, Rob Gra , Keith
- Guerin 1, Steve Habegger , Matthew P. Harrigan , Michael J.
- Hartmann 1;6, Alan Ho , Markus Homann , Trent Huang1,
- Travis S. Humble7, Sergei V. Isakov 1, Evan Jerey , Zhang
- Jiang 1, Dvir Kafri , Kostyantyn Kechedzhi , Julian Kelly ,
- Paul V. Klimov 1, Alexander Korotkov , Fedor Kostritsa1,
- David Landhuis 1, Mike Lindmark , Erik Lucero1, Dmitry
- Lyakh7, Salvatore Mandra3, Jarrod R. McClean1, Matt
- McEwen5,Anthony Megrant 1, Xiao Mi ,Kristel Michielsen8,
- Masoud Mohseni 1, Josh Mutus , Ofer Naaman , Matthew
- Neeley 1, Charles Neill , Murphy Yuezhen Niu , Eric Ostby1,
- Andre Petukhov 1, John C. Platt , Chris Quintana , Eleanor
- G. Rieel3, Pedram Roushan 1, Nicholas Rubin , Daniel
- Sank 1, Kevin J. Satzinger , Vadim Smelyanskiy , Kevin
- Sung 1, Matthew D. Trevithick , Amit Vainsencher , Ben-
- jamin Villalonga 1;9, Theodore White , Jamie Yao , Ping
- Yeh 1, Adam Zalcman , Hartmut Neven1, John M. Martinis ;5
- 1. Google Research, Mountain View, CA 94043, USA, 2.
- Department of Electrical and Computer Engineering, Uni-
- versity of Massachusetts Amherst, Amherst, MA, USA, 3.
- Quantum Articial Intelligence Lab. (QuAIL), NASA Ames
- Research Center, Moett Field, USA, 4. Institute for
- Quantum Information and Matter, Caltech, Pasadena, CA,
- USA, 5. Department of Physics, University of California,
- Santa Barbara, CA, USA, 6. Friedrich-Alexander University
- Erlangen-Nurn berg (FAU), Department of Physics, Erlangen,
- Germany, 7. Quantum Computing Institute, Oak Ridge Na-
- tional Laboratory, Oak Ridge, TN, USA, 8. Institute for
- Advanced Simulation, Julic h Supercomputing Centre, Julic h,
- Germany, 9. Department of Physics, University of Illinois
- at Urbana-Champaign, Urbana, IL, USA
- [1]Feynman, R. P. Simulating physics with computers. Int.
- J. Theor. Phys. 21, 467{488 (1982).
- [2]Devoret, M. H., Martinis, J. M. & Clarke, J. Mea-
- surements of macroscopic quantum tunneling out of the
- zero-voltage state of a current-biased josephson junction.
- Phys. Rev. Lett 55, 1908 (1985).
- [3]Nakamura, Y., Chen, C. D. & Tsai, J. S. Spectroscopy
- of energy-level splitting between two macroscopic quan-
- tum states of charge coherently superposed by josephson
- coupling. Phys. Rev. Lett. 79, 2328 (1997).
- [4]Mooij, J. et al. Josephson persistent-current qubit. Sci-
- ence 285, 1036 (1999).
- [5]Wallra, A. et al. Strong coupling of a single photon to a
- superconducting qubit using circuit quantum electrody-
- namics. Nature 431, 162 (2004).
- [6]Koch, J. et al. Charge-insensitive qubit design derived
- from the cooper pair box. Phys. Rev. A 76, 042319
- (2007).
- [7]You, J. Q. & Nori, F. Atomic physics and quantum optics
- using superconducting circuits. Nature 474, 589 (2011).
- [8]Preskill, J. Quantum computing and the entanglement
- frontier. Rapporteur talk at the 25th Solvay Conference
- on Physics, Brussels (2012).
- [9]Aaronson, S. Certied randomness from quantum
- supremacy. In preparation .
- [10]Hastings, M. B. Classical and Quantum Bounded
- Depth Approximation Algorithms. arXiv e-prints
- arXiv:1905.07047 (2019). 1905.07047.
