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Limitations of the Macaulay matrix approach for using the HHL algorithm to solve multivariate polynomial systems

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arXiv 2021年

作者： Ding, Jintai Gheorghiu, Vlad Gilyén, András Hallgren, Sean Li, Jianqiang University of Cincinnati OH United States Institute for Quantum Computing Dept. of Combinatorics & Optimization University of Waterloo ON Canada Institute for Quantum Information and Matter Caltech PasadenaCA United States Department of Computer Science and Engineering Pennsylvania State University PA United States

Recently Chen and Gao [CG21] proposed a new quantum algorithm for Boolean polynomial system solving, motivated by the cryptanalysis of some post-quantum cryptosystems. The key idea of their approach is to apply a quantum Linear System (QLS) algorithm to a Macaulay linear system over C, which is derived from the Boolean polynomial system. The efficiency of their algorithm depends on the condition number of the Macaulay matrix. In this paper, we give a strong lower bound on the condition number as a function of the Hamming weight of the Boolean solution, and show that in many (if not all) cases a Grover-based exhaustive search algorithm outperforms their algorithm. Then, we improve upon Chen and Gao’s algorithm by introducing the Boolean Macaulay linear system over C by reducing the original Macaulay linear system. This improved algorithm could potentially significantly outperform the brute-force algorithm, when the Hamming weight of the solution is logarithmic in the number of Boolean variables. © 2021, CC BY.

关键词： Linear systems

Controlled quantum Teleportation in the Presence of an Adversary

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arXiv 2022年

作者： Gangopadhyay, Sayan Wang, Tiejun Mashatan, Atefeh Ghose, Shohini Institute for Quantum Computing University of Waterloo Waterloo Canada Department of Physics and Astronomy University of Waterloo Waterloo Canada School of Science Beijing University of Posts and Telecommunications Beijing100876 China Ted Rogers School of Information Technology Management Toronto Metropolitan University Toronto Canada Department of Physics and Computer Science Wilfrid Laurier University Waterloo Canada

We present a device independent analysis of controlled quantum teleportation where the receiver is not trusted. We show that the notion of genuine tripartite nonlocality allows us to certify control power in such a scenario. By considering a specific adversarial attack strategy on a device characterized by depolarizing noise, we find that control power is a monotonically increasing function of genuine tripartite nonlocality. These results are relevant for building practical quantum communication networks and also shed light on the role of nonlocality in multipartite quantum information processing. Copyright © 2022, The Authors. All rights reserved.

关键词： Power control

Quantification of Entanglement and Coherence with Purity Detection

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arXiv 2023年

作者： Zhang, Ting Smith, Graeme Smolin, John A. Liu, Lu Peng, Xu-Jie Zhao, Qi Girolami, Davide Ma, Xiongfeng Yuan, Xiao Lu, He School of Physics State Key Laboratory of Crystal Materials Shandong University Jinan250100 China JILA University of Colorado/NIST 440 UCB BoulderCO80309 United States Center for Theory of Quantum Matter University of Colorado BoulderCO80309 United States Department of Physics University of Colorado 390 UCB BoulderCO80309 United States IBM T.J. Watson Research Center 1101 Kitchawan Road Yorktown HeightsNY10598 United States QICI Quantum Information and Computation Initiative Department of Computer Science The University of Hong Kong Pokfulam Road Hong Kong DISAT Politecnico di Torino Corso Duca degli Abruzzi 24 Torino10129 Italy Center for Quantum Information Institute for Interdisciplinary Information Sciences Tsinghua University Beijing100084 China Center on Frontiers of Computing Studies Peking University Beijing100871 China School of Computer Science Peking University Beijing100871 China Shenzhen Research Institute of Shandong University Shenzhen518057 China

