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Variational Fast Forwarding for Quantum Simulation Beyond the Coherence Time

Cristina Cirstoiu, Zoe Holmes, Joseph Iosue, Lukasz Cincio, Patrick J. Coles, Andrew Sornborger

arXiv:1910.04292v2quant-ph

TL;DR

Near-term quantum simulation is limited by finite coherence times and by the practical relevance of asymptotic fast-forwarding results to finite-depth devices. The paper introduces VFF, which variationally diagonalizes a short-time simulation unitary and reuses the learned fixed-depth circuit for longer times, reporting beyond-coherence-time simulations and eigenenergy-estimation benefits. These benefits depend on sufficiently accurate optimization and are bounded by hardware noise, ansatz expressivity, circuit depth, and no-fast-forwarding constraints.

  • Problem

    Asymptotic fast-forwarding results have limited direct impact on finite-depth, intermediate-scale near-term quantum simulations constrained by coherence time.

  • Method

    VFF variationally approximates a short-time simulation unitary by WDW † and fast-forwards evolution by replacing D with D^N in the same circuit structure.

  • Results

    VFF fast-forwarded selected Hubbard and Heisenberg simulations by approximately 30 and ∼80 timesteps, respectively, and achieved at least a factor of 6 experimentally on Rigetti hardware.

  • Takeaways & Limitations

    Lower VFF cost corresponds to longer achievable fast-forwarding and reduced variance in eigenenergy estimates.

  • Takeaways & Limitations

    VFF’s performance is limited by hardware noise, fixed-depth ansatz expressivity, circuit-depth constraints, and Hamiltonian families subject to no-fast-forwarding results.

Abstract

from arXiv · show

Trotterization-based, iterative approaches to quantum simulation are restricted to simulation times less than the coherence time of the quantum computer, which limits their utility in the near term. Here, we present a hybrid quantum-classical algorithm, called Variational Fast Forwarding (VFF), for decreasing the quantum circuit depth of quantum simulations. VFF seeks an approximate diagonalization of a short-time simulation to enable longer-time simulations using a constant number of gates. Our error analysis provides two results: (1) the simulation error of VFF scales at worst linearly in the fast-forwarded simulation time, and (2) our cost function's operational meaning as an upper bound on average-case simulation error provides a natural termination condition for VFF. We implement VFF for the Hubbard, Ising, and Heisenberg models on a simulator. Additionally, we implement VFF on Rigetti's quantum computer to demonstrate simulation beyond the coherence time. Finally, we show how to estimate energy eigenvalues using VFF.

I. INTRODUCTION

VFF addresses near-term quantum simulation limits by variationally diagonalizing a short-time unitary, then reusing a fixed-depth circuit for longer simulations. Its operationally meaningful cost function supports error guarantees and termination of the variational search.

  • Motivation: Near-term quantum computers may not benefit directly from asymptotic fast-forwarding results because finite-depth, intermediate-scale behavior determines practical feasibility.The relevant question is whether VFF can extend simulation beyond a device’s coherence time.
  • Contribution: VFF variationally searches for an approximate diagonalization of an entire simulation unitary rather than a finite set of low-lying states.This distinguishes VFF from subspace approaches such as SVQS.
  • Error control: The cost function’s operational meaning enables a termination condition that guarantees a user-defined average-case simulation-error threshold once the cost is sufficiently small.The cost is also reported as trainable without an obvious barren-plateau issue.
  • Method: VFF compiles a short-time unitary U(∆t) into V = WDW †, then simulates T = N∆t using the fixed-depth circuit W D^N W †.The diagonal unitary parameters are modified to represent longer evolution times.
  • Method: The Local Hilbert-Schmidt Test estimates the cost and its gradient with a short-depth quantum circuit during variational optimization.The cost is faithful, operationally meaningful as an upper bound on average-case compilation error, and has an efficiently estimable gradient.

3. Simulation Error Analysis

VFF’s error analysis bounds fast-forwarding error using errors from short-time implementation and approximate diagonalization. It shows at-worst linear growth with the number of time steps and connects the cost to certifiable average-case fidelity.

