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An efficient quantum algorithm for the time evolution of parameterized circuits

Stefano Barison, Filippo Vicentini, Giuseppe Carleo

arXiv:2101.04579v3quant-phcond-mat.otherphysics.comp-ph

TL;DR

Gate noise and decoherence constrain quantum algorithms requiring many operations, while existing variational dynamics methods face quadratic parameter costs. p-VQD projects each time step onto a parameterized circuit by globally optimizing its parameters. The paper reports linear hardware-cost scaling and higher accuracy than TDVA, with the small-step update matching McLachlan’s principle.

  • Problem

    Quantum simulation needs methods that work with noisy, gate-limited hardware, while TDVA incurs quadratic computational cost in the number of variational parameters.

    Source: §1 · p. 1

  • Method

    p-VQD iteratively projects the evolved state onto a parameterized ansatz by globally optimizing parameter changes at each time step.

    Source: §1 · p. 1, §2 · p. 2

  • Results

    The paper reports linear hardware-cost scaling and higher accuracy than TDVA; at fixed sample counts, integrated infidelity is up to an order of magnitude below TDVA’s for the same number of time steps.

    Source: §3 · pp. 4–6, §4 · p. 7

  • Takeaways & Limitations

    In the small-time-step limit, p-VQD’s parameter variation coincides with the solution generated by McLachlan’s variational principle.

    Source: §2.1 · p. 3, §3 · p. 4

    The number of samples depends on the required step-infidelity and optimization steps, and barren plateaus may emerge only at larger system sizes, so cost functions should be benchmarked for the intended setting.

    Source: §3 · p. 6

Abstract

from arXiv · show

We introduce a novel hybrid algorithm to simulate the real-time evolution of quantum systems using parameterized quantum circuits. The method, named "projected - Variational Quantum Dynamics" (p-VQD) realizes an iterative, global projection of the exact time evolution onto the parameterized manifold. In the small time-step limit, this is equivalent to the McLachlan's variational principle. Our approach is efficient in the sense that it exhibits an optimal linear scaling with the total number of variational parameters. Furthermore, it is global in the sense that it uses the variational principle to optimize all parameters at once. The global nature of our approach then significantly extends the scope of existing efficient variational methods, that instead typically rely on the iterative optimization of a restricted subset of variational parameters. Through numerical experiments, we also show that our approach is particularly advantageous over existing global optimization algorithms based on the time-dependent variational principle that, due to a demanding quadratic scaling with parameter numbers, are unsuitable for large parameterized quantum circuits.

1 Introduction

Hardware noise and decoherence currently limit gate counts, motivating hybrid quantum algorithms that optimize parameterized circuits. The paper proposes p-VQD, a global, linearly scaling method for real-time quantum simulation.

1 Introduction

  • Hardware noise and decoherence limit gate counts, constraining algorithms that require polynomially many operations in the qubit count.Source: §1 · p. 1
  • Hybrid variational methods optimize a tractable parameterized ansatz to approximate a target quantum state.Source: §1 · p. 1
  • TDVA has quadratic cost in the number of variational parameters, while newer resource-efficient alternatives optimize restricted subsets locally.Source: §1 · p. 1
  • p-VQD globally optimizes all parameters at once, scales linearly with parameter count, requires no ancilla qubits, and keeps circuit depth constant during simulation.Source: §1 · pp. 1–2

2 Method

p-VQD advances a parameterized ansatz by optimizing a finite-step overlap with the state evolved under the Hamiltonian. Its small-time-step limit recovers McLachlan’s variational principle, and the global cost is argued to avoid barren plateaus.

2 Method

  • At each step, p-VQD chooses parameter changes that maximize overlap between the ansatz and the Hamiltonian-evolved state, minimizing step-infidelity.The loss is scaled to remain independent of time-step size as δt approaches zero; its gradient is optimized iteratively, with a parameter-shift rule available for exact gradient estimation.Source: §2 · p. 2

2.1 Relationship with other methods

  • In the vanishing-time-step limit, p-VQD’s parameter update coincides with McLachlan’s variational principle, while TDVA’s imaginary-QGT inversion can be unstable near zero off-diagonal elements.Source: §2.1 · p. 3

2.2 Barren plateaus

  • The authors argue p-VQD avoids barren plateaus because each step optimizes infidelity between states differing by an infinitesimal transformation, whose gradient variance has a non-vanishing lower bound.They also state that replacing the global infidelity cost with a local cost having the same minimum preserves the small-step McLachlan dynamics.Source: §2.2 · p. 4

3 Results

At matched sample counts, p-VQD achieves lower integrated infidelity than TDVA, and its total measurement cost scales linearly with parameter count when optimization steps remain constant. Its accuracy and optimization behavior also depend on shot count, target precision, and system size.

