Efficient variational quantum simulator incorporating active error minimisation

Ying Li; Simon C. Benjamin

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Efficient variational quantum simulator incorporating active error minimisation

Ying Li, Simon C. Benjamin

arXiv:1611.09301v5Nov 28, 2016quant-ph

TL;DR

The paper asks whether quantum dynamics can be simulated on modest, imperfect near-term processors rather than waiting for large-scale fault-tolerant machines. It proposes a variational algorithm that closely integrates quantum and classical coprocessors, amplifies controllable errors, and extrapolates to zero error. In simulations, the hybrid protocol is much more reliable than optimized Trotterisation and suppresses error accumulation.

Problem
The paper addresses whether useful quantum simulation can run on modest near-term processors with imperfect control instead of requiring large-scale fault-tolerant quantum computing.
Method
The method uses a variational quantum–classical loop in which the quantum coprocessor evaluates required quantities while the classical computer updates parameters, with controllable errors amplified for zero-error extrapolation.
Results
The hybrid algorithm's distance is about ten times lower than Trotterisation's distance, and error reduction largely suppresses its increase over time.
Takeaways & Limitations
The authors present the algorithm as a promising candidate for early-stage nonfault-tolerant quantum computers.

Abstract

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One of the key applications for quantum computers will be the simulation of other quantum systems that arise in chemistry, materials science, etc, in order to accelerate the process of discovery. It is important to ask: Can this be achieved using near future quantum processors, of modest size and under imperfect control, or must it await the more distant era of large-scale fault-tolerant quantum computing? Here we propose a variational method involving closely integrated classical and quantum coprocessors. We presume that all operations in the quantum coprocessor are prone to error. The impact of such errors is minimised by boosting them artificially and then extrapolating to the zero-error case. In comparison to a more conventional optimised Trotterisation technique, we find that our protocol is efficient and appears to be fundamentally more robust against error accumulation.

I. INTRODUCTION

The paper asks whether quantum dynamics can be simulated on modest, imperfect near-term processors and proposes a variational hybrid quantum–classical approach with active error reduction.

Motivation: Near-term quantum processors may require useful applications before fully fault-tolerant systems become available.The paper contrasts processors of roughly fifty qubits with the millions of qubits expected for fault-tolerant computing.
Motivation: Hybrid algorithms assign a quantum coprocessor a limited subroutine while a classical computer governs the larger-scale computation.The simplified quantum task may tolerate higher per-operation error rates and potentially operate without quantum error correction.
Motivation: Variational simulation represents a target state with a tractable parameterized trial state rather than specifying an arbitrary exponentially large quantum state.Selecting useful parameters requires understanding the target molecule, material, or other quantum system.
Motivation: Quantum dynamics is needed for problems whose properties cannot be determined from static features, but existing dynamical techniques do not solve every problem.The paper situates its approach among dynamical mean-field, variational Monte Carlo, tensor-network, and density-functional methods.
Contribution: The proposed hybrid algorithm uses variational simulation for dynamics, targeting small noisy circuits instead of the many operations typically required by Trotterisation.Its distinctive feature is classical error compensation by amplifying controllable stochastic errors and extrapolating toward zero error.
Comparison: Trotter-Suzuki simulation approximates evolution by applying short-time evolutions generated by individual Hamiltonian terms.With uniform short times, approximation errors scale as T^2/Nt as Nt increases.
Method: The variational approach is useful when a quantum computer can evaluate trial-state properties that are difficult to evaluate classically.This extends variational methods beyond cases where classical algorithms can efficiently assess the trial state's fit to the target state.

III. HYBRID QUANTUM SIMULATION OF DYNAMICS

The hybrid simulator approximates Schrödinger evolution with a parameterized trial state, using a quantum computer to supply derivatives while a classical computer updates parameters iteratively.

