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Algoritmos para a informação quântica: discriminação de estados quânticos e modelo híbrido

WILSON Ricardo Matos Rabelo

Published Nov 24, 2006• 1 citation
OpenAlex

Abstract

In this thesis we consider a protocol to perform the optimal quantum state discrimination of N linearly independent non-orthogonal pure quantum states. This protocol obtains the best set of generalized measurements for the problem of the unambiguous state discrimination. Through the extension of the Hilbert space, it is possible to perform an unitary operation yielding a final configuration, which gives the best discrimination. We introduce a detailed mathematical procedure to realize this task by means of semidefinite programming and norm minimization. The former is used to fix which is the best detection conclusive probability amplitude for each quantum state of the ensemble. The latter determines the matrix which leads the input states to the final configuration. In a final step, we decompose the unitary transformation in a sequence of 1-qubit rotation matrices. Subsequently, we show applications of the protocol in the quantum cryptography and state filtering, and present a computational code in MATLAB. Our additional result deals with the computational hybrid models. In particular, we study the Semi-quantum Computer. We discuss the performance of the Search and Fourier Transform algorithms on a hybrid architecture constituted of classical and quantum processors working together. We show that the semi-quantum computer can reach an improvement over a pure classical architecture, no matter how few qubits are available and, therefore, it suggests an easier implementable technology.

