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:1602.02446

Reduction of branching graphs supporting continuous time return quantum walks

Thomas Cavin, Dmitry Solenov

Published Feb 8, 2016• 2 citations
arXivHTMLPDF

Abstract

We demonstrate that continuous time quantum walks on several types of branching graphs, including graphs with loops, are identical to quantum walks on simpler linear chain graphs. We also show graph types for which such equivalence does not exist. Several instructive examples are discussed, and a general approach to analyze more complex branching graphs is formulated. It is further illustrated with a return quantum walk solution for a cube graph with adjustable complex hopping amplitudes.

DOI
10.48550/arxiv.1602.02446

Related papers

STUDY OF CONTINUOUS-TIME QUANTUM WALKS ON QUOTIENT GRAPHS VIA QUANTUM PROBABILITY THEORY

S. Salimi

In the present paper, we study the continuous-time quantum walk on quotient graphs. On such graphs, there is a straightforward reduction of the problem to a subspace that can be considerably smaller than the original one. Along the lines of reductions, by using the idea of calculation of the probability amplitudes for continuous-time quantum walk in terms of the spectral distribution associated with the adjacency matrix of graphs [Jafarizadeh and Salimi (Ann. Phys.322 (2007))], we show that the continuous-time quantum walk on original graph Γ induces a continuous-time quantum walk on quotient graph ΓH. Finally, for example, we investigate the continuous-time quantum walk on some quotient Cayley graphs.

arXiv:0710.5813•International Journal of Quantum Information•Aug 1, 2008•17 citations

Continuous time quantum walks performing quantum gates via higher energy states

Dmitry Solenov

It has recently been noticed that quantum gates can be performed faster and can utilize interaction between qubits more effectively if performed via continuous time quantum walks through states outside of the computational basis. We formulate general requirements for such walks and illustrate them with examples of single-, two- and three-qubit gates, including Hadamard, Control-NOT, and Toffoli gates. Continuous-time quantum walks on graphs involved in these gates are investigated.

•Dec 20, 2015•1 citation

Isolated vertices in continuous-time quantum walks on dynamic graphs

Thomas G. Wong

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, $T$, and controlled-not gates, and these gates are easily extended to multiqubit systems. For example, the $T$ 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 $Z$, $S$, and $T$ 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.

arXiv:1908.00507•Physical Review A•Dec 18, 2019•11 citations

Simplification Rules for Continuous-Time Quantum Walks on Dynamic Graphs

Mostafa Atallah, Daniel Dilley, Jishnu Mahmud +2 more

Continuous-time quantum walks (CTQWs) on dynamic graphs realize quantum gates as sequences of time-evolving graph Hamiltonians, but naive constructions produce long sequences with redundancy. Simplification rules, which are graph rewrite rules that shorten a dynamic graph sequence while preserving the unitary it implements, are the CTQW analogue of circuit identities in the gate model. In this work, we give CTQW realizations of the standard single-qubit gates and introduce new graph rewrite rules. We demonstrate the simplification rules through worked circuit reductions and outline their use as transpilation primitives for converting between the circuit model and the dynamic graph framework.

quant-pharXiv:2609.18463•Sep 17, 2026•0 citations

Connection Between Continuous and Discrete Time Quantum Walks on d-Dimensional Lattices; Extensions to General Graphs

I obtain the dynamics of the continuous time quantum walk on a d-dimensional lattice, with periodic boundary conditions, as an appropriate limit of the dynamics of the discrete time quantum walk on the same lattice. This extends the main result of [8] which proved this limit for the case of the quantum walk on the infinite line and the quantum cellular automaton proposed in [1]. By highlighting the main features of the limiting procedure, I then extend it to general graphs. For a given discrete time quantum walk on a general graph, I single out the type of continuous dynamics (Hamiltonians) that can be obtained as a limit of the discrete time dynamics. 1 Generalities on quantum walks; continuous and discrete time Consider a graph G: = {V, E} with a set of vertices V of cardinality N and a set of edges E. We assume G to be undirected and without self-loops. In the interval of time ∆t a certain fraction γ∆t of a quantity pj leaves the location of the j-th vertex to move to a neighboring vertex k. The quantity pj, j = 1,..., N, at time t + ∆t is given by 1 pj(t + ∆t) = pj(t) − deg (j)γ∆tpj(t) + ∑ (k,j)∈E γ∆tpk(t). (1) Define ⃗p the N−vector whose components are the pj’s quantities, and L the Laplacian matrix defined by Ljj = − deg (j), Ljk = 1 if (j, k) ∈ E and Ljk = 0 otherwise. L is