  • Error decomposition: Algorithmic implementation error and variational compilation error combine through the triangle inequality to bound the overall simulation error.The implementation error is associated with approximating e^(-iH∆t), while the compilation error arises from approximate diagonalization.
  • Linear scaling in N: The overall simulation error scales at worst linearly with the number of time steps, N.For T = N∆t, this gives at-worst linear scaling with the fast-forwarded simulation time.
  • Linear scaling in N: The VFF cost function scales at worst quadratically in N under fast forwarding when its short-time value is small.The approximation is stated for a successfully optimized, small-cost regime.
  • Certifiable error: The Hilbert-Schmidt norm yields certifiable bounds on average-case error, while the operator norm quantifies worst-case error.The analysis emphasizes average-case error because it naturally supports a variational termination condition.
  • Certifiable error: A lower bound on average gate fidelity can be derived from the VFF cost when the short-time cost is small.The resulting threshold determines when the variational optimization can terminate while guaranteeing the desired fidelity bound.

B. Implementations

The paper evaluates fast forwarding relative to standard Trotterization using an error-tolerance-dependent ratio of achievable simulation times. Values above one represent simulation beyond the coherence-time regime.

  • Evaluation metric: Fast-forwarding is defined for a simulation error tolerance δ using R_FF^δ, the ratio of VFF-achievable time T_FF^δ to standard-Trotterization time T_Trot^δ.This ratio compares achievable simulation times at the same error tolerance.
  • Evaluation metric: R_FF^δ is an empirical coherence-time measure because it accounts for both decoherence and gate infidelity.The condition R_FF^δ > 1 captures simulation beyond the coherence time.

1. Comparing VFF to Trotterization and Compiled Trotterizations

VFF is compared with standard Trotterization and QAQC-compiled Trotterization across simulation duration and approximation quality. Although VFF is less accurate for short simulations, sufficiently low cost enables it to outperform both alternatives at longer times.

  • VFF is compared against standard Trotterization and QAQC-compiled Trotterization, which uses an unrestricted short-depth compilation of the Trotterization step.QAQC optimizes over a larger circuit space than Trotterization.
  • The reported comparisons include XY-model simulations using a five-qubit open-boundary system and a truncated diagonal ansatz.
  • Approximately 70 to 100 timesteps were fast-forwarded across all plotted Jz, Jx, and Jy values.
  • As CVFF_LHST decreases, the duration over which VFF can simulate increases, and VFF dramatically outperforms Trotterization and QAQC at costs ≲10^-2.At large cost values, diagonalization error is too high for VFF to outperform the other methods.
  • For short simulations, Trotterization and QAQC are more accurate than VFF because their circuits use fewer time steps and gates.This short-time disadvantage reflects the cost of implementing W, D, and W† in VFF.
  • Reducing CVFF_LHST to 10^-3 lowers eigenvalue error below 0.1 and yields a fast-forwarding factor of approximately 30.VFF becomes more efficient than Trotterization when RFF_δ > 1, requiring cost below approximately 0.04 for δ = 0.2.

2. Using VFF to Fast Forward Models Across a Range of Parameters

VFF was applied to Hubbard and Heisenberg models across parameter ranges, rapidly finding approximate diagonalizations and compressing long-time simulations. It was also evaluated on Rigetti hardware and used with time-series analysis for energy-eigenvalue estimation.

  • Hubbard Model: VFF kept two-site Hubbard simulation error below δ = 10^-2 for T = 30∆t while compressing the equivalent Trotterized circuit.The VFF diagonalization used 9 single-qubit gates and 7 two-qubit gates, versus 60 and 30 for the equivalent Trotterized simulation.
  • Heisenberg Model: VFF rapidly found new Heisenberg-model diagonalizations as Jz, Jx, and Jy varied across the tested parameter ranges.The tested families included antiferromagnetic classical Ising, XXZ, and XYZ Heisenberg models.
  • Heisenberg Model: For three-qubit Heisenberg models, VFF kept simulation error below δ = 10^-2 up to T ≈100∆t.Each VFF diagonalization used 111 total gates, compared with 3700 total gates for Trotterization.
  • Quantum hardware: On Rigetti Aspen-4, the true noiseless VFF cost converged to two orders of magnitude below the quantum-computer-evaluated cost.The experiment optimized a single-qubit VFF diagonalization using gradient descent and evaluated both noisy and noiseless costs.
  • Quantum hardware: VFF maintained entanglement fidelity above 0.7 through at least N_VFF = 150, whereas iterated Trotterization reached 0.586 by N = 25.The process-tomography comparison used the exact classically computed process as the reference.
  • Energy eigenvalues: VFF combined with classical time-series analysis estimated energy spectra from diagonalized Hubbard and XY-model simulations.For the Hubbard examples, longer integration times produced successively better spectral resolution, constrained by σ_λjt_max ≥ c.