Algorithm: The algorithm assumes the evolving state can be approximated by a trial state |Ψ(λ1, λ2, . . .)⟩ with variational parameters.The classical computer determines these parameters according to the Schrödinger equation, while the quantum computer supplies required derivatives.
Algorithm: At each short time step, the quantum computer evaluates quantities for the current parameters and the classical computer computes the parameters for the next time.Repeating this exchange from the initial parameters yields a final trial state intended to approximate the exact evolved state.
Limitation: Compared with Trotterisation, the hybrid method addresses an easier problem because it evolves one specified initial state rather than implementing the evolution operator for arbitrary initial states.This makes direct circuit-size comparisons an apples-to-oranges comparison.
Scope: The protocol can simplify tasks such as tracking a specified initial state, calibrating gates, and finding ground states through adiabatic evolution.Its iterative parameter updates avoid the global energy minimization required in some ground-state hybrid algorithms.
Variational theory: The variational equations can be derived from a time-dependent variational principle or McLachlan’s principle.The paper focuses on the time-dependent variational principle but states that the hybrid algorithm can be adapted to McLachlan’s formulation.
Variational theory: The parameterization must remain tractable and generally requires an informed guess about the quantum state's form.Allowing every basis amplitude would recover exact evolution but would require too many parameters for the target systems of interest.

V. VARIATIONAL ALGORITHM ON A HYBRID COMPUTER

The variational algorithm prepares parameterized trial states with quantum gates and uses shallow quantum circuits to estimate coefficients needed by the classical evolution equations.

Trial-state preparation: Trial states are prepared as |Ψ⟩ = R|¯0⟩, where R is a parameter-dependent sequence of directly implementable quantum gates.The gate sequence is determined by the variational parameters.
Trial-state preparation: When the number of variational gates is below the Hilbert-space dimension, trial states span only a sub-manifold and approximate exact evolution only when the exact state lies near it.The total number of gates is also the number of parameters in the one-parameter-per-gate formulation.
Trial-state preparation: The gate parameterization can be generalized to multiple parameters per gate and can represent both Trotter-Suzuki and unitary coupled-cluster ansätze.The paper analyzes the simpler case in which each gate depends on one parameter.
Coefficient evaluation: Quantum circuits evaluate coefficients that are intractable to obtain classically, with each parameter-associated circuit using approximately Nv gates and repeated measurements.The measurements for different parameters can be parallelized across quantum processors without a quantum link between them.
Coefficient evaluation: Hamiltonian and gate derivatives are decomposed into sums of unitary, often Pauli, operators whose expectation-related terms can be measured by quantum circuits.The relevant operators may act on one or multiple qubits.
Coefficient evaluation: The measurement circuit uses an ancilla prepared in a phase-dependent superposition, controlled gates, and an X-basis measurement to estimate each term.The measured expectation value is obtained by repeating the relatively shallow circuit and averaging outcomes.

Main program

The main program alternates quantum evaluation of differential-equation coefficients with classical numerical integration, advancing variational parameters through discrete time steps.

Main program: The algorithm starts from initial parameters, evaluates M and V on the quantum coprocessor, and solves the parameter differential equation on the classical computer.The resulting solution advances the parameters by a small time increment before the process repeats.
Main program: The quantum–classical exchange can use Euler integration and can be adapted to other numerical methods such as Runge-Kutta.The chosen integration method determines details of the information exchanged between the processors.
Main program: Time is discretized as tn = nδt, and the process repeatedly computes derivatives at tn before obtaining parameters at later times.Repeating the updates until tn+1 = T produces the final parameter set {λk(T)}.
Error accounting: The distance between the exact and hybrid states is bounded by the accumulated distances represented by the red segments in Fig. 3.The figure separates the prior trial-state distance from the error introduced by evolving that trial state over each interval.

VI. ERROR ANALYSIS

The hybrid simulation has four implementation and algorithmic error sources, whose effects can be separated using trace-distance bounds. Trial-state, integration, shot, and machine-noise imperfections affect the simulated state through distinct terms.

