We present a polynomial time exact quantum algorithm for the hidden subgroup problem in $Z_{m^k}^n$. The algorithm uses the quantum Fourier transform modulo $m$ and does not require factorization of $m$. For smooth $m$, i.e., when the prime factors of $m$ are of size $(log m)^{O(1)}$, the quantum Fourier transform can be exactly computed using the method discovered independently by Cleve and Coppersmith, while for general $m$, the algorithm of Mosca and Zalka is available. Even for $m=3$ and $k=1$ our result appears to be new. We also present applications to compute the structure of abelian and solvable groups whose order has the same (but possibly unknown) prime factors as $m$. The applications for solvable groups also rely on an exact version of a technique proposed by Watrous for computing the uniform superposition of elements of subgroups.
{"title":"An exact quantum hidden subgroup algorithm and applications to solvable groups","authors":"Muhammad Imran, G. Ivanyos","doi":"10.26421/qic22.9-10-4","DOIUrl":"https://doi.org/10.26421/qic22.9-10-4","url":null,"abstract":"We present a polynomial time exact quantum algorithm for the hidden subgroup problem in $Z_{m^k}^n$. The algorithm uses the quantum Fourier transform modulo $m$ and does not require factorization of $m$. For smooth $m$, i.e., when the prime factors of $m$ are of size $(log m)^{O(1)}$, the quantum Fourier transform can be exactly computed using the method discovered independently by Cleve and Coppersmith, while for general $m$, the algorithm of Mosca and Zalka is available. Even for $m=3$ and $k=1$ our result appears to be new. We also present applications to compute the structure of abelian and solvable groups whose order has the same (but possibly unknown) prime factors as $m$. The applications for solvable groups also rely on an exact version of a technique proposed by Watrous for computing the uniform superposition of elements of subgroups.","PeriodicalId":20904,"journal":{"name":"Quantum Inf. Comput.","volume":"194 1","pages":"770-789"},"PeriodicalIF":0.0,"publicationDate":"2022-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90459708","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 [Science 340:1205, (2013)], via entanglement polytopes Michael Walter et al. obtained a finite yet systematic classification of multi-particle entanglement. It is well known that under SLOCC, pure states of three (four) qubits are partitioned into six (nine) families. Ac'{i}n et al. proposed the generalized Schmidt decomposition for three qubits and partitioned pure states of three qubits into five types. In this paper,we present a LU invariant and an entanglement measures for the GHZ SLOCC class of three qubits, and partition states of the GHZ SLOCC class of three qubits into ten families and each family into two subfamilies under LU. We give a necessary and sufficient condition for the uniqueness of the generalized Schmidt decomposition for the GHZ SLOCC class.
{"title":"Partition GHZ SLOCC class of three qubits into ten families under LU","authors":"Dafa Li","doi":"10.2139/ssrn.4113733","DOIUrl":"https://doi.org/10.2139/ssrn.4113733","url":null,"abstract":"In [Science 340:1205, (2013)], via entanglement polytopes Michael Walter et al. obtained a finite yet systematic classification of multi-particle entanglement. It is well known that under SLOCC, pure states of three (four) qubits are partitioned into six (nine) families. Ac'{i}n et al. proposed the generalized Schmidt decomposition for three qubits and partitioned pure states of three qubits into five types. In this paper,we present a LU invariant and an entanglement measures for the GHZ SLOCC class of three qubits, and partition states of the GHZ SLOCC class of three qubits into ten families and each family into two subfamilies under LU. We give a necessary and sufficient condition for the uniqueness of the generalized Schmidt decomposition for the GHZ SLOCC class.","PeriodicalId":20904,"journal":{"name":"Quantum Inf. Comput.","volume":"11 1","pages":"402-414"},"PeriodicalIF":0.0,"publicationDate":"2022-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75250104","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}
Enrico Zardini, M. Rizzoli, Sebastiano Dissegna, E. Blanzieri, D. Pastorello
{"title":"Annealer","authors":"Enrico Zardini, M. Rizzoli, Sebastiano Dissegna, E. Blanzieri, D. Pastorello","doi":"10.26421/QIC22.15-16-4","DOIUrl":"https://doi.org/10.26421/QIC22.15-16-4","url":null,"abstract":"","PeriodicalId":20904,"journal":{"name":"Quantum Inf. Comput.","volume":"24 1","pages":"1320-1350"},"PeriodicalIF":0.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73474073","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 image representation has a significant impact in quantum image processing. In this paper, a bit-plane representation for log-polar quantum images (BRLQI) is proposed, which utilizes $(n+4)$ or $(n+6)$ qubits to store and process a grayscale or RGB color image of $2^n$ pixels. Compared to a quantum log-polar image (QUALPI), the storage capacity of BRLQI improves 16 times. Moreover, several quantum operations based on BRLQI are proposed, including color information complement operation, bit-planes reversing operation, bit-planes translation operation and conditional exchange operations between bit-planes. Combining the above operations, we designed an image scrambling circuit suitable for the BRLQI model. Furthermore, comparison results of the scrambling circuits indicate that those operations based on BRLQI have a lower quantum cost than QUALPI. In addition, simulation experiments illustrate that the proposed scrambling algorithm is effective and efficient.
