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Physics of Particles and Nuclei Letters

An algorithmic approach to solving polynomial equations associated with quantum circuits

Vladimir P. Gerdt, M. V. Zinin

Published Nov 19, 2009• 1 citation
Publisher

Abstract

In this paper we present two algorithms for reducing systems of multivariate polynomial equations over the finite field F 2 to the canonical triangular form called lexicographical Gröbner basis. This triangular form is the most appropriate for finding solutions of the system. On the other hand, the system of polynomials over F 2 whose variables also take values in F 2 (Boolean polynomials) completely describes the unitary matrix generated by a quantum circuit. In particular, the matrix itself can be computed by counting the number of solutions (roots) of the associated polynomial system. Thereby, efficient construction of the lexicographical Gröbner bases over F 2 associated with quantum circuits gives a method for computing their circuit matrices that is alternative to the direct numerical method based on linear algebra. We compare our implementation of both algorithms with some other software packages available for computing Gröbner bases over F 2.

DOI
10.1134/s154747710907005x

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In this paper we present two algorithms for reducing systems of multivariate polynomial equations over the finite field F 2 to the canonical triangular form called lexicographical Gröbner basis. This triangular form is the most appropriate for finding solutions of the system. On the other hand, the system of polynomials over F 2 whose variables also take values in F 2 (Boolean polynomials) completely describes the unitary matrix generated by a quantum circuit. In particular, the matrix itself can be computed by counting the number of solutions (roots) of the associated polynomial system. Thereby, efficient construction of the lexicographical Gröbner bases over F 2 associated with quantum circuits gives a method for computing their circuit matrices that is alternative to the direct numerical method based on linear algebra. We compare our implementation of both algorithms with some other software packages available for computing Gröbner bases over F 2.
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.
Recently it has been shown that elements of the unitary matrix determined by a quantum circuit can be computed by counting the number of common roots in the finite field ℤ 2 for a certain set of multivariate polynomials over ℤ 2 . In this paper we present a C# package that allows a user to assemble a quantum circuit and to generate the multivariate polynomial set associated with the circuit. The generated polynomial system can further be converted to the canonical triangular involutive basis form, which is appropriate for counting the number of common roots of the polynomials.
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.
In this paper we briefly describe a Mathematica program for simulation of quantum circuits and illustrate some of its facilities by simple examples. Unlike other Mathematica-based quantum simulators, our program provides a user-friendly graphical interface for generating quantum circuits and computing the circuit unitary matrices. In addition to standard linear algebra-based tools, our program implements special computer-algebra technique for constructing the multivariate polynomial system that, for a circuit composed from the Toffoli and Hadamard gates, uniquely defines the circuit matrix.
Decision of whether a Boolean equation system has a solution is an NPC problem and finding a solution is NP hard. In this paper, we present a quantum algorithm to decide whether a Boolean equation system FS has a solution and compute one if FS does have solutions with any given success probability. The runtime complexity of the algorithm is polynomial in the size of FS and the condition number of FS. As a consequence, we give a polynomial-time quantum algorithm for solving Boolean equation systems if their condition numbers are small, say polynomial in the size of FS. We apply our quantum algorithm for solving Boolean equations to the cryptanalysis of several important cryptosystems: the stream cipher Trivum, the block cipher AES, the hash function SHA-3/Keccak, and the multivariate public key cryptosystems, and show that they are secure under quantum algebraic attack only if the condition numbers of the corresponding equation systems are large. This leads to a new criterion for designing cryptosystems that can against the attack of quantum computers: their corresponding equation systems must have large condition numbers.
Polynomial transformations are basic primitives in quantum algorithms: quantum signal processing and singular value transformation compile univariate polynomials into circuits with query complexity essentially set by degree. Multivariable transformations of noncommuting matrices, however, lack a comparable synthesis theory. We develop a complete constructive theory under joint block access for all polynomials contractive on the prescribed matrix domain. Given a compact finite-state description of a degree-DDD polynomial PPP and 0<τ<10<τ<10<τ<1, we synthesize PPP with (1+τ)(1+τ)(1+τ)-optimal normalization using O(D/τ)O(D/\sqrtτ)O(D/τ​) queries, and exactly DDD under row-block access, matching a degree lower bound. The complete circuit realization is classically computable in polynomial time with polylogarithmic dependence on accuracy. The key to our construction is a finite algorithmic Schur--Agler theorem: the coefficient recurrence defines a polynomial-dimensional continuation space supporting a complete semidefinite certificate for contractivity on the prescribed domain. Factoring this certificate yields the contractive realization underlying the query algorithm. Beyond matrix transformations, our framework lifts multivariable polynomial synthesis to quantum channel transformations. Given coherent Kraus access K={Ka}aK=\{K_a\}_aK={Ka​}a​, we synthesize any finite jointly contractive family of noncommutative polynomial maps K↦{Fb(K)}bK\mapsto\{F_b(K)\}_bK↦{Fb​(K)}b​ as completely positive operations, allowing coherent interference among Kraus histories. Furthermore, channel-level transformations specified by causal Choi data can be synthesized explicitly as fixed-order quantum combs. Collectively, these results point to a broader program: multivariable approximation as a language for multi-operator quantum algorithms and higher-order quantum information processing.