We introduce a family of complex-valued edge weights on a finite simple graph $\G$ arising from a continuous-time quantum walk on the line graph $\ell\G$, packaged as the \emph{Schur state}: an $n \times n$ Hermitian matrix encoding the amplitudes of an edge-state walk. The entrywise modulus square induces a real-weighted adjacency matrix $A(e)$ and Laplacian $L(e)$, and time-averaging yields a weighted graph whose spanning-tree count we relate to that of $\G$. Our main result is \[
tn\!\left(\G, \tfrac{1}{m}\right) = \frac{1}{m^{n-1}}\, tn(\G), \] valid whenever the initial edge state is \emph{uniform commutative}, where $n=|V\G|$, $m=|E\G|$, and $tn(\G, w)$ denotes the weighted spanning-tree count. We further identify a structural mechanism -- the $-2$ eigenspace of $\ell\G$ -- providing uniform commutative states beyond the regular case, in particular for line graphs of Eulerian graphs with an even number of edges. As a side result, we establish that commutative states are precisely the states whose von Neumann entropy is preserved under average mixing.
We introduce a family of complex-valued edge weights on a finite simple graph \G arising from a continuous-time quantum walk on the line graph ℓ\G, packaged as the \emph{Schur state}: an n×n Hermitian matrix encoding the amplitudes of an edge-state walk. The entrywise modulus square induces a real-weighted adjacency matrix A(e) and Laplacian L(e), and time-averaging yields a weighted graph whose spanning-tree count we relate to that of \G. Our main result is [
tn!\left(\G, \tfrac{1}{m}\right) = \frac{1}{m^{n-1}}, tn(\G), ] valid whenever the initial edge state is \emph{uniform commutative}, where n=∣V\G∣, m=∣E\G∣, and tn(\G,w) denotes the weighted spanning-tree count. We further identify a structural mechanism -- the −2 eigenspace of ℓ\G -- providing uniform commutative states beyond the regular case, in particular for line graphs of Eulerian graphs with an even number of edges. As a side result, we establish that commutative states are precisely the states whose von Neumann entropy is preserved under average mixing.
A continuous-time quantum walk on a graph is represented by the complex matrix , where is the adjacency matrix of and is a non-negative time. If the graph models a network of interacting qubits, transfer of state among such qubits throughout time can be formalized as the action of the continuous-time quantum walk operator in the characteristic vectors of the vertices. Here, we are concerned with the problem of determining which graphs admit a perfect transfer of state. More specifically, we will study graphs whose adjacency matrix is a sum of tensor products of -matrices, focusing on the case where a graph is the tensor product of two other graphs. As a result, we will construct many new examples of perfect state transfer.
Finite Fermionic Universality from Token Graphs establishes a general finite-dimensional operator-algebra theorem for many-fermion systems. Given a finite graph G on m one-particle modes, the induced support on the k-fermion sector ∧ᵏℂᵐ is exactly the k-token graph Fₖ(G). With a diagonal occupation-label operator Dₖ having simple spectrum and a Hermitian transfer operator A_G,k supported on the token graph, the generated C*-algebra is C*(Dₖ, A_G,k) ≅ ⨁_{C ∈ π₀(Fₖ(G))} M_|C|(ℂ). In particular, when G is connected, C*(Dₖ + iA_G,k) = M_{(m choose k)}(ℂ), for every 1 ≤ k ≤ m − 1. Thus a single explicitly constructed nonnormal operator generates the complete matrix algebra on every fixed-particle-number fermionic sector. The paper proves the token-graph lift and connectivity theorem from first principles, constructs matrix units explicitly along token-graph paths, derives exact synthesis depth from graph distance, gives edge-count and diameter formulas, and classifies the superselection structure that appears when the one-particle graph is disconnected. The analysis also extends to full fermionic Fock space. For an explicit hopping-plus-pairing construction, fermion parity is the surviving superselection label and the generated algebra is M_{2ᵐ⁻¹}(ℂ) ⊕ M_{2ᵐ⁻¹}(ℂ), with commutant span{I, P}. The result is independent of superconducting phenomenology. It is a finite operator-algebra and graph-theoretic universality theorem and does not, by itself, assert a superconducting phase, select a material Hamiltonian, or establish a thermodynamic limit.
