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Quantum computation of Groebner basis through F4 and F5 algorithms

Ichio Kikuchi, Akihito Kikuchi

Published Mar 3, 2022• 1 citation
Publisher

Abstract

Our main concern in this article is how to apply quantum algorithms to compute Groebner basis through Faugere's F4 and F5 algorithms, which are regarded as the most effective algorithms for this purpose. We give examples and pseudo-codes for these algorithms and investigate where we can apply quantum algorithms. As a result, we have the designs (or the pseudo-codes) of quantum versions of the F4 and F5 algorithms by replacing classical algorithms with quantum ones. The quantum versions of F4 and F5 algorithms are, in nature, the extensions of the quantum Gaussian-Jordan elimination of Diepp, since the computations of Groebner bases are practicable through the matrix representations. In the computation of Groebner bases, quantum algorithms serve to construct the matrices as small as possible (according to the fundamental idea of F4 and F5) and to find the pivotal element as quickly as possible.

DOI
10.31219/osf.io/3b5qr

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Our main concern in this article is how to apply quantum algorithms to compute Groebner basis through Faugere's F4 and F5 algorithms, which are regarded as the most effective algorithms for this purpose. We give examples and pseudo-codes for these algorithms and investigate where we can apply quantum algorithms. As a result, we have the designs (or the pseudo-codes) of quantum versions of the F4 and F5 algorithms by replacing classical algorithms with quantum ones. The quantum versions of F4 and F5 algorithms are, in nature, the extensions of the quantum Gaussian-Jordan elimination of Diepp, since the computations of Groebner bases are practicable through the matrix representations. In the computation of Groebner bases, quantum algorithms serve to construct the matrices as small as possible (according to the fundamental idea of F4 and F5) and to find the pivotal element as quickly as possible.
In this essay, we examine the feasibility of quantum computation of Gr\"obner basis which is a fundamental tool of algebraic geometry. The classical method for computing Gr\"obner basis is based on Buchberger's algorithm, and our question is how to adopt quantum algorithm there. A Quantum algorithm for finding the maximum is usable for detecting head terms of polynomials, which are required for the computation of S-polynomials. The reduction of S-polynomials with respect to a Gr\"obner basis could be done by the quantum version of Gauss-Jordan elimination of matrices which represents polynomials. However, the frequent occurrence of zero-reductions of polynomials is an obstacle to the effective application of quantum algorithms. This is because zero-reductions of polynomials occur in non-full-rank matrices, for which quantum linear systems algorithms (through the inversion of matrices) are inadequate, as ever-known quantum linear solvers (such as Harrow-Hassidim-Lloyd) require the clandestine computations of the inverses of eigenvalues. Such algorithms should be used in limited situations with the guarantee that the matrices could be inverted. For example, the transformation from the non-reduced Gr\"obner basis to the reduced one is of this sort, and the quantum algorithms surely achieve the partial speedup of the computations.
An algorithm and its first implementation in C# are presented for assembling arbitrary quantum circuits on the base of Hadamard and Toffoli gates and for constructing multivariate polynomial systems over the finite field Z_2 arising when applying the Feynman's sum-over-paths approach to quantum circuits. The matrix elements determined by a circuit can be computed by counting the number of common roots in Z_2 for the polynomial system associated with the circuit. To determine the number of solutions in Z_2 for the output polynomial system, one can use the Groebner bases method and the relevant algorithms for computing Groebner bases.
Abstract Quantum computers are designed to outperform standard computers by running quantum algorithms. Areas in which quantum algorithms can be applied include cryptography, search and optimisation, simulation of quantum systems and solving large systems of linear equations. Here we briefly survey some known quantum algorithms, with an emphasis on a broad overview of their applications rather than their technical details. We include a discussion of recent developments and near-term applications of quantum algorithms.
Matrices are widely, and extensively, used in quantum mechanics. Indeed, one of the avenues to quantum mechanics, that of Born, Heisenberg, and Jordan, is via matrices. One additional asset of matrices is that they are the natural mathematical language to describe quantum computing. Furthermore, matrices identify themselves in the physical tangible world via optical elements such as mirrors, beam splitters, polarization rotators, and interferometers. And these are the elements that comprise the hardware of photonic quantum computers. Here, we introduce the Pauli matrices https://www.w3.org/1998/Math/MathML" display="inline"> σ x , σ y , σ z , plus the identity matrix I , via the probability amplitudes of quantum entanglement. Once this is accomplished, we show how these matrices can be used to perform mathematical transformations, on quantum entanglement states, that form the bases of quantum computing as treated in the next chapter.
This expository paper reviews some of the recent uses of computational algebraic geometry in classical and quantum optimization. The paper assumes an elementary background in algebraic geometry and adiabatic quantum computing (AQC), and concentrates on presenting concrete examples (with Python codes tested on a quantum computer) of applying algebraic geometry constructs: solving binary optimization, factoring, and compiling. Reversing the direction, we also briefly describe a novel use of quantum computers to compute Groebner bases for toric ideals. We also show how Groebner bases play a role in studying AQC at a fundamental level within a Morse theory framework. We close by placing our work in perspective, by situating this leg of the journey, as part of a marvelous intellectual expedition that began with our ancients over 4000 years ago.