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In this thesis we consider a protocol to perform the optimal quantum state discrimination of N linearly independent non-orthogonal pure quantum states. This protocol obtains the best set of generalized measurements for the problem of the unambiguous state discrimination. Through the extension of the Hilbert space, it is possible to perform an unitary operation yielding a final configuration, which gives the best discrimination. We introduce a detailed mathematical procedure to realize this task by means of semidefinite programming and norm minimization. The former is used to fix which is the best detection conclusive probability amplitude for each quantum state of the ensemble. The latter determines the matrix which leads the input states to the final configuration. In a final step, we decompose the unitary transformation in a sequence of 1-qubit rotation matrices. Subsequently, we show applications of the protocol in the quantum cryptography and state filtering, and present a computational code in MATLAB. Our additional result deals with the computational hybrid models. In particular, we study the Semi-quantum Computer. We discuss the performance of the Search and Fourier Transform algorithms on a hybrid architecture constituted of classical and quantum processors working together. We show that the semi-quantum computer can reach an improvement over a pure classical architecture, no matter how few qubits are available and, therefore, it suggests an easier implementable technology.
We consider a protocol to perform the optimal quantum state discrimination of NNN linearly independent non-orthogonal pure quantum states and present a computational code. Through the extension of the original Hilbert space, it is possible to perform an unitary operation yielding a final configuration, which gives the best discrimination without ambiguity by means of von Neumann measurements. Our goal is to introduce a detailed general mathematical procedure to realize this task by means of semidefinite programming and norm minimization. The former is used to fix which is the best detection probability amplitude for each state of the ensemble. The latter determines the matrix which leads the states to the final configuration. In a final step, we decompose the unitary transformation in a sequence of two-level rotation matrices.
We consider a protocol to perform the optimal quantum state discrimination of N linearly independent non-orthogonal pure quantum states and present a computational code. Through the extension of the original Hilbert space, it is possible to perform an unitary operation yielding a final configuration, which gives the best discrimination without ambiguity by means of von Neumann measurements. Our goal is to introduce a detailed general mathematical procedure to realize this task by means of semidefinite programming and norm minimization. The former is used to fix which is the best detection probability amplitude for each state of the ensemble. The latter determines the matrix which leads the states to the final configuration. In a final step, we decompose the unitary transformation in a sequence of two-level rotation matrices.
Abstract—In this paper, we consider the problem of unambiguous discrimination between a set of linearly independent pure quantum states. We show that the design of the optimal measurement that minimizes the probability of an inconclusive result can be formulated as a semidefinite programming problem. Based on this formulation, we develop a set of necessary and sufficient conditions for an optimal quantum measurement. We show that the optimal measurement can be computed very efficiently in polynomial time by exploiting the many well-known algorithms for solving semidefinite programs, which are guaranteed to converge to the global optimum. Using the general conditions for optimality, we derive necessary and sufficient conditions so that the measurement that results in an equal probability of an inconclusive result for each one of the quantum states is optimal. We refer to this measurement as the equal-probability measurement (EPM). We then show that for any state set, the prior probabilities of the states can be chosen such that the EPM is optimal. Finally, we consider state sets with strong symmetry properties and equal prior probabilities for which the EPM is optimal. We first consider geometrically uniform (GU) state sets that are defined over a group of unitary matrices and are generated by a single generating vector. We then consider compound GU state sets which are generated by a group of unitary matrices using multiple generating vectors, where the generating vectors satisfy a certain (weighted) norm constraint. Index Terms—Compound geometrically uniform (CGU) quantum states, equal-probability measurement (EPM), geometrically uniform (GU) quantum states, quantum detection, semidefinite programming, unambiguous discrimination. I.
We consider the problem of unambiguous discrimination between a set of linearly independent pure quantum states. We show that the design of the optimal measurement that minimizes the probability of an inconclusive result can be formulated as a semidefinite programming problem. Based on this formulation, we develop a set of necessary and sufficient conditions for an optimal quantum measurement. We show that the optimal measurement can be computed very efficiently in polynomial time by exploiting the many well-known algorithms for solving semidefinite programs, which are guaranteed to converge to the global optimum. Using the general conditions for optimality, we derive necessary and sufficient conditions so that the measurement that results in an equal probability of an inconclusive result for each one of the quantum states is optimal. We refer to this measurement as the equal-probability measurement (EPM). We then show that for any state set, the prior probabilities of the states can be chosen such that the EPM is optimal. Finally, we consider state sets with strong symmetry properties and equal prior probabilities for which the EPM is optimal. We first consider geometrically uniform (GU) state sets that are defined over a group of unitary matrices and are generated by a single generating vector. We then consider compound GU state sets which are generated by a group of unitary matrices using multiple generating vectors, where the generating vectors satisfy a certain (weighted) norm constraint.
Quantum state discrimination is a fundamental primitive in quantum information processing, underpinning tasks in quantum communication, sensing, and learning. We consider the general Bayes framework, as introduced by Helstrom, for state discrimination when, instead of a classical description of the candidate states, one has access to their \emph{source code}: the quantum circuit that prepares them. We show that the semidefinite program (SDP) for the discrimination problem can be reformulated in terms of the Gram matrix of these states, reducing the SDP variable dimensions from dLdLdL to NLNLNL, where ddd is the Hilbert space dimension, NNN is the number of candidate states, and LLL is the number of possible guesses. Importantly, we further introduce a quantum pre-processing procedure which efficiently constructs the reduced semidefinite program from the source code, enabling our method to operate directly on quantum data. We consider two applications. First, we characterize the optimal identifications for quantum changepoint problems under several reward structures, including multiple-changepoint settings that were previously computationally inaccessible. Second, we consider a quantum error classification problem and show how our reduction makes it tractable for systems of hundreds of qubits.
This thesis presents, initially, an analysis of the teleportation protocol of two qubits quantum gates, which uses an four qubits entangled state as a resource. It will be shown that the set of four qubits pure states is divided into two classes: Class 1 and Class 2, which determine, respectively, the realization of a deterministic gate teleportation, or a probabilistic remote preparation. It is still shown that when there is additional information on search problems that are not applicable to a standard quantum search algorithm, it is possible to obtain better performance using a quantum algorithm in conjunction with a classical search algorithm. Finally, a new model of image representation in quantum states is proposed, the new model has a less complex preparation and recovery compared to the other models currently proposed and allows the processing of images in both gray and colored scales.