III. DISCUSSION

VFF uses approximate diagonalization to fast-forward simulations with constant gate count, while its cost function supplies an error bound and termination condition. Simulations achieved factors of approximately 30 for Hubbard and 80 for Heisenberg, with at least a factor-of-6 hardware demonstration.

  • VFF forms an approximate fast-forwarding once a diagonalization is available, enabling simulations beyond the coherence time.
  • Approximately 30 Hubbard and 80 Heisenberg simulation timesteps were fast-forwarded for the studied models, ansätze, and thresholds.
  • At least a factor-of-6 fast-forwarding relative to Trotterization was demonstrated experimentally on Rigetti’s quantum hardware.
  • The VFF cost function bounds average-case simulation error, so lowering the cost tightens the error bound.
  • A termination condition can guarantee a desired simulation-error threshold once the cost falls below a specified value.
  • Hardware noise can limit the minimum noisy cost and loosen error bounds, although the noiseless cost was often orders of magnitude lower in the Rigetti implementation.
  • The No Fast-Forwarding Theorem limits VFF scalability for some Hamiltonian families, while many physically interesting Hamiltonians are fast-forwardable or close to fast-forwardable.
  • The diagonal-unitary ansatz can be truncated to low-locality terms, with the approximation controlled by truncation; the transverse-field Ising model is exactly diagonalizable using only 1-local terms.

2. Ansatz for W

The W ansatz uses noncommuting parameterized unitaries to generate eigenvector transformations, with layered or randomized structures and optimization strategies designed for tractability and convergence.

  • Interleaving noncommuting unitaries can generate an eigenvector unitary W(θ), although a general construction requires O(d^2) parameterized operations.
  • A fixed layered ansatz alternates single- and two-qubit unitaries, while translational invariance can reduce variational parameters by a factor of n.
  • A randomized ansatz may suit irregular Hamiltonians and potentially find shorter W(θ) circuits with fewer gates.
  • Growing the ansatz mitigates local-minimum trapping by starting shallow and adding identity-resolving unitary layers after optimization reaches a local minimum.
  • Perturbative pre-training initializes θ and γ using a known short-depth diagonalization before successively modifying the Hamiltonian toward the target.
  • Implementations use successive W layers of single-qubit gates followed by neighboring even-odd and odd-even two-qubit entangling gates.
  • The D ansatz uses commuting layers of single-qubit Z rotations and two-qubit ZZ gates, with a three-qubit layer unnecessary for the stated threshold.
  • Gradient-based optimization is recommended because gradients can be evaluated using the same quantum circuit used for cost estimation.

VI. AUTHOR CONTRIBUTIONS

The paper assigns distinct roles for algorithm formulation, analytical results, numerical results, and cost-function development. Its cost function is faithful and can be easier to train than an alternative Hilbert-Schmidt-based function.

  • AUTHOR CONTRIBUTIONS: The VFF algorithm was formulated by CC, ZH, JI, LC, PJC, and AS.
  • AUTHOR CONTRIBUTIONS: CC and PJC performed the error analysis, PJC derived the termination condition, and ZH and AS derived gradient formulas.
  • AUTHOR CONTRIBUTIONS: LC and AS performed the numerical results.
  • The entanglement fidelities F^(j)_e are defined on a 2n-qubit system partitioned into n-qubit subsystems A and B using Bell states and single-qubit channels.
  • The faithfulness argument relates CLHST to CHST, whose vanishing is equivalent to equality of the unitaries up to global phase.
  • CLHST vanishes under precisely the same conditions as CHST and is therefore faithful.
  • CLHST is proposed instead of CHST because its gradient can remain independent of n when CHST’s gradient vanishes exponentially with n.

A. Linear scaling in N

The linear-scaling analysis bounds the error of repeatedly applying an approximate unitary by the single-step error multiplied by the number of applications. Reformulating the bound through CLHST makes it certifiable on a quantum computer.

