Block-encoding-based quantum algorithm for linear systems with displacement structures
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Abstract:
Matrices with the displacement structures of circulant, Toeplitz, and Hankel types as well as matrices with structures generalizing these types are omnipresent in computations of sciences and engineering. In this paper we present efficient and memory-reduced quantum algorithms for solving linear systems with such structures by devising an approach to implement the block-encodings of these structured matrices. More specifically, by decomposing $n\ifmmode\times\else\texttimes\fi{}n$ dense matrices into linear combinations of displacement matrices, we first deduce the parametrized representations of the matrices with displacement structures so that they can be treated similarly. With such representations, we then construct $\ensuremath{\epsilon}$-approximate block-encodings of these structured matrices in two different data access models, i.e., the black-box model and the quantum random access memory (QRAM) data structure model. It is shown the quantum linear system solvers based on the proposed block-encodings provide a quadratic speedup with respect to the dimension over classical algorithms in the black-box model and an exponential speedup in the QRAM data structure model. In particular, these linear system solvers subsume known results with significant improvements and also can motivate new instances where there was no specialized quantum algorithm before. As an application, one of the quantum linear system solvers is applied to the linear prediction of time series, which justifies the claimed quantum speedup is achievable for problems of practical interest.Keywords:
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We apply our recent work on empirical estimates of quantum speedups to the practical task of community detection in complex networks. We design several quantum variants of a popular classical algorithm -- the Louvain algorithm for community detection -- and first study their complexities in the usual way, before analysing their complexities empirically across a variety of artificial and real inputs. We find that this analysis yields insights not available to us via the asymptotic analysis, further emphasising the utility in such an empirical approach. In particular, we observe that a complicated quantum algorithm with a large asymptotic speedup might not be the fastest algorithm in practice, and that a simple quantum algorithm with a modest speedup might in fact be the one that performs best. Moreover, we repeatedly find that overheads such as those arising from the need to amplify the success probabilities of quantum sub-routines such as Grover search can nullify any speedup that might have been suggested by a theoretical worst- or expected-case analysis.
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The rapid progress of Computer Science and Engineering has been accompanied by a corresponding evaluation of computation, from classical computation to quantum computation. In classical computation, computer memory made up of bits, where each bit represents either a one or a zero. In quantum computation, there are some quantum mechanical phenomena, such as superposition and entanglement, to perform operations on data. Instead of using bits, quantum computation uses qubits (quantum bits). A single qubit can represent a one, a zero, or both at the same time, which is called superposition. Because of this ability, quantum computation can perform many tasks simultaneously, faster than classical computing.
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Quantum algorithms theoretically outperform classical algorithms in solving problems of increasing size, but computational errors must be kept to a minimum to realize this potential. Despite the development of increasingly capable quantum computers (QCs), an experimental demonstration of a provable algorithmic quantum speedup employing today's non-fault-tolerant, noisy intermediate-scale quantum (NISQ) devices has remained elusive. Here, we unequivocally demonstrate such a speedup, quantified in terms of the scaling with the problem size of the time-to-solution metric. We implement the single-shot Bernstein-Vazirani algorithm, which solves the problem of identifying a hidden bitstring that changes after every oracle query, utilizing two different 27-qubit IBM Quantum (IBMQ) superconducting processors. The speedup is observed on only one of the two QCs (ibmq_montreal) when the quantum computation is protected by dynamical decoupling (DD) -- a carefully designed sequence of pulses applied to the QC that suppresses its interaction with the environment, but not without DD. In contrast to recent quantum supremacy demonstrations, the quantum speedup reported here does not rely on any additional assumptions or complexity-theoretic conjectures and solves a bona fide computational problem, in the setting of a game with an oracle and a verifier.