- [11]Kechedzhi, K. et al. Ecient population transfer via non-
- ergodic extended states in quantum spin glass. arXiv
- e-prints arXiv:1807.04792 (2018). 1807.04792.
- [12]Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quan-
- tum simulations of classical annealing processes. Phys.
- Rev. Lett. letters 101, 130504 (2008).
- [13]McClean, J. R., Boixo, S., Smelyanskiy, V. N., Babbush,
- R. & Neven, H. Barren plateaus in quantum neural net-
- work training landscapes. Nat. Comm. 9, 4812 (2018).
- [14]Cong, I., Choi, S. & Lukin, M. D. Quantum convolutional
- neural networks. arXiv:1810.03787 (2018).
- [15]Bravyi, S., Gosset, D. & Konig, R. Quantum advantage
- with shallow circuits. Science 362, 308{311 (2018).
- [16]Aspuru-Guzik, A., Dutoi, A. D., Love, P. J. & Head-
- Gordon, M. Simulated quantum computation of molecu-
- lar energies. Science 309, 1704{1707 (2005).
- [17]Peruzzo, A. et al. A variational eigenvalue solver on a
- photonic quantum processor. Nat. Commun. 5, 4213
- (2014).
- [18]Hempel, C. et al. Quantum chemistry calculations on a
- trapped-ion quantum simulator. Phys. Rev. X 8, 031022
- (2018).
- [19]Shor, P. W. Algorithms for quantum computation: dis-
- crete logarithms and factoring proceedings. Proceedings
- 35th Annual Symposium on Foundations of Computer
- Science (1994).
- [20]Fowler, A. G., Mariantoni, M., Martinis, J. M. & Cle-
- land, A. N. Surface codes: Towards practical large-scale
- quantum computation. Phys. Rev. A 86, 032324 (2012).
- [21]Barends, R. et al. Superconducting quantum circuits at
- the surface code threshold for fault tolerance. Nature
- 508, 500{503 (2014).
- [22]Corcoles, A. D. et al. Demonstration of a quantum error
- detection code using a square lattice of four supercon-
- ducting qubits. Nat. Commun. 6, 6979 (2015).
- [23]Ofek, N. et al. Extending the lifetime of a quantum bit
- with error correction in superconducting circuits. Nature
- 536, 441 (2016).
- [24]Boixo, S. et al. Characterizing quantum supremacy in
- near-term devices. Nat. Phys. 14, 595 (2018).
- [25]Aaronson, S. & Chen, L. Complexity-theoretic founda-
- tions of quantum supremacy experiments. In 32nd Com-
- putational Complexity Conference (CCC 2017) (2017).
- [26]Neill, C. et al. A blueprint for demonstrating quantum
- supremacy with superconducting qubits. Science 360,
- 195{199 (2018).
- [27]Bremner, M. J., Montanaro, A. & Shepherd, D. J.
- Average-case complexity versus approximate simulation
- of commuting quantum computations. Phys. Rev. Lett.
- 117, 080501 (2016).
- [28]Bouland, A., Feerman, B., Nirkhe, C. & Vazi-
- rani, U. Quantum supremacy and the com-
- plexity of random circuit sampling. Preprint at
- https://arxiv.org/abs/1803.04402 (2018).
- [29]See supplementary information .
- [30]Vool, U. & Devoret, M. Introduction to quantum electro-
- magnetic circuits. Int. J. Circ. Theor. Appl. 45, 897{934
- (2017).
- 8
- [31]Chen, Y. et al. Qubit architecture with high coherence
- and fast tunable coupling circuits. Phys. Rev. Lett. 113,
- 220502 (2014).
- [32]Yan, F. et al. A tunable coupling scheme for implement-
- ing high-delity two-qubit gates. Phys. Rev. Applied 10,
- 054062 (2018).
- [33]Schuster, D. I. et al. Resolving photon number states in
- a superconducting circuit. Nature 445, 515 (2007).
