For matrix A, vector b and function f, the computation of vector f(A)b arises in many scientific computing applications. We consider the problem of obtaining quantum state |f> corresponding to vector f(A)b. There is a quantum algorithm to compute state |f> using eigenvalue estimation that uses phase estimation and Hamiltonian simulation e^{im A t}. However, the algorithm based on eigenvalue estimation needs poly(1/epsilon) runtime, where epsilon is the desired accuracy of the output state. Moreover, if matrix A is not Hermitian, e^{im A t} is not unitary and we cannot run eigenvalue estimation. In this paper, we propose a quantum algorithm that uses Cauchy's integral formula and the trapezoidal rule as an approach that avoids eigenvalue estimation. We show that the runtime of the algorithm is poly(log(1/epsilon)) and the algorithm outputs state |f> even if A is not Hermitian.
{"title":"Quantum algorithm for matrix functions by Cauchy's integral formula","authors":"S. Takahira, A. Ohashi, T. Sogabe, T. Usuda","doi":"10.26421/QIC20.1-2-2","DOIUrl":"https://doi.org/10.26421/QIC20.1-2-2","url":null,"abstract":"For matrix A, vector b and function f, the computation of vector f(A)b arises in many scientific computing applications. We consider the problem of obtaining quantum state |f> corresponding to vector f(A)b. There is a quantum algorithm to compute state |f> using eigenvalue estimation that uses phase estimation and Hamiltonian simulation e^{im A t}. However, the algorithm based on eigenvalue estimation needs poly(1/epsilon) runtime, where epsilon is the desired accuracy of the output state. Moreover, if matrix A is not Hermitian, e^{im A t} is not unitary and we cannot run eigenvalue estimation. In this paper, we propose a quantum algorithm that uses Cauchy's integral formula and the trapezoidal rule as an approach that avoids eigenvalue estimation. We show that the runtime of the algorithm is poly(log(1/epsilon)) and the algorithm outputs state |f> even if A is not Hermitian.","PeriodicalId":20904,"journal":{"name":"Quantum Inf. Comput.","volume":"34 1","pages":"14-36"},"PeriodicalIF":0.0,"publicationDate":"2020-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88951384","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}
In complex vector spaces maximal sets of equiangular lines, known as SICs, are related to real quadratic number fields in a dimension dependent way. If the dimension is of the form n^2+3, the base field has a fundamental unit of negative norm, and there exists a SIC with anti-unitary symmetry. We give eight examples of exact solutions of this kind, for which we have endeavoured to make them as simple as we can---as a belated reply to the referee of an earlier publication, who claimed that our exact solution in dimension 28 was too complicated to be fit to print. An interesting feature of the simplified solutions is that the components of the fiducial vectors largely consist of algebraic units.
{"title":"Algebraic units, anti-unitary symmetries, and a small catalogue of SICs","authors":"I. Bengtsson","doi":"10.26421/QIC20.5-6-3","DOIUrl":"https://doi.org/10.26421/QIC20.5-6-3","url":null,"abstract":"In complex vector spaces maximal sets of equiangular lines, known as SICs, are related to real quadratic number fields in a dimension dependent way. If the dimension is of the form n^2+3, the base field has a fundamental unit of negative norm, and there exists a SIC with anti-unitary symmetry. We give eight examples of exact solutions of this kind, for which we have endeavoured to make them as simple as we can---as a belated reply to the referee of an earlier publication, who claimed that our exact solution in dimension 28 was too complicated to be fit to print. An interesting feature of the simplified solutions is that the components of the fiducial vectors largely consist of algebraic units.","PeriodicalId":20904,"journal":{"name":"Quantum Inf. Comput.","volume":"84 1","pages":"400-417"},"PeriodicalIF":0.0,"publicationDate":"2020-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73046862","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}
The phenomenon of quantum entanglement has a very important role in quantum mechanics. Particularly, the quantum spin chain provides a platform for theoretical and experimental investigation of many-body entanglement. In this paper, we investigate time evolution of entanglement in a four-qubit anisotropic Heisenberg XXZ chain with nearest neighboring (NN), the next nearest neighboring (NNN), and the Dzialoshinskii-Moriya (DM) interactions. Calculations of the entanglement evolution of the Werner state carried out in terms of concurrence for selected ranges of control parameters such as DM interaction, frustration, etc. The results show that for the Werner state, DM interaction and the frustration parameters play important roles. Furthermore, results show that the time evolution of the Werner state entanglement may be useful to capture the quantum phase transitions in quantum magnetic systems.
