Our previous works presented zeta functions by the Konno-Sato theorem or the Fourier analysis for one-particle models including random walks, correlated random walks, quantum walks, and open quantum random walks. This paper introduces a new zeta function for multi-particle models with probabilistic or quantum interactions, called the interacting particle system (IPS). We compute the zeta function for some tensor-type IPSs.
{"title":"/ Zeta correspondence","authors":"T. Komatsu, N. Konno, I. Sato","doi":"10.26421/qic22.3-4-4","DOIUrl":"https://doi.org/10.26421/qic22.3-4-4","url":null,"abstract":"Our previous works presented zeta functions by the Konno-Sato theorem or the Fourier analysis for one-particle models including random walks, correlated random walks, quantum walks, and open quantum random walks. This paper introduces a new zeta function for multi-particle models with probabilistic or quantum interactions, called the interacting particle system (IPS). We compute the zeta function for some tensor-type IPSs.","PeriodicalId":20904,"journal":{"name":"Quantum Inf. Comput.","volume":"98 1","pages":"251-269"},"PeriodicalIF":0.0,"publicationDate":"2021-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76476125","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 investigate the $Lambda$-polytopes, a convex-linear structure recently defined and applied to the classical simulation of quantum computation with magic states by sampling. There is one such polytope, $Lambda_n$, for every number $n$ of qubits. We establish two properties of the family ${Lambda_n, nin mathbb{N}}$, namely (i) Any extremal point (vertex) $A_alpha in Lambda_m$ can be used to construct vertices in $Lambda_n$, for all $n>m$. (ii) For vertices obtained through this mapping, the classical simulation of quantum computation with magic states can be efficiently reduced to the classical simulation based on the preimage $A_alpha$. In addition, we describe a new class of vertices in $Lambda_2$ which is outside the known classification. While the hardness of classical simulation remains an open problem for most extremal points of $Lambda_n$, the above results extend efficient classical simulation of quantum computations beyond the presently known range.
本文研究了$Lambda$ -多边形，这是最近定义的一种凸线性结构，并通过抽样将其应用于具有魔幻状态的量子计算的经典模拟。对于每一个$n$的量子位，都有一个这样的多面体$Lambda_n$。我们建立了家族${Lambda_n, nin mathbb{N}}$的两个性质，即(i)对于所有$n>m$，任何极值点(顶点)$A_alpha in Lambda_m$都可以用来构造$Lambda_n$中的顶点。(ii)对于通过这种映射得到的顶点，可以有效地将具有魔幻状态的量子计算经典模拟简化为基于预像$A_alpha$的经典模拟。此外，我们在$Lambda_2$中描述了一类新的顶点，它是已知分类之外的。虽然对于$Lambda_n$的大多数极值点，经典模拟的硬度仍然是一个开放的问题，但上述结果将量子计算的有效经典模拟扩展到目前已知的范围之外。
{"title":"On the extremal points of the Lambda polytopes and classical simulation of quantum computation with magic states","authors":"C. Okay, Michael Zurel, R. Raussendorf","doi":"10.26421/QIC21.13-14-2","DOIUrl":"https://doi.org/10.26421/QIC21.13-14-2","url":null,"abstract":"We investigate the $Lambda$-polytopes, a convex-linear structure recently defined and applied to the classical simulation of quantum computation with magic states by sampling. There is one such polytope, $Lambda_n$, for every number $n$ of qubits. We establish two properties of the family ${Lambda_n, nin mathbb{N}}$, namely (i) Any extremal point (vertex) $A_alpha in Lambda_m$ can be used to construct vertices in $Lambda_n$, for all $n>m$. (ii) For vertices obtained through this mapping, the classical simulation of quantum computation with magic states can be efficiently reduced to the classical simulation based on the preimage $A_alpha$. In addition, we describe a new class of vertices in $Lambda_2$ which is outside the known classification. While the hardness of classical simulation remains an open problem for most extremal points of $Lambda_n$, the above results extend efficient classical simulation of quantum computations beyond the presently known range.","PeriodicalId":20904,"journal":{"name":"Quantum Inf. Comput.","volume":"16 1","pages":"1091-1110"},"PeriodicalIF":0.0,"publicationDate":"2021-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87963075","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 this article, we investigate the problem of entanglement characterization by polarization measurements combined with maximum likelihood estimation (MLE). A realistic scenario is considered with measurement results distorted by random experimental errors. In particular, by imposing unitary rotations acting on the measurement operators, we can test the performance of the tomographic technique versus the amount of noise. Then, dark counts are introduced to explore the efficiency of the framework in a multi-dimensional noise scenario. The concurrence is used as a figure of merit to quantify how well entanglement is preserved through noisy measurements. Quantum fidelity is computed to quantify the accuracy of state reconstruction. The results of numerical simulations are depicted on graphs and discussed.
