We study the quantum complexity of algorithms for optimal graph traversal problems. We look at eulerian tours, optimal postman tours, approximation of travelling salesman tours and self avoiding walks. We present quan...
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(纸本)9780819466952
We study the quantum complexity of algorithms for optimal graph traversal problems. We look at eulerian tours, optimal postman tours, approximation of travelling salesman tours and self avoiding walks. We present quantum algorithms and quantum lower bounds for these problems. Our results improve the best classical algorithms for the corresponding problems.
This thesis describes quantum algorithms for Hamiltonian simulation, ordinary differential equations (ODEs), and partial differential equations (PDEs). Product formulas are used to simulate Hamiltonians which can be...
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This thesis describes quantum algorithms for Hamiltonian simulation, ordinary differential equations (ODEs), and partial differential equations (PDEs). Product formulas are used to simulate Hamiltonians which can be expressed as a sum of terms which can each be simulated individually. By simulating each of these terms in sequence, the net effect approximately simulates the total Hamiltonian. We find that the error of product formulas can be improved by randomizing over the order in which the Hamiltonian terms are simulated. We prove that this approach is asymptotically better than ordinary product formulas and present numerical comparisons for small numbers of qubits. The ODE algorithm applies to the initial value problem for time-independent first order linear ODEs. We approximate the propagator of the ODE by a truncated Taylor series, and we encode the initial value problem in a large linear system. We solve this linear system with a quantum linear system algorithm (QLSA) whose output we perform a post-selective measurement on. The resulting state encodes the solution to the initial value problem. We prove that our algorithm is asymptotically optimal with respect to several system parameters. The PDE algorithms apply the finite difference method (FDM) to Poisson's equation, the wave equation, and the Klein-Gordon equation. We use high order FDM approximations of the Laplacian operator to develop linear systems for Poisson's equation in cubic volumes under periodic, Neumann, and Dirichlet boundary conditions. Using QLSAs, we output states encoding solutions to Poisson's equation. We prove that our algorithm is exponentially faster with respect to the spatial dimension than analogous classical algorithms. We also consider how high order Laplacian approximations can be used for simulating the wave and Klein-Gordon equations. We consider under what conditions it suffices to use Hamiltonian simulation for time evolution, and we propose an algorithm for these cas
We discuss the progress (or lack of it) that has been made in discovering algorithms for computation on a quantum computer. Some possible reasons are given for the paucity of quantum algorithms so far discovered, and ...
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We discuss the progress (or lack of it) that has been made in discovering algorithms for computation on a quantum computer. Some possible reasons are given for the paucity of quantum algorithms so far discovered, and a short survey is given of the state of the field.
Least squares regression is the simplest and most widely used technique for solving overdetermined systems of linear equations Ax = b, where A is an element of R-nxP has full column rank and b is an element of R-n. Th...
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Least squares regression is the simplest and most widely used technique for solving overdetermined systems of linear equations Ax = b, where A is an element of R-nxP has full column rank and b is an element of R-n. Though there is a well known unique solution x* is an element of R-P to minimize the squared error parallel to Ax - b parallel to(2)(2), the best known classical algorithm to find x* takes time Omega(n), even for sparse and well-conditioned matrices A, a fairly large class of input instances commonly seen in practice. In this paper, we design an efficient quantum algorithm to generate a quantum state proportional to vertical bar x*>. The algorithm takes only O(logn) time for sparse and well-conditioned A. When the condition number of A is large, a canonical solution is to use regularization. We give efficient quantum algorithms for two regularized regression problems, including ridge regression and delta-truncated SVD, with similar costs and solution approximation. Given a matrix A is an element of R-nxp of rank r with SVD A = U Sigma V-T where U is an element of R-nxr, Sigma is an element of R-rxr and V is an element of R-pxr, the statistical leverage scores of A are the squared row norms of U, defined as si = parallel to U-i parallel to(2)(2), for i =1,. n. The matrix coherence is the largest statistic leverage score. These quantities play an important role in many machine learning algorithms. The best known classical algorithm to approximate these values runs in time Omega(np). In this work, we introduce an efficient quantum algorithm to approximate si in time O(logn) when A is sparse and the ratio between A's largest singular value and smallest nonzero singular value is constant. This gives an exponential speedup over the best known classical algorithms. Different than previous examples which are mainly modern algebraic or number theoretic ones, this problem is linear algebraic. It is also different than previous quantum algorithms for solving linear
There is a growing interest in quantum computers and quantum algorithm development. It has been proved that ideal quantum computers, with zero error rates and large decoherence times, can solve problems that are intra...
