Hamiltonian simulation with nearly optimal dependence on all parameters

TL;DR

The paper addresses the sparse Hamiltonian simulation problem for an $n$-qubit Hamiltonian $H$ with access to sparse-query oracles, target time $t$, and precision $\epsilon$, focusing on the parameter $\tau = d\|H\|_{\max} t$. It combines a Szegedy quantum walk with a fractional-query/LCU approach by forming a linear combination of walk steps $V_k=\sum_{m=-k}^{k} a_m U^m$ whose coefficients $a_m$ are tied to Bessel functions, enabling near-linear scaling in $\tau$ and logarithmic dependence on $1/\epsilon$. The main results include upper bounds on queries $O\left( \tau \frac{\log(\tau/\epsilon)}{\log\log(\tau/\epsilon)}\right)$ and gate counts $O\left( \tau [n + \log^{5/2}(\tau/\epsilon)] \frac{\log(\tau/\epsilon)}{\log\log(\tau/\epsilon)}\right)$, plus a tunable tradeoff giving $O\left( \tau^{1+\alpha/2} + \tau^{1-\alpha/2} \log(1/\epsilon)\right)$ for any $\alpha\in(0,1]$. A near-optimal lower bound $\Omega\left( \tau + \frac{\log(1/\epsilon)}{\log\log(1/\epsilon)} \right)$ is established, indicating the approach is close to optimal in both $\tau$ and $\epsilon$. This work advances quantum simulation by uniting two leading paradigms and providing strong scaling guarantees across all relevant parameters.

Abstract

We present an algorithm for sparse Hamiltonian simulation whose complexity is optimal (up to log factors) as a function of all parameters of interest. Previous algorithms had optimal or near-optimal scaling in some parameters at the cost of poor scaling in others. Hamiltonian simulation via a quantum walk has optimal dependence on the sparsity at the expense of poor scaling in the allowed error. In contrast, an approach based on fractional-query simulation provides optimal scaling in the error at the expense of poor scaling in the sparsity. Here we combine the two approaches, achieving the best features of both. By implementing a linear combination of quantum walk steps with coefficients given by Bessel functions, our algorithm's complexity (as measured by the number of queries and 2-qubit gates) is logarithmic in the inverse error, and nearly linear in the product $τ$ of the evolution time, the sparsity, and the magnitude of the largest entry of the Hamiltonian. Our dependence on the error is optimal, and we prove a new lower bound showing that no algorithm can have sublinear dependence on $τ$.

Theorems & Definitions (24)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 4: LCU Lemma
• Lemma 5
• proof
• Lemma 6: Robust oblivious amplitude amplification
• proof
• Lemma 7
• proof : Proof of Lemma \ref{['lem:approxV']}