{"title":"A novel enhanced quantum image representation based on bit-planes for log-polar coordinates","authors":"Xiao Chen, Zhihao Liu, Hanwu Chen, Liang Wang","doi":"10.26421/qic22.1-2-2","DOIUrl":"https://doi.org/10.26421/qic22.1-2-2","url":null,"abstract":"Quantum image representation has a significant impact in quantum image processing. In this paper, a bit-plane representation for log-polar quantum images (BRLQI) is proposed, which utilizes $(n+4)$ or $(n+6)$ qubits to store and process a grayscale or RGB color image of $2^n$ pixels. Compared to a quantum log-polar image (QUALPI), the storage capacity of BRLQI improves 16 times. Moreover, several quantum operations based on BRLQI are proposed, including color information complement operation, bit-planes reversing operation, bit-planes translation operation and conditional exchange operations between bit-planes. Combining the above operations, we designed an image scrambling circuit suitable for the BRLQI model. Furthermore, comparison results of the scrambling circuits indicate that those operations based on BRLQI have a lower quantum cost than QUALPI. In addition, simulation experiments illustrate that the proposed scrambling algorithm is effective and efficient.","PeriodicalId":20904,"journal":{"name":"Quantum Inf. Comput.","volume":"15 7 1","pages":"17-37"},"PeriodicalIF":0.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82580101","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 a recent paper by S. Pandey, V. Paulsen, J. Prakash, and M. Rahaman, the authors studied the entanglement breaking quantum channels $Phi_t:mbb{C}^{dtimes d} to mbb{C}^{d times d}$ for $t in [-frac{1}{d^2-1}, frac{1}{d+1}]$ defined by $Phi_t(X) = tX+ (1-t)tr(X) frac{1}{d}I$. They proved that Zauner's conjecture is equivalent to the statement that entanglement breaking rank of $Phi_{frac{1}{d+1}}$ is $d^2$. The authors made the extended conjecture that $ebr(Phi_t)=d^2$ for every $t in [0, frac{1}{d+1}]$ and proved it in dimensions 2 and 3. In this paper we prove that for any $t in [-frac{1}{d^2-1}, frac{1}{d+1}] setminus{0}$ the equality $ebr(Phi_t)=d^2$ is equivalent to the existence of a pair of informationally-complete unit-norm tight frames ${|x_ira}_{i=1}^{d^2}, {|y_ira}_{i=1}^{d^2}$ in $mbb{C}^d $ which are mutually unbiased in the following sense: for any $ineq j$ it holds that $|la x_i|y_jra|^2 = frac{1-t}{d}$ and $|la x_i|y_ira|^2 = frac{t(d^2-1)+1}{d}$. Moreover, it follows that $|la x_i|x_jrala y_i|y_jra|=|t|$ for $ineq j$. However, our numerical searches for solutions were not successful in dimensions 4 and 5 for values of $t$ other than $0$ or $frac{1}{d+1}$.
