Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2609.18018

Perfect state transfer under matrix powers: parity and spectral arithmetic

Xingkun Song

math.COquant-phSubmitted Sep 16, 2026• Announced Sep 17, 2026
arXivHTMLPDF

Abstract

For a real symmetric matrix $H$ and distinct vertices $a,b$, we classify exponents $k$ for which $H^k$ has perfect state transfer (PST) from $a$ to $b$. If their supported eigenvalues are integer multiples of a common positive number, every odd exponent reduces to $H$ and every positive even exponent reduces to $H^2$. We determine the minimum transfer times using a greatest common divisor of supported spectral differences. For rational symmetric matrices, symmetry of the source vertex support about zero implies the odd-power equivalence without a commensurability assumption; this includes all bipartite graphs. If the source vertex supports zero, PST under one positive even power implies PST under every positive even power. For a symmetric three-point quadratic spectrum whose outer projection signs agree and differ from the central sign, a nonzero rational shift leaves exactly one PST exponent. We classify all adjacency powers of hypercubes, cycles, and Johnson graphs, and all adjacency squares of paths. In particular, the adjacency matrix of $P_7$ has PST from vertex $2$ to vertex $6$ only at exponent $2$.

Comments
24 pages, 0 figures

Related papers

Perfect State Transfer Between Real Pure States

Chris D. Godsil, Steve Kirkland, Hermie Monterde

Abstract. Pure states correspond to one-dimensional subspaces of [Formula: see text] represented by unit vectors. In this paper, we develop the theory of perfect state transfer (PST) between real pure states with emphasis on the adjacency and Laplacian matrices as Hamiltonians of a graph representing a quantum spin network. We characterize PST between real pure states based on the spectral information of a graph and prove three fundamental results: (i) every periodic real pure state [Formula: see text] admits PST with another real pure state [Formula: see text], (ii) every connected graph admits PST between real pure states, and (iii) for any pair of real pure states [Formula: see text] and [Formula: see text] and for any time [Formula: see text], there exists a real symmetric matrix [Formula: see text] such that [Formula: see text] and [Formula: see text] admit PST relative to [Formula: see text] at time [Formula: see text]. We also determine all real pure states that admit PST in complete graphs, complete bipartite graphs, paths, and cycles. This leads to a complete characterization of pair and plus state transfer in paths and complete bipartite graphs. We give constructions of graphs that admit PST between real pure states. Finally, using results on the spread of graphs, we prove that amongst all [Formula: see text]-vertex simple unweighted graphs, the least minimum PST time between real pure states is attained by any join graph for the Laplacian case, while it is attained by the join of an empty graph and a complete graph of appropriate sizes for the adjacency case. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SIMAX and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at none . [Formula: see text]

arXiv:2502.08103•SIAM Journal on Matrix Analysis and Applications•Sep 17, 2025•2 citations

Perfect State Transfer on Oriented Circulant Graphs: A Complete Classification

Xingkun Song

The continuous-time quantum walk on an oriented circulant graph is determined by the Fourier eigenvalues of its Hermitian adjacency matrix. We classify perfect state transfer (PST) between distinct vertices in every nonempty oriented circulant graph. We show that each such graph is described by an odd primitive quadratic Dirichlet character of conductor $Δ$, a set of gcd-classes, and a choice between the two orientations of each selected class. For a graph of order $n$, we derive an explicit formula for every Fourier eigenvalue without assuming that $n/Δ$ is coprime to $Δ$. We prove that PST occurs only for $Δ\in\{3,4,8\}$ and give necessary and sufficient conditions on the connection set for each conductor. Equivalently, the square-free radicands of oriented circulant graphs with PST are exactly $1$, $2$, and $3$. More generally, when $λ_j=\sqrt{D}η_j$ with $η_j\in\mathbb{Z}$, congruences satisfied by the integers $η_j$ determine all PST pairs and times, the minimum period, and the largest vertex sets supporting multiple state transfer (MST). In this class, pretty good state transfer is equivalent to PST. We also determine the connected orders and enumerate the resulting graphs.

quant-pharXiv:2608.11992•Aug 13, 2026•0 citations

Spectral width and polynomial degree in perfect state transfer

Xingkun Song

We study the minimum time for perfect state transfer under polynomial Hamiltonians with bounded degree and spectral width. For a strongly cospectral pair and width bound $W$, the optimum, when finite, is an integer multiple of $π/W$, determined by integer interpolation with prescribed parities. For equally spaced supported eigenvalues with alternating signs, we give degree bounds under which every minimizer is affine, and sharp asymptotics for each fixed exact degree. Near-minimizing phase polynomials satisfy a quantitative Chebyshev stability estimate. We determine the optimal transfer time for every degree bound on hypercubes of odd prime dimension. For complementary vertices of $J(2m,m)$, the optimal time at fixed spectral width grows exponentially in $m$ throughout an interval of feasible degrees. We also construct polynomial Hamiltonians showing that every feasible degree $m-t$ with $t=o(m)$ admits subexponential transfer time.

