We present a quantum interior point method (IPM) for semi-definite programs that has a worst-case running time of Õ(n2.5 / ξ2 μ κ 3 log(1/ε)). The algorithm outputs a pair of matrices (S,Y) that have objective value within ε of the optimal and satisfy the constraints approximately to error xi. The parameter mu is at most √2n while kappa is an upper bound on the condition number of the intermediate solution matrices arising in the classical IPM. For the case where κ ≪ n5/6, our method provides a significant polynomial speedup over the best-known classical semi-definite program solvers that have a worst-case running time of Õ(n6). For linear programs, our algorithm has a running time of Õ(n1.5 / ξ2 μ κ 3 log (1/ε)) with the same guarantees and with parameter μ < √2n. Our technical contributions include an efficient quantum procedure for solving the Newton linear systems arising in the classical IPMs, an efficient pure state tomography algorithm, and an analysis of the IPM where the linear systems are solved approximately. Our results pave the way for the development of quantum algorithms with significant polynomial speedups for applications in optimization and machine learning.
{"title":"A Quantum Interior Point Method for LPs and SDPs","authors":"Iordanis Kerenidis, A. Prakash","doi":"10.1145/3406306","DOIUrl":"https://doi.org/10.1145/3406306","url":null,"abstract":"We present a quantum interior point method (IPM) for semi-definite programs that has a worst-case running time of Õ(n2.5 / ξ2 μ κ 3 log(1/ε)). The algorithm outputs a pair of matrices (S,Y) that have objective value within ε of the optimal and satisfy the constraints approximately to error xi. The parameter mu is at most √2n while kappa is an upper bound on the condition number of the intermediate solution matrices arising in the classical IPM. For the case where κ ≪ n5/6, our method provides a significant polynomial speedup over the best-known classical semi-definite program solvers that have a worst-case running time of Õ(n6). For linear programs, our algorithm has a running time of Õ(n1.5 / ξ2 μ κ 3 log (1/ε)) with the same guarantees and with parameter μ < √2n. Our technical contributions include an efficient quantum procedure for solving the Newton linear systems arising in the classical IPMs, an efficient pure state tomography algorithm, and an analysis of the IPM where the linear systems are solved approximately. Our results pave the way for the development of quantum algorithms with significant polynomial speedups for applications in optimization and machine learning.","PeriodicalId":365166,"journal":{"name":"ACM Transactions on Quantum Computing","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115641408","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Mapping functions on bits to Hamiltonians acting on qubits has many applications in quantum computing. In particular, Hamiltonians representing Boolean functions are required for applications of quantum annealing or the quantum approximate optimization algorithm to combinatorial optimization problems. We show how such functions are naturally represented by Hamiltonians given as sums of Pauli Z operators (Ising spin operators) with the terms of the sum corresponding to the function’s Fourier expansion. For many classes of Boolean functions which are given by a compact description, such as a Boolean formula in conjunctive normal form that gives an instance of the satisfiability problem, it is #P-hard to compute its Hamiltonian representation, i.e., as hard as computing its number of satisfying assignments. On the other hand, no such difficulty exists generally for constructing Hamiltonians representing a real function such as a sum of local Boolean clauses each acting on a fixed number of bits as is common in constraint satisfaction problems. We show composition rules for explicitly constructing Hamiltonians representing a wide variety of Boolean and real functions by combining Hamiltonians representing simpler clauses as building blocks, which are particularly suitable for direct implementation as classical software. We further apply our results to the construction of controlled-unitary operators, and to the special case of operators that compute function values in an ancilla qubit register. Finally, we outline several additional applications and extensions of our results to quantum algorithms for optimization. A goal of this work is to provide a design toolkit for quantum optimization which may be utilized by experts and practitioners alike in the construction and analysis of new quantum algorithms, and at the same time to provide a unified framework for the various constructions appearing in the literature.
