scholarly journals Deciding unitary equivalence between matrix polynomials and sets of bipartite quantum states

2011 ◽  
Vol 11 (9&10) ◽  
pp. 813-819
Author(s):  
Eric Chitambar ◽  
Carl Miller ◽  
Yaoyun Shi
Keyword(s):  
Success Probability ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Quantum States ◽  
Unitary Equivalence ◽  
Matrix Polynomials ◽  
Unitary Transformations ◽  
The Matrix ◽  
Pure States ◽  
High Success Probability

In this brief report, we consider the equivalence between two sets of $m+1$ bipartite quantum states under local unitary transformations. For pure states, this problem corresponds to the matrix algebra question of whether two degree $m$ matrix polynomials are unitarily equivalent; i.e. $UA_iV^\dagger=B_i$ for $0\leq i\leq m$ where $U$ and $V$ are unitary and $(A_i, B_i)$ are arbitrary pairs of rectangular matrices. We present a randomized polynomial-time algorithm that solves this problem with an arbitrarily high success probability and outputs transforming matrices $U$ and $V$.

scholarly journals Robustness Computation of Dynamic Controllability in Probabilistic Temporal Networks with Ordinary Distributions

2020 ◽  
Author(s):  
Michael Saint-Guillain ◽  
Tiago Stegun Vaquero ◽  
Jagriti Agrawal ◽  
Steve Chien
Keyword(s):  
Probability Distributions ◽  
Dynamic Control ◽  
Success Probability ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Temporal Networks ◽  
Fixed Parameter ◽  
Probability Of Success ◽  
Dynamic Policy ◽  
Simple Temporal Networks

Most existing works in Probabilistic Simple Temporal Networks (PSTNs) base their frameworks on well-defined probability distributions. This paper addresses on PSTN Dynamic Controllability (DC) robustness measure, i.e. the execution success probability of a network under dynamic control. We consider PSTNs where the probability distributions of the contingent edges are ordinary distributed (e.g. non-parametric, non-symmetric). We introduce the concepts of dispatching protocol (DP) as well as DP-robustness, the probability of success under a predefined dynamic policy. We propose a fixed-parameter pseudo-polynomial time algorithm to compute the exact DP-robustness of any PSTN under NextFirst protocol, and apply to various PSTN datasets, including the real case of planetary exploration in the context of the Mars 2020 rover, and propose an original structural analysis.

scholarly journals On Partially Trace Distance Preserving Maps and Reversible Quantum Channels

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Long Jian ◽  
Kan He ◽  
Qing Yuan ◽  
Fei Wang
Keyword(s):  
Quantum States ◽  
Quantum Channels ◽  
Trace Distance ◽  
Linear Maps ◽  
Unitary Transformations ◽  
Pure States ◽  
Positive Linear Maps

We give a characterization of trace-preserving and positive linear maps preserving trace distance partially, that is, preservers of trace distance of quantum states or pure states rather than all matrices. Applying such results, we give a characterization of quantum channels leaving Helstrom's measure of distinguishability of quantum states or pure states invariant and show that such quantum channels are fully reversible, which are unitary transformations.

scholarly journals EQUIVALENCE OF TRIPARTITE QUANTUM STATES UNDER LOCAL UNITARY TRANSFORMATIONS

2005 ◽  
Vol 03 (04) ◽  
pp. 603-609 ◽  
Author(s):  
SERGIO ALBEVERIO ◽  
LAURA CATTANEO ◽  
SHAO-MING FEI ◽  
XIAO-HONG WANG
Keyword(s):  
Quantum States ◽  
Composite Systems ◽  
Unitary Transformations ◽  
Complete Set ◽  
Pure States

The equivalence of tripartite pure states under local unitary transformations is investigated. The nonlocal properties for a class of tripartite quantum states in ℂK ⊗ ℂM ⊗ ℂN composite systems are investigated and a complete set of invariants under local unitary transformations for these states is presented. It is shown that two of these states are locally equivalent if and only if all these invariants have the same values.

scholarly journals Quantum Algorithm for Attacking RSA Based on Fourier Transform and Fixed-Point

2021 ◽  
Vol 26 (6) ◽  
pp. 489-494
Author(s):  
Yahui WANG ◽  
Huanguo ZHANG
Keyword(s):  
Fixed Point ◽  
Polynomial Time ◽  
Multiplicative Group ◽  
Success Probability ◽  
Quantum Algorithm ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Rsa Cryptosystem ◽  
Shor's Algorithm ◽  
Quantum Polynomial

Shor in 1994 proposed a quantum polynomial-time algorithm for finding the order r of an element a in the multiplicative group Zn*, which can be used to factor the integer n by computing [see formula in PDF]and hence break the famous RSA cryptosystem. However, the order r must be even. This restriction can be removed. So in this paper, we propose a quantum polynomial-time fixed-point attack for directly recovering the RSA plaintext M from the ciphertext C, without explicitly factoring the modulus n. Compared to Shor’s algorithm, the order r of the fixed-point C for RSA(e, n) satisfying [see formula in PDF] does not need to be even. Moreover, the success probability of the new algorithm is at least [see formula in PDF] and higher than that of Shor’s algorithm, though the time complexity for both algorithms is about the same.

