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International Conference on Next-Gen Quantum and Advanced Computing: Algorithms, Security, and Beyond (NQComp)

Graph-Decomposition Based Shift Matrices: Applications to Quantum Channels and Circuit Implementations

Banita Katuwal, M S Srinath, Y. Lakshmi Naidu

Published Mar 5, 2026• 0 citations
Publisher

Abstract

Perfect state transfer (PST) is a key requirement for high-fidelity quantum communication. In discrete-time quantum walks, the shift matrix governs the system dynamics and plays a central role in enabling PST. This work presents a shift-matrix construction based on shunt decompositions of d-regular directed graphs, under which path, cycle, and complete graphs exhibit both perfect state transfer and periodic dynamics. Unlike existing permutation-based approaches that yield periodicity without PST, the proposed method achieves both properties. A SHAP-like attribution analysis is performed to identify the arcs essential for PST. In addition, quantum-circuit implementations of the shift matrices and benchmark results under ideal and noisy conditions are reported. The framework is further extended to a unitary construction of a reset channel, whose reduced Kraus representation leverages the underlying graph structure to minimize information loss.

DOI
10.1109/nqcomp68334.2026.11497753

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•Apr 15, 2024•0 citations
Perfect state transfer (PST) is a key requirement for high-fidelity quantum communication. In discrete-time quantum walks, the shift matrix governs the system dynamics and plays a central role in enabling PST. This work presents a shift-matrix construction based on shunt decompositions of d-regular directed graphs, under which path, cycle, and complete graphs exhibit both perfect state transfer and periodic dynamics. Unlike existing permutation-based approaches that yield periodicity without PST, the proposed method achieves both properties. A SHAP-like attribution analysis is performed to identify the arcs essential for PST. In addition, quantum-circuit implementations of the shift matrices and benchmark results under ideal and noisy conditions are reported. The framework is further extended to a unitary construction of a reset channel, whose reduced Kraus representation leverages the underlying graph structure to minimize information loss.
This paper investigates perfect state transfer (PST) in discrete-time quantum walks constructed via the shunt decomposition method. The walks are defined on a graph GGG and its associated quotient graph G/πG/πG/π, induced by an equitable partition πππ. Through the shunt decomposition of GGG, we derive an explicit relation between the shift operator of the parent graph GGG and that of its quotient graph G/πG/πG/π. We construct a reflection operator based on the characteristic matrix, which establishes a connection between the transition operator of the parent graph and that of its lower-dimensional quotient graph. We then prove that PST occurs on GGG if and only if it occurs on G/πG/πG/π. Furthermore, we express the unitary evolution operator of the quotient graph in terms of Chebyshev polynomials of the first kind, from which we derive explicit criteria for PST. As an application, we establish PST on the cycle graph CnC_{n}Cn​ at time k=n/2k = n/2k=n/2, and lift the result to the parent graph C2nC_{2n}C2n​ via the equitable partition πππ. We further show that if an equitable partition πππ of GGG induces a quotient isomorphic to Kn↺K_n^{\circlearrowleft}Kn↺​, the complete digraph on nnn vertices with a loop at every vertex, then PST occurs at step k=nk = nk=n, and the walk is periodic at k=2nk = 2nk=2n. This framework is applied to two families of graphs, which are the complete bipartite digraph Kn,n⇌K_{n,n}^{\rightleftharpoons}Kn,n⇌​ and the circulant graph Circ⁡(2n,S)\operatorname{Circ}(2n, S)Circ(2n,S), where SSS consists of all odd residues modulo 2n2n2n and n=2sn = 2^sn=2s for some s≥1s \geq 1s≥1, establishing PST in their respective line digraphs. Collectively, these results also answer the question posed by Godsil and Zhan concerning which shunt decompositions or embeddings of a graph admit PST.
In this paper, we define directed partial join graphs with signed couplings and construct discrete-time quantum-walk transition operators for these graphs using the shunt-decomposition framework. The resulting transition operators apply to several important graph families, including complete graphs with loops, circulant partial joins, complete bipartite graphs, tensor powers of complete bipartite graphs, all with signed couplings. For each family, we identify the corresponding structure of the transition operator and derive necessary and sufficient conditions for periodicity and perfect state transfer (PST), when one of the two directed regular graphs admits PST. Based on these results, we identify two types of state transfer; internal PST, which occurs between vertices within the same graph, and coupling PST, which occurs between two components of join graphs. We further develop a double-cover construction for directed partial join graphs and derive conditions for periodicity and PST when the associated transition operators do not necessarily commute. Using this construction, we establish PST results for double covers of complete graphs with loops. In particular, we provide an example in which the complete graph KnK_nKn​ does not exhibit PST for n≥4n\geq4n≥4, whereas a suitable partial join of KnK_nKn​ exhibits PST when n=2mn=2^mn=2m, m≥2m\geq2m≥2. Hence, these results extend the class of graph families admitting PST in shunt-decomposition-based quantum walks and provide a unified framework for studying quantum state transfer in graph joins, products, and covers.
