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
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
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.
Estimating nonlinear properties such as Rényi entropies and observable-weighted moments serves as a central strategy for spectrum spectroscopy, which is fundamental to property prediction and analysis in quantum information science, statistical mechanics, and many-body physics. However, existing approaches are susceptible to noise and require significant resources, making them challenging for near-term quantum hardware. In this work, we introduce a framework for resource-efficient simultaneous estimation of quantum state moments via qubit reuse. For an m-qubit quantum state ρ, our method achieves the simultaneous estimation of the full hierarchy of moments Tr(ρ2),…,Tr(ρk), as well as arbitrary polynomial functionals and their observable-weighted counterparts. By leveraging qubit reset operations, our core circuit for simultaneous moment estimation requires only 2m+1 physical qubits and O(k) CSWAP gates, achieving a near-optimal sample complexity of O(klogk/ε2). We demonstrate this protocol's utility by showing that the estimated moments yield tight bounds on a state's maximum eigenvalue and present applications in quantum virtual cooling to access low-energy states of the Heisenberg model. Furthermore, we show the protocol's viability on near-term quantum hardware by experimentally measuring higher-order Rényi entropy on a superconducting quantum processor. Our method provides a scalable and resource-efficient route to quantum system characterization and spectroscopy on near-term quantum hardware.
We address how one can empirically infer properties of quantum states generated by dynamics involving measurements. Our focus is on many-body settings where the number of measurements is extensive, making brute-force approaches based on postselection intractable due to their exponential sample complexity. We introduce a general-purpose scheme that can be used to infer any property of the postmeasurement ensemble of states (e.g., the average entanglement entropy, or frame potential) using a scalable number of experimental repetitions. We first identify a general class of “estimable properties” that can be directly extracted from experimental data. Then, based on empirical observations of such quantities, we show how one can indirectly infer information about any particular given nonestimable quantity of interest through classical postprocessing. Our approach is based on an optimization task, where one asks what are the minimum and maximum values that the desired quantity could possibly take, while ensuring consistency with observations. The true value of this quantity must then lie within a feasible range between these extrema, resulting in two-sided bounds. Narrow feasible ranges can be obtained by using a classical simulation of the device to determine which estimable properties one should measure. Even in cases where this simulation is inaccurate, unambiguous information about the true value of a given quantity realized on the quantum device can be learned. As an immediate application, we show that our method can be used to verify the emergence of quantum state designs in experiments. We identify some fundamental obstructions that in some cases prevent sharp knowledge of a given quantity from being inferred, and discuss what can be learned in cases where classical simulation is too computationally demanding to be feasible. In particular, we prove that any observer who cannot perform a classical simulation cannot distinguish the output states from those sampled from a maximally structureless ensemble. Published by the American Physical Society 2024
Quantum mechanics features a variety of distinct properties such as coherence and entanglement, which could be explored to showcase potential advantages over classical counterparts in information processing. In general, legitimate quantum operations must adhere to principles of quantum mechanics, particularly the requirements of complete positivity and trace preservation. Nonetheless, non-physical maps, especially Hermitian-preserving maps, play a crucial role in quantum information science. To date, there exists no effective method for implementing these non-physical maps with quantum devices. In this work, we introduce the Hermitian-preserving map exponentiation algorithm, which can effectively realize the action of an arbitrary Hermitian-preserving map by encoding its output into a quantum process. We analyze the performances of this algorithm, including its sample complexity and robustness, and prove its optimality in certain cases. When combined with algorithms such as the Hadamard test and quantum phase estimation, it allows for the extraction of information and generation of states from outputs of Hermitian-preserving maps, enabling various applications. Utilizing positive but not completely positive maps, this algorithm provides exponential advantages in entanglement detection and quantification compared to protocols based on single-copy operations. In addition, it facilitates the recovery of noiseless quantum states from multiple copies of noisy states by implementing the inverse map of the corresponding noise channel, offering an intriguing approach to handling quantum errors. Our findings present a pathway for systematically and efficiently implementing non-physical actions with quantum devices, thereby boosting the exploration of potential quantum advantages across a wide range of information processing tasks.
Efficiently estimating properties of large and strongly coupled quantum systems is a central focus in many-body physics and quantum information theory.While quantum computers promise speedups for many of these tasks, near-term devices are prone to noise that will generally reduce the accuracy of such estimates.Here, we propose a sample-efficient and noise-resilient protocol for learning properties of quantum states building on the shadow estimation scheme [Huang et al., Nature Physics 16, 1050-1057(2020)].By introducing an experimentally friendly calibration procedure, our protocol can efficiently characterize and mitigate noises in the shadow estimation scheme, given only minimal assumptions on the experimental conditions.When the strength of noises can be bounded, our protocol approximately retains the same order of sample efficiency as the standard shadow estimation scheme, while also possessing a provable noise resilience.We give rigorous bounds on the sample complexity of our protocol and demonstrate its performance with several numerical experiments, including estimations of quantum fidelity, correlation functions and energy expectations, etc., which highlight a wide spectrum of potential applications of our protocol on near-term devices.
Quantum computing holds the promise of enabling algorithms that are beyond the reach of classical computation, hence demonstrating quantum computational advantage. However, current quantum devices face significant limitations, primarily due to the lack of error correction. In this context, the following question arises: can quantum computational advantage be demonstrated with today's noisy hardware? Recently, Gaussian boson sampling (GBS) has emerged as a leading candidate to achieve this milestone, due to its relative experimental feasibility. The first part of this thesis focuses on two central challenges associated with GBS. The first one is the impact of noise on the computational complexity of the task. Using a phase-space approach, we establish a necessary non-classicality condition that any experimental proof of quantum advantage must satisfy, and demonstrate that there exists a threshold temperature at which quantum sampling experiments become classically simulable. We then investigate the possibility of introducing higher-order non-linearities as a mean to enhance the protocol’s robustness against noise. Then, we turn our attention to the problem of validating a GBS experiment, and propose the use binned-detector probability distributions as a suitable quantity to perform efficient statistical validation. In the second part of the thesis, we present a novel approach to protect a quantum observable from a given noise model.We start by introducing the concept of quantum observables over time, an operator that jointly describes two observables at two distinct time points, and use the latter to establish a notion of time-reversal for non-unitary quantum channels with respect to a reference observable, enabling the systematic construction of recovery maps that preserve the latter. These recovery maps, although generally non-physical, can still be used in noiseless expectation value estimation tasks and can achieve optimal performance in terms of sampling overhead, outperforming known error mitigation protocols.