Extracting classical information from quantum systems is an essential step of many quantum algorithms. However, this information could be corrupted as the systems are prone to quantum noises, and its distortion under quantum dynamics has not been adequately investigated. In this work, we introduce a systematic framework to study how well we can retrieve information from noisy quantum states. Given a noisy quantum channel, we fully characterize the range of recoverable classical information. This condition allows a natural measure quantifying the information recoverability of a channel. Moreover, we resolve the minimum information retrieving cost, which, along with the corresponding optimal protocol, is efficiently computable by semidefinite programming. As applications, we establish the limits on the information retrieving cost for practical quantum noises and employ the corresponding protocols to mitigate errors in ground state energy estimation. Our work gives the first full characterization of information recoverability of noisy quantum states from the recoverable range to the recovering cost, revealing the ultimate limit of probabilistic error cancellation.
Extracting classical information from quantum systems is an essential step of many quantum algorithms. However, this information could be corrupted as the systems are prone to quantum noises, and its distortion under quantum dynamics has not been adequately investigated. In this work, we introduce a systematic framework to study how well we can retrieve information from noisy quantum states. Given a noisy quantum channel, we fully characterize the range of recoverable classical information. This condition allows a natural measure quantifying the information recoverability of a channel. Moreover, we resolve the minimum information retrieving cost, which, along with the corresponding optimal protocol, is efficiently computable by semidefinite programming. As applications, we establish the limits on the information retrieving cost for practical quantum noises and employ the corresponding protocols to mitigate errors in ground state energy estimation. Our work gives the first full characterization of information recoverability of noisy quantum states from the recoverable range to the recovering cost, revealing the ultimate limit of probabilistic error cancellation.
The rapid advancement of quantum computing has led to an extensive demand for effective techniques to extract classical information from quantum systems, particularly in fields like quantum machine learning and quantum chemistry. However, quantum systems are inherently susceptible to noises, which adversely corrupt the information encoded in quantum systems. In this work, we introduce an efficient algorithm that can recover information from quantum states under Pauli noise. The core idea is to learn the necessary information of the unknown Pauli channel by post-processing the classical shadows of the channel. For a local and bounded-degree observable, only partial knowledge of the channel is required rather than its complete classical description to recover the ideal information, resulting in a polynomial-time algorithm. This contrasts with conventional methods such as probabilistic error cancellation, which requires the full information of the channel and exhibits exponential scaling with the number of qubits. We also prove that this scalable method is optimal on the sample complexity and generalise the algorithm to the weight contracting channel. Furthermore, we demonstrate the validity of the algorithm on the 1D anisotropic Heisenberg-type model via numerical simulations. As a notable application, our method can be severed as a sample-efficient error mitigation scheme for Clifford circuits.
Quantum computing promises transformative advances in computation, communication, sensing, and machine learning. Yet the realization of large-scale fault-tolerant quantum computers remains hindered by the enormous overhead required for quantum error correction. This challenge raises a fundamental question: Must useful quantum computing wait until fully fault-tolerant quantum hardware becomes available? In this Perspective, we propose Recoverable Quantum Computation (RQC), an information-centric paradigm for quantum computing with errors. Rather than requiring faithful preservation of the complete quantum state, RQC focuses on preserving the computational information required to accomplish a given task. A quantum computation is considered recoverable if the desired computational information can be extracted from noisy quantum outputs with an overhead that preserves quantum advantage relative to the best known classical method. We introduce recoverability as an operational principle for evaluating noisy quantum computations and propose practical metrics based on recovery overhead and recoverability efficiency. We illustrate the framework using quantum Fourier transform period estimation on IBM quantum hardware and a conceptual example from quantum machine learning, demonstrating that useful computational information may remain recoverable despite significant physical errors. Building on these examples, we propose a preliminary classification of quantum applications according to their expected recoverability and outline a research roadmap toward a predictive theory of recoverability. RQC is intended not as an alternative to fault-tolerant quantum computing, but as a complementary paradigm for understanding and evaluating useful quantum computation in the broad intermediate regime between today's noisy quantum processors and tomorrow's fault-tolerant quantum computers.
