Randomized Algorithms and Lower Bounds for Quantum Simulation

TL;DR

This paper analyzes randomized and deterministic quantum algorithms for simulating exponential of Hamiltonians, providing bounds on their accuracy and complexity, and demonstrating the efficiency of certain randomized approaches.

Contribution

It introduces a scheme to bound the trace distance in randomized quantum algorithms and establishes lower bounds on the number of exponentials needed for simulation.

Findings

Randomized algorithms can match deterministic efficiency but are simpler.

Both algorithm types require at least (t^{3/2}\u03b5^{-1/2}) exponentials for simulation.

A scheme to bound the trace distance of randomized algorithms' final states.

Abstract

We consider deterministic and {\em randomized} quantum algorithms simulating $e^{-iHt}$ by a product of unitary operators $e^{-iA_jt_j}$, $j=1,...,N$, where $A_j\in\{H_1,...,H_m\}$, $H=\sum_{i=1}^m H_i$ and $t_j > 0$ for every $j$. Randomized algorithms are algorithms approximating the final state of the system by a mixed quantum state. First, we provide a scheme to bound the trace distance of the final quantum states of randomized algorithms. Then, we show some randomized algorithms, which have the same efficiency as certain deterministic algorithms, but are less complicated than their opponentes. Moreover, we prove that both deterministic and randomized algorithms simulating $e^{-iHt}$ with error $\e$ at least have $\Omega(t^{3/2}\e^{-1/2})$ exponentials.