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quantum algorithms for Testing Hamiltonian Symmetry

Physical Review Letters 2022年第16期129卷 160503-160503页

作者： Margarite L. LaBorde Mark M. Wilde Hearne Institute for Theoretical Physics Department of Physics and Astronomy and Center for Computation and Technology Louisiana State University Baton Rouge Louisiana 70803 USA

Symmetries in a Hamiltonian play an important role in quantum physics because they correspond directly with conserved quantities of the related system. In this Letter, we propose quantum algorithms capable of testing whether a Hamiltonian exhibits symmetry with respect to a group. We demonstrate that familiar expressions of Hamiltonian symmetry in quantum mechanics correspond directly with the acceptance probabilities of our algorithms. We execute one of our symmetry-testing algorithms on existing quantum computers for simple examples of both symmetric and asymmetric cases.

关键词： quantum computation quantum information processing Group theory quantum algorithms

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Robustness of quantum algorithms against coherent control errors

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Physical Review A 2024年第1期109卷 012417-012417页

作者： Julian Berberich Daniel Fink Christian Holm Institute for Computational Physics University of Stuttgart 70569 Stuttgart Germany

Coherent control errors, for which ideal Hamiltonians are perturbed by unknown multiplicative noise terms, are a major obstacle for reliable quantum computing. In this paper we present a framework for analyzing the robustness of quantum algorithms against coherent control errors using Lipschitz bounds. We derive worst-case fidelity bounds which show that the resilience against coherent control errors is mainly influenced by the norms of the Hamiltonians generating the individual gates. These bounds are explicitly computable even for large circuits and they can be used to guarantee fault tolerance via threshold theorems. Moreover, we apply our theoretical framework to derive a guideline for robust quantum algorithm design and transpilation, which amounts to reducing the norms of the Hamiltonians. Using the three-qubit quantum Fourier transform as an example application, we demonstrate that this guideline targets robustness more effectively than existing ones based on circuit depth or gate count. Furthermore, we apply our framework to study the effect of parameter regularization in variational quantum algorithms. The practicality of the theoretical results is demonstrated via implementations in simulation and on a quantum computer.

关键词： quantum algorithms quantum circuits quantum error correction quantum gates

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Variational quantum algorithms for simulation of Lindblad dynamics

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quantum SCIENCE AND TECHNOLOGY 2024年第2期9卷 025015-025015页

作者： Watad, Tasneem M. Lindner, Netanel H. Phys Dept Technion Haifa Israel

We introduce variational hybrid classical-quantum algorithms to simulate the Lindblad master equation and its adjoint for time-evolving Markovian open quantum systems and quantum observables. Our methods are based on a direct representation of density matrices and quantum observables as quantum superstates. We design and optimize low-depth variational quantum circuits that efficiently capture the unitary and non-unitary dynamics of the solutions. We benchmark and test the algorithms on different models and system sizes, showing their potential for utility with near-future hardware.

关键词： quantum simulation quantum algorithms quantum computation variational hybrid classical-quantum algorithm Lindblad dynamics

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Near-Optimal quantum algorithms for Multivariate Mean Estimation 2022

Near-Optimal Quantum Algorithms for Multivariate Mean Estima...

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54th Annual ACM SIGACT Symposium on Theory of Computing (STOC)

作者： Cornelissen, Arjan Hamoudi, Yassine Jerbi, Sofiene Univ Amsterdam QuSoft Amsterdam Netherlands Univ Calif Berkeley Simons Inst Theory Comp Berkeley CA 94720 USA Univ Innsbruck Inst Theoret Phys Innsbruck Austria

ISBN: (纸本)9781450392648

We propose the first near-optimal quantum algorithm for estimating in Euclidean norm the mean of a vector-valued random variable with finite mean and covariance. Our result aims at extending the theory of multivariate sub-Gaussian estimators [Lugosi and Mendelson, 2019] to the quantum setting. Unlike classically, where any univariate estimator can be turned into a multivariate estimator with at most a logarithmic overhead in the dimension, no similar result can be proved in the quantum setting. Indeed, [Heinrich, 2004] ruled out the existence of a quantum advantage for the mean estimation problem when the sample complexity is smaller than the dimension. Our main result is to show that, outside this low-precision regime, there does exist a quantum estimator that outperforms any classical estimator. More precisely, we prove that the approximation error can be decreased by a factor of about the square root of the ratio between the dimension and the sample complexity. Our approach is substantially more involved than in the univariate setting, where most quantum estimators rely only on phase estimation. We exploit a variety of additional algorithmic techniques such as linear amplitude amplification, the BernsteinVazirani algorithm, and quantum singular value transformation. Our analysis is also deeply rooted in proving concentration inequalities for multivariate truncated statistics. Finally, we describe several applications of our algorithms, notably in measuring the expectation values of commuting observables and in the field of machine learning.