Accurately estimating high-order moments of quantum states is an elementary precondition for many crucial tasks in quantum computing, such as entanglement spectroscopy, entropy estimation, spectrum estimation, and predicting nonlinear features from quantum states. But in reality, inevitable quantum noise prevents us from accessing the desired value. In this paper, we address this issue by systematically analyzing the feasibility and efficiency of extracting high-order moments from noisy states. We first show that there exists a quantum protocol capable of accomplishing this task if and only if the underlying noise channel is invertible. We then establish a method for deriving protocols that attain optimal sample complexity using quantum operations and classical postprocessing only. Our protocols, in contrast to conventional ones, incur lower overheads and avoid sampling different quantum operations due to a novel technique called the “‘observable shift method,” making the protocols strong candidates for practical use on current quantum devices. The proposed method also indicates the power of entangled protocols in retrieving high-order information, whereas in the existing methods, entanglement does not help. We further construct the protocol for large quantum systems to retrieve the depolarizing channels, making the proposed method scalable. Our work contributes to a deeper understanding of how quantum noise could affect high-order information extraction and provides guidance on how to tackle it. Published by the American Physical Society 2024
This thesis studies the capacities of quantum channels under repeated noisy evolution, with a focus on the long-time behavior of classical and quantum information transmission. The main goal is to understand how much information can still be preserved or recovered when a quantum channel is applied many times, and to develop structural, analytic, and algorithmic tools for characterizing these capacities.The core results are presented in Chapter 2. First, the thesis investigates the infinite-time regime and shows that the asymptotic capacities of quantum channels are determined by the structure of their peripheral subspace. This leads to explicit characterizations of the infinite-time classical capacity and bounds on the infinite-time quantum capacity, together with results on additivity and extensions to more general settings. The thesis then studies the finite-time behavior for GNS-symmetric channels and proves explicit exponential convergence bounds, showing that the capacities approach their infinite-time values rapidly. In addition, algorithmic methods are developed for computing the peripheral structure and the associated capacity quantities efficiently.Chapter 3 explores a related optimization-based direction motivated by Sum-of-Squares methods and semidefinite programming. Using ideas inspired by separability and SDP hierarchies, this part investigates whether machine-learning-assisted ansatz discovery can help construct more efficient SOS certificates. Although this direction is not yet complete in the specific capacity setting, it provides promising intermediate results and suggests new computational approaches for related problems in quantum information theory. Overall, this thesis provides a structural and quantitative understanding of the long-time information-carrying capabilities of noisy quantum dynamical systems, while also opening new perspectives through optimization and machine-learning-based methods.
Randomized protocols are procedures that incorporate probabilistic choices during their execution and they play a central role in quantum algorithms, spanning Hamiltonian simulation, noise mitigation, and measurement tasks. In practical implementations, the dominant cost of such protocols typically arises from circuit execution and measurement, and depends on hardware-specific resources such as gate counts, circuit depth, runtime, or dissipated energy. We introduce a general framework for applying classical importance sampling to randomized quantum protocols. Given a cost function for running quantum circuits, the proposed approach minimizes a net-cost figure of merit that jointly captures the computational expense per circuit and the estimator variance. We further extend the framework to scenarios where the quantum computation is subject to errors arising either from algorithmic approximations or from physical noise, proving that importance sampling preserves estimator bias despite altering the sampling distribution, and to settings with error-detection schemes, where we characterize the resulting changes in the optimal sampling strategy and achievable net-cost reduction. Representative applications include the Qdrift protocol, dephasing channels, mixed-states simulation, composite observables estimation, classical shadows, and probabilistic error cancellation. Overall, our results establish a principled approach for reducing the computational resources required by randomized quantum protocols through classical sampling optimization.
Quantum error correction (QEC) is an essential element of physical quantum information processing systems. Most QEC efforts focus on extending classical error correction schemes to the quantum regime. The input to a noisy system is embedded in a coded subspace, and error recovery is performed via an operation designed to perfectly correct for a set of errors, presumably a large subset of the physical noise process. In this paper, we examine the choice of recovery operation. Rather than seeking perfect correction on a subset of errors, we seek a recovery operation to maximize the entanglement fidelity for a given input state and noise model. In this way, the recovery operation is optimal for the given encoding and noise process. This optimization is shown to be calculable via a semidefinite program, a well-established form of convex optimization with efficient algorithms for its solution. The error recovery operation may also be interpreted as a combining operation following a quantum spreading channel, thus providing a quantum analogy to the classical diversity combining operation.