Parallel Quantum Algorithm for Hamiltonian Simulation

TL;DR

This paper introduces a parallel quantum algorithm for Hamiltonian simulation that significantly reduces the dependence on simulation precision, achieving an exponential improvement over previous methods by leveraging parallel quantum walks and a truncated Taylor series.

Contribution

The paper presents a novel parallel quantum algorithm for simulating Hamiltonians with polylogarithmic log(1/ε) dependence, improving efficiency over prior algorithms without parallelism.

Findings

Achieves doubly logarithmic dependence on precision in circuit depth.

Introduces a new notion of parallel quantum walk based on Childs' quantum walk.

Demonstrates effectiveness on physical models like Heisenberg, Sachdev-Ye-Kitaev, and quantum chemistry models.

Abstract

We study how parallelism can speed up quantum simulation. A parallel quantum algorithm is proposed for simulating the dynamics of a large class of Hamiltonians with good sparse structures, called uniform-structured Hamiltonians, including various Hamiltonians of practical interest like local Hamiltonians and Pauli sums. Given the oracle access to the target sparse Hamiltonian, in both query and gate complexity, the running time of our parallel quantum simulation algorithm measured by the quantum circuit depth has a doubly (poly-)logarithmic dependence $\operatorname{polylog}\log(1/\epsilon)$ on the simulation precision $\epsilon$. This presents an exponential improvement over the dependence $\operatorname{polylog}(1/\epsilon)$ of previous optimal sparse Hamiltonian simulation algorithm without parallelism. To obtain this result, we introduce a novel notion of parallel quantum walk, based on Childs' quantum walk. The target evolution unitary is approximated by a truncated Taylor series, which is obtained by combining these quantum walks in a parallel way. A lower bound $\Omega(\log \log (1/\epsilon))$ is established, showing that the $\epsilon$-dependence of the gate depth achieved in this work cannot be significantly improved. Our algorithm is applied to simulating three physical models: the Heisenberg model, the Sachdev-Ye-Kitaev model and a quantum chemistry model in second quantization. By explicitly calculating the gate complexity for implementing the oracles, we show that on all these models, the total gate depth of our algorithm has a $\operatorname{polylog}\log(1/\epsilon)$ dependence in the parallel setting.