在S. Pandey, V. Paulsen, J. Prakash和M. Rahaman最近的一篇论文中，作者研究了$Phi_t(X) = tX+ (1-t)tr(X) frac{1}{d}I$定义的$t in [-frac{1}{d^2-1}, frac{1}{d+1}]$的量子通道$Phi_t:mbb{C}^{dtimes d} to mbb{C}^{d times d}$的纠缠破坏。他们证明了Zauner猜想等价于$Phi_{frac{1}{d+1}}$的纠缠破秩为$d^2$的表述。作者对每个$t in [0, frac{1}{d+1}]$都提出了扩展猜想$ebr(Phi_t)=d^2$，并在2维和3维上进行了证明。在本文中，我们证明了对于任意$t in [-frac{1}{d^2-1}, frac{1}{d+1}] setminus{0}$等式$ebr(Phi_t)=d^2$等价于$mbb{C}^d $中存在一对信息完备的单位范数紧框架${|x_ira}_{i=1}^{d^2}, {|y_ira}_{i=1}^{d^2}$，它们在以下意义上是相互无偏的:对于任意$ineq j$，它持有$|la x_i|y_jra|^2 = frac{1-t}{d}$和$|la x_i|y_ira|^2 = frac{t(d^2-1)+1}{d}$。此外，可以得出$ineq j$等于$|la x_i|x_jrala y_i|y_jra|=|t|$。然而，我们的数值搜索解决方案是不成功的，在维度4和5的值$t$除了$0$或$frac{1}{d+1}$。
{"title":"On the continuous Zauner conjecture","authors":"D. Yakymenko","doi":"10.26421/QIC22.9-10-1","DOIUrl":"https://doi.org/10.26421/QIC22.9-10-1","url":null,"abstract":"In a recent paper by S. Pandey, V. Paulsen, J. Prakash, and M. Rahaman, the authors studied the entanglement breaking quantum channels $Phi_t:mbb{C}^{dtimes d} to mbb{C}^{d times d}$ for $t in [-frac{1}{d^2-1}, frac{1}{d+1}]$ defined by $Phi_t(X) = tX+ (1-t)tr(X) frac{1}{d}I$. They proved that Zauner's conjecture is equivalent to the statement that entanglement breaking rank of $Phi_{frac{1}{d+1}}$ is $d^2$. The authors made the extended conjecture that $ebr(Phi_t)=d^2$ for every $t in [0, frac{1}{d+1}]$ and proved it in dimensions 2 and 3. In this paper we prove that for any $t in [-frac{1}{d^2-1}, frac{1}{d+1}] setminus{0}$ the equality $ebr(Phi_t)=d^2$ is equivalent to the existence of a pair of informationally-complete unit-norm tight frames ${|x_ira}_{i=1}^{d^2}, {|y_ira}_{i=1}^{d^2}$ in $mbb{C}^d $ which are mutually unbiased in the following sense: for any $ineq j$ it holds that $|la x_i|y_jra|^2 = frac{1-t}{d}$ and $|la x_i|y_ira|^2 = frac{t(d^2-1)+1}{d}$. Moreover, it follows that $|la x_i|x_jrala y_i|y_jra|=|t|$ for $ineq j$. However, our numerical searches for solutions were not successful in dimensions 4 and 5 for values of $t$ other than $0$ or $frac{1}{d+1}$.","PeriodicalId":20904,"journal":{"name":"Quantum Inf. Comput.","volume":"1 1","pages":"721-732"},"PeriodicalIF":0.0,"publicationDate":"2021-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77645857","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 quantum guesswork quantifies the minimum number of queries needed to guess the state of a quantum ensemble if one is allowed to query only one state at a time. Previous approaches to the computation of the guesswork were based on standard semi-definite programming techniques and therefore lead to approximated results. In contrast, we show that computing the quantum guesswork of qubit ensembles with uniform probability distribution corresponds to solving a quadratic assignment problem and we provide an algorithm that, upon the input of any qubit ensemble over a discrete ring, after finitely many steps outputs the exact closed-form expression of its guesswork. While in general the complexity of our guesswork-computing algorithm is factorial in the number of states, our main result consists of showing a more-than-quadratic speedup for symmetric ensembles, a scenario corresponding to the three-dimensional analog of the maximization version of the turbine-balancing problem. To find such symmetries, we provide an algorithm that, upon the input of any point set over a discrete ring, after finitely many steps outputs its exact symmetries. The complexity of our symmetries-finding algorithm is polynomial in the number of points. As examples, we compute the guesswork of regular and quasi-regular sets of qubit states.