Entanglement and coherence are fundamental properties of quantum systems, promising to power the near future quantum technologies. Yet, their quantification, rather than mere detection, generally requires reconstructing the spectrum of quantum states, i.e., experimentally challenging measurement sets that increase exponentially with the system size. Here, we demonstrate quantitative bounds to operationally useful entanglement and coherence that are universally valid, analytically computable, and experimentally friendly. Specifically, our main theoretical results are lower and upper bounds to the coherent information and the relative entropy of coherence in terms of local and global purities of quantum states. To validate our proposal, we experimentally implement two purity detection methods in an optical system: shadow estimation with random measurements and collective measurements on pairs of state copies. The experiment shows that both the coherent information and the relative entropy of coherence of pure and mixed unknown quantum states can be bounded by purity functions. Our research offers an efficient means of verifying large-scale quantum information processing without spectrum reconstruction. © 2023, CC BY.

关键词： Optical systems

Conceptual Understanding through Efficient Automated Design of quantum Optical Experiments

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Physical Review X 2021年第3期11卷 031044-031044页

作者： Mario Krenn Jakob S. Kottmann Nora Tischler Alán Aspuru-Guzik Chemical Physics Theory Group Department of Chemistry University of Toronto Canada Department of Computer Science University of Toronto Canada Vector Institute for Artificial Intelligence Toronto Canada Institute of Advanced Research in Artificial Intelligence (IARAI) Vienna Austria Centre for Quantum Computation and Communication Technology (Australian Research Council) Centre for Quantum Dynamics Griffith University Brisbane Australia Dahlem Center for Complex Quantum Systems Freie Universität Berlin Germany Lebovic Fellow Canadian Institute for Advanced Research (CIFAR) Toronto Canada

Artificial intelligence (AI) is a potentially disruptive tool for physics and science in general. One crucial question is how this technology can contribute at a conceptual level to help acquire new scientific understanding. Scientists have used AI techniques to rediscover previously known concepts. So far, no examples of that kind have been reported that are applied to open problems for getting new scientific concepts and ideas. Here, we present Theseus, an algorithm that can provide new conceptual understanding, and we demonstrate its applications in the field of experimental quantum optics. To do so, we make four crucial contributions. (i) We introduce a graph-based representation of quantum optical experiments that can be interpreted and used algorithmically. (ii) We develop an automated design approach for new quantum experiments, which is orders of magnitude faster than the best previous algorithms at concrete design tasks for experimental configuration. (iii) We solve several crucial open questions in experimental quantum optics which involve practical blueprints of resource states in photonic quantum technology and quantum states and transformations that allow for new foundational quantum experiments. Finally, and most importantly, (iv) the interpretable representation and enormous speed-up allow us to produce solutions that a human scientist can interpret and gain new scientific concepts from outright. We anticipate that Theseus will become an essential tool in quantum optics for developing new experiments and photonic hardware. It can further be generalized to answer open questions and provide new concepts in a large number of other quantum physical questions beyond quantum optical experiments. Theseus is a demonstration of explainable AI (XAI) in physics that shows how AI algorithms can contribute to science on a conceptual level.

关键词： Entanglement manipulation quantum optics quantum state engineering quantum states of light

Efficient Verification of Stabilizer Code Subspaces with Local Measurements

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arXiv 2024年

作者： Zheng, Congcong Yu, Xutao Zhang, Zaichen Xu, Ping Wang, Kun State Key Lab of Millimeter Waves Southeast University Nanjing211189 China Frontiers Science Center for Mobile Information Communication and Security Southeast University Nanjing210096 China Purple Mountain Laboratories Nanjing211111 China National Mobile Communications Research Laboratory Southeast University Nanjing210096 China Institute for Quantum Information State Key Laboratory of High Performance Computing College of Computer Science and Technology National University of Defense Technology Changsha410073 China Hefei National Laboratory Hefei230088 China