quant-pharXiv:2609.23633•Sep 22, 2026•0 citations

Characterization of circulant graphs having perfect state transfer

Milan Bašić

In this paper we answer the question of when circulant quantum spin networks with nearest-neighbor couplings can give perfect state transfer. The network is described by a circulant graph $G$, which is characterized by its circulant adjacency matrix $A$. Formally, we say that there exists a {\it perfect state transfer} (PST) between vertices $a,b\in V(G)$ if $|F(τ)_{ab}|=1$, for some positive real number $τ$, where $F(t)=\exp(ıAt)$. Saxena, Severini and Shparlinski ({\it International Journal of Quantum Information} 5 (2007), 417--430) proved that $|F(τ)_{aa}|=1$ for some $a\in V(G)$ and $τ\in \R^+$ if and only if all eigenvalues of $G$ are integer (that is, the graph is integral). The integral circulant graph $\ICG_n (D)$ has the vertex set $Z_n = \{0, 1, 2, ..., n - 1\}$ and vertices $a$ and $b$ are adjacent if $\gcd(a-b,n)\in D$, where $D \subseteq \{d : d \mid n,\ 1\leq d

arXiv:1104.1825•Apr 11, 2011•12 citations

Quantum State Transfer in Graphs

Gabriel Coutinho

Let X be a graph, A its adjacency matrix, and t a non-negative real number. The matrix exp(i t A) determines the evolution in time of a certain quantum system defined on the graph. It represents a continuous-time quantum walk in X. We say that X admits perfect state transfer from a vertex u to a vertex v if there is a time t such that \n| exp(i t A)_(u,v) | = 1. \n \nThe main problem we study in this thesis is that of determining which simple graphs admit perfect state transfer. For some classes of graphs the problem is solved. For example, a path on n vertices admits perfect state transfer if and only if n=2 or n=3. However, the general problem of determining all graphs that admit perfect state transfer is substantially hard. \n \nIn this thesis, we focus on some special cases. We provide necessary and sufficient conditions for a distance-regular graph to admit perfect state transfer. In particular, we provide a detailed account of which distance-regular graphs of diameter three do so. \n \nA graph is said to be spectrally extremal if the number of distinct eigenvalues is equal to the diameter plus one. Distance-regular graphs are examples of such graphs. We study perfect state transfer in spectrally extremal graphs and explore rich connections to the topic of orthogonal polynomials. We characterize perfect state transfer in such graphs. \n \nWe also provide a general framework in which perfect state transfer in graph products can be studied. We use this to determine when direct products and double covers of graphs admit perfect state transfer. As a consequence, we provide many new examples of simple graphs admitting perfect state transfer. We also provide some advances in the understanding of perfect state transfer in Cayley graphs for the groups (Z_2)^d and Z_n. \n \nFinally, we consider the problem of determining which trees admit perfect state transfer. We show more generally that, except for the path on two vertices, if a connected bipartite graph contains a unique perfect matching, then it cannot admit perfect state transfer. We also consider this problem in the context of another model of quantum walks determined by the matrix exp(i t L), where L is the Laplacian matrix of the graph. In particular, we show that no tree on an odd number of vertices admits perfect state transfer according to this model.

•Aug 13, 2014•34 citations

Perfect state transfer on mixed graphs: complete classes and transfer times

Xingkun Song

For perfect state transfer (PST) on unweighted mixed graphs, we classify the normalized transfer times of complete PST classes. A finite set $Λ\subset\mathbb R/\mathbb Z$ containing zero occurs at a nonstationary periodic vertex if and only if $\cos(2π(x-y))\in\mathbb Q$ for all $x,y\inΛ$. Every admissible set has a connected oriented realization. We also classify the possible return phases of oriented realizations at the minimum vertex period. Transfers at rational multiples of the common minimum vertex period partition a complete class into sets of size at most six, or at most three in an oriented graph with return phase $-1$; both bounds are sharp. We construct complete classes of every finite size, including classes in which all transfers between distinct vertices occur at irrational multiples of the period and no switching automorphism maps a class vertex to a distinct class vertex. We also characterize simultaneous realization in connected oriented graphs with prescribed relative minimum vertex periods and return phases. The proof combines an imaginary quadratic field restriction with an unweighted construction that selects the complete target set.