{"title":"Time evolution of entanglement in a four-qubit Heisenberg chain","authors":"H. Pakarzadeh, Zahra Norouzi, J. Vahedi","doi":"10.26421/QIC20.9-10-2","DOIUrl":"https://doi.org/10.26421/QIC20.9-10-2","url":null,"abstract":"The phenomenon of quantum entanglement has a very important role in quantum mechanics. Particularly, the quantum spin chain provides a platform for theoretical and experimental investigation of many-body entanglement. In this paper, we investigate time evolution of entanglement in a four-qubit anisotropic Heisenberg XXZ chain with nearest neighboring (NN), the next nearest neighboring (NNN), and the Dzialoshinskii-Moriya (DM) interactions. Calculations of the entanglement evolution of the Werner state carried out in terms of concurrence for selected ranges of control parameters such as DM interaction, frustration, etc. The results show that for the Werner state, DM interaction and the frustration parameters play important roles. Furthermore, results show that the time evolution of the Werner state entanglement may be useful to capture the quantum phase transitions in quantum magnetic systems.","PeriodicalId":20904,"journal":{"name":"Quantum Inf. Comput.","volume":"30 1","pages":"736-746"},"PeriodicalIF":0.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86000185","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}
We show that Connes' embedding problem is equivalent to the weak Tsirelson problem in the setting of two-outcome synchronous correlation sets. We further show that the extreme points of two-outcome synchronous correlation sets can be realized using a certain class of universal C*-algebras. We examine these algebras in the three-experiment case and verify that the strong and weak Tsirelson problems have affirmative answers in that setting.
{"title":"Connes' embedding problem","authors":"Travis B. Russell","doi":"10.26421/QIC20.5-6-1","DOIUrl":"https://doi.org/10.26421/QIC20.5-6-1","url":null,"abstract":"We show that Connes' embedding problem is equivalent to the weak Tsirelson problem in the setting of two-outcome synchronous correlation sets. We further show that the extreme points of two-outcome synchronous correlation sets can be realized using a certain class of universal C*-algebras. We examine these algebras in the three-experiment case and verify that the strong and weak Tsirelson problems have affirmative answers in that setting.","PeriodicalId":20904,"journal":{"name":"Quantum Inf. Comput.","volume":"208 1","pages":"361-374"},"PeriodicalIF":0.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76100780","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}
The Knapsack Problem is a prominent problem that is used in resource allocation and cryptography. This paper presents an oracle and a circuit design that verifies solutions to the decision problem form of the Bounded Knapsack Problem. This oracle can be used by Grover Search to solve the optimization problem form of the Bounded Knapsack Problem. This algorithm leverages the quadratic speed-up offered by Grover Search to achieve a quantum algorithm for the Knapsack Problem that shows improvement with regard to classical algorithms. The quantum circuits were designed using the Microsoft Q# Programming Language and verified on its local quantum simulator. The paper also provides analyses of the complexity and gate cost of the proposed oracle. The work in this paper is the first such proposed method for the Knapsack Optimization Problem.
{"title":"Quantum-based algorithm and circuit design for bounded Knapsack optimization problem","authors":"Wenjun Hou, M. Perkowski","doi":"10.26421/QIC20.9-10-4","DOIUrl":"https://doi.org/10.26421/QIC20.9-10-4","url":null,"abstract":"The Knapsack Problem is a prominent problem that is used in resource allocation and cryptography. This paper presents an oracle and a circuit design that verifies solutions to the decision problem form of the Bounded Knapsack Problem. This oracle can be used by Grover Search to solve the optimization problem form of the Bounded Knapsack Problem. This algorithm leverages the quadratic speed-up offered by Grover Search to achieve a quantum algorithm for the Knapsack Problem that shows improvement with regard to classical algorithms. The quantum circuits were designed using the Microsoft Q# Programming Language and verified on its local quantum simulator. The paper also provides analyses of the complexity and gate cost of the proposed oracle. The work in this paper is the first such proposed method for the Knapsack Optimization Problem.","PeriodicalId":20904,"journal":{"name":"Quantum Inf. Comput.","volume":"294 1","pages":"766-786"},"PeriodicalIF":0.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74779257","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}
We analyse different Bayesian games where payoffs of players depend on the types of players involved in a two-player game. The dependence is assumed to commensurate with the CHSH game setting. For this, we consider two different types of each player (Alice and Bob) in the game, thus resulting in four different games clubbed together as one Bayesian game. Considering different combinations of common interest, and conflicting interest coordination and anti-coordination games, we find that quantum strategies are always preferred over classical strategies if the shared resource is a pure non-maximally entangled state. However, when the shared resource is a class of mixed state, then quantum strategies are useful only for a given range of the state parameter. Surprisingly, when all conflicting interest games (Battle of the Sexes game and Chicken game) are merged into the Bayesian game picture, then the best strategy for Alice and Bob is to share a set of non-maximally entangled pure states. We demonstrate that this set not only gives higher payoff than any classical strategy, but also outperforms a maximally entangled pure Bell state, mixed Werner states, and Horodecki states. We further propose the representation of a special class of Bell inequalitytilted Bell inequality, as a common as well as conflicting interest Bayesian game. We thereafter, study the effect of sharing an arbitrary two-qubit pure state and a class of mixed state as quantum resource in those games; thus verifying that non-maximally entangled states with high randomness help attain maximum quantum benefit. Additionally, we propose a general framework of a two-player Bayesian game for d-dimensions Bell-CHSH inequality, with and without the tilt factor.
