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Hamiltonian Simulation Using Linear Combinations of Unitary Operations

Andrew M. Childs, Nathan Wiebe

TL;DR

The paper introduces a non-deterministic framework to implement linear combinations of unitaries (LCU) for quantum simulation, enabling multi-product formulas to be realized on quantum hardware. This approach yields improved error scaling and competitive or better complexity than traditional product-formula methods, with a detailed analysis of success probabilities and resource counts. The authors provide an optimality result for the LCU protocol within a broad class of circuits and develop comprehensive error bounds and iteration strategies to simulate $e^{-iHt}$ with controllable accuracy $\epsilon$ and failure probability $\beta$. Overall, the work proposes a new paradigm for quantum Hamiltonian simulation that can influence both quantum algorithms and numerical analysis by enabling coherent averaging of approximations.

Abstract

We present a new approach to simulating Hamiltonian dynamics based on implementing linear combinations of unitary operations rather than products of unitary operations. The resulting algorithm has superior performance to existing simulation algorithms based on product formulas and, most notably, scales better with the simulation error than any known Hamiltonian simulation technique. Our main tool is a general method to nearly deterministically implement linear combinations of nearby unitary operations, which we show is optimal among a large class of methods.

Hamiltonian Simulation Using Linear Combinations of Unitary Operations

TL;DR

The paper introduces a non-deterministic framework to implement linear combinations of unitaries (LCU) for quantum simulation, enabling multi-product formulas to be realized on quantum hardware. This approach yields improved error scaling and competitive or better complexity than traditional product-formula methods, with a detailed analysis of success probabilities and resource counts. The authors provide an optimality result for the LCU protocol within a broad class of circuits and develop comprehensive error bounds and iteration strategies to simulate with controllable accuracy and failure probability . Overall, the work proposes a new paradigm for quantum Hamiltonian simulation that can influence both quantum algorithms and numerical analysis by enabling coherent averaging of approximations.

Abstract

We present a new approach to simulating Hamiltonian dynamics based on implementing linear combinations of unitary operations rather than products of unitary operations. The resulting algorithm has superior performance to existing simulation algorithms based on product formulas and, most notably, scales better with the simulation error than any known Hamiltonian simulation technique. Our main tool is a general method to nearly deterministically implement linear combinations of nearby unitary operations, which we show is optimal among a large class of methods.

Paper Structure

This paper contains 8 sections, 13 theorems, 89 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Adding and Subtracting Unitary Operations Using Quantum Computers
  3. Implementing Multi-Product Formulas on Quantum Computers
  4. Review of Lie--Trotter--Suzuki and Multi-Product Formulas
  5. Implementing Multi-Product Formulas Using Quantum Computers
  6. Analysis of Simulation and Errors in Multi-product Formulas
  7. Conclusions
  8. Optimality of the linear combination procedure

Key Result

Theorem 1

Let the system Hamiltonian be $H=\sum_{j=1}^m H_j$ where each $H_j\in \mathbb{C}^{2^n\times 2^n}$ is Hermitian and satisfies $\|{H_j}\| \le h$ for a given constant $h$. Then the Hamiltonian evolution $e^{-iHt}$ can be simulated on a quantum computer with failure probability and error at most $\epsil elementary operations and exponentials of the $H_j$s.

Figures (3)

  • Figure 1: Quantum circuit for non-deterministically performing an operator proportional to $\kappa U_a + U_b$ given a measurement outcome of zero.
  • Figure 2: Scaling of $\kappa$ with $k$ for three values of $\gamma$ centered around $\gamma_c \colonequals 1+\log(\eta)/2$. The data show that $\kappa$ approaches $1$ polynomially quickly if $\gamma=\gamma_c$, whereas a slight increase in $\gamma$ causes $\kappa$ to grow exponentially and a slight decrease causes $\kappa$ to converge to $1$ exponentially with $k$.
  • Figure 3: A general circuit for implementing a linear combination of $k+1$ unitary operators using unitary operations $A$ and $B$. We assume for simplicity that $k+1$ is an integer power of $2$. This circuit corresponds to preparing the ancilla states in an arbitrary state (specified by $A$) and measuring them in an arbitrary basis (specified by $B$).

Theorems & Definitions (27)

  • Theorem 1
  • Lemma 2
  • proof
  • Theorem 3
  • proof
  • Definition 1
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • ...and 17 more