{"title":"Entanglement characterization by single-photon counting with random noise","authors":"A. Czerwinski","doi":"10.26421/QIC22.1-2-1","DOIUrl":"https://doi.org/10.26421/QIC22.1-2-1","url":null,"abstract":"In this article, we investigate the problem of entanglement characterization by polarization measurements combined with maximum likelihood estimation (MLE). A realistic scenario is considered with measurement results distorted by random experimental errors. In particular, by imposing unitary rotations acting on the measurement operators, we can test the performance of the tomographic technique versus the amount of noise. Then, dark counts are introduced to explore the efficiency of the framework in a multi-dimensional noise scenario. The concurrence is used as a figure of merit to quantify how well entanglement is preserved through noisy measurements. Quantum fidelity is computed to quantify the accuracy of state reconstruction. The results of numerical simulations are depicted on graphs and discussed.","PeriodicalId":20904,"journal":{"name":"Quantum Inf. Comput.","volume":"27 1","pages":"1-16"},"PeriodicalIF":0.0,"publicationDate":"2021-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74324809","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}
Giacomo Baggio, F. Ticozzi, Peter D. Johnson, L. Viola
We formalize the problem of dissipative quantum encoding, and explore the advantages of using Markovian evolution to prepare a quantum code in the desired logical space, with emphasis on discrete-time dynamics and the possibility of exact finite-time convergence. In particular, we investigate robustness of the encoding dynamics and their ability to tolerate initialization errors, thanks to the existence of non-trivial basins of attraction. As a key application, we show that for stabilizer quantum codes on qubits, a finite-time dissipative encoder may always be constructed, by using at most a number of quantum maps determined by the number of stabilizer generators. We find that even in situations where the target code lacks gauge degrees of freedom in its subsystem form, dissipative encoders afford nontrivial robustness against initialization errors, thus overcoming a limitation of purely unitary encoding procedures. Our general results are illustrated in a number of relevant examples, including Kitaev’s toric code.
{"title":"Dissipative encoding of quantum information","authors":"Giacomo Baggio, F. Ticozzi, Peter D. Johnson, L. Viola","doi":"10.26421/QIC21.9-10-2","DOIUrl":"https://doi.org/10.26421/QIC21.9-10-2","url":null,"abstract":"We formalize the problem of dissipative quantum encoding, and explore the advantages of using Markovian evolution to prepare a quantum code in the desired logical space, with emphasis on discrete-time dynamics and the possibility of exact finite-time convergence. In particular, we investigate robustness of the encoding dynamics and their ability to tolerate initialization errors, thanks to the existence of non-trivial basins of attraction. As a key application, we show that for stabilizer quantum codes on qubits, a finite-time dissipative encoder may always be constructed, by using at most a number of quantum maps determined by the number of stabilizer generators. We find that even in situations where the target code lacks gauge degrees of freedom in its subsystem form, dissipative encoders afford nontrivial robustness against initialization errors, thus overcoming a limitation of purely unitary encoding procedures. Our general results are illustrated in a number of relevant examples, including Kitaev’s toric code.","PeriodicalId":20904,"journal":{"name":"Quantum Inf. Comput.","volume":"27 1","pages":"737-770"},"PeriodicalIF":0.0,"publicationDate":"2021-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88453612","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 this work, the application of continuous time quantum walks (CTQW) to the Maximum Clique (MC) problem was studied. Performing CTQW on graphs can generate distinct periodic probability amplitudes for different vertices. We found that the intensities of the probability amplitudes at some frequencies imply the clique structure of special kinds of graphs. Recursive algorithms with time complexity O(N^6) in classical computers were proposed to determine the maximum clique. We have experimented on random graphs where each edge exists with different probabilities. Although counter examples were not found for random graphs, whether these algorithms are universal is beyond the scope of this work.
