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arXiv:2105.01239 · Physical Review A

Dual-state purification for practical quantum error mitigation

Mingxia Huo, Ying Li

Published Feb 18, 2022• 49 citations
arXivHTMLPDF

Abstract

Quantum error mitigation is essential for many computing tasks on the noisy quantum computer with a limited number of qubits. In this paper, we propose a practical protocol of error mitigation by virtually purifying the quantum state without qubit overhead or requiring only one ancilla qubit. In dual-state purification, we effectively generate a purified state with increased fidelity using the erroneous state and its dual state, respectively, prepared with the noisy quantum circuit and the dual map of its inverse circuit. Combined with tomography purification, we can make sure that the final estimate of an observable is obtained from a pure state. While the absolute error increases, the numerical result, based on a depolarizing model, suggests that our protocol reduces the error compared to the case without the protocol by a rescaling factor decreasing with the qubit number and circuit depth; i.e., the relative performance of purification is better for larger circuits. On a cloud quantum computer, we successfully demonstrate the reduced error with a quantum variational eigensolver circuit.

DOI
10.1103/physreva.105.022427

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Quantum error mitigation is essential for many computing tasks on the noisy quantum computer with a limited number of qubits. In this paper, we propose a practical protocol of error mitigation by virtually purifying the quantum state without qubit overhead or requiring only one ancilla qubit. In dual-state purification, we effectively generate a purified state with increased fidelity using the erroneous state and its dual state, respectively, prepared with the noisy quantum circuit and the dual map of its inverse circuit. Combined with tomography purification, we can make sure that the final estimate of an observable is obtained from a pure state. While the absolute error increases, the numerical result, based on a depolarizing model, suggests that our protocol reduces the error compared to the case without the protocol by a rescaling factor decreasing with the qubit number and circuit depth; i.e., the relative performance of purification is better for larger circuits. On a cloud quantum computer, we successfully demonstrate the reduced error with a quantum variational eigensolver circuit.
Quantum error correction physically removes errors from a quantum state, while quantum error mitigation improves observable estimates by processing noisy measurement data. We introduce \emph{purification quantum error suppression} (PQES), a hybrid approach that uses multiple noisy copies of an unknown state to combine these two ideas. The protocol uses SWAP tests to physically reduce errors by purification, while the full outcome record is used to combine all branches without postselection. In this way, PQES avoids the fixed-success-outcome requirement of standard SWAP-test purification while still accessing the power-purified state ρNρ^NρN. The SWAP identities allow purification steps to be interleaved with unitary circuit blocks, so errors can be suppressed during a computation rather than only at the final measurement. We provide both a parallel binary-tree implementation and a more compact register-recycled implementation using O(Mℓ)O(M\ell)O(Mℓ) coherent data qubits for an MMM-qubit register and N=2ℓN=2^\ellN=2ℓ input copies. We analyze the resulting error thresholds under representative noise models. For local depolarizing noise on the product-state family studied here, the threshold is pth=3/4p_{\mathrm{th}}=3/4pth​=3/4 for any register size, while local dephasing of ∣+⟩⊗M|+\rangle^{\otimes M}∣+⟩⊗M has a threshold of pth=1/2p_{\mathrm{th}}=1/2pth​=1/2. Local Clifford twirling can be used to convert dephasing into a depolarization channel and restore the higher threshold.
Quantum annealing (QA) is one of the methods to prepare a ground state of a problem Hamiltonian. In the absence of noise, QA can accurately estimate the ground-state energy if the adiabatic condition is satisfied. However, in practice, systems are known to suffer from decoherence. Meanwhile, considerable research effort has been devoted to noisy intermediate-scale quantum (NISQ) computation. For practical NISQ computation, many error-mitigation (EM) methods have been proposed to mitigate the effects of noise. This paper proposes a QA strategy combined with an EM method, namely, dual-state purification, to suppress the effects of decoherence. Our protocol consists of four parts: the conventional dynamics, single-qubit projective measurements, the Hamiltonian dynamics corresponding to an inverse map of the first dynamics, and postprocessing of the measurement results. Importantly, our protocol works without two-qubit gates that require pulse operations; hence, it is suitable for devices designed for practical QA. In addition, we present numerical calculations to show that our protocol is more accurate than the conventional QA in estimating the ground-state energy under decoherence.
