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A unifying framework for quantum simulation algorithms for time-dependent Hamiltonian dynamics

Yu Cao, Shi Jin, Nana Liu

Abstract

Recently, there has been growing interest in simulating time-dependent Hamiltonians using quantum algorithms, driven by diverse applications, such as quantum adiabatic computing. While techniques for simulating time-independent Hamiltonian dynamics are well-established, time-dependent Hamiltonian dynamics is less explored and it is unclear how to systematically organize existing methods and to find new methods. Sambe-Howland's continuous clock elegantly transforms time-dependent Hamiltonian dynamics into time-independent Hamiltonian dynamics, which means that by taking different discretizations, existing methods for time-independent Hamiltonian dynamics can be exploited for time-dependent dynamics. In this work, we systemically investigate how Sambe-Howland's clock can serve as a unifying framework for simulating time-dependent Hamiltonian dynamics. Firstly, we demonstrate the versatility of this approach by showcasing its compatibility with analog quantum computing and digital quantum computing. Secondly, for digital quantum computers, we illustrate how this framework, combined with time-independent methods (e.g., product formulas, multi-product formulas, qDrift, and LCU-Taylor), can facilitate the development of efficient algorithms for simulating time-dependent dynamics. This framework allows us to (a) resolve the problem of finding minimum-gate time-dependent product formulas; (b) establish a unified picture of both Suzuki's and Huyghebaert and De Raedt's approaches; (c) generalize Huyghebaert and De Raedt's first and second-order formula to arbitrary orders; (d) answer an unsolved question in establishing time-dependent multi-product formulas; (e) and recover continuous qDrift on the same footing as time-independent qDrift. Thirdly, we demonstrate the efficacy of our newly developed higher-order Huyghebaert and De Raedt's algorithm through digital adiabatic simulation.

A unifying framework for quantum simulation algorithms for time-dependent Hamiltonian dynamics

Abstract

Recently, there has been growing interest in simulating time-dependent Hamiltonians using quantum algorithms, driven by diverse applications, such as quantum adiabatic computing. While techniques for simulating time-independent Hamiltonian dynamics are well-established, time-dependent Hamiltonian dynamics is less explored and it is unclear how to systematically organize existing methods and to find new methods. Sambe-Howland's continuous clock elegantly transforms time-dependent Hamiltonian dynamics into time-independent Hamiltonian dynamics, which means that by taking different discretizations, existing methods for time-independent Hamiltonian dynamics can be exploited for time-dependent dynamics. In this work, we systemically investigate how Sambe-Howland's clock can serve as a unifying framework for simulating time-dependent Hamiltonian dynamics. Firstly, we demonstrate the versatility of this approach by showcasing its compatibility with analog quantum computing and digital quantum computing. Secondly, for digital quantum computers, we illustrate how this framework, combined with time-independent methods (e.g., product formulas, multi-product formulas, qDrift, and LCU-Taylor), can facilitate the development of efficient algorithms for simulating time-dependent dynamics. This framework allows us to (a) resolve the problem of finding minimum-gate time-dependent product formulas; (b) establish a unified picture of both Suzuki's and Huyghebaert and De Raedt's approaches; (c) generalize Huyghebaert and De Raedt's first and second-order formula to arbitrary orders; (d) answer an unsolved question in establishing time-dependent multi-product formulas; (e) and recover continuous qDrift on the same footing as time-independent qDrift. Thirdly, we demonstrate the efficacy of our newly developed higher-order Huyghebaert and De Raedt's algorithm through digital adiabatic simulation.

Paper Structure

This paper contains 52 sections, 13 theorems, 130 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Problem setup
  3. Sambe-Howland’s clock
  4. Related works for time-dependent Hamiltonian simulation
  5. Summary of results
  6. Notations
  7. Preliminaries: computational models, applications, and more on Sambe-Howland's clock
  8. Computational models
  9. The idealized case ($\omega\to 0$)
  10. For analog quantum computing ($\omega>0$)
  11. Digitized adiabatic simulation
  12. The functional space of Sambe-Howland's clock
  13. Sambe-Howland's clock and the perspective of numerical PDEs
  14. Analog quantum computing
  15. High-order approximation via Richardson's extrapolation
  16. ...and 37 more sections

Key Result

Theorem 3

Recall $\rho_\omega$ from eqn::trace_Psi and suppose $G$ is given in eqn::G. For any observable $\widehat{O}$ with $\norm{\widehat{O} }_{\infty} \le 1$, and suppose coefficients $\{\alpha_j\}_{j=1}^{M} \subseteq \mathbb{R}$ and integer-valued time steps $\{k_j\}_{j=1}^{M} \subseteq \mathbb{N}$ are w Then where $C_m$ is a constant whose expression can be found in eqn::Cm.

Figures (3)

  • Figure 1: Averaged simulation error (in trace distance) for Grover's search algorithm with respect to the gate counts $N$ for various time-dependent schemes. "MIN-(I)" refers to the scheme in \ref{['eqn::product_td']} with the ordering $H_1, \hat{p}, H_2$. HDR refers to the scheme in \ref{['eqn::product_td_v2']}. IACS refers to the scheme in ikeda_minimum_2023; see \ref{['eqn::FRS_Magnus']}. The blue dashed line indicates the scaling $\order{N^{-4}}$ and is the same for all subplots.
  • Figure 2: Averaged simulation error in trace distance for adiabatic Google's PageRank algorithm with respect to the gate counts $N$ for various time-dependent schemes. "MIN-(I)" refers to the scheme \ref{['eqn::product_td']} with the ordering $H_1, \hat{p}, H_2$. HDR refers to the scheme in \ref{['eqn::product_td_v2']}. IACS refers to the scheme in ikeda_minimum_2023; see also \ref{['eqn::FRS_Magnus']}. The blue dashed line indicates the scaling $\order{N^{-4}}$ and is the same for all subplots.
  • Figure 3: Averaged simulation error in trace distance for the quantum Ising model with respect to the gate counts $N$ for various time-dependent schemes. "MIN-(I)" refers to the scheme \ref{['eqn::product_td']} with the ordering $H_1, \hat{p}, H_2$. HDR refers to the scheme in \ref{['eqn::product_td_v2']}. IACS refers to the scheme in ikeda_minimum_2023; see also \ref{['eqn::FRS_Magnus']}. The blue dashed line indicates the scaling $\order{N^{-4}}$ and is the same for all subplots.

Theorems & Definitions (24)

  • Remark 1
  • Theorem 3
  • Theorem 4: Choice of $\omega$ for fully analog digitized adiabatic simulation
  • Remark 2
  • Theorem 5: ostmeyer_optimised_2023
  • Theorem 6
  • proof
  • Lemma 7
  • Remark 3
  • Theorem 8
  • ...and 14 more