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A blueprint for demonstrating quantum supremacy with superconducting qubits

C. Neill, P. Roushan, K. Kechedzhi, S. Boixo, S. V. Isakov, V. Smelyanskiy, R. Barends, B. Burkett, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, A. Fowler, B. Foxen, R. Graff, E. Jeffrey, J. Kelly, E. Lucero, A. Megrant, J. Mutus, M. Neeley, C. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White, H. Neven, J. M. Martinis

arXiv:1709.06678v1quant-ph

TL;DR

The paper addresses how superconducting-qubit experiments can demonstrate computations whose sampling complexity challenges classical simulation. It characterizes complexity through Hilbert-space exploration and develops calibrated control and simulation approaches, finding rapidly growing effective state spaces and computational hardness indications. These results support extending the experiments toward larger systems, while the supplied passages do not state a specific limitation.

  • Problem

    The paper asks how engineered quantum systems can perform sampling tasks whose computational complexity may exceed classical capabilities.

  • Method

    The study uses superconducting gmon dynamics, calibrated control models, Hilbert-space truncation analyses, and complexity arguments based on mapping sampling amplitudes to a complex-temperature partition function.

  • Results

    The effective Hilbert-space dimension fits D ∼ 2.4^N/0.14 for 20–40 gmons, while classical memory comparisons limit a supercomputer to 47, 44, or 37 qubits under different truncations.

  • Takeaways & Limitations

    The results provide a foundation for comparing larger superconducting-qubit experiments with classical algorithms and for pursuing quantum supremacy.

Abstract

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Fundamental questions in chemistry and physics may never be answered due to the exponential complexity of the underlying quantum phenomena. A desire to overcome this challenge has sparked a new industry of quantum technologies with the promise that engineered quantum systems can address these hard problems. A key step towards demonstrating such a system will be performing a computation beyond the capabilities of any classical computer, achieving so-called quantum supremacy. Here, using 9 superconducting qubits, we demonstrate an immediate path towards quantum supremacy. By individually tuning the qubit parameters, we are able to generate thousands of unique Hamiltonian evolutions and probe the output probabilities. The measured probabilities obey a universal distribution, consistent with uniformly sampling the full Hilbert-space. As the number of qubits in the algorithm is varied, the system continues to explore the exponentially growing number of states. Combining these large datasets with techniques from machine learning allows us to construct a model which accurately predicts the measured probabilities. We demonstrate an application of these algorithms by systematically increasing the disorder and observing a transition from delocalized states to localized states. By extending these results to a system of 50 qubits, we hope to address scientific questions that are beyond the capabilities of any classical computer.

Supplementary Information for "A blueprint for demonstrating quantum supremacy

This section is identified as supplementary information for the paper “A blueprint for demonstrating quantum supremacy.”

  • The material is supplementary information for the paper.
  • The paper concerns a blueprint for demonstrating quantum supremacy.
  • The supplied section identifier does not state additional scientific content.

I. QUBIT ARCHITECTURE

The device uses superconducting gmon qubits with tunable frequencies and interactions, supported by a circuit design involving SQUIDs, couplers, and control hardware.

  • The device uses superconducting gmon qubits with tunable frequencies and interactions.
  • The qubits combine a capacitor, flux-tunable DC SQUID, and shunt inductor.
  • The qubits have energy-relaxation times near 10–15 µs and Ramsey dephasing times around 5 µs near the flux-insensitive point.

II. RAW DATA WITH PREDICTIONS

The supplementary figures compare measured output probabilities with model predictions and show raw probability histograms without state-number normalization.

  • The measured probabilities strongly resemble the predictions, providing a visual demonstration of an accurate control model.
  • Measured probabilities for two 5-qubit experiments after 10 cycles are overlaid with expected probabilities.
  • The plotted error bars represent 3 standard deviations from 50,000 samples.
  • Raw output probabilities for 5- to 9-qubit experiments after 5 cycles are shown in histograms without normalization by the number of states.

