Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Dual channel multi-product formulas

Seung Park, Sangjin Lee, Kyunghyun Baek

TL;DR

This work advances quantum simulation accuracy by introducing the dual-channel multi-product formula (DCMPF), which pairs a PF circuit with its inverse to double the Trotter error suppression from $O(t^{\alpha+K})$ to $O(t^{\alpha+2K})$ while reducing required circuit depth by roughly a factor of two. The authors provide theoretical proofs (via operator- and expectation-value analyses) and numerical demonstrations on the 1D transverse Ising model and the XXZ spin chain, showing DCMPF outperforms conventional MPF under fixed circuit depth and noise. Key results include a formal proof that DCMPF preserves conditioning and variance properties and practical evidence that a shallower circuit with more foldings yields smaller algorithmic errors without increasing sampling error. The work suggests significant practical gains for near-term quantum devices by mitigating algorithmic errors more efficiently, potentially lowering hardware error mitigation overhead and enabling higher-precision simulations within constrained resources.

Abstract

Product-formula (PF) based quantum simulation is a promising approach for simulating quantum systems on near-term quantum computers. Achieving a desired simulation precision typically requires a polynomially increasing number of Trotter steps, which remains challenging due to the limited performance of current quantum hardware. To alleviate this issue, post-processing techniques such as the multi-product formula (MPF) have been introduced to suppress algorithmic errors within restricted hardware resources. In this work, we propose a dual-channel multi-product formula that achieves a two-fold improvement in Trotter error scaling. As a result, our method enables the target simulation precision to be reached with approximately half the circuit depth compared to conventional MPF schemes. Importantly, the reduced circuit depth directly translates into lower physical error mitigation overhead when implemented on real quantum hardware. We demonstrate that, for a fixed CNOT count as a measure of quantum circuit, our proposal yields significantly smaller algorithmic errors, while the sampling error remains essentially unchanged.

Dual channel multi-product formulas

TL;DR

This work advances quantum simulation accuracy by introducing the dual-channel multi-product formula (DCMPF), which pairs a PF circuit with its inverse to double the Trotter error suppression from to while reducing required circuit depth by roughly a factor of two. The authors provide theoretical proofs (via operator- and expectation-value analyses) and numerical demonstrations on the 1D transverse Ising model and the XXZ spin chain, showing DCMPF outperforms conventional MPF under fixed circuit depth and noise. Key results include a formal proof that DCMPF preserves conditioning and variance properties and practical evidence that a shallower circuit with more foldings yields smaller algorithmic errors without increasing sampling error. The work suggests significant practical gains for near-term quantum devices by mitigating algorithmic errors more efficiently, potentially lowering hardware error mitigation overhead and enabling higher-precision simulations within constrained resources.

Abstract

Product-formula (PF) based quantum simulation is a promising approach for simulating quantum systems on near-term quantum computers. Achieving a desired simulation precision typically requires a polynomially increasing number of Trotter steps, which remains challenging due to the limited performance of current quantum hardware. To alleviate this issue, post-processing techniques such as the multi-product formula (MPF) have been introduced to suppress algorithmic errors within restricted hardware resources. In this work, we propose a dual-channel multi-product formula that achieves a two-fold improvement in Trotter error scaling. As a result, our method enables the target simulation precision to be reached with approximately half the circuit depth compared to conventional MPF schemes. Importantly, the reduced circuit depth directly translates into lower physical error mitigation overhead when implemented on real quantum hardware. We demonstrate that, for a fixed CNOT count as a measure of quantum circuit, our proposal yields significantly smaller algorithmic errors, while the sampling error remains essentially unchanged.
Paper Structure (15 sections, 38 equations, 2 figures)

This paper contains 15 sections, 38 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Brief review of MPF
  3. Dual channel MPF protocol
  4. Numerical demonstration
  5. Theoretical performance : 1D Transverse Ising model
  6. Noisy simulation: XXZ spin chain
  7. Conclusion
  8. Acknowledgement
  9. Data availability statement:
  10. Detailed calculation of conventional MPF
  11. Conventional MPF: operator level, Eq. \ref{['eq:mpf_basic']}
  12. Conventional MPF: expectation value level, Eq. \ref{['eq:mpf_extra']}
  13. Proofs of dual channel MPF
  14. Dual channel MPF: operator level, Eq. \ref{['eq:mpf_dual_basic']}
  15. Dual channel MPF: expectation value level, Eq. \ref{['eq:mpf_dual_extra']}

Figures (2)

  • Figure 1: Absolute error of the global $z$-magnetization simulated using the MPF based on regular first-order PFs (blue dots), the DCMPF based on first-order PFs (red dots), and the DCMPF based on third-order PFs (orange dots), shown with respect to the exact value. The corresponding fitting results are plotted as dashed lines.
  • Figure 2: Relative error of the simulated average $z$-magnetization with respect to the exact value, plotted against the number of CNOT gates (in the deepest circuit when implemented within the MPFs). The dynamics are simulated using the second-order Suzuki-Trotter PF (green dashed lines), the MPF method based on the second-order symmetric Trotter PFs (blue dots), and our MPF method using the first-order Trotter PFs (red stars). Depolarizing noise is applied to each CNOT gate with probabilities $10^{-8},10^{-7},4\times10^{-7},7\times10^{-7}$ and $10^{-6}$, corresponding to data points shown from vivid to gradually transparent colors. The folding numbers chosen for each MPF simulation are displayed together.