Smith Normal Form Research Articles

Extracting Topological Orders of Generalized Pauli Stabilizer Codes in Two Dimensions

In this paper, we introduce an algorithm for extracting topological data from translation invariant generalized Pauli stabilizer codes in two-dimensional systems, focusing on the analysis of anyon excitations and string operators. The algorithm applies to Zd qudits, including instances where d is a nonprime number. This capability allows the identification of topological orders that differ from the Zd toric codes. It extends our understanding beyond the established theorem that Pauli stabilizer codes for Zp qudits (with p being a prime) are equivalent to finite copies of Zp toric codes and trivial stabilizers. The algorithm is designed to determine all anyons and their string operators, enabling the computation of their fusion rules, topological spins, and braiding statistics. The method converts the identification of topological orders into computational tasks, including Gaussian elimination, the Hermite normal form, and the Smith normal form of truncated Laurent polynomials. Furthermore, the algorithm provides a systematic approach for studying quantum error-correcting codes. We apply it to various codes, such as self-dual CSS quantum codes modified from the two-dimensional honeycomb color code and non-CSS quantum codes that contain the double semion topological order or the six-semion topological order. Published by the American Physical Society 2024

The Smith normal form of the walk matrix of the Dynkin graph An

In this paper, we give the rank of the walk matrix of the Dynkin graph An, and prove that its Smith normal form isdiag(1,…,1︸⌈n2⌉,0,…,0).

The qudit Pauli group: non-commuting pairs, non-commuting sets, and structure theorems

Qudits with local dimension d>2 can have unique structure and uses that qubits (d=2) cannot. Qudit Pauli operators provide a very useful basis of the space of qudit states and operators. We study the structure of the qudit Pauli group for any, including composite, d in several ways. To cover composite values of d, we work with modules over commutative rings, which generalize the notion of vector spaces over fields. For any specified set of commutation relations, we construct a set of qudit Paulis satisfying those relations. We also study the maximum size of sets of Paulis that mutually non-commute and sets that non-commute in pairs. Finally, we give methods to find near minimal generating sets of Pauli subgroups, calculate the sizes of Pauli subgroups, and find bases of logical operators for qudit stabilizer codes. Useful tools in this study are normal forms from linear algebra over commutative rings, including the Smith normal form, alternating Smith normal form, and Howell normal form of matrices. Possible applications of this work include the construction and analysis of qudit stabilizer codes, entanglement assisted codes, parafermion codes, and fermionic Hamiltonian simulation.

The maximum four point condition matrix of a tree

The Four point condition (4PC henceforth) is a well known condition characterising distances in trees T. Let w,x,y,z be four vertices in T and let dx,y denote the distance between vertices x,y in T. The 4PC condition says that among the three terms dw,x+dy,z, dw,y+dx,z and dw,z+dx,y the maximum value equals the second maximum value.We define an (n2)×(n2) sized matrix Max4PCT from a tree T where the rows and columns are indexed by size-2 subsets. The entry of Max4PCT corresponding to the row indexed by {w,x} and column {y,z} is the maximum value among the three terms dw,x+dy,z, dw,y+dx,z and dw,z+dx,y. In this work, we determine basic properties of this matrix like rank, give an algorithm that outputs a family of bases, and find the determinant of Max4PCT when restricted to our basis. We further determine the inertia and the Smith Normal Form (SNF) of Max4PCT.

Bicrystallography-informed Frenkel–Kontorova model for interlayer dislocations in strained 2D heterostructures

