Concentration for random product formulas

TL;DR

This paper analyzes the qDRIFT randomized quantum simulation method, showing that a typical realization approximates the ideal evolution with small error, and that the gate complexity depends on system size and interaction strengths, not on the number of Hamiltonian terms.

Contribution

The work provides a comprehensive analysis of a single realization of qDRIFT, revealing its efficiency and conditions under which it approximates the ideal quantum evolution.

Findings

Typical realization approximates ideal evolution with small diamond-norm error.

Gate complexity depends on system size and interaction strengths, not on Hamiltonian term count.

Randomized evolution from a fixed input state yields shorter circuits, unlike deterministic methods.

Abstract

Quantum simulation has wide applications in quantum chemistry and physics. Recently, scientists have begun exploring the use of randomized methods for accelerating quantum simulation. Among them, a simple and powerful technique, called qDRIFT, is known to generate random product formulas for which the average quantum channel approximates the ideal evolution. qDRIFT achieves a gate count that does not explicitly depend on the number of terms in the Hamiltonian, which contrasts with Suzuki formulas. This work aims to understand the origin of this speed-up by comprehensively analyzing a single realization of the random product formula produced by qDRIFT. The main results prove that a typical realization of the randomized product formula approximates the ideal unitary evolution up to a small diamond-norm error. The gate complexity is already independent of the number of terms in the Hamiltonian, but it depends on the system size and the sum of the interaction strengths in the Hamiltonian. Remarkably, the same random evolution starting from an arbitrary, but fixed, input state yields a much shorter circuit suitable for that input state. In contrast, in deterministic settings, such an improvement usually requires initial state knowledge. The proofs depend on concentration inequalities for vector and matrix martingales, and the framework is applicable to other randomized product formulas. Our bounds are saturated by certain commuting Hamiltonians.