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arXiv:2111.02345

NISQ: Error Correction, Mitigation, and Noise Simulation

Ningping Cao, Junan Lin, David W. Kribs, Yiu‐Tung Poon, Bei Zeng, Raymond Laflamme

Published Nov 3, 2021• 14 citations
arXivHTMLPDF

Abstract

Error-correcting codes were invented to correct errors on noisy communication channels. Quantum error correction (QEC), however, may have a wider range of uses, including information transmission, quantum simulation/computation, and fault-tolerance. These invite us to rethink QEC, in particular, about the role that quantum physics plays in terms of encoding and decoding. The fact that many quantum algorithms, especially near-term hybrid quantum-classical algorithms, only use limited types of local measurements on quantum states, leads to various new techniques called Quantum Error Mitigation (QEM). This work examines the task of QEM from several perspectives. Using some intuitions built upon classical and quantum communication scenarios, we clarify some fundamental distinctions between QEC and QEM. We then discuss the implications of noise invertibility for QEM, and give an explicit construction called Drazin-inverse for non-invertible noise, which is trace preserving while the commonly-used Moore-Penrose pseudoinverse may not be. Finally, we study the consequences of having an imperfect knowledge about the noise, and derive conditions when noise can be reduced using QEM.

DOI
10.48550/arxiv.2111.02345

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Error-correcting codes were invented to correct errors on noisy communication channels. Quantum error correction (QEC), however, may have a wider range of uses, including information transmission, quantum simulation/computation, and fault-tolerance. These invite us to rethink QEC, in particular, about the role that quantum physics plays in terms of encoding and decoding. The fact that many quantum algorithms, especially near-term hybrid quantum-classical algorithms, only use limited types of local measurements on quantum states, leads to various new techniques called Quantum Error Mitigation (QEM). This work examines the task of QEM from several perspectives. Using some intuitions built upon classical and quantum communication scenarios, we clarify some fundamental distinctions between QEC and QEM. We then discuss the implications of noise invertibility for QEM, and give an explicit construction called Drazin-inverse for non-invertible noise, which is trace preserving while the commonly-used Moore-Penrose pseudoinverse may not be. Finally, we study the consequences of having an imperfect knowledge about the noise, and derive conditions when noise can be reduced using QEM.
Error-correcting codes were originally developed to protect both classical computation and communication from noise. Quantum error correction (QEC) adapts these ideas to the quantum setting, where the no-cloning theorem and continuous-variable error models raise qualitatively new difficulties and underlie schemes for fault-tolerant quantum computation. In contrast, quantum error mitigation (QEM) offers practical techniques to reduce noise effects in quantum devices without the significant hardware overhead required by QEC. This work examines the task of QEM from several perspectives. Using some intuitions built upon classical and quantum communication scenarios, we clarify some fundamental distinctions between QEC and QEM. We then discuss the implications of noise invertibility for QEM and give an explicit construction called Drazin inverse for non-invertible noise, which is trace preserving while the commonly used Moore–Penrose pseudoinverse may not be. Finally, we study the consequences of having an imperfect knowledge about the noise and derive conditions when noise can be reduced using QEM.
Quantum computation is expected to have significant practical applications. The main problem in quantum computing is susceptibility to noise. Quantum error correction was developed to allow large-scale applications of quantum computing by achieving fault tolerance. However, as large-scale fault-tolerant quantum computation is not available, the question of applications of near-term hardware remains important. Current noisy intermediate-scale (NISQ) quantum devices can be used for small applications or as testbeds for larger applications; noise suppression techniques are of significant importance to their applicability. This thesis addresses noise suppression via quantum error mitigation (QEM) and quantum error correction (QEC) for near-term applications. The work aims to improve the understanding of noise-suppression techniques and their performance for exploring the physical systems of interest. The results presented in this thesis are divided into QEM and QEC chapters. The majority of the work is focused on QEM. The main interest of this thesis is the application of QEM to Hamiltonian simulations. Two simulation methods: the Trotter method and a quantum signal processing-based method, are explored. Both simulation methods, enhanced by zero-noise extrapolation, are employed to simulate the transverse-field Ising model. The obtained numerical results are promising, as the QEM allows a good recovery of noiseless values in low-noise shallow circuits, even with fixed sampling resources. A further result obtained in the QEM domain is a definition of a mitigation technique based on optimal measurement (in a metrological sense). The presentation includes analytical calculations of implementation costs and error propagation, as well as a numerical application to Hamiltonian simulation, yielding promising results: noise suppression by several orders of magnitude. The QEC part of the thesis explores its application in quantum metrology. The focus is on parameter sensing in many-body Hamiltonians. The results are purely analytical and extend a known lemma to mmm-local homogeneous Pauli Hamiltonian parameter sensing. This allows discussion of the scaling achieved for sensing the Hamiltonian parameters with respect to the code properties. The results in this thesis establish applications of QEM and QEC in selected near-term applications, quantify their performance, and discuss caveats and potential future applications. This thesis presents a theoretical and numerical study with implications for future experimental implementations. The relative success of the QEM and QEC shown in this thesis relies on tailoring the techniques to a chosen problem, highlighting the importance of technique adaptation in near-term applications.
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Quantum systems operate using quantum bits (qubits), which are highly sensitive to external noises and disturbances and they are easily prone to errors. In the Noisy Intermediate-Scale Quantum (NISQ) era, these errors make difficult to build reliable, fault tolerant quantum circuits for real world deployment. This work provides an overview of need for Quantum Error Correction (QEC) in NISQ devices, as qubits commonly get affected by bit-flip, phase-flip, and combined errors that degrades system performance and examines Quantum Error Correction Code (QECC) techniques usage in communication, computation, and quantum multimedia applications such as image and video transmission. In addition, a simple three-qubit repetition circuit is introduced to demonstrate an error-mitigation approach. These findings, also show how QEC improves fidelity, reduces logical errors, and improves noise tolerance. The analysis also identifies challenges and outlines the future research direction aimed at achieving efficient and scalable fault tolerant quantum circuits.
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Quantum computing is entering a transitional regime between noisy intermediate-scale quantum (NISQ) processing and early fault-tolerant quantum computation (FTQC), in which increasingly capable hardware is beginning to support repeated syndrome measurements, partial error correction, and logical-qubit operations, while residual physical and logical errors remain non-negligible. In this regime, error suppression, error mitigation, and quantum error correction are increasingly better viewed as complementary layers of a unified error-reduction strategy rather than as separate approaches, with each acting at a different stage of the quantum computation to improve simulation reliability. Thus, in this review, we provide a practical and forward-looking overview of the principal hardware error sources and the corresponding error suppression and mitigation methods for reducing their impact across the current NISQ-FTQC transition. We discuss hardware-aware circuit design, coherent-error suppression, readout mitigation, noise extrapolation, classical inference, and software-supported workflows, with particular emphasis on their implementation on actual quantum processors. We further examine how error mitigation techniques can be adapted to encoded and logical-qubit settings so that they can operate alongside quantum error correction to suppress residual logical errors and improve the accuracy of computation in the early fault-tolerant regime.