{"title":"Algorithms for finding the maximum clique based on continuous time quantum walks","authors":"Xi Li, Mingyou Wu, Hanwu Chen, Zhibao Liu","doi":"10.26421/QIC21.1-2-4","DOIUrl":"https://doi.org/10.26421/QIC21.1-2-4","url":null,"abstract":"In this work, the application of continuous time quantum walks (CTQW) to the Maximum Clique (MC) problem was studied. Performing CTQW on graphs can generate distinct periodic probability amplitudes for different vertices. We found that the intensities of the probability amplitudes at some frequencies imply the clique structure of special kinds of graphs. Recursive algorithms with time complexity O(N^6) in classical computers were proposed to determine the maximum clique. We have experimented on random graphs where each edge exists with different probabilities. Although counter examples were not found for random graphs, whether these algorithms are universal is beyond the scope of this work.","PeriodicalId":20904,"journal":{"name":"Quantum Inf. Comput.","volume":"36 1","pages":"59-79"},"PeriodicalIF":0.0,"publicationDate":"2021-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85849980","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 existence of localization for the Grover walk on the multi-dimensional lattice is known. This paper gives some conditions for the existence of localization for the space-homogeneous quantum walks. We also prove that localization does not occur for the Fourier walk on the multi-dimensional lattice.
{"title":"The Multi-dimensional Lattice","authors":"Akihiro Narimatsu","doi":"10.26421/QIC21.5-6-3","DOIUrl":"https://doi.org/10.26421/QIC21.5-6-3","url":null,"abstract":"The existence of localization for the Grover walk on the multi-dimensional lattice is known. This paper gives some conditions for the existence of localization for the space-homogeneous quantum walks. We also prove that localization does not occur for the Fourier walk on the multi-dimensional lattice.","PeriodicalId":20904,"journal":{"name":"Quantum Inf. Comput.","volume":"6 1","pages":"387-394"},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80902675","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 this paper, we work on a quantum walk whose system is manipulated by a five-diagonal unitary matrix, and present long-time limit distributions. The quantum walk launches off a location and delocalizes in distribution as its system is getting updated. The fivediagonal matrix contains a phase term and the quantum walk becomes a standard coined walk when the phase term is fixed at special values. Or, the phase term gives an effect on the quantum walk. As a result, we will see an explicit form of a long-time limit distribution for a quantum walk driven by the matrix, and thanks to the exact form, we understand how the quantum walker approximately distributes in space after the long-time evolution has been executed on the walk.
{"title":"A limit distribution for a quantum walk driven by a five-diagonal unitary matrix","authors":"T. Machida","doi":"10.26421/QIC21.1-2-2","DOIUrl":"https://doi.org/10.26421/QIC21.1-2-2","url":null,"abstract":"In this paper, we work on a quantum walk whose system is manipulated by a five-diagonal unitary matrix, and present long-time limit distributions. The quantum walk launches off a location and delocalizes in distribution as its system is getting updated. The fivediagonal matrix contains a phase term and the quantum walk becomes a standard coined walk when the phase term is fixed at special values. Or, the phase term gives an effect on the quantum walk. As a result, we will see an explicit form of a long-time limit distribution for a quantum walk driven by the matrix, and thanks to the exact form, we understand how the quantum walker approximately distributes in space after the long-time evolution has been executed on the walk.","PeriodicalId":20904,"journal":{"name":"Quantum Inf. Comput.","volume":"70 1","pages":"19-36"},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85801909","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}
Quantum key distribution (QKD) offers a very strong property called everlasting security, which says if authentication is unbroken during the execution of QKD, the generated key remains information-theoretically secure indefinitely. For this purpose, we propose the use of certain universal hashing based MACs for use in QKD, which are fast, very efficient with key material, and are shown to be highly secure. Universal hash functions are ubiquitous in computer science with many applications ranging from quantum key distribution and information security to data structures and parallel computing. In QKD, they are used at least for authentication, error correction, and privacy amplification. Using results from Cohen [Duke Math. J., 1954], we also construct some new families of ε-almost-∆-universal hash function families which have much better collision bounds than the well-known Polynomial Hash. Then we propose a general method for converting any such family to an ε-almost-strongly universal hash function family, which makes them useful in a wide range of applications, including authentication in QKD.