Four error sources are identified: trial-wavefunction, numerical-integration, shot-noise, and quantum-machine errors.
Shot and machine noise are implementation errors, whereas trial-wavefunction and integration errors arise from the algorithm itself.
When the coefficient matrix is locally invertible, derivative errors depend linearly on deviations in measured M and V.
Trace distance separates total simulation error into algorithm-error and implementation-error contributions using the triangle inequality.
Finite-step integration error is controlled by reducing δt, while implementation errors arise from inaccurate derivatives obtained from noisy M and V.

B. Cost of the hybrid algorithm

The hybrid algorithm uses many small circuits whose total computational cost scales polynomially with integration error, shot noise, and simulated time. Several error contributions can be monitored or controlled during execution.

The initial-state and final-state preparation contributions are not T-dependent and may become relatively small for long simulations.
Trial-wavefunction errors can be continually estimated during execution because their relevant quantities are measurable with circuits analogous to those used for M and V.
δt ∼ ε2/(∆(3)T 2) limits finite-integration-step error to ε, requiring N ∼ ∆(3)T 3/ε2 time steps.
Shot-noise mitigation requires Nr ∼ ∥A∥∆2T 2/ε′2 circuit repetitions to limit its overall effect to ε′.
The total gate cost is polynomial in the integration error ε, shot-noise error ε′, and simulated time T.
Each calculation consists of many isolated small circuits, which can in principle be performed in parallel using separate quantum computers.

VII. EFFECT OF MACHINE NOISE AND ERROR REDUCTION

Machine noise is the main implementation challenge, but some noise rescales all measured coefficients uniformly and therefore does not alter the inferred variational derivatives. Inhomogeneous scaling, by contrast, produces computing errors.

Machine noise is more problematic than shot noise because repetition suppresses shot noise, while machine noise is harder to dismiss.
Uniform depolarising noise scales both M and V by the same nonzero factor, leaving the solution for the variational derivatives unchanged.
Only inhomogeneous scaling of quantum outputs causes computing errors in this analysis.
Balanced measurement errors transform an ideal output through an additive offset and multiplicative factor determined by p0 and p1.
Unbalanced measurement errors can be corrected from benchmarked p0 and p1, though higher error probabilities require more accurate output estimation.

A. Error reduction

The error-reduction protocol estimates the zero-noise expectation value by tuning stochastic error probabilities and extrapolating measured outputs. Its reliability depends on small circuits, low enough total error, and stable calibrated noise.

Stochastic operation noise is modeled as an ideal operation followed by an error map occurring with probability ϵ.
The noisy circuit state is expanded around the ideal state by scaling all error probabilities with a tunable factor r.
The protocol measures ⟨X⟩(r) at several nonzero error factors and fits ⟨X⟩(r)=⟨X⟩(0)+χr to estimate the zero-error value.
Second-order fitting can correct higher-order machine-noise contributions, but extrapolation error remains dependent on shot noise.
The protocol only works for small-size circuits, because high-order error terms become significant when Ngϵ ≳1.
With two-qubit infidelities of 10−3 to 10−4, the protocol could support hundreds of gates.
Noise must be benchmarked beforehand or monitored during execution, and its nature should not vary significantly during simulation.

B. Error twirling and simulation

The method converts non-stochastic two-qubit-gate errors into stochastic errors using redundant Pauli gates, then tunes their probabilities by adding randomized Pauli operations. This enables controllable error amplification for the algorithm’s error-reduction procedure.

Error conversion: Non-stochastic errors in two-qubit gates can be converted into stochastic Pauli errors using redundant Pauli gates.Two-qubit gates are identified as the main error source, and the conversion does not introduce significant additional operations.
Error conversion: Random Pauli gates applied before and after a controlled-phase gate cancel algebraically while transforming its effective noise.The post-gate operation is restricted to ΛUΛ† when the pre-gate operation is U.
Effective noise: The resulting effective noise is stochastic when the Pauli gates are ideal.If Pauli-gate errors are non-stochastic, the effective controlled-phase-gate noise may no longer be fully stochastic.
Error tuning: Error probabilities can be increased by randomly applying Pauli gates after a controlled-phase gate.For target scaling rϵe,f, the added Pauli error probability is approximately (r−1)ϵe,f when rϵe,f ≪ 1.