{"title":"Nonlocality, entanglement, and randomness in different conflicting interest Bayesian games","authors":"Hargeet Kaur, Atul Kumar","doi":"10.26421/QIC20.11-12-1","DOIUrl":"https://doi.org/10.26421/QIC20.11-12-1","url":null,"abstract":"We analyse different Bayesian games where payoffs of players depend on the types of players involved in a two-player game. The dependence is assumed to commensurate with the CHSH game setting. For this, we consider two different types of each player (Alice and Bob) in the game, thus resulting in four different games clubbed together as one Bayesian game. Considering different combinations of common interest, and conflicting interest coordination and anti-coordination games, we find that quantum strategies are always preferred over classical strategies if the shared resource is a pure non-maximally entangled state. However, when the shared resource is a class of mixed state, then quantum strategies are useful only for a given range of the state parameter. Surprisingly, when all conflicting interest games (Battle of the Sexes game and Chicken game) are merged into the Bayesian game picture, then the best strategy for Alice and Bob is to share a set of non-maximally entangled pure states. We demonstrate that this set not only gives higher payoff than any classical strategy, but also outperforms a maximally entangled pure Bell state, mixed Werner states, and Horodecki states. We further propose the representation of a special class of Bell inequalitytilted Bell inequality, as a common as well as conflicting interest Bayesian game. We thereafter, study the effect of sharing an arbitrary two-qubit pure state and a class of mixed state as quantum resource in those games; thus verifying that non-maximally entangled states with high randomness help attain maximum quantum benefit. Additionally, we propose a general framework of a two-player Bayesian game for d-dimensions Bell-CHSH inequality, with and without the tilt factor.","PeriodicalId":20904,"journal":{"name":"Quantum Inf. Comput.","volume":"1 1","pages":"901-934"},"PeriodicalIF":0.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89261576","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}
We study the localization and the discrete probability function of a quantum search on the one-dimensional (1D) cycle with self-loops for n vertices and m marked vertices. First, unmarked vertices have no localization since the quantum search on unmarked vertices behaves like the 1D three-state quantum walk (3QW) and localization does not occur with nonlocal initial states on a 3QW, according to residue calculations and the Riemann-Lebesgue theorem. Second, we show that localization does occur on the marked vertices and derive an analytic expression for localization by the degenerate 1eigenvalues contributing to marked vertices. Therefore localization can contribute to a quantum search. Furthermore, we emphasize that localization comes from the self-loops. Third, using the localization of a quantum search, the asymptotic average probability distribution (AAPD) and the discrete probability function (DPF) of a quantum search are obtained. The DPF shows that Szegedys quantum search on the 1D cycle with self-loops spreads ballistically.
{"title":"Localization and discrete probability function of Szegedy's quantum search one-dimensional cycle with self-loops","authors":"Mengke Xu, Zhihao Liu, Hanwu Chen, S. Zheng","doi":"10.26421/QIC20.15-16-2","DOIUrl":"https://doi.org/10.26421/QIC20.15-16-2","url":null,"abstract":"We study the localization and the discrete probability function of a quantum search on the one-dimensional (1D) cycle with self-loops for n vertices and m marked vertices. First, unmarked vertices have no localization since the quantum search on unmarked vertices behaves like the 1D three-state quantum walk (3QW) and localization does not occur with nonlocal initial states on a 3QW, according to residue calculations and the Riemann-Lebesgue theorem. Second, we show that localization does occur on the marked vertices and derive an analytic expression for localization by the degenerate 1eigenvalues contributing to marked vertices. Therefore localization can contribute to a quantum search. Furthermore, we emphasize that localization comes from the self-loops. Third, using the localization of a quantum search, the asymptotic average probability distribution (AAPD) and the discrete probability function (DPF) of a quantum search are obtained. The DPF shows that Szegedys quantum search on the 1D cycle with self-loops spreads ballistically.","PeriodicalId":20904,"journal":{"name":"Quantum Inf. Comput.","volume":"56 1","pages":"1281-1303"},"PeriodicalIF":0.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79679678","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}
Francisco José Orts Gómez, Gloria Ortega López, E. M. Garzón
Despite the great interest that the scientific community has in quantum computing, the scarcity and high cost of resources prevent to advance in this field. Specifically, qubits are very expensive to build, causing the few available quantum computers are tremendously limited in their number of qubits and delaying their progress. This work presents new reversible circuits that optimize the necessary resources for the conversion of a sign binary number into two's complement of N digits. The benefits of our work are two: on the one hand, the proposed two's complement converters are fault tolerant circuits and also are more efficient in terms of resources (essentially, quantum cost, number of qubits, and T-count) than the described in the literature. On the other hand, valuable information about available converters and, what is more, quantum adders, is summarized in tables for interested researchers. The converters have been measured using robust metrics and have been compared with the state-of-the-art circuits. The code to build them in a real quantum computer is given.