IV. HILBERT-SPACE DIMENSION

The effective Hilbert-space dimension grows exponentially with qubit number, and higher-level truncation schemes increase the state count beyond the fixed-excitation two-level baseline.

  • IV. HILBERT-SPACE DIMENSION: The fixed-excitation two-level subspace provides a lower-bound complexity scaling of approximately 2^N/N.
  • IV. HILBERT-SPACE DIMENSION: The truncation schemes compare two-level, fixed-excitation, single-doublon, and qutrit state spaces.
  • IV. HILBERT-SPACE DIMENSION: Single-doublon states are included because they are closest in energy to the qubit subspace and most significantly modify the evolution.
  • IV. HILBERT-SPACE DIMENSION: A supercomputer is limited to approximately 47, 44, or 37 qubits depending on the truncation scheme.
  • IV. HILBERT-SPACE DIMENSION: The plotted memory comparisons assume the state is represented using complex 64-bit numbers.

V. POST-SELECTION

Post-selection removes outcomes inconsistent with excitation-number conservation, reducing experimental errors while also reducing the accessible Hilbert-space and data rate. The broader calibration effort addresses pulse imperfections before Hamiltonian control.

  • V. POST-SELECTION: Excitation-number conservation identifies erroneous outcomes because the experiment starts and should end with half the qubits excited.Measurement error and photon loss can change excitation number, enabling rejection of inconsistent outcomes.
  • V. POST-SELECTION: 5.0%/qubit erroneous outcomes occur initially at 0 cycles, consistent with measurement infidelity.The rejected fraction is measured across 5-to-9-qubit experiments as a function of cycle count.
  • V. POST-SELECTION: Arbitrary time-dependent Hamiltonian control requires calibrating pulse distortion, relative timing, crosstalk, and device physics.The calibration pipeline converts room-temperature signals into target-device Hamiltonian matrix elements.

A. Calibration 1: Pulse distortion

The pulse-distortion calibration models frequency-dependent attenuation in control lines, infers the transfer function from qubit phase dynamics, and uses it to correct the delivered pulses.

  • A. Calibration 1: Pulse distortion: Control pulses undergo frequency-dependent attenuation between room temperature and the target device.The transfer function captures this nonideal response.
  • A. Calibration 1: Pulse distortion: The distortion model uses ϵ for the distorted pulse fraction and τ for its characteristic time-scale.Typical distortion is ϵ ≈1%, with τ values of 10 ns and 70 ns; the model is consistent with the data although its physical origin is not completely understood.
  • A. Calibration 1: Pulse distortion: A square flux-bias pulse is followed by qubit phase measurements over delay time to infer the transfer function.The ideal phase is time-independent, whereas the observed settling response is fit by the distortion model across all nine qubits.
  • A. Calibration 1: Pulse distortion: The inferred transfer function corrects distortion by dividing output signals in the frequency domain by that transfer function.The corrected phase response is shown in black.

B. Calibration 2: Timing

Timing calibration synchronizes microwave, qubit flux-bias, and coupler flux-bias lines by measuring timing-dependent excitation and correcting inferred offsets.

  • B. Calibration 2: Timing: Cabling and filtering cause control pulses to arrive at different times, so all lines are synchronized to the center-qubit flux-bias line Q5.Synchronization proceeds outward from the center toward both array edges.
  • B. Calibration 2: Timing: The calibration varies the relative timing of a microwave pi-pulse and a qubit or coupler flux-bias pulse while measuring qubit excitation.The same procedure is applied to qubit and coupler flux-bias lines.
  • B. Calibration 2: Timing: A flux-bias pulse overlapping the pi-pulse shifts the qubit frequency and makes the microwave pulse off-resonance.Pulses arriving before or after the pi-pulse leave the qubit excited, while overlap prevents complete excitation.
  • B. Calibration 2: Timing: Fitting the excitation data determines timing offsets that are used to correct subsequent pulses.This converts the measured timing mismatch into a control correction.

C. Calibration 3: Crosstalk

Crosstalk correction and physical modeling convert imperfect control signals into calibrated Hamiltonian parameters. The model captures nonlinear qubit, coupler, and resonator effects while remaining calibratable through single-qubit spectroscopy.