{"title":"Computing the quantumguesswork: a quadratic assignment problem","authors":"M. Dall’Arno, F. Buscemi, Takeshi Koshiba","doi":"10.26421/QIC23.9-10-1","DOIUrl":"https://doi.org/10.26421/QIC23.9-10-1","url":null,"abstract":"The quantum guesswork quantifies the minimum number of queries needed to guess the state of a quantum ensemble if one is allowed to query only one state at a time. Previous approaches to the computation of the guesswork were based on standard semi-definite programming techniques and therefore lead to approximated results. In contrast, we show that computing the quantum guesswork of qubit ensembles with uniform probability distribution corresponds to solving a quadratic assignment problem and we provide an algorithm that, upon the input of any qubit ensemble over a discrete ring, after finitely many steps outputs the exact closed-form expression of its guesswork. While in general the complexity of our guesswork-computing algorithm is factorial in the number of states, our main result consists of showing a more-than-quadratic speedup for symmetric ensembles, a scenario corresponding to the three-dimensional analog of the maximization version of the turbine-balancing problem. To find such symmetries, we provide an algorithm that, upon the input of any point set over a discrete ring, after finitely many steps outputs its exact symmetries. The complexity of our symmetries-finding algorithm is polynomial in the number of points. As examples, we compute the guesswork of regular and quasi-regular sets of qubit states.","PeriodicalId":20904,"journal":{"name":"Quantum Inf. Comput.","volume":"57 1","pages":"721-732"},"PeriodicalIF":0.0,"publicationDate":"2021-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83899762","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 unitary group acting on the Hilbert space ${cal H}:=(C^2)^{otimes 3}$ of three quantum bits admits a Lie subgroup, $U^{S_3}(8)$, of elements which permute with the symmetric group of permutations of three objects. Under the action of such a Lie subgroup, the Hilbert space ${cal H}$ splits into three invariant subspaces of dimensions $4$, $2$ and $2$ respectively, each corresponding to an irreducible representation of $su(2)$. The subspace of dimension $4$ is uniquely determined and corresponds to states that are themselves invariant under the action of the symmetric group. This is the so called {it symmetric sector.} The subspaces of dimension two are not uniquely determined and we parametrize them all. We provide an analysis of pure states that are in the subspaces invariant under $U^{S_3}(8)$. This concerns their entanglement properties, separability criteria and dynamics under the Lie subgroup $U^{S_3}(8)$. As a physical motivation for the states and dynamics we study, we propose a physical set-up which consists of a symmetric network of three spin $frac{1}{2}$ particles under a common driving electro-magnetic field. {For such system, we solve the control theoretic problem of driving a separable state to a state with maximal distributed entanglement.
{"title":"Symmetric states and dynamics of three quantum bits","authors":"F. Albertini, D. D’Alessandro","doi":"10.26421/QIC22.7-8-1","DOIUrl":"https://doi.org/10.26421/QIC22.7-8-1","url":null,"abstract":"The unitary group acting on the Hilbert space ${cal H}:=(C^2)^{otimes 3}$ of three quantum bits admits a Lie subgroup, $U^{S_3}(8)$, of elements which permute with the symmetric group of permutations of three objects. Under the action of such a Lie subgroup, the Hilbert space ${cal H}$ splits into three invariant subspaces of dimensions $4$, $2$ and $2$ respectively, each corresponding to an irreducible representation of $su(2)$. The subspace of dimension $4$ is uniquely determined and corresponds to states that are themselves invariant under the action of the symmetric group. This is the so called {it symmetric sector.} The subspaces of dimension two are not uniquely determined and we parametrize them all. We provide an analysis of pure states that are in the subspaces invariant under $U^{S_3}(8)$. This concerns their entanglement properties, separability criteria and dynamics under the Lie subgroup $U^{S_3}(8)$. As a physical motivation for the states and dynamics we study, we propose a physical set-up which consists of a symmetric network of three spin $frac{1}{2}$ particles under a common driving electro-magnetic field. {For such system, we solve the control theoretic problem of driving a separable state to a state with maximal distributed entanglement.","PeriodicalId":20904,"journal":{"name":"Quantum Inf. Comput.","volume":"33 1","pages":"541-568"},"PeriodicalIF":0.0,"publicationDate":"2021-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89555161","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}
Our computers today, from sophisticated servers to small smartphones, operate based on the same computing model, which requires running a sequence of discrete instructions, specified as an algorithm. This sequential computing paradigm has not yet led to a fast algorithm for an NP-complete problem despite numerous attempts over the past half a century. Unfortunately, even after the introduction of quantum mechanics to the world of computing, we still followed a similar sequential paradigm, which has not yet helped us obtain such an algorithm either. Here a completely different model of computing is proposed to replace the sequential paradigm of algorithms with inherent parallelism of physical processes. Using the proposed model, instead of writing algorithms to solve NP-complete problems, we construct physical systems whose equilibrium states correspond to the desired solutions and let them evolve to search for the solutions. The main requirements of the model are identified and quantum circuits are proposed for its potential implementation.
