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Amplitude-Phase Separation toward Optimal and Fast-Forwardable Simulation of Non-Unitary Dynamics

Qitong Hu, Shi Jin

TL;DR

A generic framework is established, referred to as the Amplitude-Phase Separation methods, which formulates any non-unitary evolution into separate simulation of a unitary operator and a Hermitian operator, to allow one to take best advantage of, and to even improve existing algorithms, developed for unitary or Hermitian evolution respectively.

Abstract

Quantum simulation of the linear non-unitary dynamics is crucial in scientific computing. In this work, we establish a generic framework, referred to as the Amplitude-Phase Separation (APS) methods, which formulates any non-unitary evolution into separate simulation of a unitary operator and a Hermitian operator, thus allow one to take best advantage of, and to even improve existing algorithms, developed for unitary or Hermitian evolution respectively. We utilize two techniques: the first achieves a provably optimal query complexity via a shifted Dyson series; the second breaks the conventional linear dependency, achieving fast-forwarding by exhibiting a square-root dependence on the norm of the dissipative part. Furthermore, one can derive existing methods such as the LCHS (Linear Combination of Hamiltonian Simulation) and the NDME (Non-Diagonal Density Matrix Encoding) methods from APS. The APS provides an effective and generic pathway for developing efficient quantum algorithms for general non-unitary dynamics to achieve either optimal query complexity or fast-forwarding property, outperforming the existing algorithms for the same problems.

Amplitude-Phase Separation toward Optimal and Fast-Forwardable Simulation of Non-Unitary Dynamics

TL;DR

A generic framework is established, referred to as the Amplitude-Phase Separation methods, which formulates any non-unitary evolution into separate simulation of a unitary operator and a Hermitian operator, to allow one to take best advantage of, and to even improve existing algorithms, developed for unitary or Hermitian evolution respectively.

Abstract

Quantum simulation of the linear non-unitary dynamics is crucial in scientific computing. In this work, we establish a generic framework, referred to as the Amplitude-Phase Separation (APS) methods, which formulates any non-unitary evolution into separate simulation of a unitary operator and a Hermitian operator, thus allow one to take best advantage of, and to even improve existing algorithms, developed for unitary or Hermitian evolution respectively. We utilize two techniques: the first achieves a provably optimal query complexity via a shifted Dyson series; the second breaks the conventional linear dependency, achieving fast-forwarding by exhibiting a square-root dependence on the norm of the dissipative part. Furthermore, one can derive existing methods such as the LCHS (Linear Combination of Hamiltonian Simulation) and the NDME (Non-Diagonal Density Matrix Encoding) methods from APS. The APS provides an effective and generic pathway for developing efficient quantum algorithms for general non-unitary dynamics to achieve either optimal query complexity or fast-forwarding property, outperforming the existing algorithms for the same problems.
Paper Structure (23 sections, 20 theorems, 145 equations, 1 figure, 1 table)

This paper contains 23 sections, 20 theorems, 145 equations, 1 figure, 1 table.

Table of Contents

  1. Amplitude-Phase Separation
  2. Phase-Driven APS
  3. Amplitude-Driven APS
  4. Relation with Product Formula
  5. Optimality for Non-Unitary Dynamics
  6. Shifted Dyson Series for Dissipative Operators
  7. Non-Unitary Optimality = Phase-Driven APS + Shifted Dyson Series
  8. Improved Query Complexity for Time-Dependent Case
  9. Extensions of the APS Framework
  10. Ill-Posed Problems
  11. Inhomogeneous Term
  12. Fast-Forwarding for Time-Independent Non-Unitary Dynamics
  13. Fast-Forwarding for Dissipative Operators
  14. Time-Independent Case
  15. Piecewise Time-Independent Case
  16. ...and 8 more sections

Key Result

Theorem 1

There exists a quantum algorithm for simulating $e^{-At}$ with $A_1\succeq 0$ to within error tolerance $\varepsilon$, whose query complexity to $\mathrm{HAM_A}$ is where $\mathcal{O}\left(\frac{\|u_0\|}{\|u(t)\|}\right)$ arises from the dissipative term, corresponding to the number of queries to $u_0$ and the number of repetitions, and whose gate complexity scales as $\mathcal{O}(\log \varepsilo

Figures (1)

  • Figure 1: Amplitude-Phase Separation (APS) for Non-Unitary Dynamics.

Theorems & Definitions (35)

  • Theorem 1
  • Theorem 2
  • Theorem A.1
  • proof
  • Theorem A.2
  • proof
  • Theorem B.1
  • proof
  • Lemma B.1
  • proof
  • ...and 25 more