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Certifying When Temporal Order Can Be Ignored in Continuous-Time Quantum Walks

Madjid Eshaghi Gordji, Mohamadali Berahman

Published Oct 28, 2026• 0 citations
PDF (CC BY)Publisher

Abstract

Abstract Continuous-time quantum walks on dynamic graphs are generated by an ordered sequence of Hamiltonians. Replacing that sequence by one average Hamiltonian can simplify analysis and control, but it may change the transported quantum state. We study this replacement in the single-excitation sector for sequential, undirected edge couplings. Nonzero edge Hamiltonians commute precisely when their edges coincide or are vertex-disjoint, giving a pulse-independent exact reduction for schedules supported on a matching. For general schedules, we derive a nonperturbative spectral-norm bound between the ordered unitary and the exponential of the integrated Hamiltonian. In the edge model, the bound becomes a weighted count of temporal wedges—pairs of distinct event edges meeting at one vertex—and is computable in expected linear time. The same certificate controls state-transfer amplitudes and probabilities, process fidelity, average gate fidelity, and sensitivity to event reordering. A weak-pulse expansion shows that its quadratic scaling is generally sharp. An exact three-vertex calculation and reproducible numerical tests illustrate when the certificate is informative and when cancellation makes it conservative. The result is a directly checkable sufficient condition for replacing a dynamic quantum walk by its weighted aggregate.