{"title":"Towards algorithm-free physical equilibrium model of computing","authors":"S. Mousavi","doi":"10.26421/QIC21.15-16-3","DOIUrl":"https://doi.org/10.26421/QIC21.15-16-3","url":null,"abstract":"Our computers today, from sophisticated servers to small smartphones, operate based on the same computing model, which requires running a sequence of discrete instructions, specified as an algorithm. This sequential computing paradigm has not yet led to a fast algorithm for an NP-complete problem despite numerous attempts over the past half a century. Unfortunately, even after the introduction of quantum mechanics to the world of computing, we still followed a similar sequential paradigm, which has not yet helped us obtain such an algorithm either. Here a completely different model of computing is proposed to replace the sequential paradigm of algorithms with inherent parallelism of physical processes. Using the proposed model, instead of writing algorithms to solve NP-complete problems, we construct physical systems whose equilibrium states correspond to the desired solutions and let them evolve to search for the solutions. The main requirements of the model are identified and quantum circuits are proposed for its potential implementation.","PeriodicalId":20904,"journal":{"name":"Quantum Inf. Comput.","volume":"71 1","pages":"1296-1306"},"PeriodicalIF":0.0,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74887818","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 correlations and entanglement in identical-particle systems have been a puzzling question which has attracted vast interest and widely different approaches. Witness formalism developed first for entanglement measurement can be adopted to other kind of correlations. An approach is introduced by Kraus emph{et al.}, [Phys. Rev. A textbf{79}, 012306 (2009)] based on pairing correlations in fermionic systems and the use of witness formalism to detect pairing. In this contribution, a two-particle-annihilation operator is used for constructing a two-particle observable as a candidate witness for pairing correlations of both fermionic and bosonic systems. The corresponding separability bounds are also obtained. Two different types of separability definition are introduced for bosonic systems and the separability bounds associated with each type are discussed.
{"title":"Witnessing pairing correlations in identical-particle systems","authors":"C. Aksak, S. Turgut","doi":"10.26421/qic21.15-16-4","DOIUrl":"https://doi.org/10.26421/qic21.15-16-4","url":null,"abstract":"Quantum correlations and entanglement in identical-particle systems have been a puzzling question which has attracted vast interest and widely different approaches. Witness formalism developed first for entanglement measurement can be adopted to other kind of correlations. An approach is introduced by Kraus emph{et al.}, [Phys. Rev. A textbf{79}, 012306 (2009)] based on pairing correlations in fermionic systems and the use of witness formalism to detect pairing. In this contribution, a two-particle-annihilation operator is used for constructing a two-particle observable as a candidate witness for pairing correlations of both fermionic and bosonic systems. The corresponding separability bounds are also obtained. Two different types of separability definition are introduced for bosonic systems and the separability bounds associated with each type are discussed.","PeriodicalId":20904,"journal":{"name":"Quantum Inf. Comput.","volume":"7 1","pages":"1307-1319"},"PeriodicalIF":0.0,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89583966","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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