SHARE

TWEET

# Untitled

a guest
Sep 20th, 2019
21,233
Never

**Not a member of Pastebin yet?**

**, it unlocks many cool features!**

__Sign Up__- You have reached the cached page for https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/20190030475.pdf
- Below is a snapshot of the Web page as it appeared on 10/29/2006 (the last time our crawler visited it). This is the version of the page that was used for ranking your search results. The page may have changed since we last cached it. To see what might have changed (without the highlights), go to the current page.
- You searched for: "quantum supremacy using a programmable superconducting processor" We have highlighted matching words that appear in the page below.
- Bing is not responsible for the content of this page.
- 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
- NASA STI Program ... in Profile
- Since its founding, NASA has been dedicated
- to the advancement of aeronautics and space
- science. The NASA scientific and technical
- information (STI) program plays a key part in
- helping NASA maintain this important role.
- The NASA STI program operates under the
- auspices of the Agency Chief Information Officer.
- It collects, organizes, provides for archiving, and
- disseminates NASA’s STI. The NASA STI
- program provides access to the NTRS Registered
- and its public interface, the NASA Technical
- Reports Server, thus providing one of the largest
- collections of aeronautical and space science STI
- in the world. Results are published in both non-
- NASA channels and by NASA in the NASA STI
- Report Series, which includes the following report
- types:
- • TECHNICAL PUBLICATION. Reports of
- completed research or a major significant
- phase of research that present the results of
- NASA Programs and include extensive data
- or theoretical analysis. Includes compila-
- tions of significant scientific and technical
- data and information deemed to be of
- continuing reference value. NASA counter-
- part of peer-reviewed formal professional
- papers but has less stringent limitations on
- manuscript length and extent of graphic
- presentations.
- • TECHNICAL MEMORANDUM.
- Scientific and technical findings that are
- preliminary or of specialized interest,
- e.g., quick release reports, working
- papers, and bibliographies that contain
- minimal annotation. Does not contain
- extensive analysis.
- • CONTRACTOR REPORT. Scientific and
- technical findings by NASA-sponsored
- contractors and grantees.
- • CONFERENCE PUBLICATION.
- Collected papers from scientific and
- technical conferences, symposia, seminars,
- or other meetings sponsored or
- co-sponsored by NASA.
- • SPECIAL PUBLICATION. Scientific,
- technical, or historical information from
- NASA programs, projects, and missions,
- often concerned with subjects having
- substantial public interest.
- • TECHNICAL TRANSLATION.
- English-language translations of foreign
- scientific and technical material pertinent to
- NASA’s mission.
- Specialized services also include organizing
- and publishing research results, distributing
- specialized research announcements and
- feeds, providing information desk and personal
- search support, and enabling data exchange
- services.
- For more information about the NASA STI
- program, see the following:
- • Access the NASA STI program home page
- at http://www.sti.nasa.gov
- • E-mail your question to help@sti.nasa.gov
- • Phone the NASA STI Information Desk at
- 757-864-9658
- • Write to:
- NASA STI Information Desk
- Mail Stop 148
- NASA Langley Research Center
- Hampton, VA 23681-2199
- 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

RAW Paste Data

We use cookies for various purposes including analytics. By continuing to use Pastebin, you agree to our use of cookies as described in the Cookies Policy.