To achieve the practical applications of near-term noisy quantum devices, low-cost ways to mitigate the noise damages in the devices are essential. In many applications, the noiseless state we want to prepare is often a pure state, which has recently inspired a range of purification-based quantum error mitigation proposals. The existing proposals either are limited to the suppressions of only the leading-order state preparation errors, or require a large number of long-range gates that might be challenging to implement depending on the qubit architecture. This article will provide an overview of the different purification-based quantum error mitigation schemes and propose a resource-efficient scheme that can correct state preparation errors up to any order while requiring only half of the qubits and less than half of the long-range gates compared to before.
Quantum error mitigation is a key approach for extracting target state properties on state-of-the-art noisy machines and early fault-tolerant devices. Using the ideas from flag fault tolerance and virtual state purification, we develop the virtual-channel-purification (VCP) protocol, which consumes similar qubit and gate resources as virtual state purification but offers stronger error suppression with increased system size and more noisy operation copies. The application of VCP does not require specific knowledge about the target quantum state, the target problem and the gate noise model in the target circuit, and can still offer rigorous performance guarantees for practical noise regimes as long as the noise is incoherent. Further connections are made between VCP and quantum error correction to produce the virtual error-correction (VEC) protocol, one of the first protocols that combine quantum error correction (QEC) and quantum error mitigation beyond directly applying error-mitigation protocols on top of logical qubits. Assuming perfect syndrome extraction, VEC can virtually remove all correctable noise in the channel while paying only the same sampling cost as low-order purification. It can achieve QEC-level protection on an unencoded register when transmitting it through a noisy channel, removing the associated encoding qubit overhead. Another variant of VEC can mimic the error-suppression power of the surface code by inputting only a bit-flip and a phase-flip code. Our protocol can also be adapted to key tasks in quantum networks like channel capacity activation and entanglement distribution.
Abstract Quantum state purification is crucial in quantum communication and computation, aiming to recover a purified state from multiple copies of an unknown noisy state. This work introduces a general state purification framework designed to achieve the highest fidelity with a specified probability and characterize the associated trade-offs. For i.i.d. quantum states under depolarizing noise, our framework can replicate the purification protocol proposed by Barenco et al (1997 SIAM J. Comput. 26 1541–57) and further provide exact formulas for the purification fidelity and probability with explicit trade-offs. We prove the protocols’ optimality for two copies of noisy states with any dimension and confirm its optimality for higher numbers of copies and dimensions through numerical analysis. Our methodological approach paves the way for proving the protocol’s optimality in more general scenarios and leads to optimal protocols for other noise models. Furthermore, we present a systematic implementation method via block encoding and parameterized quantum circuits, providing explicit circuits for purifying three-copy and four-copy states under depolarizing noise. Finally, we estimate the sample complexity and generalize the protocol to a recursive form, demonstrating its practicality for quantum computers with limited memory.
Fighting against noise is crucial for Noisy Intermediate Scale Quantum (NISQ) devices to demonstrate practical quantum applications. In this work, we give a paradigm of quantum error mitigation based on the vectorization of density matrices for variational quantum algorithms. Different from the ideas of existing quantum error mitigation methods that try to distill noiseless information from noisy quantum states, our proposal directly changes the way of encoding information and maps the density matrices of noisy quantum states to noiseless virtual pure states, which is realized by an NISQ-friendly measurement protocol and a classical postprocessing procedure. Our protocol requires no knowledge of the noise model, no ability to tune the noise strength, and no ancilla qubits for complicated controlled unitaries. Under our encoding, NISQ devices are always preparing virtual pure quantum states that are highly desired resources for variational quantum algorithms to have good performance in many tasks. We show how this protocol can be well fitted into variational quantum algorithms. We give several concrete ansatz constructions that are suitable for our proposal and do theoretical analysis on the sampling complexity, the expressibility, and the trainability. We also give a discussion on how this protocol is influenced by large noise and how it can be well combined with other quantum error mitigation protocols. The effectiveness of our proposal is demonstrated by various numerical experiments.