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In ensemble (or bulk) quantum computation, all computations are performed on an ensemble of computers rather than on a single computer. Measurements of qubits in an individual computer cannot be performed; instead, only expectation values (over the complete ensemble of computers) can be measured. As a result of this limitation on the model of computation, many algorithms cannot be processed directly on such computers, and must be modified, as the common strategy of delaying the measurements usually does not resolve this ensemble-measurement problem. Here we present several new strategies for resolving this problem. Based on these strategies we provide new versions of some of the most important quantum algorithms, versions that are suitable for implementing on ensemble quantum computers, e.g., on liquid NMR quantum computers. These algorithms are Shor’s factorization algorithm, Grover’s search algorithm (with several marked items), and an algorithm for quantum fault-tolerant computation. The first two algorithms are simply modified using a randomizing and a sorting strategies. For the last algorithm, we develop a classical-quantum hybrid strategy for removing measurements. We use it to present a novel quantum fault-tolerant scheme. More explicitly, we present schemes for fault-tolerant measurement-free implementation of Toffoli and $$\sigma_{z}^{1/4},$$ as these operations cannot be implemented “bitwise”, and their standard fault-tolerant implementations require measurement.
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The rapid progress of computer science has been accompanied by a corresponding evolution of computation, from classical computation to quantum computation. As quantum computing is on its way to becoming an established discipline of computing science, much effort is being put into the development of new quantum algorithms. One of quantum algorithms is Grover's algorithm, which is used for searching an element in an unstructured list of N elements with quadratic speed-up over classical algorithms. In this work, Quantum Computer Language (QCL) is used to make a Grover's quantum search simulation in a classical computer document.
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In the past few decades, there has been an acute problem of creating a quantum computer that uses quantum mechanical effects such as quantum parallelism and quantum entanglement for computations. Using these mechanisms, the quantum computer is able to solve some of the NP-class problems in polynomial time. A hardware approach to modeling quantum computing is considered, and a general mathematical model of the operation of a quantum computer is described, and a technique for mathematical modeling of quantum computing is presented. At the stage of consideration of the method of mathematical modeling, the most resource-intensive parts of the model are indicated. Also, issues related to data parallelization on a hardware accelerator, which are planned to be used to simulate quantum computing, were considered, and algorithms for the operation of this type of accelerator were given. A more compact data format is proposed, which is recommended to be used when implementing the accelerator. Using the proposed format, it is possible to reduce the resources spent on elementary arithmetic operations. The possibility of introducing an optimization algorithm to minimize the time spent in computations, as well as to speed up the execution of quantum algorithms in simulators of a quantum computer at the stage of the effect of quantum gates on the quantum register model, is proposed. The results of the optimization algorithm and its comparison with the classical mathematical approach using a software model are presented.
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In this paper, we briefly review the basic concepts of quantum computation, entanglement, quantum cryptography and quantum fourier transform.  Quantum algorithms like Deutsch Jozsa, Shor’s  factorization and Grover’s data search are developed using fourier transform and quantum computation concepts to build quantum computers. Researchers are finding a way to build quantum computer that works more efficiently than classical computer. Among the standard well known algorithms in the field of quantum computation and communication we describe mathematically Deutsch Jozsa algorithm in detail for 2 and 3 qubits. Calculation of balanced and unbalanced states is shown in the mathematical description of the algorithm.
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Quantum computing simulation platform can simulate the computation results of the quantum computer based on traditional computers, which is an effective way to promote the development of quantum computing software, algorithms and hardware at the current immature stage of the real quantum computer. Since quantum computers have exponential calculation acceleration compared with traditional computers, the main problems in implementing quantum computing simulation on traditional computers are low computational efficiency and long time-consuming. A quantum circuit which is a sequence of quantum gates acting on a collection of qubits is the general quantum computing model. So by the means of quantum circuit optimization, the calculation speed can be significantly increased while keeping the calculation result unchanged. The existing empirical rules of quantum circuit optimization methods have limitations and there is no common and automatic quantum circuit optimization method. In this paper, a general and automatic quantum circuit optimization method based on the genetic algorithm is proposed, by which the equivalent optimal quantum circuit is obtained through a finite number of searching in a large searching space. This method is not limited by the hardware of the quantum computing simulation and the composition of the quantum circuit. The experimental results show that for the QFT algorithm of 29 qubits, the running time can be shortened by 41.4% and for the variational circuit of 6 qubits, the running time can be shortened by 18.8% compared with the state-of-the-art quantum circuit optimization method. So this method can improve the quantum computing simulation capability and operating efficiency and provide a rapid development way for quantum algorithms and applications.
Quantum circuit
Quantum sort
Quantum Fourier transform
One-way quantum computer
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