Quantum simulation at fixed particle number admits two equivalent descriptions, a first-quantized (particle) representation and a second-quantized (occupation-number) representation. Their quantum resource costs differ sharply across computational tasks, so the ability to convert coherently between them is valuable. We construct an explicit unitary Q, with inverse Q†, that maps a first-quantized state to its fixed-N occupation-number form while diagnosing the input's particle-exchange symmetry. The conversion is therefore symmetry-agnostic at the input yet fully resolved at the output, and it applies uniformly to bosonic, fermionic, and parastatistical sectors. At its foundation lies a structural identification that we place at the center of this work: the quantum Schur transform supplied by Schur-Weyl duality is the non-abelian Fourier transform of the commuting pair (SN,U(d)), and the occupation-number representation is its weight basis, retaining only the labels shared by both factors, the irrep λ and the u(d) weight. This reduction is lossless for bosons and fermions, while a canonical Gelfand-Tsetlin promise renders it one-to-one for the remaining sectors. Algorithmically, Q composes the strong Schur transform with reversible arithmetic that computes occupations as successive row-sum differences of the Gelfand-Tsetlin pattern, yielding gate complexity poly(N,d,log(1/ε)). The converted state is prepared efficiently in quantum memory. Any classical algorithm that outputs it explicitly, however, pays a cost set by the sector dimension, which is polynomial of degree N in d at fixed N and exponential in N when d=Θ(N). Finally, an efficient classical sampler for the induced occupation-number distribution would yield one for arbitrary quantum circuits, contrary to standard complexity assumptions.
Many problems in applied mathematics and physics are formulated most naturally in terms of matrices, and can be solved by computing functions of these matrices. For example, in quantum mechanics, the coherent dynamics of physical systems is described by the matrix exponential of their Hamiltonian. In state of the art experiments, one can now observe such unitary evolution of many-body systems, which is of fundamental interest in the study of many-body quantum phenomena. On the other hand the theoretical simulation of such non-equilibrium many-body dynamics is very challenging. In this thesis, we develop a symbolic approach to matrix functions and quantum dynamics based on a novel algebraic structure we identify for sets of walks on graphs. We begin by establishing the graph theoretic equivalent to the fundamental theorem of arithmetic: all the walks on any finite digraph uniquely factorise into products of prime elements. These are the simple paths and simple cycles, walks forbidden from visiting any vertex more than once. We give an algorithm that efficiently factorises individual walks and obtain a recursive formula to factorise sets of walks. This yields a universal continued fraction representation for the formal series of all walks on digraphs. It only involves simple paths and simple cycles and is thus called a path-sum. In the second part, we recast matrix functions into path-sums. We present explicit results for a matrix raised to a complex power, the matrix exponential, matrix inverse, and matrix logarithm. We introduce generalised matrix powers which extend desirable properties of the Drazin inverse to all powers of a matrix. In the third part, we derive an intermediary form of path-sum, called walk-sum, relying solely on physical considerations. Walk-sum describes the dynamics of a quantum system as resulting from the coherent superposition of its histories, a discrete analogue to the Feynman path-integrals. Using walk-sum we simulate the dynamics of quantum random walks and of Rydberg-excited Mott insulators. Using path-sum, we demonstrate many-body Anderson localisation in an interacting disordered spin system. We give two observable signatures of this phenomenon: localisation of the system magnetisation and of the linear magnetic response function. Lastly we return to the study of sets of walks. We show that one can construct as many representations of series of walks as there are ways to define a walk product such that the factorisation of a walk always exist and is unique. Illustrating this result we briefly present three further methods to evaluate functions of matrices. Regardless of the method used, we show that graphs are uniquely characterised, up to an isomorphism, by the prime walks they sustain.
Description:A reconstruction theory for complex quantum coherence encoded on graphs using controlled quantum Fisher information. Finite-order data determine edge magnitudes and cycle holonomies up to an explicit integral-lattice ambiguity, revealing continuous and torsion obstructions to identifiability. With physical labelled Schur controls, exact finite-value measurements reconstruct the full coherence matrix up to vertex gauge using at most |E| + 2β scalar QFI values, with explicit injectivity and stability bounds. .
A continuous time quantum walk consists of a particle moving among the vertices of a graph G. Its movement is governed by the structure of the graph. More formally, the adjacency matrix A is the Hamiltonian that determines the movement of our particle. Quantum walks have found a number of algorithmic applications, including unstructured search, element distinctness and Boolean formula evaluation. We will examine the properties of periodicity and state transfer. In particular, we will prove a result of the author along with Godsil, Kirkland and Severini, which states that pretty good state transfer occurs in a path of length n if and only if the n+1 is a power of two, a prime, or twice a prime. We will then examine the property of strong cospectrality, a necessary condition for pretty good state transfer from u to v. \n \nWe will then consider quantum walks involving more than one particle. In addition to moving around the graph, these particles interact when they encounter one another. Varying the nature of the interaction term gives rise to a range of different behaviours. We will introduce two graph invariants, one using a continuous-time multi-particle quantum walk, and the other using a discrete-time quantum walk. Using cellular algebras, we will prove several results which characterize the strength of these two graph invariants. \n \nLet A be an association scheme of n × n matrices. Then, any element of A can act on the space of n × n matrices by left multiplication, right multiplication, and Schur multiplication. The set containing these three linear mappings for all elements of A generates an algebra. This is an example of a Jaeger algebra. Although these algebras were initially developed by Francois Jaeger in the context of spin models and knot invariants, they prove to be useful in describing multi-particle walks as well. We will focus on triply-regular association schemes, proving several new results regarding the representation of their Jaeger algebras. As an example, we present the simple modules of a Jaeger algebra for the 4-cube.