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There is a growing interest in quantum computers and quantum algorithm development. It has been proved that ideal quantum computers, with zero error rates and large decoherence times, can solve problems that are intractable for today's classical computers. quantum computers use two resources, superposition and entanglement, that have no classical analog. Since quantum computer platforms that are currently available comprise only a few dozen of qubits, the use of quantum simulators is essential in developing and testing new quantum algorithms. We present a novel quantum simulator based on memristor crossbar circuits and use them to simulate well-known quantum algorithms, namely the Deutsch and Grover quantum algorithms. In quantum computing the dominant algebraic operations are matrix-vector multiplications. The execution time grows exponentially with the simulated number of qubits, causing an exponential slowdown in quantum algorithm execution using classical computers. In this work, we show that the inherent characteristics of memristor arrays can be used to overcome this problem and that memristor arrays can be used not only as independent quantum simulators but also as a part of a quantum computer stack where classical computers accelerators are connected. Our memristive crossbar circuits are re-configurable and can be programmed to simulate any quantum algorithm.
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...
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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
In quantum information, trace distance is a basic metric of distinguishability between quantum states. However, there is no known efficient approach to estimate the value of trace distance in general. In this paper, w...
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In quantum information, trace distance is a basic metric of distinguishability between quantum states. However, there is no known efficient approach to estimate the value of trace distance in general. In this paper, we propose efficient quantum algorithms for estimating the trace distance within additive error epsilon between mixed quantum states of rank r . Specifically, we first provide a quantum algorithm using r & sdot;O (1/epsilon(2)) queries to the quantum circuits that prepare the purifications of quantum states. Then, we modify this quantum algorithm to obtain another algorithm using O (r(2)/epsilon(5)) samples of quantum states, which can be applied to quantum state certification. These algorithms have query/sample complexities that are independent of the dimension N of quantum states, and their time complexities only incur an extra O(log(N)) factor. In addition, we show that the decision version of low-rank trace distance estimation is BQP -complete.
We investigate the possibility to calculate the ground-state energy of the atomic systems on a quantum computer. For this purpose we evaluate the lowest binding energy of the moscovium atom with the use of the iterati...
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We investigate the possibility to calculate the ground-state energy of the atomic systems on a quantum computer. For this purpose we evaluate the lowest binding energy of the moscovium atom with the use of the iterative phase estimation and variational quantum eigensolver (VQE). The calculations by the VQE are performed with a disentangled unitary coupled cluster ansatz and with various types of hardware-efficient ansatze. The optimization is performed with the use of the Adam and quantum natural gradients procedures. The scalability of the ansatze and optimizers is tested by increasing the size of the basis set and the number of active electrons. The number of gates required for the iterative phase estimation and VQE is also estimated.
There exist quantum algorithms that are more efficient than their classical counterparts;such algorithms were invented by Shor in 1994 and then Grover in 1996. A lack of invention since Grover's algorithm has been...
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There exist quantum algorithms that are more efficient than their classical counterparts;such algorithms were invented by Shor in 1994 and then Grover in 1996. A lack of invention since Grover's algorithm has been commonly attributed to the non-intuitive nature of quantum algorithms to the classically trained person. Thus, the idea of using computers to automatically generate quantum algorithms based on an evolutionary model emerged. A limitation of this approach is that quantum computers do not yet exist and quantum simulation on a classical machine has an exponential order overhead. Nevertheless, early research into evolving quantum algorithms has shown promise. This paper provides an introduction into quantum and evolutionary algorithms for the computer scientist not familiar with these fields. The exciting field of using evolutionary algorithms to evolve quantum algorithms is then reviewed.
Longest common substring (LCS), longest palindrome substring (LPS), and Ulam distance (UL) are three fundamental string problems that can be classically solved in near linear time. In this work, we present sublinear t...
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Longest common substring (LCS), longest palindrome substring (LPS), and Ulam distance (UL) are three fundamental string problems that can be classically solved in near linear time. In this work, we present sublinear time quantum algorithms for these problems along with quantum lower bounds. Our results shed light on a very surprising fact: Although the classic solutions for LCS and LPS are almost identical (via suffix trees), their quantum computational complexities are different. While we give an exact o (root n) time algoritham for LPS, we prove that LCS needs at least time omega(sic) (n(2/3 )) even for 0/1 strings.
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