scholarly journals A SUFFICIENT AND NECESSARY CONDITION FOR SUPERDENSE CODING OF QUANTUM STATES

2008 ◽  
Vol 06 (05) ◽  
pp. 1115-1125 ◽  
Author(s):  
DAOWEN QIU
Keyword(s):  
Lower Bound ◽  
Success Probability ◽  
Quantum States ◽  
Entangled States ◽  
Necessary Condition ◽  
Entangled State ◽  
Maximally Entangled State ◽  
High Success Probability ◽  
Sufficient And Necessary Condition ◽  
Arbitrary Quantum

Recently, Harrow et al. [Phys. Rev. Lett.92 (2004) 187901] gave a method for preparing an arbitrary quantum state with high success probability by physically transmitting some qubits, and by consuming a maximally entangled state, together with exhausting some shared random bits. In this paper, we discover that some states are impossible to be perfectly prepared by Alice and Bob initially sharing some entangled states. In particular, we present a sufficient and necessary condition for the states being enabled to be exactly prepared with probability equal to unity, in terms of the initial entangled states (maybe nonmaximally). In contrast, if the initially shared entanglement is maximal, then the probabilities for preparing these quantum states are smaller than unity. Furthermore, the lower bound on the probability for preparing some states are derived.

Learning Importance of Preferences

10.29007/v68w ◽  
2018 ◽  
Author(s):  
Ying Zhu ◽  
Mirek Truszczynski
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Scoring Rules ◽  
Np Hard ◽  
Individual Preferences

We study the problem of learning the importance of preferences in preference profiles in two important cases: when individual preferences are aggregated by the ranked Pareto rule, and when they are aggregated by positional scoring rules. For the ranked Pareto rule, we provide a polynomial-time algorithm that finds a ranking of preferences such that the ranked profile correctly decides all the examples, whenever such a ranking exists. We also show that the problem to learn a ranking maximizing the number of correctly decided examples (also under the ranked Pareto rule) is NP-hard. We obtain similar results for the case of weighted profiles when positional scoring rules are used for aggregation.

scholarly journals Metric Dimension Parameterized By Treewidth

Algorithmica ◽  
2021 ◽  
Author(s):  
Édouard Bonnet ◽  
Nidhi Purohit
Keyword(s):  
Computable Function ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Minimum Size ◽  
Metric Dimension ◽  
Resolving Set ◽  
Input Graph ◽  
Outerplanar Graphs ◽  
Bounded Treewidth ◽  
Fpt Algorithm

AbstractA resolving set S of a graph G is a subset of its vertices such that no two vertices of G have the same distance vector to S. The Metric Dimension problem asks for a resolving set of minimum size, and in its decision form, a resolving set of size at most some specified integer. This problem is NP-complete, and remains so in very restricted classes of graphs. It is also W[2]-complete with respect to the size of the solution. Metric Dimension has proven elusive on graphs of bounded treewidth. On the algorithmic side, a polynomial time algorithm is known for trees, and even for outerplanar graphs, but the general case of treewidth at most two is open. On the complexity side, no parameterized hardness is known. This has led several papers on the topic to ask for the parameterized complexity of Metric Dimension with respect to treewidth. We provide a first answer to the question. We show that Metric Dimension parameterized by the treewidth of the input graph is W[1]-hard. More refinedly we prove that, unless the Exponential Time Hypothesis fails, there is no algorithm solving Metric Dimension in time $$f(\text {pw})n^{o(\text {pw})}$$ f ( pw ) n o ( pw ) on n-vertex graphs of constant degree, with $$\text {pw}$$ pw the pathwidth of the input graph, and f any computable function. This is in stark contrast with an FPT algorithm of Belmonte et al. (SIAM J Discrete Math 31(2):1217–1243, 2017) with respect to the combined parameter $$\text {tl}+\Delta$$ tl + Δ , where $$\text {tl}$$ tl is the tree-length and $$\Delta$$ Δ the maximum-degree of the input graph.

A general inverse matrix series relation and associated polynomials – II

2021 ◽  
Vol 71 (2) ◽  
pp. 301-316
Author(s):  
Reshma Sanjhira
Keyword(s):  
Matrix Polynomial ◽  
Combinatorial Identities ◽  
Inverse Matrix ◽  
Matrix Polynomials ◽  
Special Cases ◽  
Series Representations ◽  
The Matrix ◽  
Associated Polynomials ◽  
Matrix Pair ◽  
Matrix Series

Abstract We propose a matrix analogue of a general inverse series relation with an objective to introduce the generalized Humbert matrix polynomial, Wilson matrix polynomial, and the Rach matrix polynomial together with their inverse series representations. The matrix polynomials of Kiney, Pincherle, Gegenbauer, Hahn, Meixner-Pollaczek etc. occur as the special cases. It is also shown that the general inverse matrix pair provides the extension to several inverse pairs due to John Riordan [An Introduction to Combinatorial Identities, Wiley, 1968].

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