In recent years, van der Waals (vdW) heterostructures and homostructures, which consist of stacks of two-dimensional (2D) materials, have risen to prominence due to their association with exotic quantum phenomena originating from correlated electronic states harbored by them. Atomistic scale relaxation effects play an extremely important role in the electronic scale quantum physics of these systems, providing means of manipulation of these materials and allowing them to be tailored for emergent technologies. We investigate such structural relaxation effects in this work using atomistic and mesoscale models, within the context of twisted bilayer graphene — a well-known heterostructure system that features moiré patterns arising from the lattices of the two graphene layers. For small twist angles, atomic relaxation effects in this system are associated with the natural emergence of interface dislocations or strain solitons, which result from the cyclic nature of the generalized stacking fault energy (GSFE), that measures the interface energy based on the relative movement of the two layers. In this work, we first demonstrate using atomistic simulations that atomic reconstruction in bilayer graphene under a large twist also results from interface dislocations, although the Burgers vectors of such dislocations are considerably smaller than those observed in small-twist systems. To reveal the translational invariance of the heterointerface responsible for the formation of such dislocations, we derive the translational symmetry of the GSFE of a 2D heterostructure using the notions of coincident site lattices (CSLs) and displacement shift complete lattices (DSCLs). The workhorse for this exercise is a recently developed Smith normal form bicrystallography framework. Next, we construct a bicrystallography-informed and frame-invariant Frenkel–Kontorova model, which can predict the formation of strain solitons in arbitrary 2D heterostructures, and apply it to study a heterostrained, large-twist bilayer graphene system. Our mesoscale model is found to produce results consistent with atomistic simulations. We anticipate that the model will be invaluable in predicting structural relaxation and for providing insights into various heterostructure systems, especially in cases where the fundamental unit cell is large and therefore, atomistic simulations are computationally expensive.

On the Smith Normal Form of the Least Common Multiple of Matrices from some Class of Matrices

On the Smith Normal Form of the Least Common Multiple of Matrices from some Class of Matrices

The Smith normal form of the walk matrix of the extended Dynkin graph [formula omitted

In this paper, we provide an explicit formula for the rank of the walk matrix of the extended Dynkin graph D˜n. Moreover, we prove that the Smith normal form of the walk matrix of the extended Dynkin graph D˜n is{diag(1,…,1︸r−1,n−12,0,…,0),ifnis odd,diag(1,…,1︸r−1,n−1,0,…,0),ifnis even, where r=⌊n2⌋.

New Results on the Unimodular Equivalence of Multivariate Polynomial Matrices

The equivalence of systems is a crucial concept in multidimensional systems. The Smith normal forms of multivariate polynomial matrices play important roles in the theory of polynomial matrices. In this paper, we mainly study the unimodular equivalence of some special kinds of multivariate polynomial matrices and obtain some tractable criteria under which such matrices are unimodular equivalent to their Smith normal forms. We propose an algorithm for reducing such nD polynomial matrices to their Smith normal forms and present an example to illustrate the availability of the algorithm. Furthermore, we extend the results to the non-square case.

The Smith normal form and reduction of weakly linear matrices

The reduction of a multidimensional system is closely related to the reduction of a multivariate polynomial matrix, for which the Smith normal form of the matrix plays a key role. In this paper, we investigate the reduction of weakly linear multivariate polynomial matrices to their Smith normal forms. Using hierarchical-recursive method and Quillen-Suslin Theorem, we derive some necessary and sufficient conditions ensuring that such matrices can be reduced to their Smith normal forms, and these conditions are easily checked by computing the reduced Gröbner bases of the relevant polynomial ideals. Based on the new results, we propose an algorithm for reducing weakly linear multivariate polynomial matrices to their Smith normal forms.

Using Local Smith Normal Form for Numerical Implementation of the Generalized Frobenius Method

Using Local Smith Normal Form for Numerical Implementation of the Generalized Frobenius Method

Rational points of lattice ideals on a toric variety and toric codes

We show that the number of rational points of a subgroup inside a toric variety over a finite field defined by a homogeneous lattice ideal can be computed via Smith normal form of the matrix whose columns constitute a basis of the lattice. This generalizes and yields a concise toric geometric proof of the same fact proven purely algebraically by Lopez and Villarreal for the case of a projective space and a standard homogeneous lattice ideal of dimension one. We also prove a Nullstellensatz type theorem over a finite field establishing a one to one correspondence between subgroups of the dense split torus and certain homogeneous lattice ideals. As application, we compute the main parameters of generalized toric codes on subgroups of the torus of Hirzebruch surfaces, generalizing the existing literature.