  • C. Calibration 3: Crosstalk: Unwanted geometric coupling sends a fraction of each control line’s flux to neighboring devices, producing linear flux crosstalk.The correction procedure measures source-to-target effects and compensates them with target-device pulses.
  • C. Calibration 3: Crosstalk: 0.1-0.3% crosstalk occurs between physically adjacent lines, while qubit-to-neighboring-coupler crosstalk reaches 4%.Desired output fluxes are multiplied by the crosstalk matrix to generate corrected control fluxes.
  • D. Calibration 4: Control model: The physical model represents qubits and couplers with nonlinear flux-dependent frequencies, anharmonicities, and couplings.The qubits are approximated by a Bose-Hubbard model with frequency, anharmonicity, and coupling parameters.
  • D. Calibration 4: Control model: 20 oscillator levels provide 1 Hz accuracy for a 5 GHz qubit with -200 MHz nonlinearity at β = 0.Keeping more than about 60 levels causes the numerical method to break down as states enter neighboring cosine-potential minima.
  • D. Calibration 4: Control model: 24 hours calibrates 9 qubits and 8 couplers using fits to each qubit’s two lowest transition energies versus qubit and coupler flux.This strategy requires only single-qubit experiments and is described as likely to scale to larger systems.

1. Sampling amplitude as a classical partition function

The sampling amplitude is rewritten as a partition function over particle-conserving trajectories, each carrying a phase and non-negative weight. In the chaotic regime, exponentially large trajectory weights cancel to produce an exponentially small amplitude, making classical evaluation hard.

  • Trajectory representation: The time evolution is Trotter-decomposed into a sum over occupation-number trajectories on a discrete space-time lattice.Each trajectory records boson occupations at discrete times, while swap operators enforce particle-number conservation.
  • Trajectory representation: Each trajectory contributes a phase factor and a real non-negative weight to the partition function.The phase includes swap and diagonal-Hamiltonian contributions, while non-conserving trajectories have zero weight.
  • Classical partition function: The resulting partition function is a discrete classical model generalizing the Potts model with four-site interactions and complex parameters.The phase factors can be binned into a polynomial number of phase values, with each bin summing the weights of corresponding trajectories.
  • Classical partition function: Trajectory multiplicities compensate swap-weight suppression, so the dominant binned weights scale exponentially with the number of qubits.For trajectories with ν swaps, the entropic arrangement factor offsets the suppression as the number of time points grows.
  • Computational hardness: In the chaotic regime, exponentially large weights cancel to yield an amplitude exponentially small in the number of qubits.Computing this cancellation resembles a sign-problem partition-function estimate and is argued to require exponential accuracy for polynomial resources.
  • Computational hardness: The two-level limit maps to free fermions and permits efficient determinant evaluation, whereas higher levels introduce interactions and chaotic evolution.Including the second excited level breaks the strictly antisymmetric free-fermion structure; random parameters are expected to produce chaos.

1. Direct simulation with truncated bosons

Direct simulation truncates each gmon’s Fock space to a finite number of excitation levels and evaluates accuracy with fourth-order Runge-Kutta integration. Three levels suffice in the studied regime, while the qubit-only approximation fails and simulation cost grows exponentially.

  • Truncation scheme: The truncation parameter m sets the highest retained excitation level for each gmon, with m = 1 denoting the qubit subspace and m = 2 including |2⟩.The truncated Bose-Hubbard dynamics are integrated numerically and compared using cross-entropy difference after projection to the qubit subspace.
  • Truncation accuracy: The cross-entropy difference between m = 3 and m = 4 is exactly 1, indicating that m = 3 suffices in the studied parameter regime.Adding more energy levels does not change the resulting state at this truncation depth.
  • Truncation accuracy: The m = 1 truncation gives approximately zero fidelity because it maps to free fermions rather than chaotic dynamics.Including the second excited level is necessary to capture the interactions relevant to the studied evolution.
  • Scaling cost: At half filling, the effective Hilbert-space dimension for N = 20–40 gmons fits D ∼2.4^N/0.14.This estimate accounts for boson-number conservation and targets the regime relevant to future experiment–simulation comparisons.
  • Band structure: For g ≪η, the spectrum separates into excitation-number bands, but the bands begin overlapping at g ∼20 MHz.This overlap limits perturbative approaches that assume well-separated bands.