关键词： Mean Estimation Multivariate Statistics quantum algorithms Lower Bounds

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Efficient quantum algorithms for Stabilizer Entropies

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Physical Review Letters 2024年第24期132卷 240602-240602页

作者： Tobias Haug Soovin Lee M. S. Kim Quantum Research Center Blackett Laboratory

Stabilizer entropies (SEs) are measures of nonstabilizerness or “magic” that quantify the degree to which a state is described by stabilizers. SEs are especially interesting due to their connections to scrambling, localization and property testing. However, applications have been limited so far as previously known measurement protocols for SEs scale exponentially with the number of qubits. Here, we efficiently measure SEs for integer Rényi index n>1 via Bell measurements. The SE of N-qubit quantum states can be measured with O(n) copies and O(nN) classical computational time, where for even n we additionally require the complex conjugate of the state. We provide efficient bounds of various nonstabilizerness monotones that are intractable to compute beyond a few qubits. Using the IonQ quantum computer, we measure SEs of random Clifford circuits doped with non-Clifford gates and give bounds for the stabilizer fidelity, stabilizer extent, and robustness of magic. We provide efficient algorithms to measure Clifford-averaged 4n-point out-of-time-order correlators and multifractal flatness. With these measures we study the scrambling time of doped Clifford circuits and random Hamiltonian evolution depending on nonstabilizerness. Counterintuitively, random Hamiltonian evolution becomes less scrambled at long times, which we reveal with the multifractal flatness. Our results open up the exploration of nonstabilizerness with quantum computers.

关键词： Entropy quantum algorithms quantum algorithms & computation quantum computation quantum correlations, foundations & formalism quantum information theory Resource theories Many-body techniques

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Iterative quantum algorithms for maximum independent set

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Physical Review A 2024年第5期110卷 052435-052435页

作者： Lucas T. Brady Stuart Hadfield Quantum Artificial Intelligence Laboratory NASA Ames Research Center Moffett Field California 94035 USA USRA Research Institute for Advanced Computer Science (RIACS) Mountain View California 94043 USA

quantum algorithms have been widely studied in the context of combinatorial optimization problems. While this endeavor can often analytically and practically achieve quadratic speedups, theoretical and numeric studies remain limited, especially compared to the study of classical algorithms. We propose and study an alternative class of hybrid approaches to quantum optimization, termed iterative quantum algorithms, which in particular generalizes the recursive quantum approximate optimization algorithm (QAOA). This paradigm can incorporate hard problem constraints, which we demonstrate by considering the maximum independent set (MIS) problem. We show that, for QAOA with p=1 circuit layers, this algorithm performs exactly the same operations and selections as the classical greedy algorithm for MIS. We then turn to deeper p>1 circuits and other ways to modify the quantum algorithm that can no longer be easily mimicked by classical algorithms, and empirically confirm improved performance. Our work demonstrates the practical importance of incorporating proven classical techniques into more effective hybrid quantum-classical algorithms.

关键词： quantum algorithms quantum algorithms & computation

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Improved quantum algorithms for the k-XOR Problem 28th

Improved Quantum Algorithms for the k-XOR Problem

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28th International Conference on Selected Areas in Cryptography (SAC)

作者： Schrottenloher, Andre CWI Cryptol Grp Amsterdam Netherlands

ISBN: (纸本)9783030992774;9783030992767

The k-XOR problem can be generically formulated as the following: given many n-bit strings generated uniformly at random, find k distinct of them which XOR to zero. This generalizes collision search (two equal elements) to a k-tuple of inputs. This problem has become ubiquitous in cryptanalytic algorithms, including variants in which the XOR operation is replaced by a modular addition (k-SUM) or other non-commutative operations (e.g., the composition of permutations). The case where a single solution exists on average is of special importance. At EUROCRYPT 2020, Naya-Plasencia and Schrottenloher defined a class of quantum merging algorithms for the k-XOR problem, obtained by combining quantum search. They represented these algorithms by a set of merging trees and obtained the best ones through linear optimization of their parameters. In this paper, we give a simplified representation of merging trees that makes their analysis easier. We give better quantum algorithms for the Single-solution k-XOR problem by relaxing one of the previous constraints, and making use of quantum walks. Our algorithms subsume or improve over all previous quantum algorithms for Single-solution k-XOR. For example, we give an algorithm for 4-XOR (or 4-SUM) in quantum time (O) over tilde (2(7n/24)).