We provide a brief survey of perfect state transfer in quantum walks on finite graphs. The ability to transfer a quantum state from one part of a quantum computer to another is a key ingredient of scalable architectures. Transfer through structures that do not require locally varying dynamic control simplifies the design and hence reduces the opportunities for errors to arise. Continuous time walks quantum walks on highly structured graphs exhibit perfect state transfer for the complete graph of size 2, the path of length 3, and the cycle of size 4. From these, larger graphs can be constructed, and the use of edge weights widens this set considerably. Discrete-time quantum walks have more flexibility through exploiting the coin degrees of freedom, but with the disadvantage that local control of the coin is required if the degree of the vertices varies. The closely related property of periodicity (exact return to the starting state) is also mentioned.
The Unitary Coin Discrete-Time Quantum Walk (UCDTQW) serves as a universal model for quantum computers. In this paper, we conduct matrix analysis based on the unitary operator of UCDTQW, constructing the system’s shift operator as the unitary form of the adjacency matrix corresponding to the complete graph [Formula: see text]. Through a set of transformation equations, we realize the reciprocal mapping between the adjacency matrix and the shift operator. Specifically, by splitting the undirected edges of [Formula: see text] into multiple directed arcs, we construct a directed multigraph structure. Our study reveals that the unitary evolution operator of this multigraph exhibits properties analogous to those of the shift operator, particularly uncovers the periodic characteristics of the wave function. Furthermore, under another edge connection scheme, we achieve the construction of the quantum circuit representation for the shift operator.
In this paper, we analyze state transfer in quantum walks by using combinatorial methods. We generalize perfect state transfer in two-reflection discrete-time quantum walks to a notion that we call peak state transfer; we define peak state transfer as the highest state transfer that can be achieved between an initial and a target state under unitary evolution, even when perfect state transfer is unattainable. We give a spectral characterization of peak state transfer that allows us to fully characterize peak state transfer in the arc-reversal (Grover) walk on various families of graphs, including strongly regular graphs and incidence graphs of block designs (assuming that the walk starts at a point of the design). In addition, we provide many examples of peak state transfer, including an infinite family where the amount of peak state transfer tends to 111 as the number of vertices grows. We further demonstrate that peak state transfer properties extend to infinite families of graphs generated by vertex blow-ups, and we characterize periodicity in the vertex-face walk on toroidal grids. In our analysis, we make extensive use of the spectral decomposition of a matrix that is obtained by projecting the transition matrix down onto a subspace. Though we are motivated by a problem in quantum computing, we identify several open problems that are purely combinatorial, arising from the spectral conditions required for peak state transfer in discrete-time quantum walks.
Abstract Continuous-time quantum walks (CTQW) becoming more significant for quantum information and quantum computation. The perfect state transfer (PST) on graphs is a specific phenomena of CTQW which is widely used in quantum communication of small system, and be found critical for quantum algorithm design recently, such as quantum approximate optimization algorithms and quantum search. If a network GGG contains 2N2N2N vertices, and PST occurs between NNN pairs of vertices at the same time TTT, then we say GGG admits symmetric PST (SPST). In this work, we will study the SPST on bipartite graphs via spectra. And then, we will present how to set couplings of special bipartite graphs such as cycles, grids and chains to admit SPST is given. For cycles with 6, 8, 10, 12 vertices, we give the analytical method to obtain the couplings to admit SPST. Analogously, the couplings for chains and grids to admit SPST are analytical obtained. We also present a typical approach by using the graph tensor product to construct new quantum networks that admit SPST, in which the new quantum networks are derived from special kinds of integral graphs, such as complete graphs, strongly regular graphs and distance regular graphs.