- [34]Jerey, E. et al. Fast accurate state measurement with
- superconducting qubits. Phys. Rev. Lett. 112, 190504
- (2014).
- [35]Chen, Z. et al. Measuring and suppressing quantum state
- leakage in a superconducting qubit. Phys. Rev. Lett. 116,
- 020501 (2016).
- [36]Klimov, P. V. et al. Fluctuations of energy-relaxation
- times in superconducting qubits. Phys. Rev. Lett. 121,
- 090502 (2018).
- [37]Yan, F. et al. The
- ux qubit revisited to enhance coher-
- ence and reproducibility. Nat. Commun. 7, 12964 (2016).
- [38]Knill, E. et al. Randomized benchmarking of quantum
- gates. Phys. Rev. A 77, 012307 (2008).
- [39]Magesan, E., Gambetta, J. M. & Emerson, J. Scalable
- and robust randomized benchmarking of quantum pro-
- cesses. Phys. Rev. Lett. 106, 180504 (2011).
- [40]Cross, A. W., Magesan, E., Bishop, L. S., Smolin, J. A. &
- Gambetta, J. M. Scalable randomised benchmarking of
- non-cliord gates. NPJ Quantum Information 2, 16012
- (2016).
- [41]Wallra, A. et al. Approaching unit visibility for control
- of a superconducting qubit with dispersive readout. Phys.
- Rev. Lett. 95, 060501 (2005).
- [42]De Raedt, H. et al. Massively parallel quantum computer
- simulator, eleven years later. Comput. Phys. Commun.
- 237, 47 { 61 (2019).
- [43]Markov, I. L., Fatima, A., Isakov, S. V. & Boixo, S.
- Quantum supremacy is both closer and farther than it
- appears. Preprint at https://arxiv.org/abs/1807.10749
- (2018).
- [44]Villalonga, B. et al. A
- exible high-performance sim-
- ulator for the verication and benchmarking of quan-
- tum circuits implemented on real hardware. Preprint at
- https://arxiv.org/abs/1811.09599 (2018).
- [45]Boixo, S., Isakov, S. V., Smelyanskiy, V. N. &
- Neven, H. Simulation of low-depth quantum circuits
- as complex undirected graphical models. Preprint at
- https://arxiv.org/abs/1712.05384 (2017).
- [46]Chen, J., Zhang, F., Huang, C., Newman, M. & Shi,
- Y. Classical simulation of intermediate-size quantum
- circuits. Preprint at https://arxiv.org/abs/1805.01450
- (2018).
- [47]Villalonga, B. et al. Establishing the quantum supremacy
- frontier with a 281 p
- op/s simulation. Preprint at
- https://arxiv.org/abs/1905.00444 (2019).
- [48]Pednault, E. et al. Breaking the 49-qubit barrier
- in the simulation of quantum circuits. Preprint at
- https://arxiv.org/abs/1710.05867 (2017).
- [49]Chen, Z. Y. et al. 64-qubit quantum circuit simulation.
- Sci. Bull. 63, 964{971 (2018).
- [50]Chen, M.-C. et al. Quantum teleportation-inspired al-
- gorithm for sampling large random quantum circuits.
- Preprint at https://arxiv.org/abs/1901.05003 (2019).
- [51]Shor, P. W. Scheme for reducing decoherence in quan-
- tum computer memory. Phys. Rev. A 52, R2493{R2496
- (1995).
- [52]Devoret, M. H. & Schoelkopf, R. J. Superconducting
- circuits for quantum information: An outlook. Science
- 339, 1169{1174 (2013).
- [53]Mohseni, M. et al. Commercialize quantum technologies
- in ve years. Nature 543, 171 (2017).
- [54]Grover, L. K. Quantum mechanics helps in searching for
- a needle in a haystack. letters 79, 325 (1997).
- [55]Bernstein, E. & Vazirani, U. Quantum complexity the-
- ory. Proc. 25th Annual ACM Symposium on Theory of
- Computing (1993).
- Title BEFORE YOU CONTINUE
- Author n.l.heimerl
- Created Date 9/4/2019 11:05:03 AM

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