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Robustness of quantum algorithms against coherent control errors

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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Performance Study of Variational quantum algorithms for Solving the Poisson Equation on a quantum Computer

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Physical Review Applied 2023年第1期20卷 014054-014054页

作者： Mazen Ali Matthias Kabel Fraunhofer ITWM Kaiserslautern 67663 Germany

Recent advances in quantum computing and their increased availability has led to a growing interest in possible applications. Among those is the solution of partial differential equations (PDEs) for, e.g., material or flow simulation. Currently, the most promising route to useful deployment of quantum processors in the short to near term are so-called hybrid variational quantum algorithms (VQAs). Thus, variational methods for PDEs have been proposed as a candidate for quantum advantage in the noisy intermediate-scale quantum (NISQ) era. In this work, we conduct an extensive study of utilizing VQAs on real quantum devices to solve the simplest prototype of a PDE—the Poisson equation. Although results on noiseless simulators for small problem sizes may seem deceivingly promising, the performance on quantum computers is very poor. We argue that direct resolution of PDEs via an amplitude encoding of the solution is not a good use case within reach of today’s quantum devices—especially when considering large system sizes and more complicated nonlinear PDEs that are required in order to be competitive with classical high-end solvers.

关键词： quantum algorithms quantum computation quantum simulation Differential equations Variational approach

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Surrogate-based optimization for variational quantum algorithms

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Physical Review A 2023年第3期107卷 032415-032415页

作者： Ryan Shaffer Lucas Kocia Mohan Sarovar Department of Physics University of California Berkeley California 94720 USA Quantum Algorithms and Applications Collaboratory Sandia National Laboratories Livermore California 94550 USA

Variational quantum algorithms are a class of techniques intended to be used on near-term quantum computers. The goal of these algorithms is to perform large quantum computations by breaking the problem down into a large number of shallow quantum circuits, complemented by classical optimization and feedback between each circuit execution. One path for improving the performance of these algorithms is to enhance the classical optimization technique. Given the relative ease and abundance of classical computing resources, there is ample opportunity to do so. In this work, we introduce the idea of learning surrogate models for variational circuits using a few experimental measurements, and then performing parameter optimization using these models as opposed to the original data. We demonstrate this idea using a surrogate model based on kernel approximations, through which we reconstruct local patches of variational cost functions using batches of noisy quantum circuit results. Through application to the quantum approximate optimization algorithm and preparation of ground states for molecules, we demonstrate the superiority of surrogate-based optimization over commonly used optimization techniques for variational algorithms.

关键词： Optimization problems quantum algorithms quantum theory quantum spin models Data analysis Numerical approximation & analysis Probability theory Variational approach

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Demonstrating a Continuous Set of Two-Qubit Gates for Near-Term quantum algorithms

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Physical Review Letters 2020年第12期125卷 120504-120504页

作者： B. Foxen C. Neill A. Dunsworth P. Roushan B. Chiaro A. Megrant J. Kelly Zijun Chen K. Satzinger R. Barends F. Arute K. Arya R. Babbush D. Bacon J. C. Bardin S. Boixo D. Buell B. Burkett Yu Chen R. Collins E. Farhi A. Fowler C. Gidney M. Giustina R. Graff M. Harrigan T. Huang S. V. Isakov E. Jeffrey Z. Jiang D. Kafri K. Kechedzhi P. Klimov A. Korotkov F. Kostritsa D. Landhuis E. Lucero J. McClean M. McEwen X. Mi M. Mohseni J. Y. Mutus O. Naaman M. Neeley M. Niu A. Petukhov C. Quintana N. Rubin D. Sank V. Smelyanskiy A. Vainsencher T. C. White Z. Yao P. Yeh A. Zalcman H. Neven J. M. Martinis Department of Physics University of California Santa Barbara California 93106 USA Google Research Santa Barbara California 93117 USA Department of Electrical and Computer Engineering University of Massachusetts–Amherst Amherst Massachusetts 01003 USA

quantum algorithms offer a dramatic speedup for computational problems in material science and chemistry. However, any near-term realizations of these algorithms will need to be optimized to fit within the finite resources offered by existing noisy hardware. Here, taking advantage of the adjustable coupling of gmon qubits, we demonstrate a continuous two-qubit gate set that can provide a threefold reduction in circuit depth as compared to a standard decomposition. We implement two gate families: an imaginary swap-like (iSWAP-like) gate to attain an arbitrary swap angle, θ, and a controlled-phase gate that generates an arbitrary conditional phase, ϕ. Using one of each of these gates, we can perform an arbitrary two-qubit gate within the excitation-preserving subspace allowing for a complete implementation of the so-called Fermionic simulation (fSim) gate set. We benchmark the fidelity of the iSWAP-like and controlled-phase gate families as well as 525 other fSim gates spread evenly across the entire fSim(θ,ϕ) parameter space, achieving a purity-limited average two-qubit Pauli error of 3.8×10−3 per fSim gate.