DOI
10.21203/rs.3.rs-10777819/v1

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Abstract Continuous-time quantum walks on dynamic graphs are generated by an ordered sequence of Hamiltonians. Replacing that sequence by one average Hamiltonian can simplify analysis and control, but it may change the transported quantum state. We study this replacement in the single-excitation sector for sequential, undirected edge couplings. Nonzero edge Hamiltonians commute precisely when their edges coincide or are vertex-disjoint, giving a pulse-independent exact reduction for schedules supported on a matching. For general schedules, we derive a nonperturbative spectral-norm bound between the ordered unitary and the exponential of the integrated Hamiltonian. In the edge model, the bound becomes a weighted count of temporal wedges—pairs of distinct event edges meeting at one vertex—and is computable in expected linear time. The same certificate controls state-transfer amplitudes and probabilities, process fidelity, average gate fidelity, and sensitivity to event reordering. A weak-pulse expansion shows that its quadratic scaling is generally sharp. An exact three-vertex calculation and reproducible numerical tests illustrate when the certificate is informative and when cancellation makes it conservative. The result is a directly checkable sufficient condition for replacing a dynamic quantum walk by its weighted aggregate.
Abstract Continuous-time quantum walks (CTQWs) provide a versatile framework for exploring quantum transport on graphs. In this work, we investigate how the introduction of edge-weight modulation at a single vertex can suppress its occupation probability. We show that when the edges connected to the root vertex are enhanced by a factor J , the probability of detecting the walker at this vertex decays as 1/J21/J^21/J2 1 / J 2 , provided the initial state has no components on the vertex itself or its nearest neighbors. We derive the full eigenvalue and eigenvector structure of this system, revealing that the suppression arises from the decoupling of two symmetric line subgraphs and the destructive interference of higher-order contributions. The analysis is extended to tree graphs, where we demonstrate the same scaling behavior and identify the role of local graph geometry in controlling vertex probabilities. These results suggest edge-weight modulation as a mechanism for manipulating transport pathways in CTQWs, with potential applications in quantum information transfer and state engineering, and may serve as a probe of decoherence effects in open quantum systems.
It was recently shown that continuous-time quantum walks on dynamic graphs, i.e., sequences of static graphs whose edges change at specific times, can implement a universal set of quantum gates. This result treated all isolated vertices as having self-loops, so they all evolved by a phase under the quantum walk. In this paper, we permit isolated vertices to be loopless or looped, and loopless isolated vertices do not evolve at all under the quantum walk. Using this distinction, we construct simpler dynamic graphs that implement the Pauli gates and a set of universal quantum gates consisting of the Hadamard, TTT, and controlled-not gates, and these gates are easily extended to multiqubit systems. For example, the TTT gate is simplified from a sequence of six graphs to a single graph, and the number of vertices is reduced by a factor of 4. We also construct a generalized phase gate, of which ZZZ, SSS, and TTT are specific instances. Finally, we validate our implementations by numerically simulating a quantum circuit consisting of layers of one- and two-qubit gates, similar to those in recent quantum supremacy experiments, using a quantum walk.
In this work, we present a new algorithm for generating quantum circuits that efficiently implement continuous time quantum walks on arbitrary simple sparse graphs. The algorithm, called matching decomposition, works by decomposing a continuous-time quantum walk Hamiltonian into a collection of exactly implementable Hamiltonians corresponding to matchings in the underlying graph followed by a novel graph compression algorithm that merges edges in the graph. We develop a greedy matching heuristic and a compression-aware matching heuristic, both of which can be used in the quantum circuit algorithm. Lastly, we convert the walks to a circuit and Trotterize over these components. The dynamics of the walker on each edge in the matching can be implemented in the circuit model as sequences of CX and CRx gates. We do not use Pauli decomposition when implementing walks along each matching. Furthermore, we compare greedy (compression-aware) matching decomposition to a standard Pauli-based simulation pipeline and find that greedy (compression-aware) matching decomposition consistently yields substantial resource reductions, requiring up to 43%\%% (70\%) fewer controlled gates and up to 54%\%% (75\%) shallower circuits than Pauli decomposition across multiple graph families. Finally, we also present examples and theoretical results for when matching decomposition can exactly simulate a continuous-time quantum walk on a graph.
Quantum walks generated by the adjacency matrix or the Laplacian are known to exhibit low transfer fidelity on general graphs. In this paper, we study continuous-time quantum walks governed by the generalized Laplacian operator L_k = A+kD, where A is the adjacency matrix, D is the degree matrix, and k is a real-valued parameter. Recent work of Duda, McLaughlin, and Wong showed that in the single-excitation Heisenberg (XYZ) spin model, one can realize walks generated by this family of operators on signed weighted graphs. Motivated by earlier studies on vertex-weighted graphs, we demonstrate that for certain graphs, tuning the parameter k can significantly enhance the fidelity of state transfer between endpoints.
This paper investigates continuous-time quantum walks on directed bipartite graphs based on a graph's adjacency matrix. We prove that on bipartite graphs, probability transport between the two node partitions can be completely suppressed by tuning a model parameter $\ensuremath{\alpha}$. We provide analytic solutions to the quantum walks for the star and circulant graph classes that are valid for an arbitrary value of the number of nodes NNN, time ttt, and the model parameter $\ensuremath{\alpha}$. We discuss quantitative and qualitative aspects of quantum walks based on directed graphs and their undirected counterparts. Numerical simulations of quantum walks on circulant graphs show complex interference phenomena and how complete suppression of transport is achieved near $\ensuremath{\alpha}=\ensuremath{\pi}/2$. By proving two mirror symmetries around $\ensuremath{\alpha}=0$ and $\ensuremath{\pi}/2$ we show that these quantum walks have a period of $\ensuremath{\pi}$ in $\ensuremath{\alpha}$. We show that undirected edges lose their effect on the quantum walk at $\ensuremath{\alpha}=\ensuremath{\pi}/2$ and present non-bipartite graphs that exhibit suppression of transport. Finally, we analytically compute the Hamiltonians of quantum walks on the directed ring graph.
Continuous-time quantum walks (CTQWs) on dynamic graphs, referred to as dynamic CTQWs, are a recently introduced universal model of computation that offers a new paradigm in which to envision quantum algorithms. In this work we develop a mapping from dynamic CTQWs to the gate model of computation in the form of an algorithm to convert arbitrary single edge walks and single self loop walks, which are the fundamental building blocks of dynamic CTQWs, to their circuit model counterparts. We use this mapping to introduce an arbitrary quantum state preparation framework based on dynamic CTQWs. Our approach utilizes global information about the target state, relates state preparation to finding the optimal path in a graph, and leads to optimizations in the reduction of controls that are not as obvious in other approaches. Interestingly, classical optimization problems such as the minimal hitting set, minimum spanning tree, and shortest Hamiltonian path problems arise in our framework. We test our methods against uniformly controlled rotations methods, used by Qiskit, and find ours requires fewer CX gates when the target state has a polynomial number of non-zero amplitudes.