[Paper Review] A Partially Random Trotter Algorithm for Quantum Hamiltonian Simulations

Shi Jin, Xiantao Li|arXiv (Cornell University)|Sep 16, 2021

Quantum Computing Algorithms and Architecture41 references4 citations

TL;DR

This paper proposes a hybrid Trotter-Suzuki algorithm for quantum Hamiltonian simulation that deterministically treats large-amplitude Hamiltonian terms while stochastically sampling the remaining terms. By balancing bias from operator splitting and variance from random sampling, the method achieves lower mean square error than purely deterministic or random approaches, with theoretical error bounds and gate count estimates validated numerically.

ABSTRACT

Given the Hamiltonian, the evaluation of unitary operators has been at the heart of many quantum algorithms. Motivated by existing deterministic and random methods, we present a hybrid approach, where Hamiltonians with large amplitude are evaluated at each time step, while the remaining terms are evaluated at random. The bound for the mean square error is obtained, together with a concentration bound. The mean square error consists of a variance term and a bias term, arising respectively from the random sampling of the Hamiltonian terms and the operator splitting error. Leveraging on the bias/variance trade-off, the error can be minimized by balancing the two. The concentration bound provides an estimate on the number of gates. The estimates are verified by using numerical experiments on classical computers.

Motivation & Objective

• To address the high gate count and error in fully deterministic Trotter-Suzuki methods for large-scale quantum Hamiltonians.
• To reduce variance in random sampling methods by excluding large-amplitude terms from stochastic evaluation.
• To establish a theoretical framework for mean square error that captures the bias-variance trade-off in hybrid quantum simulation.
• To derive concentration bounds for estimating required gate counts to achieve target accuracy.
• To validate the error model and performance gains through numerical simulations on quantum spin chains.

Proposed method

• The Hamiltonian is partitioned into two groups: large-amplitude terms treated deterministically via standard Trotter splitting, and smaller terms treated stochastically via random sampling.
• The algorithm uses a hybrid Trotter step where deterministic evolution of dominant terms is combined with random selection of remaining terms at each time step.
• Error analysis decomposes the mean square error into a bias term (from operator splitting) and a variance term (from random sampling), following a standard statistical bias-variance trade-off.
• Concentration inequalities are derived to estimate the number of gates required to achieve a desired error probability with high confidence.
• The method allows flexible random sampling strategies, including both uniform and importance sampling, with the latter shown to reduce variance.
• Numerical experiments on the Heisenberg spin chain validate the theoretical error bounds and demonstrate improved accuracy over purely deterministic or random approaches.

Experimental results

Research questions

• RQ1 Can a hybrid approach that treats large-amplitude Hamiltonian terms deterministically and small ones stochastically reduce overall simulation error in quantum Hamiltonian dynamics?
• RQ2 How does the mean square error in the hybrid algorithm decompose into bias and variance components, and can this decomposition be used to optimize performance?
• RQ3 What is the optimal partitioning of Hamiltonian terms between deterministic and stochastic treatment to minimize total error?
• RQ4 How do different sampling strategies—uniform vs. importance sampling—affect the variance and gate count in the hybrid algorithm?
• RQ5 Can concentration bounds be derived to estimate the number of quantum gates required to achieve a given accuracy with high probability?

Key findings

• The hybrid algorithm achieves lower mean square error than both fully deterministic and fully random Trotter methods on the Heisenberg chain with n=12 and 2^12 gates.
• Importance sampling for random terms yields lower variance and better accuracy than uniform sampling, confirming its superiority in variance reduction.
• The mean square error exhibits a clear bias-variance trade-off, with optimal performance achieved when the two error components are balanced.
• Theoretical error bounds are validated numerically, showing good agreement with observed mean square error across different system sizes and gate counts.
• Using batch sizes K=4 or K=8 for random sampling yields similar performance to K=1, suggesting that multiple random terms per step can be used without significant loss in accuracy.
• The optimal partitioning of Hamiltonian terms—where to set the cutoff between deterministic and stochastic treatment—tends to occur at the point where coefficient magnitudes drop significantly, though a systematic method remains an open challenge.