math.COCross-listedarXiv:2610.08333•Oct 7, 2026•0 citations
For a real symmetric matrix HHH and distinct vertices a,ba,ba,b, we classify exponents kkk for which HkH^kHk has perfect state transfer (PST) from aaa to bbb. If their supported eigenvalues are integer multiples of a common positive number, every odd exponent reduces to HHH and every positive even exponent reduces to H2H^2H2. We determine the minimum transfer times using a greatest common divisor of supported spectral differences. For rational symmetric matrices, symmetry of the source vertex support about zero implies the odd-power equivalence without a commensurability assumption; this includes all bipartite graphs. If the source vertex supports zero, PST under one positive even power implies PST under every positive even power. For a symmetric three-point quadratic spectrum whose outer projection signs agree and differ from the central sign, a nonzero rational shift leaves exactly one PST exponent. We classify all adjacency powers of hypercubes, cycles, and Johnson graphs, and all adjacency squares of paths. In particular, the adjacency matrix of P7P_7P7​ has PST from vertex 222 to vertex 666 only at exponent 222.
Abstract. Pure states correspond to one-dimensional subspaces of [Formula: see text] represented by unit vectors. In this paper, we develop the theory of perfect state transfer (PST) between real pure states with emphasis on the adjacency and Laplacian matrices as Hamiltonians of a graph representing a quantum spin network. We characterize PST between real pure states based on the spectral information of a graph and prove three fundamental results: (i) every periodic real pure state [Formula: see text] admits PST with another real pure state [Formula: see text], (ii) every connected graph admits PST between real pure states, and (iii) for any pair of real pure states [Formula: see text] and [Formula: see text] and for any time [Formula: see text], there exists a real symmetric matrix [Formula: see text] such that [Formula: see text] and [Formula: see text] admit PST relative to [Formula: see text] at time [Formula: see text]. We also determine all real pure states that admit PST in complete graphs, complete bipartite graphs, paths, and cycles. This leads to a complete characterization of pair and plus state transfer in paths and complete bipartite graphs. We give constructions of graphs that admit PST between real pure states. Finally, using results on the spread of graphs, we prove that amongst all [Formula: see text]-vertex simple unweighted graphs, the least minimum PST time between real pure states is attained by any join graph for the Laplacian case, while it is attained by the join of an empty graph and a complete graph of appropriate sizes for the adjacency case. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SIMAX and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at none . [Formula: see text]
The continuous-time quantum walk on an oriented circulant graph is determined by the Fourier eigenvalues of its Hermitian adjacency matrix. We classify perfect state transfer (PST) between distinct vertices in every nonempty oriented circulant graph. We show that each such graph is described by an odd primitive quadratic Dirichlet character of conductor ΔΔΔ, a set of gcd-classes, and a choice between the two orientations of each selected class. For a graph of order nnn, we derive an explicit formula for every Fourier eigenvalue without assuming that n/Δn/Δn/Δ is coprime to ΔΔΔ. We prove that PST occurs only for Δ∈{3,4,8}Δ\in\{3,4,8\}Δ∈{3,4,8} and give necessary and sufficient conditions on the connection set for each conductor. Equivalently, the square-free radicands of oriented circulant graphs with PST are exactly 111, 222, and 333. More generally, when λj=Dηjλ_j=\sqrt{D}η_jλj​=D​ηj​ with ηj∈Zη_j\in\mathbb{Z}ηj​∈Z, congruences satisfied by the integers ηjη_jηj​ determine all PST pairs and times, the minimum period, and the largest vertex sets supporting multiple state transfer (MST). In this class, pretty good state transfer is equivalent to PST. We also determine the connected orders and enumerate the resulting graphs.
We study the minimum time for perfect state transfer under polynomial Hamiltonians with bounded degree and spectral width. For a strongly cospectral pair and width bound WWW, the optimum, when finite, is an integer multiple of π/Wπ/Wπ/W, determined by integer interpolation with prescribed parities. For equally spaced supported eigenvalues with alternating signs, we give degree bounds under which every minimizer is affine, and sharp asymptotics for each fixed exact degree. Near-minimizing phase polynomials satisfy a quantitative Chebyshev stability estimate. We determine the optimal transfer time for every degree bound on hypercubes of odd prime dimension. For complementary vertices of J(2m,m)J(2m,m)J(2m,m), the optimal time at fixed spectral width grows exponentially in mmm throughout an interval of feasible degrees. We also construct polynomial Hamiltonians showing that every feasible degree m−tm-tm−t with t=o(m)t=o(m)t=o(m) admits subexponential transfer time.