关键词： quantum algorithms quantum computation quantum gates quantum simulation Superconducting qubits quantum chemistry methods

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Recursive multiport schemes for implementing quantum algorithms with photonic integrated circuits

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Physical Review A 2016年第1期93卷 012323-012323页

作者： Gelo Noel M. Tabia []Institute of Computer Science University of Tartu J. Livii 2 50409 Tartu Estonia

We present recursive multiport schemes for implementing quantum Fourier transforms and the inversion step in Grover's algorithm on an integrated linear optics device. In particular, each scheme shows how to execute a quantum operation on 2d modes using a pair of circuits for the same operation on d modes. The circuits operate on path-encoded qudits and realize d-dimensional unitary transformations on these states using linear optical networks with Od2 optical elements. To evaluate the schemes against realistic errors, we ran simulations of proof-of-principle experiments using a simple fabrication model of silicon-based photonic integrated devices that employ directional couplers and thermo-optic modulators for beam splitters and phase shifters, respectively. We find that high-fidelity performance is achievable with our multiport circuits for 2-qubit and 3-qubit quantum Fourier transforms, and for quantum search on four-item and eight-item databases.

关键词： Integrated optics quantum algorithms quantum information processing

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Complexity-Theoretic Limitations on quantum algorithms for Topological Data Analysis

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PRX quantum 2023年第4期4卷 040349-040349页

作者： Alexander Schmidhuber Seth Lloyd Institute for Theoretical Physics ETH Zurich Zurich 8092 Switzerland Center for Theoretical Physics Massachusetts Institute of Technology Cambridge Massachusetts 02139 USA Department of Mechanical Engineering Massachusetts Institute of Technology Cambridge Massachusetts 02139 USA Turing Inc. Cambridge Massachusetts 02139 USA

quantum algorithms for topological data analysis (TDA) seem to provide an exponential advantage over the best classical approach while remaining immune to dequantization procedures and the data-loading problem. In this paper, we give complexity-theoretic evidence that the central task of TDA—estimating Betti numbers—is intractable even for quantum computers. Specifically, we prove that the problem of computing Betti numbers exactly is #P-hard, while the problem of approximating Betti numbers up to multiplicative error is NP-hard. Moreover, both problems retain their hardness if restricted to the regime where quantum algorithms for TDA perform best. Because quantum computers are not expected to solve #P-hard or NP-hard problems in subexponential time, our results imply that quantum algorithms for TDA offer only a polynomial advantage in the worst case. We support our claim by showing that the seminal quantum algorithm for TDA developed by Lloyd, Garnerone, and Zanardi achieves a quadratic speedup over the best-known classical approach in asymptotically almost all cases. Finally, we argue that an exponential quantum advantage can be recovered if the input data is given as a specification of simplices rather than as a list of vertices and edges.

关键词： quantum algorithms quantum algorithms & computation quantum computation quantum information theory

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Efficient Measure for the Expressivity of Variational quantum algorithms

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Physical Review Letters 2022年第8期128卷 080506-080506页

作者： Yuxuan Du Zhuozhuo Tu Xiao Yuan Dacheng Tao JD Explore Academy Beijing 101111 China School of Computer Science Faculty of Engineering The University of Sydney Darlington NSW 2008 Australia Center on Frontiers of Computing Studies Department of Computer Science Peking University Beijing 100871 China

The superiority of variational quantum algorithms (VQAs) such as quantum neural networks (QNNs) and variational quantum eigensolvers (VQEs) heavily depends on the expressivity of the employed Ansätze. Namely, a simple Ansatz is insufficient to capture the optimal solution, while an intricate Ansatz leads to the hardness of trainability. Despite its fundamental importance, an effective strategy of measuring the expressivity of VQAs remains largely unknown. Here, we exploit an advanced tool in statistical learning theory, i.e., covering number, to study the expressivity of VQAs. Particularly, we first exhibit how the expressivity of VQAs with an arbitrary Ansätze is upper bounded by the number of quantum gates and the measurement observable. We next explore the expressivity of VQAs on near-term quantum chips, where the system noise is considered. We observe an exponential decay of the expressivity with increasing circuit depth. We also utilize the achieved expressivity to analyze the generalization of QNNs and the accuracy of VQE. We numerically verify our theory employing VQAs with different levels of expressivity. Our Letter opens the avenue for quantitative understanding of the expressivity of VQAs.