{"title":"Efficient reversible quantum design of sig-magnitude to two's complement converters","authors":"Francisco José Orts Gómez, Gloria Ortega López, E. M. Garzón","doi":"10.26421/QIC20.9-10-3","DOIUrl":"https://doi.org/10.26421/QIC20.9-10-3","url":null,"abstract":"Despite the great interest that the scientific community has in quantum computing, the scarcity and high cost of resources prevent to advance in this field. Specifically, qubits are very expensive to build, causing the few available quantum computers are tremendously limited in their number of qubits and delaying their progress. This work presents new reversible circuits that optimize the necessary resources for the conversion of a sign binary number into two's complement of N digits. The benefits of our work are two: on the one hand, the proposed two's complement converters are fault tolerant circuits and also are more efficient in terms of resources (essentially, quantum cost, number of qubits, and T-count) than the described in the literature. On the other hand, valuable information about available converters and, what is more, quantum adders, is summarized in tables for interested researchers. The converters have been measured using robust metrics and have been compared with the state-of-the-art circuits. The code to build them in a real quantum computer is given.","PeriodicalId":20904,"journal":{"name":"Quantum Inf. Comput.","volume":"81 1","pages":"747-765"},"PeriodicalIF":0.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79318855","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}
Designing a quantum oracle is an important step in practical realization of Grover algorithm, therefore it is useful to create methodologies to design oracles. Lattice diagrams are regular two-dimensional structures that can be directly mapped onto a quantum circuit. We present a quantum oracle design methodology based on lattices. The oracles are designed with a proposed method using generalized Boolean symmetric functions realized with lattice diagrams. We also present a decomposition-based algorithm that transforms non-symmetric functions into symmetric or partially symmetric functions. Our method, which combines logic minimization, logic decomposition, and mapping, has lower quantum cost with fewer ancilla qubits. Overall, we obtain encouraging synthesis results superior to previously published data.
{"title":"Realization of Quantum Oracles using Symmetries of Boolean Functions","authors":"Peng Gao, Yiwei Li, M. Perkowski, Xiaoyu Song","doi":"10.26421/QIC20.5-6-4","DOIUrl":"https://doi.org/10.26421/QIC20.5-6-4","url":null,"abstract":"Designing a quantum oracle is an important step in practical realization of Grover algorithm, therefore it is useful to create methodologies to design oracles. Lattice diagrams are regular two-dimensional structures that can be directly mapped onto a quantum circuit. We present a quantum oracle design methodology based on lattices. The oracles are designed with a proposed method using generalized Boolean symmetric functions realized with lattice diagrams. We also present a decomposition-based algorithm that transforms non-symmetric functions into symmetric or partially symmetric functions. Our method, which combines logic minimization, logic decomposition, and mapping, has lower quantum cost with fewer ancilla qubits. Overall, we obtain encouraging synthesis results superior to previously published data.","PeriodicalId":20904,"journal":{"name":"Quantum Inf. Comput.","volume":"57 1","pages":"418-448"},"PeriodicalIF":0.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81353987","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}
Execution of Grover's quantum search algorithm needs rather limited resources without much fine tuning. Consequently, the algorithm can be implemented in a wide variety of physical set-ups, which involve wave dynamics but may not need other quantum features. Several of these set-ups are described, pointing out that some of them occur quite naturally. In particular, it is entirely possible that the algorithm played a key role in selection of the universal structure of genetic languages.
{"title":"Grover's algorithm in natural settings","authors":"Apoorva D. Patel","doi":"10.26421/QIC21.11-12-3","DOIUrl":"https://doi.org/10.26421/QIC21.11-12-3","url":null,"abstract":"Execution of Grover's quantum search algorithm needs rather limited resources without much fine tuning. Consequently, the algorithm can be implemented in a wide variety of physical set-ups, which involve wave dynamics but may not need other quantum features. Several of these set-ups are described, pointing out that some of them occur quite naturally. In particular, it is entirely possible that the algorithm played a key role in selection of the universal structure of genetic languages.","PeriodicalId":20904,"journal":{"name":"Quantum Inf. Comput.","volume":"52 1","pages":"945-954"},"PeriodicalIF":0.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89362189","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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