We address the task of verifying whether a quantum computer, designed to be protected by a specific stabilizer code, correctly encodes the corresponding logical qubits. To achieve this, we develop a general framework for subspace verification and explore several stabilizer code subspaces of practical significance. First, we present two efficient verification strategies for general stabilizer code subspaces, utilizing measurements of their stabilizer generators and stabilizer groups, respectively. Then, building on the observation that certain tests can be conducted in parallel when the subspace exhibits specific structural properties, we propose a coloring strategy tailored to graph code subspaces and an XZ strategy tailored to Calderbank-Shor-Steane (CSS) code subspaces. Compared to stabilizer-based strategies, these new strategies require significantly fewer measurement settings and consume fewer state copies, approaching near-global optimality. Notably, all the strategies employ a limited number of Pauli measurements, are non-adaptive, and work on mixed states, enabling efficient experimental certification of both logical qubits and logical operations in noisy quantum computers. This work contributes to the first systematic study of efficient verification of stabilizer code subspaces with local measurements. © 2024, CC BY.

关键词： Qubits

Higher cup products on hypercubic lattices: application to lattice models of topological phases

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arXiv 2021年

作者： Chen, Yu-An Tata, Sri Condensed Matter Theory Center Joint Quantum Institute Department of Physics NIST University of Maryland College ParkMD20472 United States Joint Center for Quantum Information and Computer Science University of Maryland College ParkMD20742 United States

In this paper, we derive the explicit formula for higher cup products on hypercubic lattices, based on the recently developed geometrical interpretation on the simplicial complexes. We illustrate how this formalism can elucidate lattice constructions on hypercubic lattices for various models and deriving them from spacetime actions. In particular, we demonstrate explicitly that the (3+1)D SPT (Equation presented) (where w1 and w2 are the first and second Stiefel-Whitney classes) is dual to the 3-fermion Walker-Wang model constructed on the cubic lattice. Other examples include the double-semion model, and also the "fermionic" toric code in arbitrary dimensions on hypercubic lattices. In addition, we extend previous constructions of exact boson-fermion dualities and the Gu-Wen Grassmann Integral to arbitrary dimensions. Another result which may be of independent interest is a derivation of a cochain-level action for the generalized double-semion model, reproducing a recently derived action on the cohomology level. Copyright © 2021, The Authors. All rights reserved.

关键词： Topology

Variational secure cloud quantum computing

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Physical Review A 2022年第2期105卷 022603-022603页

作者： Yuta Shingu Yuki Takeuchi Suguru Endo Shiro Kawabata Shohei Watabe Tetsuro Nikuni Hideaki Hakoshima Yuichiro Matsuzaki Department of Physics Faculty of Science Division I Tokyo University of Science Shinjuku Tokyo 162-8601 Japan Research Center for Emerging Computing Technologies National Institute of Advanced Industrial Science and Technology (AIST) 1-1-1 Umezono Tsukuba Ibaraki 305-8568 Japan NTT Communication Science Laboratories NTT Corporation3-1 Morinosato Wakamiya Atsugi Kanagawa 243-0198 Japan NTT Computer and Data Science Laboratories NTT corporation Musashino Tokyo 180-8585 Japan NEC-AIST Quantum Technology Cooperative Research Laboratory National Institute of Advanced Industrial Science and Technology (AIST) Tsukuba Ibaraki 305-8568 Japan

Variational quantum algorithms (VQAs) have been considered to be useful applications of noisy intermediate-scale quantum (NISQ) devices. Typically, in VQAs, a parametrized ansatz circuit is used to generate a trial wave function, and the parameters are optimized to minimize a cost function. On the other hand, blind quantum computing (BQC) has been studied in order to provide a quantum algorithm with security by using cloud networks. A client with a limited ability to perform quantum operations hopes to have access to a quantum computer of a server, and BQC allows the client to use the server's computer without leakage of the client's information (such as input, running quantum algorithms, and output) to the server. However, BQC is designed for fault-tolerant quantum computing, and this requires many ancillary qubits, which may not be suitable for NISQ devices. Here, we propose an efficient way to implement the NISQ computing with guaranteed security for the client. In our architecture, only N+1 qubits are required, under an assumption that the form of ansätze is known to the server, where N denotes the necessary number of the qubits in the original NISQ algorithms. The client only performs single-qubit measurements on an ancillary qubit sent from the server, and the measurement angles can specify the parameters for the ansätze of the NISQ algorithms. The no-signaling principle guarantees that neither parameters chosen by the client nor the outputs of the algorithm are leaked to the server. This work paves the way for new applications of NISQ devices.