关键词： quantum algorithms Merging algorithms k-XOR k-SUM Bicomposite search

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quantum algorithms for group convolution, cross-correlation, and equivariant transformations

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Physical Review A 2022年第3期106卷 032402-032402页

作者： Grecia Castelazo Quynh T. Nguyen Giacomo De Palma Dirk Englund Seth Lloyd Bobak T. Kiani Department of Electrical Engineering and Computer Science Massachusetts Institute of Technology Cambridge Massachusetts 02139 USA Scuola Normale Superiore Pisa Italy Department of Mathematics University of Bologna Bologna Italy Research Laboratory for Electronics Massachusetts Institute of Technology Cambridge Massachusetts 02139 USA Department of Mechanical Engineering Massachusetts Institute of Technology Cambridge Massachusetts 02139 USA

Group convolutions and cross-correlations, which are equivariant to the actions of group elements, are commonly used to analyze or take advantage of symmetries inherent in a given problem setting. Here, we provide efficient quantum algorithms for performing linear group convolutions and cross-correlations on data stored as quantum states. Runtimes for our algorithms are poly-logarithmic in the dimension of the group and the desired error of the operation. Motivated by the rich literature on quantum algorithms for solving algebraic problems, our theoretical framework opens a path for quantizing many algorithms in machine learning and numerical methods that employ group operations.

关键词： Machine learning quantum algorithms quantum computation quantum information processing

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Qubit-Efficient Randomized quantum algorithms for Linear Algebra

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PRX quantum 2024年第2期5卷 020324-020324页

作者： Samson Wang Sam McArdle Mario Berta Department of Physics Imperial College London London SW7 2BW United Kingdom AWS Center for Quantum Computing Pasadena CA 91106 USA Institute for Quantum Information and Matter Caltech Pasadena CA 91125 USA Department of Computing Imperial College London London SW7 2BW United Kingdom Institute for Quantum Information RWTH Aachen University 52056 Aachen Germany

We propose a class of randomized quantum algorithms for the task of sampling from matrix functions, without the use of quantum block encodings or any other coherent oracle access to the matrix elements. As such, our use of qubits is purely algorithmic and no additional qubits are required for quantum data structures. Our algorithms start from a classical data structure in which the matrix of interest is specified in the Pauli basis. For N×N Hermitian matrices, the space cost is log (N)+1 qubits and, depending on the structure of the matrices, the gate complexity can be comparable to state-of-the-art methods that use quantum data structures of up to size O(N2), when considering equivalent end-to-end problems. Within our framework, we present a quantum linear system solver that allows one to sample properties of the solution vector, as well as algorithms for sampling properties of ground states and Gibbs states of Hamiltonians. As a concrete application, we combine these subroutines to present a scheme for calculating Green’s functions of quantum many-body systems.

关键词： quantum algorithms quantum algorithms & computation quantum algorithms for chemical calculations

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quantum algorithms and Their Cryptographic Applications

Quantum Algorithms and Their Cryptographic Applications

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作者： McCaig, Xavier P. University of Nebraska at Omaha

学位级别：M.S., Master of Science/Master of Surgery

This work aims to demonstrate the abilities of modern quantum computers and quantum programming languages through implementations and practical usages of two of the most famous quantum algorithms: Shor's algorithm and Grover's algorithm, as well as through general quantum computing operations. These implementations will further demonstrate a proof-of-concept of relatively few-qubit quantum computers to encode and transmit data in a secure, tamper-proof manner using the Chinese Remainder Theorem. Lastly, this work discusses an implementation of Grover's algorithm requiring fewer qubits, but which may be limited in overall applicability

关键词： Computer science Cryptography Cybersecurity quantum algorithms quantum computing

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