In this paper we answer the question of when circulant quantum spin networks with nearest-neighbor couplings can give perfect state transfer. The network is described by a circulant graph GGG, which is characterized by its circulant adjacency matrix AAA. Formally, we say that there exists a {\it perfect state transfer} (PST) between vertices a,b∈V(G)a,b\in V(G)a,b∈V(G) if ∣F(τ)ab∣=1|F(τ)_{ab}|=1∣F(τ)ab​∣=1, for some positive real number τττ, where F(t)=exp⁡(ıAt)F(t)=\exp(ıAt)F(t)=exp(ıAt). Saxena, Severini and Shparlinski ({\it International Journal of Quantum Information} 5 (2007), 417--430) proved that ∣F(τ)aa∣=1|F(τ)_{aa}|=1∣F(τ)aa​∣=1 for some a∈V(G)a\in V(G)a∈V(G) and τ∈R+τ\in \R^+τ∈R+ if and only if all eigenvalues of GGG are integer (that is, the graph is integral). The integral circulant graph $\ICG_n (D)$ has the vertex set Zn={0,1,2,...,n−1}Z_n = \{0, 1, 2, ..., n - 1\}Zn​={0,1,2,...,n−1} and vertices aaa and bbb are adjacent if gcd⁡(a−b,n)∈D\gcd(a-b,n)\in Dgcd(a−b,n)∈D, where $D \subseteq \{d : d \mid n,\ 1\leq d
Let X be a graph, A its adjacency matrix, and t a non-negative real number. The matrix exp(i t A) determines the evolution in time of a certain quantum system defined on the graph. It represents a continuous-time quantum walk in X. We say that X admits perfect state transfer from a vertex u to a vertex v if there is a time t such that \n| exp(i t A)_(u,v) | = 1. \n \nThe main problem we study in this thesis is that of determining which simple graphs admit perfect state transfer. For some classes of graphs the problem is solved. For example, a path on n vertices admits perfect state transfer if and only if n=2 or n=3. However, the general problem of determining all graphs that admit perfect state transfer is substantially hard. \n \nIn this thesis, we focus on some special cases. We provide necessary and sufficient conditions for a distance-regular graph to admit perfect state transfer. In particular, we provide a detailed account of which distance-regular graphs of diameter three do so. \n \nA graph is said to be spectrally extremal if the number of distinct eigenvalues is equal to the diameter plus one. Distance-regular graphs are examples of such graphs. We study perfect state transfer in spectrally extremal graphs and explore rich connections to the topic of orthogonal polynomials. We characterize perfect state transfer in such graphs. \n \nWe also provide a general framework in which perfect state transfer in graph products can be studied. We use this to determine when direct products and double covers of graphs admit perfect state transfer. As a consequence, we provide many new examples of simple graphs admitting perfect state transfer. We also provide some advances in the understanding of perfect state transfer in Cayley graphs for the groups (Z_2)^d and Z_n. \n \nFinally, we consider the problem of determining which trees admit perfect state transfer. We show more generally that, except for the path on two vertices, if a connected bipartite graph contains a unique perfect matching, then it cannot admit perfect state transfer. We also consider this problem in the context of another model of quantum walks determined by the matrix exp(i t L), where L is the Laplacian matrix of the graph. In particular, we show that no tree on an odd number of vertices admits perfect state transfer according to this model.
For perfect state transfer (PST) on unweighted mixed graphs, we classify the normalized transfer times of complete PST classes. A finite set Λ⊂R/ZΛ\subset\mathbb R/\mathbb ZΛ⊂R/Z containing zero occurs at a nonstationary periodic vertex if and only if cos⁡(2π(x−y))∈Q\cos(2π(x-y))\in\mathbb Qcos(2π(x−y))∈Q for all x,y∈Λx,y\inΛx,y∈Λ. Every admissible set has a connected oriented realization. We also classify the possible return phases of oriented realizations at the minimum vertex period. Transfers at rational multiples of the common minimum vertex period partition a complete class into sets of size at most six, or at most three in an oriented graph with return phase −1-1−1; both bounds are sharp. We construct complete classes of every finite size, including classes in which all transfers between distinct vertices occur at irrational multiples of the period and no switching automorphism maps a class vertex to a distinct class vertex. We also characterize simultaneous realization in connected oriented graphs with prescribed relative minimum vertex periods and return phases. The proof combines an imaginary quadratic field restriction with an unweighted construction that selects the complete target set.