Composite QDrift-Product Formulas for Quantum and Classical Simulations in Real and Imaginary Time

TL;DR

This paper introduces a unified composite channel approach for quantum and classical simulations in real and imaginary time, demonstrating significant efficiency improvements for various Hamiltonians including a 20-fold speedup for Jellium.

Contribution

It extends the composite channel method to imaginary time and local Hamiltonians, unifying quantum and classical simulation techniques with rigorous error bounds.

Findings

Achieves constant factor speedups for multiple Hamiltonians

Provides upper bounds on simulation norms and errors

Demonstrates a 20-fold speedup for Jellium

Abstract

Recent work has shown that it can be advantageous to implement a composite channel that partitions the Hamiltonian $H$ for a given simulation problem into subsets $A$ and $B$ such that $H=A+B$, where the terms in $A$ are simulated with a Trotter-Suzuki channel and the $B$ terms are randomly sampled via the QDrift algorithm. Here we show that this approach holds in imaginary time, making it a candidate classical algorithm for quantum Monte-Carlo calculations. We upper-bound the induced Schatten-$1 \to 1$ norm on both imaginary-time QDrift and Composite channels. Another recent result demonstrated that simulations of Hamiltonians containing geometrically-local interactions for systems defined on finite lattices can be improved by decomposing $H$ into subsets that contain only terms supported on that subset of the lattice using a Lieb-Robinson argument. Here, we provide a quantum algorithm by unifying this result with the composite approach into ``local composite channels" and we upper bound the diamond distance. We provide exact numerical simulations of algorithmic cost by counting the number of gates of the form $e^{-iH_j t}$ and $e^{-H_j \beta}$ to meet a certain error tolerance $\epsilon$. We show constant factor advantages for a variety of interesting Hamiltonians, the maximum of which is a $\approx 20$ fold speedup that occurs for a simulation of Jellium.