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Ensemble-Based Quantum Signal Processing for Error Mitigation

Suying Liu, Yulong Dong, Dong An, Murphy Yuezhen Niu

TL;DR

This work tackles the challenge of noise in quantum signal processing (QSP) on near-term devices by introducing Ensemble-based QSP (EnQSP), which averages across ensembles of noisy QSP Circuits to suppress random Z-phase misrotations without increasing circuit depth or ancilla requirements. The authors rigorously derive how ensemble averaging yields a scaled version of the desired polynomial transformation on a block-encoded matrix and provide concrete resource bounds, including how repetition and depth scale with noise strength and problem parameters. They apply EnQSP to Hamiltonian simulation, quantum linear systems, and ground-state preparation, offering balanced error strategies that trade polynomial approximation error against hardware noise, and they also present a classical ensemble-average generalization for observable estimation. The framework integrates error mitigation directly into the algorithm design, offering practical pathways for robust QSP-based quantum computing on near- to mid-term devices, while acknowledging limitations such as the focus on Z-noise and opportunities for extension to broader error models and device architectures.

Abstract

Despite rapid advances in quantum hardware, noise remains a central obstacle to deploying quantum algorithms on near-term devices. In particular, random coherent errors that accumulate during circuit execution constitute a dominant and fundamentally challenging noise source. We introduce a noise-resilient framework for Quantum Signal Processing (QSP) that mitigates such coherent errors without increasing circuit depth or ancillary qubit requirements. Our approach uses ensembles of noisy QSP circuits combined with measurement-level averaging to suppress random phase errors in Z rotations. Building on this framework, we develop robust QSP algorithms for implementing polynomial functions of Hermitian matrices and for estimating observables, with applications to Hamiltonian simulation, quantum linear systems, and ground-state preparation. We analyze the trade-off between approximation error and hardware noise, which is essential for practical implementation under the stringent depth and coherence constraints of current quantum hardware. Our results establish a practical pathway for integrating error mitigation seamlessly into algorithmic design, advancing the development of robust quantum computing, and enabling the discovery of scientific applications with near- and mid-term quantum devices.

Ensemble-Based Quantum Signal Processing for Error Mitigation

TL;DR

This work tackles the challenge of noise in quantum signal processing (QSP) on near-term devices by introducing Ensemble-based QSP (EnQSP), which averages across ensembles of noisy QSP Circuits to suppress random Z-phase misrotations without increasing circuit depth or ancilla requirements. The authors rigorously derive how ensemble averaging yields a scaled version of the desired polynomial transformation on a block-encoded matrix and provide concrete resource bounds, including how repetition and depth scale with noise strength and problem parameters. They apply EnQSP to Hamiltonian simulation, quantum linear systems, and ground-state preparation, offering balanced error strategies that trade polynomial approximation error against hardware noise, and they also present a classical ensemble-average generalization for observable estimation. The framework integrates error mitigation directly into the algorithm design, offering practical pathways for robust QSP-based quantum computing on near- to mid-term devices, while acknowledging limitations such as the focus on Z-noise and opportunities for extension to broader error models and device architectures.

Abstract

Despite rapid advances in quantum hardware, noise remains a central obstacle to deploying quantum algorithms on near-term devices. In particular, random coherent errors that accumulate during circuit execution constitute a dominant and fundamentally challenging noise source. We introduce a noise-resilient framework for Quantum Signal Processing (QSP) that mitigates such coherent errors without increasing circuit depth or ancillary qubit requirements. Our approach uses ensembles of noisy QSP circuits combined with measurement-level averaging to suppress random phase errors in Z rotations. Building on this framework, we develop robust QSP algorithms for implementing polynomial functions of Hermitian matrices and for estimating observables, with applications to Hamiltonian simulation, quantum linear systems, and ground-state preparation. We analyze the trade-off between approximation error and hardware noise, which is essential for practical implementation under the stringent depth and coherence constraints of current quantum hardware. Our results establish a practical pathway for integrating error mitigation seamlessly into algorithmic design, advancing the development of robust quantum computing, and enabling the discovery of scientific applications with near- and mid-term quantum devices.
Paper Structure (23 sections, 12 theorems, 74 equations, 2 figures, 4 tables)

This paper contains 23 sections, 12 theorems, 74 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Results
  3. Quantum Signal Processing and Matrix Polynomials
  4. An Ensemble Method against Random Coherent Errors
  5. Balance between Algorithmic Errors and Implementation Errors
  6. Applications in Scientific Computing
  7. Generalizations of the Ensemble QSP
  8. Discussion
  9. Preliminaries
  10. Block-encoding
  11. Quantum signal processing and qubitization
  12. Noisy QSP
  13. Error model
  14. Expectation of the noisy circuit
  15. Algorithms
  16. ...and 8 more sections

Key Result

Lemma 8

Let $A = \sum_{j=0}^{m-1} y_j A_j$ and $\|y\|_1 \leq \beta$. Suppose that $(P_L,P_R)$ is a pair of unitaries such that $P_L \ket{0} = \sum_{j=0}^{m-1} c_j\ket{j}$ and $P_R \ket{0} = \sum_{j=0}^{m-1} d_j\ket{j}$ with $y_j = \beta c_j^* d_j$ for all $j$, and $W = \sum_{j=0}^{m-1} \ket{j}\bra{j} \otime

Figures (2)

  • Figure 1: Quantum circuit for block-encoding $P(A)$ using QSP and qubitization.
  • Figure 2: Quantum circuit for estimating $\braket{\psi|\mathbb{E}\widetilde{O}|\psi}$ of a random matrix $\widetilde{O}$ with Hermitian expectation. Here $O_{\psi}$ is the state preparation oracle of $\ket{\psi}$, $U_{\widetilde{O}}$ is a block-encoding of $\widetilde{O}$, and $\mathrm{H}$ is the Hadamard gate.

Theorems & Definitions (20)

  • Definition 7: Block-encoding
  • Lemma 8
  • Lemma 9
  • Lemma 10: GilyenSuLowEtAl2019
  • Lemma 11: GilyenSuLowEtAl2019
  • Theorem 12
  • proof
  • Theorem 13
  • proof
  • Theorem 14
  • ...and 10 more