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MarQSim: Reconciling Determinism and Randomness in Compiler Optimization for Quantum Simulation

Xiuqi Cao, Junyu Zhou, Yuhao Liu, Yunong Shi, Gushu Li

TL;DR

This paper tackles the challenge of efficiently compiling quantum Hamiltonian simulations by reconciling deterministic and randomized approaches. It introduces MarQSim, which casts Hamiltonian-term ordering as sampling from a Markov chain over a novel HTT Graph IR, and proves sufficient conditions on transition matrices to guarantee correct $e^{i\mathcal{H}t}$ simulation with an $\epsilon$ error bound. A Min-Cost Flow formulation is developed to construct transition matrices that preserve the required stationary distribution while enabling objectives like CNOT-gate cancellation, and this can be further enhanced by random perturbations to balance convergence speed and optimization gains. Empirical results across molecular and spin models demonstrate substantial CNOT and total-gate reductions with maintained fidelity, validating MarQSim as a scalable, high-level compiler for quantum Hamiltonian simulation with broad applicability to quantum algorithms.

Abstract

Quantum simulation, fundamental in quantum algorithm design, extends far beyond its foundational roots, powering diverse quantum computing applications. However, optimizing the compilation of quantum Hamiltonian simulation poses significant challenges. Existing approaches fall short in reconciling deterministic and randomized compilation, lack appropriate intermediate representations, and struggle to guarantee correctness. Addressing these challenges, we present MarQSim, a novel compilation framework. MarQSim leverages a Markov chain-based approach, encapsulated in the Hamiltonian Term Transition Graph, adeptly reconciling deterministic and randomized compilation benefits. We rigorously prove its algorithmic efficiency and correctness criteria. Furthermore, we formulate a Min-Cost Flow model that can tune transition matrices to enforce correctness while accommodating various optimization objectives. Experimental results demonstrate MarQSim's superiority in generating more efficient quantum circuits for simulating various quantum Hamiltonians while maintaining precision.

MarQSim: Reconciling Determinism and Randomness in Compiler Optimization for Quantum Simulation

TL;DR

This paper tackles the challenge of efficiently compiling quantum Hamiltonian simulations by reconciling deterministic and randomized approaches. It introduces MarQSim, which casts Hamiltonian-term ordering as sampling from a Markov chain over a novel HTT Graph IR, and proves sufficient conditions on transition matrices to guarantee correct simulation with an error bound. A Min-Cost Flow formulation is developed to construct transition matrices that preserve the required stationary distribution while enabling objectives like CNOT-gate cancellation, and this can be further enhanced by random perturbations to balance convergence speed and optimization gains. Empirical results across molecular and spin models demonstrate substantial CNOT and total-gate reductions with maintained fidelity, validating MarQSim as a scalable, high-level compiler for quantum Hamiltonian simulation with broad applicability to quantum algorithms.

Abstract

Quantum simulation, fundamental in quantum algorithm design, extends far beyond its foundational roots, powering diverse quantum computing applications. However, optimizing the compilation of quantum Hamiltonian simulation poses significant challenges. Existing approaches fall short in reconciling deterministic and randomized compilation, lack appropriate intermediate representations, and struggle to guarantee correctness. Addressing these challenges, we present MarQSim, a novel compilation framework. MarQSim leverages a Markov chain-based approach, encapsulated in the Hamiltonian Term Transition Graph, adeptly reconciling deterministic and randomized compilation benefits. We rigorously prove its algorithmic efficiency and correctness criteria. Furthermore, we formulate a Min-Cost Flow model that can tune transition matrices to enforce correctness while accommodating various optimization objectives. Experimental results demonstrate MarQSim's superiority in generating more efficient quantum circuits for simulating various quantum Hamiltonians while maintaining precision.
Paper Structure (36 sections, 5 theorems, 30 equations, 16 figures, 2 tables, 2 algorithms)

This paper contains 36 sections, 5 theorems, 30 equations, 16 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Qubit and Quantum State
  4. Quantum Program and Circuit
  5. Quantum Simulation
  6. Markov Stochastic Process
  7. Minimum-Cost Flow Problem
  8. Previous Works and Motivation
  9. Deterministic Compilation with Fixed Hamiltonian Term Order
  10. Randomized Compilation with Sampling the Hamiltonian Terms
  11. Comparison of the Two Types of Approaches and Our Motivation
  12. MarQSim Intermediate Representation and Compilation Algorithm
  13. Hamiltonian Term Transition Graph as Intermediate Representation
  14. Compiling the Hamiltonian Term Transition Graph to Quantum Circuit
  15. Compiler Optimization via Min-cost Flow
  16. ...and 21 more sections

Key Result

Theorem 4.1

Given a Hamiltonian $\mathcal{H}=\sum_{j=1}^n h_j H_j$, the quantum circuit compiled by Algorithm alg:ARQSC_FT correctly approximates the operator $e^{i\mathcal{H}t}$ if the HTT graph and the corresponding transition matrix $\ \mathbf{P}$ satisfies the following two conditions: The approximation error $\epsilon$ is bounded by where $\lambda$ is the sum of the absolute values of the Hamiltonian t

Figures (16)

  • Figure 1: Overview of MarQSim compiler framework
  • Figure 2: Quantum circuit example
  • Figure 3: Decomposition of ${\rm exp}({\rm i}X_4Y_3Z_2I_1\frac{\theta}{2})$
  • Figure 4: A Markov chain example. Left: state transition graph. Right: state transition matrix.
  • Figure 5: An MCFP example. The minimum cost solution with flow amount 1 is obtained and the actual flow on each edge is denoted by the red arrows attached to the edges.
  • ...and 11 more figures

Theorems & Definitions (21)

  • Definition 2.1: Markov Chain
  • Definition 2.2: Homogeneous Chain
  • Definition 2.3: Transition Matrix
  • Definition 2.4: State Transition Graph
  • Definition 2.5: Recurrent Class
  • Definition 2.6: Stationary Distribution
  • Remark 2.1: Properties of Stationary Distribution
  • Example 2.1
  • Definition 2.7: Flow Network
  • Definition 2.8: Minimum-Cost Flow Problem
  • ...and 11 more