将位元上的函数映射到作用于量子位上的哈密顿量在量子计算中有许多应用。特别是，量子退火或量子近似优化算法在组合优化问题中的应用需要哈密顿量来表示布尔函数。我们展示了这样的函数是如何被哈密顿算子自然地表示为泡利Z算子(伊辛自旋算子)的和，其和的项对应于函数的傅立叶展开。对于许多由紧凑描述给出的布尔函数类，例如给出可满足性问题实例的合取范式布尔公式，计算其哈密顿表示是# p -困难的，即与计算其满足赋值的数量一样困难。另一方面，对于构造表示实函数的哈密顿量，例如每个作用于固定位数的局部布尔子句的和，通常不存在这样的困难，这在约束满足问题中是常见的。通过将表示简单子句的哈密顿量组合为构建块，我们展示了用于显式构造表示各种布尔函数和实函数的哈密顿量的组合规则，这些规则特别适合作为经典软件的直接实现。我们进一步将我们的结果应用于控制酉算子的构造，以及在辅助量子位寄存器中计算函数值的算子的特殊情况。最后，我们概述了我们的结果在量子算法优化中的几个额外应用和扩展。这项工作的目标是提供一个量子优化的设计工具包，供专家和从业者在构建和分析新的量子算法时使用，同时为文献中出现的各种结构提供一个统一的框架。
{"title":"On the Representation of Boolean and Real Functions as Hamiltonians for Quantum Computing","authors":"Stuart Hadfield","doi":"10.1145/3478519","DOIUrl":"https://doi.org/10.1145/3478519","url":null,"abstract":"\u0000 Mapping functions on bits to Hamiltonians acting on qubits has many applications in quantum computing. In particular, Hamiltonians representing Boolean functions are required for applications of quantum annealing or the quantum approximate optimization algorithm to combinatorial optimization problems. We show how such functions are naturally represented by Hamiltonians given as sums of Pauli\u0000 Z\u0000 operators (Ising spin operators) with the terms of the sum corresponding to the function’s Fourier expansion. For many classes of Boolean functions which are given by a compact description, such as a Boolean formula in conjunctive normal form that gives an instance of the satisfiability problem, it is #P-hard to compute its Hamiltonian representation, i.e., as hard as computing its number of satisfying assignments. On the other hand, no such difficulty exists generally for constructing Hamiltonians representing a real function such as a sum of local Boolean clauses each acting on a fixed number of bits as is common in constraint satisfaction problems. We show composition rules for explicitly constructing Hamiltonians representing a wide variety of Boolean and real functions by combining Hamiltonians representing simpler clauses as building blocks, which are particularly suitable for direct implementation as classical software. We further apply our results to the construction of controlled-unitary operators, and to the special case of operators that compute function values in an ancilla qubit register. Finally, we outline several additional applications and extensions of our results to quantum algorithms for optimization. A goal of this work is to provide a\u0000 design toolkit for quantum optimization\u0000 which may be utilized by experts and practitioners alike in the construction and analysis of new quantum algorithms, and at the same time to provide a unified framework for the various constructions appearing in the literature.\u0000","PeriodicalId":365166,"journal":{"name":"ACM Transactions on Quantum Computing","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123901940","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Patrick J. Coles, S. Eidenbenz, S. Pakin, A. Adedoyin, J. Ambrosiano, P. Anisimov, W. Casper, Gopinath Chennupati, Carleton Coffrin, H. Djidjev, David Gunter, S. Karra, N. Lemons, Shizeng Lin, A. Lokhov, A. Malyzhenkov, D. Mascarenas, S. Mniszewski, B. Nadiga, D. O’Malley, D. Oyen, Lakshman Prasad, Randy M. Roberts, Philip Romero, N. Santhi, N. Sinitsyn, P. Swart, Marc Vuffray, J. Wendelberger, B. Yoon, R. J. Zamora, Wei Zhu
As quantum computers become available to the general public, the need has arisen to train a cohort of quantum programmers, many of whom have been developing classical computer programs for most of their careers. While currently available quantum computers have less than 100 qubits, quantum computing hardware is widely expected to grow in terms of qubit count, quality, and connectivity. This review aims at explaining the principles of quantum programming, which are quite different from classical programming, with straightforward algebra that makes understanding of the underlying fascinating quantum mechanical principles optional. We give an introduction to quantum computing algorithms and their implementation on real quantum hardware. We survey 20 different quantum algorithms, attempting to describe each in a succinct and self-contained fashion. We show how these algorithms can be implemented on IBM’s quantum computer, and in each case, we discuss the results of the implementation with respect to differences between the simulator and the actual hardware runs. This article introduces computer scientists, physicists, and engineers to quantum algorithms and provides a blueprint for their implementations.
{"title":"Quantum Algorithm Implementations for Beginners","authors":"Patrick J. Coles, S. Eidenbenz, S. Pakin, A. Adedoyin, J. Ambrosiano, P. Anisimov, W. Casper, Gopinath Chennupati, Carleton Coffrin, H. Djidjev, David Gunter, S. Karra, N. Lemons, Shizeng Lin, A. Lokhov, A. Malyzhenkov, D. Mascarenas, S. Mniszewski, B. Nadiga, D. O’Malley, D. Oyen, Lakshman Prasad, Randy M. Roberts, Philip Romero, N. Santhi, N. Sinitsyn, P. Swart, Marc Vuffray, J. Wendelberger, B. Yoon, R. J. Zamora, Wei Zhu","doi":"10.1145/3517340","DOIUrl":"https://doi.org/10.1145/3517340","url":null,"abstract":"As quantum computers become available to the general public, the need has arisen to train a cohort of quantum programmers, many of whom have been developing classical computer programs for most of their careers. While currently available quantum computers have less than 100 qubits, quantum computing hardware is widely expected to grow in terms of qubit count, quality, and connectivity. This review aims at explaining the principles of quantum programming, which are quite different from classical programming, with straightforward algebra that makes understanding of the underlying fascinating quantum mechanical principles optional. We give an introduction to quantum computing algorithms and their implementation on real quantum hardware. We survey 20 different quantum algorithms, attempting to describe each in a succinct and self-contained fashion. We show how these algorithms can be implemented on IBM’s quantum computer, and in each case, we discuss the results of the implementation with respect to differences between the simulator and the actual hardware runs. This article introduces computer scientists, physicists, and engineers to quantum algorithms and provides a blueprint for their implementations.","PeriodicalId":365166,"journal":{"name":"ACM Transactions on Quantum Computing","volume":"32 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134028751","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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