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Optimal Hamiltonian Simulation by Quantum Signal Processing

Guang Hao Low, Isaac L. Chuang

TL;DR

It is argued that physical intuition can lead to optimal simulation methods by showing that a focus on simple single-qubit rotations elegantly furnishes an optimal algorithm for Hamiltonian simulation, a universal problem that encapsulates all the power of quantum computation.

Abstract

The physics of quantum mechanics is the inspiration for, and underlies, quantum computation. As such, one expects physical intuition to be highly influential in the understanding and design of many quantum algorithms, particularly simulation of physical systems. Surprisingly, this has been challenging, with current Hamiltonian simulation algorithms remaining abstract and often the result of sophisticated but unintuitive constructions. We contend that physical intuition can lead to optimal simulation methods by showing that a focus on simple single-qubit rotations elegantly furnishes an optimal algorithm for Hamiltonian simulation, a universal problem that encapsulates all the power of quantum computation. Specifically, we show that the query complexity of implementing time evolution by a $d$-sparse Hamiltonian $\hat{H}$ for time-interval $t$ with error $ε$ is $\mathcal{O}(td\|\hat{H}\|_{\text{max}}+\frac{\log{(1/ε)}}{\log{\log{(1/ε)}}})$, which matches lower bounds in all parameters. This connection is made through general three-step "quantum signal processing" methodology, comprised of (1) transducing eigenvalues of $\hat{H}$ into a single ancilla qubit, (2) transforming these eigenvalues through an optimal-length sequence of single-qubit rotations, and (3) projecting this ancilla with near unity success probability.

Optimal Hamiltonian Simulation by Quantum Signal Processing

TL;DR

It is argued that physical intuition can lead to optimal simulation methods by showing that a focus on simple single-qubit rotations elegantly furnishes an optimal algorithm for Hamiltonian simulation, a universal problem that encapsulates all the power of quantum computation.

Abstract

The physics of quantum mechanics is the inspiration for, and underlies, quantum computation. As such, one expects physical intuition to be highly influential in the understanding and design of many quantum algorithms, particularly simulation of physical systems. Surprisingly, this has been challenging, with current Hamiltonian simulation algorithms remaining abstract and often the result of sophisticated but unintuitive constructions. We contend that physical intuition can lead to optimal simulation methods by showing that a focus on simple single-qubit rotations elegantly furnishes an optimal algorithm for Hamiltonian simulation, a universal problem that encapsulates all the power of quantum computation. Specifically, we show that the query complexity of implementing time evolution by a -sparse Hamiltonian for time-interval with error is , which matches lower bounds in all parameters. This connection is made through general three-step "quantum signal processing" methodology, comprised of (1) transducing eigenvalues of into a single ancilla qubit, (2) transforming these eigenvalues through an optimal-length sequence of single-qubit rotations, and (3) projecting this ancilla with near unity success probability.

Paper Structure

This paper contains 3 theorems, 18 equations, 1 figure.

Key Result

Theorem 1

$\forall$ even $N>0$, a choice of real functions $A(\theta),C(\theta)$ can be implemented by some $\vec{\phi}\in\mathbb{R}^N$ if and only if all these are true: (1) $\forall \theta\in\mathbb{R},\;A^2(\theta)+C^2(\theta)\le 1$.$\quad$ (2) $A(0)=1$. (3) $A(\theta)=\sum^{N/2}_{k=0}a_k \cos{(k\theta)},

Figures (1)

  • Figure 1: Quantum circuits mapping (a) a sequence of single-qubit rotations $\hat{V}(\theta)$ to (d) quantum signal processing $\hat{V}$. Each single-qubit rotation $\hat{R}_\phi(\theta)$ is replaced by (b) $\hat{U}_\phi$, built from Hadamard gates and controlled-$W$ with eigenstates $\hat{W}|u_\lambda\rangle=e^{i\theta_\lambda}|u_\lambda\rangle$. Thus (c) $\hat{U}_\phi$ on input $|u_\lambda\rangle$ reduces to a single-qubit rotation $\hat{R}_\phi(\theta_\lambda)$. By linearity, $\hat{V}$ on an arbitrary input $|\psi\rangle$ may be understood as rotations $\hat{V}(\theta_\lambda)$ controlled by a superposition of $|u_\lambda\rangle$. By some choice of single-qubit input state and measurement basis, coefficients of the $|u_\lambda\rangle$ are then rescaled by the components of the function $\hat{V}(\theta_\lambda)$ programmed by $\vec{\phi}$.

Theorems & Definitions (3)

  • Theorem 1: Achievable ($A$,$C$)
  • Theorem 2: Quantum Signal Processing
  • Theorem 3: Optimal sparse Hamiltonian simulation