VIII. NUMERICAL RESULTS — QUANTUM ISING MODEL

The quantum Ising model is used to numerically evaluate the hybrid algorithm under a depolarising-noise model. At a two-qubit-gate error rate of 0.1%, the hybrid method is substantially more reliable than optimized Trotterisation, including when numerical-integration and shot-noise effects are included.

Model and setup: The numerical experiment simulates quantum Ising-model dynamics using a trial state, with at least four qubits required for the chosen parameters.The simulation uses J = B = 1/2 and ns = 3, and the trial state can exactly match the true state for correct parameters.
Noise model: The machine-noise model assumes depolarising errors, equal error rates within single- and two-qubit operations, and single-qubit errors one tenth as large.Memory errors are neglected because the hybrid algorithm’s circuits are small, and the modeled operation errors are stochastic.
Noise model: 0.1% two-qubit-gate error rate is used in the numerical simulations.This is identified as the state-of-the-art error rate in the cited simulation setup.
Results: The hybrid algorithm’s trace distance is about ten times lower than Trotterisation’s, while error reduction largely suppresses its growth over time.The comparison uses a Trotterisation time interval optimized to minimize errors; symmetric Trotter-Suzuki decomposition is also considered.
Results: Including numerical-integration error and shot noise increases the hybrid algorithm’s distance but leaves it much lower than Trotterisation’s.The simulations use δt = 2π × 10^-4 and repeat circuits Nr = 10^4 or 10^6 times for measurements with error reduction.

IX. SUMMARY

The proposed algorithm combines a classical processor with a quantum coprocessor to simulate quantum-system time evolution using shallow circuits and active error reduction. The authors report suppression of some noise effects and identify the method as promising for early-stage nonfault-tolerant quantum computers.

Summary: The hybrid algorithm uses closely integrated classical and quantum processors to simulate quantum-system time evolution with fewer quantum operations than canonical Trotterisation.The quantum coprocessor repeatedly implements small-size circuits, allowing independent coprocessors to operate in parallel.
Summary: The algorithm can automatically correct some noise-induced errors and estimate the zero-error limit by deliberately amplifying stochastic noise.This error reduction is performed without code-based quantum error correction and its associated resource overheads.
Conclusion: The authors identify the algorithm as a promising candidate for early-stage nonfault-tolerant quantum computers.This conclusion follows the reported combination of shallow circuits, noise robustness, and parallelizable coprocessor execution.

Appendix A: Simulated model of Fig. 1(b)

The appendix specifies a two-qubit simulated model and numerical procedures for comparing the hybrid algorithm with Trotterisation under noise and shot noise.

Simulated model: The simulated qubit evolution uses H = −(σy + σz cos t −σy sin t)/2 and a two-parameter trial state, requiring two qubits.The initial parameters are λ1 = 3/4 and λ2 = −1/2, with T = 2π.
Error reduction: The hybrid algorithm estimates ⟨X⟩(0) by fitting noisy measurements at error-rate scale factors r1 = 1 and r2 = 2.The fitted function is ⟨X⟩ = ⟨X⟩(0) + χr.
Trotterisation comparison: For the quantum Ising model, Trotterisation decomposes the Hamiltonian into HZ and HX terms and uses parameters selected to minimise errors.The final time step is adjusted so the total simulated time is T.
Trotterisation comparison: The average trace distance between the true state and the Trotterised state is used to select δt, with T = 4π for the plotted comparison.The reported choice is δt = 2π × 10^-1.4 because it minimises the average distance.
Trotterisation comparison: A lowest-order symmetric Trotter-Suzuki decomposition performs worse than basic Trotterisation because its larger gate count increases errors from 0.1% physical gate noise.The additional gates negate the potential reduction in Trotterisation error.