3 Results

  • At fixed sample counts, p-VQD’s integrated infidelity is up to an order of magnitude below TDVA’s for the same number of time steps.Figure 2 reports accumulated fidelity error over the full evolution as a function of total samples.Source: §3 · p. 4, Figure 2 · p. 5
  • Avoiding matrix inversion makes p-VQD more stable against shot noise, which can perturb ill-conditioned Euler–Lagrange matrices.The paper notes that shot noise is unavoidable even for fault-tolerant devices.Source: §3 · pp. 4–5
  • With increasing shot counts, both algorithms converge toward exact magnetization values; p-VQD gives comparable results to TDVA with about ten times fewer shots.The x-axis magnetization was hardest for both methods to capture, suggesting the ansatz choice may matter.Source: §3 · p. 5, Figure 3 · p. 5
  • Using the previous optimal parameter shift as the next initial guess often reduces optimization to one iteration.Figure 4 tracks optimization steps and per-step accuracy error for a 10^-5 threshold at δte = 0.05.Source: §3 · p. 5, Figure 4 · p. 6
  • p-VQD’s total computational cost scales as O(p) when optimization steps remain O(1); tighter per-step accuracy can increase shots and optimization steps.The reported measurement cost is linear in parameter count, versus quadratic for TDVA.Source: §3 · pp. 5–6
  • Global and local cost functions require similar iteration counts in the tested size range, but the authors advise benchmarking each system and size because barren plateaus may appear at larger scales.The comparison uses a state-vector simulator, ansatz depth 3, and a 10^-5 per-step accuracy target.Source: Figure 6 · p. 6, §3 · pp. 6–7

4 Discussion

The discussion presents p-VQD as more hardware-efficient and accurate than TDVA, while noting the importance of ansatz choice and a circuit-depth drawback.

4 Discussion

  • p-VQD is asymptotically more hardware efficient than TDVA while retaining higher accuracy.The authors also identify gradient estimation as a possible efficiency improvement and say controlling two-qubit gates may support small-system device simulations.Source: §4 · p. 7
  • Projected time evolution avoids numerical instabilities due to matrix inversion combined with statistical fluctuations from finite-shot measurements.Source: §4 · p. 7
  • The authors numerically investigate the absence of barren plateaus, which they say paves the way toward simulating larger quantum systems.Source: §4 · p. 7
  • The choice of parametrization is fundamental, and the device circuit is approximately twice as deep as the ansatz.Source: §4 · p. 7
  • Possible applications include two-dimensional interacting systems and comparisons with variational states based on tensor networks or neural networks.Source: §4 · p. 7

Data availability

The simulation code is available as open-source Python code using Qiskit on GitHub.

Data availability

A The projected real time evolution

The supplied passage consists of a bibliographic citation.

A The projected real time evolution

B Relationship of step-infidelity with time step and parameter variation

The appendix uses Taylor expansions of the overlap to explain the time-step scaling of the step-infidelity. In the small-step limit, the 1/δt^2 factor removes dependence on the time-step size.

B Relationship of step-infidelity with time step and parameter variation

C Equivalence between the p-VQD and the McLachlan’s variational principle

In the small-time-step limit, p-VQD yields the same parameter evolution as McLachlan’s variational principle. The derivation uses the overlap’s expansion and the quantum geometric tensor.

C Equivalence between the p-VQD and the McLachlan’s variational principle

D Optimisation routine for a single time-step

For a single time step, optimization reduces step-infidelity by nearly an order of magnitude at each step; initializing from the preceding update reduces the steps required.

D Optimisation routine for a single time-step

E Relationship between step infidelity and number of shots

With few gradient-sampling shots, Adam reaches infidelities up to two orders of magnitude below the minimum resolvable from those shots, while the optimizers become comparable as shots increase. The reported behavior depends on gradient variance and optimizer choice.

E Relationship between step infidelity and number of shots

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