Quantum simulation suggests a path forward for the efficient simulation of problems in condensed-matter physics, quantum chemistry and materials science. While the majority of quantum simulation algorithms is deterministic, a recent surge of ideas has shown that randomization can greatly benefit algorithmic performance. In this work, we introduce a scheme for quantum simulation that unites the advantages of randomized compiling and higher-order multi-product formulas as they are used in linear-combination-of-unitaries (LCU) algorithms. In doing so, we propose a framework of randomized sampling that could prove useful for programmable quantum simulators and present two new algorithms tailored to it. Our framework greatly reduces the circuit depth by circumventing the need for oblivious amplitude amplification required by standard LCU methods, rendering it especially useful for near-term quantum computing. Our algorithms achieve a simulation error that shrinks exponentially with the circuit depth. To corroborate their functioning, we prove rigorous performance bounds as well as the concentration of the randomized sampling procedure. We also discuss examples at hand of a Hamiltonian comprising pairwise anticommuting terms.Quantum simulation suggests a path forward for the efficient simulation of problems in condensed-matter physics, quantum chemistry and materials science. While the majority of quantum simulation algorithms is deterministic, a recent surge of ideas has shown that randomization can greatly benefit algorithmic performance. In this work, we introduce a scheme for quantum simulation that unites the advantages of randomized compiling and higher-order multi-product formulas as they are used in linear-combination-of-unitaries (LCU) algorithms. In doing so, we propose a framework of randomized sampling that could prove useful for programmable quantum simulators and present two new algorithms tailored to it. Our framework greatly reduces the circuit depth by circumventing the need for oblivious amplitude amplification required by standard LCU methods, rendering it especially useful for near-term quantum computing. Our algorithms achieve a simulation error that shrinks exponentially with the circuit depth. To corroborate their functioning, we prove rigorous performance bounds as well as the concentration of the randomized sampling procedure. We also discuss examples at hand of a Hamiltonian comprising pairwise anticommuting terms.

Jan 19, 20217 citations

A perspective on quantum simulations

E. Solano

Proceedings of the International School of Physics "Enrico Fermi"Jan 1, 20180 citations

Randomized Algorithms and Lower Bounds for Quantum Simulation

Zhang, Chi

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 $Ω(t^{3/2}\e^{-1/2})$ exponentials.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 $Ω(t^{3/2}\e^{-1/2})$ exponentials.

arXiv:0910.4145Oct 21, 20091 citation

The Simulation of Quantum Systems

David M. Ceperley

Recent Progress in Many-Body TheoriesJan 1, 19953 citations

Title: The Geometry of Randomness – Characterizing Fractal Behavior with Quantum Simulation

Jincheng Zhang

This paper investigates the geometry of randomness, specifically focusing on characterizing fractal behavior using quantum simulation. The core claim is to develop a novel quantum simulation technique to unveil the underlying fractal structure of random systems, thereby providing a deeper understanding of their complex dynamics. The method aims to emulate the evolution of a random system through quantum mechanics, allowing us to directly observe and quantify the emergence of fractal patterns. The research will explore the potential of this approach to surpass current approximation methods, offering a fundamentally new pathway to analyze and quantify fractal geometry.This paper investigates the geometry of randomness, specifically focusing on characterizing fractal behavior using quantum simulation. The core claim is to develop a novel quantum simulation technique to unveil the underlying fractal structure of random systems, thereby providing a deeper understanding of their complex dynamics. The method aims to emulate the evolution of a random system through quantum mechanics, allowing us to directly observe and quantify the emergence of fractal patterns. The research will explore the potential of this approach to surpass current approximation methods, offering a fundamentally new pathway to analyze and quantify fractal geometry.

Sep 1, 20260 citations

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 $Ω(t^{3/2}\e^{-1/2})$ exponentials.