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arXiv:2404.09973

Quantum Error Suppression with Subgroup Stabilisation

Bo Yang, Elham Kashefi, Dominik Leichtle, Ollivier, Harold

Published Apr 15, 2024• 1 citation
arXivHTMLPDF

Abstract

Quantum state purification is the functionality that, given multiple copies of an unknown state, outputs a state with increased purity. This will be an essential building block for near- and middle-term quantum ecosystems before the availability of full fault tolerance, where one may want to suppress errors not only in expectation values but also in quantum states. We propose an effective state purification gadget with a moderate quantum overhead by projecting $M$ noisy quantum inputs to their symmetric subspace defined by a set of projectors forming a symmetric subgroup with order $M$. Our method, applied in every short evolution over $M$ redundant copies of noisy states, can suppress both coherent and stochastic errors by a factor of $1/M$, respectively. This reduces the circuit implementation cost $M$ times smaller than the state projection to the full symmetric subspace proposed by Barenco et al. more than two decades ago. We also show that our gadget purifies the depolarised inputs with probability $p$ to asymptotically $O\left(p^{2}\right)$ with an optimal choice of $M$ when $p$ is small. The sampling cost scales $O\left(p^{-1}\right)$ for small $p$, which is also shown to be asymptotically optimal. Our method provides flexible choices of state purification depending on the hardware restrictions before fully fault-tolerant computation is available.

DOI
10.48550/arxiv.2404.09973

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Protecting quantum information from noise is essential for reliable quantum computation and quantum memories. Quantum state purification addresses this challenge by processing multiple noisy copies of an unknown quantum state to produce an output state of higher fidelity. Here we establish nearly tight upper and lower bounds on the non-Clifford resources required to implement the symmetric-projection purification protocol. For nnn copies of a ddd-dimensional system, we construct a Clifford+T circuit that block-encodes the symmetric projector with operator-norm error at most εεε using O~(n)\widetilde{O}(n)O(n) T gates, where the tilde-OOO notation suppresses polylogarithmic factors in nnn, ddd, and 1/ε1/ε1/ε. Conversely, using stabilizer nullity, we prove that any Clifford+T implementation of such a block encoding with error ε<121−1/nε<\frac{1}{2}\sqrt{1-1/n}ε<21​1−1/n​ requires at least 2n−22n-22n−2 T gates, even allowing arbitrary positive normalization and clean stabilizer ancillas. Taken together, these results establish the nearly optimal scaling of non-Clifford cost with the number of copies and uncover the fundamental resource requirements for coherent symmetry projection in quantum state purification.
Quantum state purification is the task of recovering a nearly pure copy of an unknown pure quantum state using multiple noisy copies of the state. This basic task has applications to quantum communication over noisy channels and quantum computation with imperfect devices, but has only been studied previously for the case of qubits. We derive an efficient purification procedure based on the swap test for qudits of any dimension, starting with any initial error parameter. Treating the initial error parameter and the dimension as constants, we show that our procedure has sample complexity asymptotically optimal in the final error parameter. Our protocol has a simple recursive structure that can be applied when the states are provided one at a time in a streaming fashion, requiring only a small quantum memory to implement.
One of the major challenges for erroneous quantum computers is undoubtedly the control over the effect of noise. Considering the rapid growth of available quantum resources that are not fully fault tolerant, it is crucial to develop practical hardware-friendly quantum error mitigation (QEM) techniques to suppress unwanted errors. Here, we propose a novel generalized quantum subspace expansion method which can handle stochastic, coherent, and algorithmic errors in quantum computers. By fully exploiting the substantially extended subspace, we can efficiently mitigate the noise present in the spectra of a given Hamiltonian, without relying on any information of noise. The performance of our method is discussed under two highly practical setups: the quantum subspaces are mainly spanned by powers of the noisy state ρ^{m} and a set of error-boosted states, respectively. We numerically demonstrate in both situations that we can suppress errors by orders of magnitude, and show that our protocol inherits the advantages of previous error-agnostic QEM techniques as well as overcoming their drawbacks.
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.
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.