In this paper we present two algorithms for reducing systems of multivariate polynomial equations over the finite field F2 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 F2 whose variables also take values in F2 (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 F2 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 F2. 相似文献
In this paper, we study Abelian groups that are small with respect to different classes of groups. Completely decomposable
torsion free groups that are small with respect to an arbitrary class of torsion free groups are described completely. Direct
products of groups small with respect to the class of slender groups are derived. 相似文献
The entanglement characteristics of two qubits are encoded in the invariants of the adjoint action of the group SU(2) ⊗ SU(2)
on the space of density matrices
\mathfrakP+ {\mathfrak{P}_{+} } , defined as the space of 4 × 4 positive semidefinite Hermitian matrices. The corresponding ring
\textC[ \mathfrakP+ ]\textSU( 2 ) ?\textSU ?( 2 ) {\text{C}}{\left[ {{\mathfrak{P}_{+} }} \right]^{{\text{SU}}\left( {2} \right) \otimes {\text{SU}} \otimes \left( {2} \right)}} of polynomial invariants is studied. A special integrity basis for
\textC[ \mathfrakP+ ]\textSU( 2 ) ?\textSU ?( 2 ) {\text{C}}{\left[ {{\mathfrak{P}_{+} }} \right]^{{\text{SU}}\left( {2} \right) \otimes {\text{SU}} \otimes \left( {2} \right)}} is described, and the constraints on its elements imposed by the positive semidefiniteness of density matrices are given
explicitly in the form of polynomial inequalities. The suggested basis is characterized by the property that the minimum number
of invariants, namely, two primary invariants of degree 2, 3 and one secondary invariant of degree 4 appearing in the Hironaka
decomposition of
\textC[ \mathfrakP+ ]\textSU( 2 ) ?\textSU ?( 2 ) {\text{C}}{\left[ {{\mathfrak{P}_{+} }} \right]^{{\text{SU}}\left( {2} \right) \otimes {\text{SU}} \otimes \left( {2} \right)}} , are subject to the polynomial inequalities. Bibliography: 32 titles. 相似文献
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 ℤ2 for a certain set of multivariate polynomials over ℤ2. 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.
The text was submitted by the authors in English. 相似文献
A symbolic algorithm to generate the multilayer operator-difference schemes for solving the evolution problem of the time-dependent Schrödinger equation is elaborated. An additional gauge transformation of operator-difference schemes to make good use of the finite-element discretization is applied. The efficiency of the generated numerical schemes until the sixth order with respect to the time step and until the seventh order with respect to the spatial step is demonstrated by calculations of some finite-dimensional quantum systems in external fields. 相似文献