4. Simulation with a fixed number of bands

An approximate classical simulation retains a fixed number of excitation bands rather than the full truncated Hilbert space. These band-based spaces can remain accurate relative to expected experimental errors, while their memory and runtime still grow rapidly with system size.

  • Fixed-band truncation: The effective space [d, t] retains bands with at most d doublons and t triplons.For example, [1, 0] includes the qubit band and one-doublon band, while includes four bands.
  • Accuracy: The truncation has numerical error below expected experimental error even for fairly large maximum coupling and many pulses.The [2, 0] and [1, 0] spaces have larger errors but remain valid or likely comparable methods for the stated comparison.
  • Resource estimates: The effective Hilbert-space dimension is evaluated through log2(D) to compare simulation size with an equivalent qubit count.The analysis plots this quantity for two- and three-band truncations as the number of gmons increases.
  • Resource estimates: Distributed fourth-order Runge-Kutta simulation time is constrained mainly by network bandwidth between supercomputer nodes.Each integration step requires multiple vector communications when the quantum state is distributed across nodes.
  • Quantum loss of memory: Disorder-driven chaotic dynamics spread wave-function amplitudes over the available Hilbert space, while time-cross-entropy measures loss of memory between evolution times.The measured statistic rapidly approaches the value expected for uncorrelated times after the first two pulses.

2. Entanglement spread in the driven system

The driven system’s entanglement spread depends on disorder: it is ballistic at low disorder but becomes sublinear and dramatically slower at stronger disorder, without entering a many-body localized phase.

  • Entanglement spread: Quantum entanglement is relevant to classical simulability because MPS- and DMRG-type algorithms incur costs exponential in entanglement entropy.The protocol therefore checks whether entanglement reaches volume-law scaling within the evolution time.
  • Low disorder: At low disorder, entanglement spreads ballistically, consistent with the ergodic phase.This behavior persists even with diffusive particle transport.
  • Strong disorder: At stronger disorder, entanglement crosses over to sublinear spreading and slows dramatically.The reported entropy and disorder-strength measurements are shown in Figs. S23 and S24.
  • Strong disorder: Despite the slowdown, the broadband drive is not expected to produce a many-body localized phase because the system continues to absorb energy.The contrast is with periodically driven systems that undergo a transition to a many-body localized phase.
  • Strong disorder: Rare regions with strongly fluctuating disorder potentials likely contribute significantly to the slowed entanglement transport in one dimension.This mechanism is identified as the likely reason for the strong-disorder behavior.

3. Convergence to Porter-Thomas

The output probabilities approach the Porter-Thomas distribution under gmon evolution as pulses increase, whereas the free-fermion approximation does not show this convergence.

  • Distribution: The Porter-Thomas distribution is the exponential distribution expected when a quantum system uniformly explores all accessible states.For half filling, the distribution is written as f(pNstates) = e−pNstates.
  • Gmon evolution: 18 gmons after 10 pulses show a good approximation of the numerical output distribution to Porter-Thomas.The output probabilities are measured at half filling after projection into the qubit subspace.
  • Free-fermion approximation: The 18-gmon free-fermion evolution does not approximate Porter-Thomas and is characterized as nonchaotic.This contrasts with the gmon evolution shown for the same pulse count.
  • Gmon evolution: With 12 gmons, output-distribution entropy approaches the Porter-Thomas entropy as the number of pulses increases.The convergence is shown for different maximum values of g in the pulses.
  • Free-fermion approximation: With 16 gmons under the free-fermion approximation, output-distribution entropy does not converge to Porter-Thomas as pulses increase.The corresponding figure uses m = 1 and Tpulse = 20.5 ± 4.5 ns.
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