The Smith normal form of the walk matrix of the Dynkin graph Dn for n ≡ 0 (mod 4)

We prove that the walk matrix of the Dynkin graph Dn has the Smith normal formdiag[1,1,…,1︸n2−1,2,2,…,2︸n2−1,0,0] when n≡0(mod4). This gives an affirmative answer to a question in (Wang et al., 2022 [22]).

Smith Normal Form and the generalized spectral characterization of oriented graphs

Spectral characterization of graphs is an important topic in spectral graph theory. An oriented graph Gσ is obtained from a simple undirected graph G by assigning to every edge of G a direction so that Gσ becomes a directed graph. The skew-adjacency matrix of an oriented graph Gσ is a real skew-symmetric matrix S(Gσ)=(sij), where sij=−sji=1 if (i,j) is an arc; sij=sji=0 otherwise. Let Gσ and Hτ be two oriented graphs whose skew-adjacency matrices are S(Gσ) and S(Hτ), respectively. We say Gσ is R-cospectral to Hτ if tJ−S(Gσ) and tJ−S(Hτ) have the same spectrum for any t∈R, where J is the all-ones matrix. An oriented graph Gσ is said to be determined by the generalized skew spectrum (DGSS for short), if any oriented graph which is R-cospectral to Gσ is isomorphic to Gσ. Let W(Gσ)=[e,S(Gσ)e,S2(Gσ)e,…,Sn−1(Gσ)e] be the skew-walk-matrix of Gσ, where e is the all-ones vector. A theorem of Qiu, Wang and Wang [9] states that if Gσ is a self-converse oriented graph and 2−⌊n2⌋det⁡W(Gσ) is odd and square-free, then Gσ is DGSS. In this paper, based on the Smith Normal Form of the skew-walk-matrix of Gσ we obtain our main result: Let q be a prime and Gσ be a self-converse oriented graph on n vertices with det⁡W(Gσ)≠0. Assume that rankq(W(Gσ))=n−1 if q is an odd prime, and rankq(W(Gσ))=⌈n2⌉ if q=2. If Q is a regular rational orthogonal matrix satisfying QTS(Gσ)Q=S(Hτ) for some oriented graph Hτ, then the level of Q divides dn(W(Gσ))q, where dn(W(Gσ)) is the n-th invariant factor of W(Gσ). Consequently, it leads to an easier way to prove Qiu, Wang and Wang's theorem above.

One-dimensional symmetric phases protected by frieze symmetries

We make a systematic study of symmetry-protected topological gapped phases of quantum spin chains in the presence of the frieze space groups in one dimension using matrix product states. Here, the spatial symmetries of the one-dimensional lattice are considered together with an additional ``vertical reflection,'' which we take to be an on-site ${\mathbb{Z}}_{2}$ symmetry. We identify seventeen distinct non-trivial phases, define canonical forms, and compare the topological indices obtained from the MPS analysis with the group cohomological predictions. We furthermore construct explicit renormalization group fixed-point wave functions for symmetry-protected topological phases with global on-site symmetries, possibly combined with time reversal and parity symmetry. En route, we demonstrate how group cohomology can be computed using the Smith normal form.

The cotype zeta function of [formula omitted

We give an asymptotic formula for the number of sublattices Λ⊆Zd of index at most X for which Zd/Λ has rank at most m, answering a question of Nguyen and Shparlinski. We compare this result to work of Stanley and Wang on Smith normal forms of random integral matrices and discuss connections to the Cohen–Lenstra heuristics. Our arguments are based on Petrogradsky’s formulas for the cotype zeta function of Zd, a multivariable generalization of the subgroup growth zeta function of Zd.