关键词： Learning theory Machine learning quantum algorithms quantum computation Resource theories

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Efficient quantum algorithms for the hidden subgroup problem over semi-direct product groups

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quantum INFORMATION & COMPUTATION 2007年第5-6期7卷 559-570页

作者： Inui, Yoshifumi Le Gall, Francois Univ Tokyo Dept Comp Sci Bunkyo Ku Tokyo 1130033 Japan JST SORST Quantum Computat & Informat Project ERATO Bunkyo Ku Tokyo 1130033 Japan

In this paper, we consider the hidden subgroup problem (HSP) over the class of semi-direct product groups Z(p)(r) X Z(q), for p and q prime. We first present a classification of these groups in five classes. Then, we describe a polynomial-time quantum algorithm solving the HSP over all the groups of one of these classes: the groups of the form Z(p)(r) X Z(p), where p is an odd prime. Our algorithm works even in the most general case where the group is presented as a black-box group with not necessarily unique encoding. Finally, we extend this result and present an efficient algorithm solving the HSP over the groups Z(pr)(m) x Z(p)

关键词： quantum algorithms hidden subgroup problem black-box groups

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Near-Term Efficient quantum algorithms for Entanglement Analysis

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Physical Review Applied 2023年第2期20卷 024071-024071页

作者： Ranyiliu Chen Benchi Zhao Xin Wang Institute for Quantum Computing Baidu Research Beijing 100193 China QMATH Department of Mathematical Sciences University of Copenhagen Universitetsparken 5 Copenhagen 2100 Denmark Thrust of Artificial Intelligence Information Hub Hong Kong University of Science and Technology (Guangzhou) Nansha China

Entanglement plays a crucial role in quantum physics and is the key resource in quantum information processing. However, entanglement detection and quantification are believed to be hard due to the operational impracticality of existing methods. This work proposes three near-term efficient algorithms that exploit the hybrid quantum-classical technique to address this difficulty. The first algorithm finds the Schmidt decomposition—a powerful tool to analyze the properties and structure of entanglement—for bipartite pure states. While the logarithm negativity can be calculated from the Schmidt decomposition, we propose the second algorithm to estimate the logarithm negativity for bipartite pure states, where the width of the parameterized quantum circuits is further reduced. Finally, we generalize our framework for mixed states, leading to our third algorithm that detects entanglement on specific families of states, and determines distillability in general. All three algorithms share a similar framework where the optimizations are accomplished by maximizing a cost function utilizing local parameterized quantum circuits, with better hardware efficiency and practicality compared to existing methods. The experimental implementation on quantum Leaf using the Institute of Physics, Chinese Academy of Sciences superconducting quantum processor exhibits the validity and practicality of our methods for analyzing and quantifying entanglement on near-term quantum devices.

关键词： Entanglement detection Entanglement measures quantum algorithms quantum entanglement

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Thermodynamic optimization of quantum algorithms: On-the-go erasure of qubit registers

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Physical Review A 2022年第6期106卷 062426-062426页

作者： Florian Meier Lídia del Rio Institute for Theoretical Physics ETH Zurich 8093 Zurich Switzerland Atominstitut TU Wien 1020 Vienna Austria

We consider two bottlenecks in quantum computing: limited memory size and noise caused by heat dissipation. Trying to optimize both, we investigate “on-the-go erasure” of quantum registers that are no longer needed for a given algorithm: freeing up auxiliary qubits as they stop being useful would facilitate the parallelization of computations. We study the minimal thermodynamic cost of erasure in these scenarios, applying results on the Landauer erasure of entangled quantum registers. For the class of algorithms solving the Abelian hidden subgroup problem, we find optimal on-the-go erasure protocols. We conclude that there is a trade-off: if we have enough partial information about a problem to build efficient on-the-go erasure, we can use it to instead simplify the algorithm, so that fewer qubits are needed to run the computation in the first place. We provide explicit protocols for these two approaches.

关键词： quantum algorithms quantum entanglement quantum thermodynamics Resource theories Thermodynamics of computation

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