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Quantum classification and search algorithms using spinorial representations

Lauro Mascarenhas, Vinicius N. A. Lula-Rocha, Marco A. S. Trindade

Abstract

We propose an algebraic formulation for two distinct quantum algorithms: a quantum classification algorithm and a quantum search algorithm with a non-uniform initial distribution, both based on Clifford algebras and spinorial representations. In the classification algorithm, we exploit properties of spinorial representations to construct orthogonal quantum states associated with different classes, allowing the identification of an item's class through the evaluation of expectation values of operators derived from the generators of the Clifford algebra. In the quantum search algorithm, we consider a database with prior information in which the oracle is implemented directly using generators of the Clifford algebra, simplifying its realization. The proposed approach provides a unified algebraic description for both algorithms, employing spinorial representations in the construction of quantum states and operators. Computational implementations are presented.

Quantum classification and search algorithms using spinorial representations

Abstract

We propose an algebraic formulation for two distinct quantum algorithms: a quantum classification algorithm and a quantum search algorithm with a non-uniform initial distribution, both based on Clifford algebras and spinorial representations. In the classification algorithm, we exploit properties of spinorial representations to construct orthogonal quantum states associated with different classes, allowing the identification of an item's class through the evaluation of expectation values of operators derived from the generators of the Clifford algebra. In the quantum search algorithm, we consider a database with prior information in which the oracle is implemented directly using generators of the Clifford algebra, simplifying its realization. The proposed approach provides a unified algebraic description for both algorithms, employing spinorial representations in the construction of quantum states and operators. Computational implementations are presented.
Paper Structure (5 sections, 8 theorems, 124 equations, 2 figures, 2 algorithms)

This paper contains 5 sections, 8 theorems, 124 equations, 2 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Algebraic formulation
  3. Spinorial quantum algorithms for classification and search
  4. Conclusions and Perspectives
  5. Appendix

Key Result

Lemma 1

Given a representation of a Clifford algebra $Cl(2n)$ defined by: where $\sigma_1, \sigma_2, \sigma_3$ are Pauli matrices, $I$ is the identity matrix and $1 \leq j \leq n$Gilbert, there is an eigenvector $\vert \Gamma_j \rangle$ of $\Gamma_{j}$ orthogonal to the vector $\Gamma_i \Gamma_j \vert \Gamma_j \rangle$, with $i\neq j$ . The eigenvectors of $\Gamma_j$ and and respectively, with eigenval

Figures (2)

  • Figure 1: Comparative probability distributions for Algorithm \ref{['alg.:Algoritmo1']} with 2, 3, 4, and 5 qubits. Each panel corresponds to a different classification operator. The histograms show the theoretical and experimental probabilities of the computational basis states after applying the operator O.
  • Figure 2: Probability distributions of Algorithm \ref{['alg:Algoritmo2']} for two qubits. One observes amplitude amplification in the solution subspace and suppression by destructive interference in the orthogonal complement. Solid bars represent experimental data and hatched bars represent theoretical predictions, showing the high fidelity of Clifford transformations in the segmentation of the state space.

Theorems & Definitions (13)

  • Lemma 1
  • Definition 1
  • Theorem 1
  • Theorem 2
  • Definition 2
  • Proposition 1
  • Proposition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • ...and 3 more