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Quantum computing for machine learning attracts increasing attention and recent technological developments suggest that especially adiabatic quantum computing may soon be of practical interest. In this paper, we therefore consider this paradigm and discuss how to adopt it to the problem of binary clustering. Numerical simulations demonstrate the feasibility of our approach and illustrate how systems of qubits adiabatically evolve towards a solution.
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The graph-theoretic Ramsey numbers are notoriously difficult to calculate. In fact, for the two-color Ramsey numbers $R(m,n)$ with $m,ngeq 3$, only nine are currently known. We present a quantum algorithm for the computation of the Ramsey numbers $R(m,n)$. We show how the computation of $R(m,n)$ can be mapped to a combinatorial optimization problem whose solution can be found using adiabatic quantum evolution. We numerically simulate this adiabatic quantum algorithm and show that it correctly determines the Ramsey numbers R(3,3) and R(2,s) for $5leq sleq 7$. We then discuss the algorithms experimental implementation, and close by showing that Ramsey number computation belongs to the quantum complexity class QMA.
Farhi and others have introduced the notion of solving NP problems using adiabatic quantum com- puters. We discuss an application of this idea to the problem of integer factorization, together with a technique we call gluing which can be used to build adiabatic models of interesting problems. Although adiabatic quantum computers already exist, they are likely to be too small to directly tackle problems of interesting practical sizes for the foreseeable future. Therefore, we discuss techniques for decomposition of large problems, which permits us to fully exploit such hardware as may be available. Numerical re- sults suggest that even simple decomposition techniques may yield acceptable results with subexponential overhead, independent of the performance of the underlying device.
In the Graph Isomorphism problem two N-vertex graphs G and G are given and the task is to determine whether there exists a permutation of the vertices of G that preserves adjacency and transforms G into G. If yes, then G and G are said to be isomorphic; otherwise they are non-isomorphic. The GI problem is an important problem in computer science and is thought to be of comparable difficulty to integer factorization. In this paper we present a quantum algorithm that solves arbitrary instances of GI and can also determine all automorphisms of a given graph. We show how the GI problem can be converted to a combinatorial optimization problem that can be solved using adiabatic quantum evolution. We numerically simulate the algorithms quantum dynamics and show that it correctly: (i) distinguishes non-isomorphic graphs; (ii) recognizes isomorphic graphs; and (iii) finds all automorphisms of a given graph G. We then discuss the GI quantum algorithms experimental implementation, and close by showing how it can be leveraged to give a quantum algorithm that solves arbitrary instances of the NP-Complete Sub-Graph Isomorphism problem.
Adiabatic quantum computing and optimization have garnered much attention recently as possible models for achieving a quantum advantage over classical approaches to optimization and other special purpose computations. Both techniques are probabilistic in nature and the minimum gap between the ground state and first excited state of the system during evolution is a major factor in determining the success probability. In this work we investigate a strategy for increasing the minimum gap and success probability by introducing intermediate Hamiltonians that modify the evolution path between initial and final Hamiltonians. We focus on an optimization problem relevant to recent hardware implementations and present numerical evidence for the existence of a purely local intermediate Hamiltonian that achieve the optimum performance in terms of pushing the minimum gap to one of the end points of the evolution. As a part of this study we develop a convex optimization formulation of the search for optimal adiabatic schedules that makes this computation more tractable, and which may be of independent interest. We further study the effectiveness of random intermediate Hamiltonians on the minimum gap and success probability, and empirically find that random Hamiltonians have a significant probability of increasing the success probability, but only by a modest amount.
Recent years have witnessed the fast development of quantum computing. Researchers around the world are eager to run larger and larger quantum algorithms that promise speedups impossible to any classical algorithm. However, the available quantum computers are still volatile and error-prone. Thus, layout synthesis, which transforms quantum programs to meet these hardware limitations, is a crucial step in the realization of quantum computing. In this paper, we present two synthesizers, one optimal and one approximate but nearly optimal. Although a few optimal approaches to this problem have been published, our optimal synthesizer explores a larger solution space, thus is optimal in a stronger sense. In addition, it reduces time and space complexity exponentially compared to some leading optimal approaches. The key to this success is a more efficient spacetime-based variable encoding of the layout synthesis problem as a mathematical programming problem. By slightly changing our formulation, we arrive at an approximate synthesizer that is even more efficient and outperforms some leading heuristic approaches, in terms of additional gate cost, by up to 100%, and also fidelity by up to 10x on a comprehensive set of benchmark programs and architectures. For a specific family of quantum programs named QAOA, which is deemed to be a promising application for near-term quantum computers, we further adjust the approximate synthesizer by taking commutation into consideration, achieving up to 75% reduction in depth and up to 65% reduction in additional cost compared to the tool used in a leading QAOA study.
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