  • For unitary matrices U1 and U2, the difference between their Nth powers is bounded by N times the single-step difference in any Schatten norm.
  • The bound follows because the telescoping expansion contains N terms and Schatten norms are unitarily invariant.
  • The lemma is reformulated using the VFF cost function and the Hilbert-Schmidt norm, making the bound efficiently estimable on a quantum computer.
  • The phase-independent error quantity depends only on the Hilbert-Schmidt inner product, because global phases are unphysical for direct n-qubit implementation.
  • CLHST(U^N, V^N) is bounded approximately by nN^2 CLHST(U, V) under the stated normalization and small-cost conditions.

C. An operational termination condition

VFF’s cost function bounds average-case simulation error, so reducing it tightens the error guarantee and yields a principled optimization termination condition.

  • C. An operational termination condition: The VFF cost function is related to average-case diagonalization error and bounds the average simulation error.This relation makes the cost operationally meaningful for certifying simulation accuracy.
  • C. An operational termination condition: The derivation rewrites total and diagonalization errors using Hilbert-Schmidt norms and average fidelity to connect the cost function with simulation accuracy.The analysis also relates the Hilbert-Schmidt treatment to operator-norm error analysis.
  • C. An operational termination condition: For sufficiently small VFF cost, the resulting average-fidelity bound provides a certifiable error guarantee for the fast-forwarded simulation.The bound is expressed through the Hilbert-Schmidt-based cost and its associated error quantity.
  • C. An operational termination condition: A target average fidelity can be guaranteed by terminating the variational optimization once the VFF cost falls below a prescribed threshold.The threshold is determined from the desired fast-forwarding time, simulation fidelity, and fixed initial Trotter error.

ESTIMATION OF ENERGY EIGENVALUES

VFF can support energy-eigenvalue estimation by comparing target-unitary energies with estimates extracted from the approximate unitary.

  • ESTIMATION OF ENERGY EIGENVALUES: The target unitary’s energies are compared with estimates extracted from the approximate VFF unitary, for example through time-series analysis.The comparison uses an ordering of approximate energies supplied by the Hoffmann-Wielandt theorem.
  • ESTIMATION OF ENERGY EIGENVALUES: The analysis introduces an arbitrary global phase when relating the approximate unitary’s eigenvalue information to the target energies.This phase is included while expanding the energy-comparison expression.
  • ESTIMATION OF ENERGY EIGENVALUES: The resulting bound is connected to the local VFF cost function through the Hilbert-Schmidt-test cost.The relation links eigenvalue-estimation accuracy to the approximate diagonalization quality.

COST FUNCTION GRADIENT DERIVATION

This section derives VFF’s cost-function gradient and places the method in the context of fast-forwarding, finite-size behavior, and fixed-depth implementation limits.

  • COST FUNCTION GRADIENT DERIVATION: The gradient derivation expresses how the local Hilbert-Schmidt-test cost changes with ansatz parameters in the eigenvector circuit.The derivation decomposes the eigenvector operator into Pauli rotations and differentiates with respect to one rotation angle.
  • COST FUNCTION GRADIENT DERIVATION: Fast-forwarding concerns Hamiltonian families whose simulation time grows faster than the required circuit resources, while quantum diagonalizability provides one route to exponential fast forwarding.The text also notes that the broader relationship between fast-forwardable and quantum-diagonalizable Hamiltonians remains open.
  • COST FUNCTION GRADIENT DERIVATION: Finite-size and finite-depth behavior makes asymptotic no-fast-forwarding results insufficient to determine whether VFF can simulate beyond coherence time.The paper treats asymptotic fast-forwardability and finite-regime VFF feasibility as largely independent.
  • COST FUNCTION GRADIENT DERIVATION: Fixed-depth circuits impose an approximation error and may limit VFF scalability for Hamiltonians whose diagonalization requires unfavorable resources at larger system sizes.The expected scaling is better for quantum-diagonalizable Hamiltonians than for Hamiltonians requiring exponential diagonalization resources.
  • COST FUNCTION GRADIENT DERIVATION: A non-quantum-diagonalizable Hamiltonian may still admit a good fixed-depth diagonalization approximation at small n, whereas large constant-factor overheads can make another Hamiltonian unsuitable.Thus asymptotic polynomial scaling alone does not guarantee a practical fixed-depth ansatz.
  • COST FUNCTION GRADIENT DERIVATION: The schematic distinguishes simulated time from required gate depth and marks the regime where time exceeds coherence time while gate depth remains below it.It presents this distinction for both fast-forwardable and non-fast-forwardable Hamiltonian families.
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