关键词： Measurement-based quantum computing quantum algorithms quantum networks

quantum Algorithm for Petz Recovery Channels and Pretty Good Measurements

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Physical Review Letters 2022年第22期128卷 220502-220502页

作者： András Gilyén Seth Lloyd Iman Marvian Yihui Quek Mark M. Wilde Institute for Quantum Information and Matter California Institute of Technology Pasadena California 91125 USA Simons Institute for the Theory of Computing Berkeley California 94720 USA Department of Mechanical Engineering and Research Laboratory of Electronics Massachusetts Institute of Technology Cambridge Massachusetts 02139 USA Department of Physics and Department of Electrical and Computer Engineering Duke University Durham North Carolina 27708 USA Information Systems Laboratory Stanford University Stanford California 94305 USA Hearne Institute for Theoretical Physics Department of Physics and Astronomy and Center for Computation and Technology Louisiana State University Baton Rouge Louisiana 70803 USA Stanford Institute for Theoretical Physics Stanford University Stanford California 94305 USA

The Petz recovery channel plays an important role in quantum information science as an operation that approximately reverses the effect of a quantum channel. The pretty good measurement is a special case of the Petz recovery channel, and it allows for near-optimal state discrimination. A hurdle to the experimental realization of these vaunted theoretical tools is the lack of a systematic and efficient method to implement them. This Letter sets out to rectify this lack: Using the recently developed tools of quantum singular value transformation and oblivious amplitude amplification, we provide a quantum algorithm to implement the Petz recovery channel when given the ability to perform the channel that one wishes to reverse. Moreover, we prove that, in some sense, our quantum algorithm’s usage of the channel implementation cannot be improved by more than a quadratic factor. Our quantum algorithm also provides a procedure to perform pretty good measurements when given multiple copies of the states that one is trying to distinguish.

关键词： quantum algorithms quantum channels quantum error correction

Efficient algorithms to solve atom reconfiguration problems. III. The bird and batching algorithms and other parallel implementations on GPUs

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arXiv 2025年

作者： Afiouni, Fouad El Sabeh, Remy Nishimura, Naomi El Hajj, Izzat Mouawad, Amer E. Cooper, Alexandre Department of Computer Science American University of Beirut Lebanon School of Electrical and Computer Engineering Purdue University United States David R. Cheriton School of Computer Science University of Waterloo Canada Institute for Quantum Computing University of Waterloo Canada Department of Physics and Astronomy University of Waterloo Canada

We present efficient implementations of atom reconfiguration algorithms for both CPUs and GPUs, along with a batching routine to merge displacement operations for parallel execution. Leveraging graph-theoretic methods, our approach derives improved algorithms that achieve reduced time complexity and faster operational running times. First, we introduce an enhanced version of the redistribution-reconfiguration (red-rec) algorithm, which offers superior operational and runtime performance. We detail its efficient implementation on a GPU using a parallel approach. Next, we present an optimized version of the assignment-reconfiguration-ordering (aro) algorithm, specifically tailored for unweighted grid graphs. Finally, we introduce the bird algorithm to solve reconfiguration problems on grids, achieving performance gains over both red-rec and aro. These algorithms can be used to prepare defect-free configurations of neutral atoms in arrays of optical traps, serve as subroutines in more complex algorithms, or cross-benchmark the operational and runtime performance of new algorithms. They are suitable for realizing quantum circuits incorporating displacement operations and are optimized for real-time operation on increasingly large system sizes. © 2025, CC BY.

关键词： Trees (mathematics)