The paper is a continuation of arXiv:2012.10364, where the approach was developed to constructing the exact matrix model for any generalized Ising system, and such model was constructed for certain 2d system. In this paper, the properties of the model are specified for light block diagonalization. A corresponding example is considered. For the example, general exact partition function is obtained and analysed. The analysis shows that the free energy does not depend on the amount of rows with a large amount of cells. For the example with light boundary conditions, the partition function is obtained and the specific free energy per spin is plotted.
The 16th-order transfer matrix of the three-dimensional Ising model in the particular case n = m = 2 (n × m is number of spins in a layer) is specified by the interaction parameters of three basis vectors. The matrix eigenvectors are divided into two classes, even and odd. Using the symmetry of the eigenvectors, we find their corresponding eigenvalues in general form. Eight of the sixteen eigenvalues related to odd eigenvectors are found from quadratic equations. Four eigenvalues related to even eigenvectors are found from a fourth-degree equation with symmetric coefficients. Each of the remaining four eigenvalues is equal to unity.
Abstract In this chapter we study the solution of the two-dimensional Ising model by means of the transfer matrix. Unlike the methods discussed in the previous chapter, the transfer matrix approach has greater generality and can used to solve exactly other two-dimensional models. Even if the general ideas behind this approach have been explained in Chapter 2 by means of the one-dimensional case, their application to the two-dimensional cases requires more powerful and sophisticated mathematical tools: for instance, the study the eigenvalues of the transfer matrix in the Ising model for T = Tcneeds to employ elliptic functions. The same is also true for other models. In order to present in the simplest possible way the main lines of this method, in the following we focus attention only on the solution of the model at T = Tcbecause this case can be analyzed in terms of simple trigonometric functions.
Abstract The connection matrix of the Ising model on a
d-dimensional hypercube is investigated. In addition to the interactions between the nearest neighbors, the interactions between the next-nearest neighbors are taken into account. For such a matrix, the exact relations for the eigenvalues and eigenvectors, which are reasonably simply expressed through the corresponding characteristics of the one-dimensional Ising model, are obtained. Both periodic and free boundary conditions are considered.
Quantum Signal Processing (QSP) and Quantum Singular Value Transformation (QSVT) provide an efficient framework for implementing polynomials of block-encoded matrices, and thus offer a systematic approach to quantum algorithm design. However, despite a number of recent advances, important limitations remain. In particular, QSP can only transform unitary matrices, by applying a polynomial to their eigenvalues, while QSVT is a singular-value transformation and thus one can only obtain the polynomial of Hermitian matrices. As a consequence, these techniques do not directly apply to an arbitrary non-Hermitian matrix that is not diagonalizable. In this work, we propose a simple yet powerful method to extend these ideas to arbitrary square matrices by acting on their eigenvalues. To this end, we introduce the notion of an
n-regular block encoding, namely, a block encoding whose
k-th power reproduces the
k-th power of the encoded matrix for every
0<k<n. We show that applying QSP to any unitary with this property is equivalent to applying a polynomial of degree at most
n to the block-encoded matrix, independently of its internal structure. Moreover, we provide a simple construction that transforms any block encoding into an
n-regular one using only
O(logn) ancillary qubits and operations. Finally, we show that this construction induces the desired transformation on the eigenvalues associated with the Jordan normal form of the matrix; for non-diagonalizable matrices this yields the full matrix function
P(A), whose action on each Jordan block involves the derivatives of
P, and not merely the map
λ↦P(λ).