The structure of the Smith normal form of the greatest common divisor and the least common multiple of the third order matrices over Bezout domains of stable range 1.5

The structure of the Smith normal form of the greatest common divisor and the least common multiple of the third order matrices over Bezout domains of stable range 1.5

A fast algorithm for computing the Smith normal form with multipliers for a nonsingular integer matrix

A Las Vegas randomized algorithm is given to compute the Smith multipliers for a nonsingular integer matrix A, that is, unimodular matrices U and V such that AV=US, with S the Smith normal form of A. The expected running time of the algorithm is about the same as required to multiply together two matrices of the same dimension and size of entries as A. Explicit bounds are given for the size of the entries in both unimodular multipliers. The main tool used by the algorithm is the Smith massager, a relaxed version of V, the unimodular matrix specifying the column operations of the Smith computation. From the perspective of efficiency, the main tools used are fast linear system solving and partial linearization of integer matrices. As an application of the Smith with multipliers algorithm, a fast algorithm is given to find the fractional part of the inverse of the input matrix.

Smith Normal Form and the generalized spectral characterization of graphs

Spectral characterizations of graphs are an important topic in spectral graph theory, which has received a lot of attention in recent years. It is generally very hard to show that a given graph is determined by its spectrum. Recently, Wang [9] gave a simple arithmetic condition for graphs being determined by their generalized spectra. Let G be a graph on n vertices with adjacency matrix A, and W=[e,Ae,…,An−1e] (e is the all-one vector) be the walk-matrix of G. A theorem of Wang [9] states that if 2−⌊n/2⌋det⁡W (which is always an integer) is odd and square-free, then G is determined by the generalized spectrum. In this paper, we find a new and short route which leads to a stronger version of the above theorem. The result is achieved by using the Smith Normal Form of the walk-matrix of G. The proposed method gives a new insight in dealing with the problem of generalized spectral characterization of graphs.

Interface dislocations and grain boundary disconnections using Smith normal bicrystallography

The CSL/DSCL model for interfaces in crystalline materials offers a unified framework to study interface dislocations in phase boundaries and disconnections in grain boundaries. The model relies on the existence of a coincidence relation between the two lattices that meet at an interface. The model’s ability to quantitatively predict the thermodynamics and kinetics of interfaces has been demonstrated for a limited set of symmetric tilt grain boundaries (STGBs) in cubic materials and twin boundaries. However, the lack of a general framework of interface defects prevents its applicability to arbitrary rational boundaries. In this paper, we present a mathematical framework based on the Smith normal form (SNF) for integer matrices to study the bicrystallography of rational phase and grain boundaries. One of the main results of the paper is a constructive proof of the invariance of the coincident site lattice (CSL) under discrete relative displacements of the parent lattices (of possibly different kind) by a displacement shift complete lattice (DSCL) vector. In addition, we obtain necessary and sufficient conditions on two lattices, related by not only rotations but also lattice distortions, for the existence of a coincidence relation. We first apply these results to explore coincidence relations in arbitrary phase boundaries, and study interface dislocations. In particular, we demonstrate the framework for a phase boundary formed by a strained hexagonal lattice and a square lattice. As a second application, we show how to enumerate all possible (geometric) disconnection modes in arbitrary rational grain boundaries, including glide and non-glide modes in both STGBs and asymmetric-tilt grain boundaries (ATGBs). The constructive nature of the framework lends itself to an algorithmic implementation based exclusively on integer matrix algebra to construct all interfaces that admit CSLs up to a prescribed size, and determine disconnection modes in grain boundaries. We demonstrate the use of SNF bicrystallography on selected bicrystal misorientation axes/angles in face-centered cubic (fcc), body-centered cubic (bcc), and hexagonal (hex) lattices.

On the walk matrix of the Dynkin graph Dn

Let W(Dn) denote the walk matrix of the Dynkin graph Dn (n≥4), a tree obtained from the path of order n−1 by adding a pendant edge at the second vertex. We prove that rankW(Dn)=n−2 if 4|n and rankW(Dn)=n−1 otherwise. Furthermore, we prove that the Smith normal form of W(Dn) isdiag[1,1,…,1︸⌈n2⌉,2,2,…,2︸⌊n2⌋−1,0] when 4∤n. This confirms a recent conjecture in [W. Wang, F. Liu, W. Wang, Generalized spectral characterizations of almost controllable graphs, European J. Combin. 96(2021): 103348].