{"title":"Everlasting security of quantum key distribution with 1K-DWCDM and quadratic hash","authors":"Khodakhast Bibak, Robert Ritchie, B. Zolfaghari","doi":"10.26421/QIC21.3-4-1","DOIUrl":"https://doi.org/10.26421/QIC21.3-4-1","url":null,"abstract":"Quantum key distribution (QKD) offers a very strong property called everlasting security, which says if authentication is unbroken during the execution of QKD, the generated key remains information-theoretically secure indefinitely. For this purpose, we propose the use of certain universal hashing based MACs for use in QKD, which are fast, very efficient with key material, and are shown to be highly secure. Universal hash functions are ubiquitous in computer science with many applications ranging from quantum key distribution and information security to data structures and parallel computing. In QKD, they are used at least for authentication, error correction, and privacy amplification. Using results from Cohen [Duke Math. J., 1954], we also construct some new families of ε-almost-∆-universal hash function families which have much better collision bounds than the well-known Polynomial Hash. Then we propose a general method for converting any such family to an ε-almost-strongly universal hash function family, which makes them useful in a wide range of applications, including authentication in QKD.","PeriodicalId":20904,"journal":{"name":"Quantum Inf. Comput.","volume":"351 1","pages":"181-202"},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75103197","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 introduce a variant of quantum signatures in which nonbinary symbols are signed instead of bits. The public keys are fingerprinting states, just as in the scheme of Gottesman and Chuang [1], but we allow for multiple ways to reveal the private key partially. The effect of this modification is a reduction of the number of qubits expended per message bit. Asymptotically the expenditure becomes as low as one qubit per message bit. We give a security proof, and we present numerical results that show how the improvement in public key size depends on the message length.
{"title":"Quantum digital signatures with smaller public keys","authors":"B. Škorić","doi":"10.26421/QIC21.11-12-4","DOIUrl":"https://doi.org/10.26421/QIC21.11-12-4","url":null,"abstract":"We introduce a variant of quantum signatures in which nonbinary symbols are signed instead of bits. The public keys are fingerprinting states, just as in the scheme of Gottesman and Chuang [1], but we allow for multiple ways to reveal the private key partially. The effect of this modification is a reduction of the number of qubits expended per message bit. Asymptotically the expenditure becomes as low as one qubit per message bit. We give a security proof, and we present numerical results that show how the improvement in public key size depends on the message length.","PeriodicalId":20904,"journal":{"name":"Quantum Inf. Comput.","volume":"11 1","pages":"955-973"},"PeriodicalIF":0.0,"publicationDate":"2020-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73111089","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}
Parzanchevski and Sarnak recently adapted an algorithm of Ross and Selinger for factorization of PU(2)-diagonal elements to within distance $varepsilon$ into an efficient probabilistic algorithm for any PU(2)-element, using at most $3log_pfrac{1}{varepsilon^3}$ factors from certain well-chosen sets. The Clifford+$T$ gates are one such set arising from $p=2$. In that setting, we leverage recent work of Carvalho Pinto and Petit to improve this to $frac{7}{3}log_2frac{1}{varepsilon^3}$, and implement the algorithm in Haskell.
{"title":"Short paths in PU(2)","authors":"Zachary Stier","doi":"10.26421/QIC21.9-10-3","DOIUrl":"https://doi.org/10.26421/QIC21.9-10-3","url":null,"abstract":"Parzanchevski and Sarnak recently adapted an algorithm of Ross and Selinger for factorization of PU(2)-diagonal elements to within distance $varepsilon$ into an efficient probabilistic algorithm for any PU(2)-element, using at most $3log_pfrac{1}{varepsilon^3}$ factors from certain well-chosen sets. The Clifford+$T$ gates are one such set arising from $p=2$. In that setting, we leverage recent work of Carvalho Pinto and Petit to improve this to $frac{7}{3}log_2frac{1}{varepsilon^3}$, and implement the algorithm in Haskell.","PeriodicalId":20904,"journal":{"name":"Quantum Inf. Comput.","volume":"26 1","pages":"2"},"PeriodicalIF":0.0,"publicationDate":"2020-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86945289","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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