•Jan 1, 2009•11 citations

Quantum Algorithms and Covering Spaces

Tobias J. Osborne, Simone Severini B

In this paper we isolate the combinatorial property responsible (at least in part) for the computational speedups recently observed in some quantum walk algorithms. We find that continuous-time quantum walks can exploit the covering space property of certain graphs. We demonstrate that a quantum walk on a graph Y which covers a smaller graph X can be equivalent to a quantum walk on the smaller graph X. This equivalence occurs only when the walk begins on certain initial states, fibre-constant states, which respect the graph covering space structure. We illustrate these observations with walks on Cayley graphs; we show that walks on fibre-constant initial states for Cayley graphs are equivalent to walks on the induced Schreier graph. We also consider the problem of constructing efficient gate sequences simulating the time evolution of a continuous-time quantum walk. For the case of the walk on the m-torus graph T^m on 2^n vertices we construct a gate sequence which uses O(\poly(n)) gates which is independent of the time t the walk is simulated for (and so the sequence can simulate the walk for exponential times). We argue that there exists a wide class of nontrivial operators based on quantum walks on graphs which can be measured efficiently. We introduce a new general class of computational problems, HiddenCover, which includes a variant of the general hidden subgroup problem as a subclass. We argue that quantum computers ought to be able to utilise covering space structures to efficiently solve HiddenCover problems.

arXiv:quant-ph/0403127•Mar 17, 2004•1 citation
We demonstrate that continuous time quantum walks on several types of branching graphs, including graphs with loops, are identical to quantum walks on simpler linear chain graphs. We also show graph types for which such equivalence does not exist. Several instructive examples are discussed, and a general approach to analyze more complex branching graphs is formulated. It is further illustrated with a return quantum walk solution for a cube graph with adjustable complex hopping amplitudes.
In the present paper, we study the continuous-time quantum walk on quotient graphs. On such graphs, there is a straightforward reduction of the problem to a subspace that can be considerably smaller than the original one. Along the lines of reductions, by using the idea of calculation of the probability amplitudes for continuous-time quantum walk in terms of the spectral distribution associated with the adjacency matrix of graphs [Jafarizadeh and Salimi (Ann. Phys.322 (2007))], we show that the continuous-time quantum walk on original graph Γ induces a continuous-time quantum walk on quotient graph ΓH. Finally, for example, we investigate the continuous-time quantum walk on some quotient Cayley graphs.
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.
Continuous-time quantum walks (CTQWs) on dynamic graphs realize quantum gates as sequences of time-evolving graph Hamiltonians, but naive constructions produce long sequences with redundancy. Simplification rules, which are graph rewrite rules that shorten a dynamic graph sequence while preserving the unitary it implements, are the CTQW analogue of circuit identities in the gate model. In this work, we give CTQW realizations of the standard single-qubit gates and introduce new graph rewrite rules. We demonstrate the simplification rules through worked circuit reductions and outline their use as transpilation primitives for converting between the circuit model and the dynamic graph framework.
I obtain the dynamics of the continuous time quantum walk on a d-dimensional lattice, with periodic boundary conditions, as an appropriate limit of the dynamics of the discrete time quantum walk on the same lattice. This extends the main result of [8] which proved this limit for the case of the quantum walk on the infinite line and the quantum cellular automaton proposed in [1]. By highlighting the main features of the limiting procedure, I then extend it to general graphs. For a given discrete time quantum walk on a general graph, I single out the type of continuous dynamics (Hamiltonians) that can be obtained as a limit of the discrete time dynamics. 1 Generalities on quantum walks; continuous and discrete time Consider a graph G: = {V, E} with a set of vertices V of cardinality N and a set of edges E. We assume G to be undirected and without self-loops. In the interval of time ∆t a certain fraction γ∆t of a quantity pj leaves the location of the j-th vertex to move to a neighboring vertex k. The quantity pj, j = 1,..., N, at time t + ∆t is given by 1 pj(t + ∆t) = pj(t) − deg (j)γ∆tpj(t) + ∑ (k,j)∈E γ∆tpk(t). (1) Define ⃗p the N−vector whose components are the pj’s quantities, and L the Laplacian matrix defined by Ljj = − deg (j), Ljk = 1 if (j, k) ∈ E and Ljk = 0 otherwise. L is
In this paper we isolate the combinatorial property responsible (at least in part) for the computational speedups recently observed in some quantum walk algorithms. We find that continuous-time quantum walks can exploit the covering space property of certain graphs. We demonstrate that a quantum walk on a graph Y which covers a smaller graph X can be equivalent to a quantum walk on the smaller graph X. This equivalence occurs only when the walk begins on certain initial states, fibre-constant states, which respect the graph covering space structure. We illustrate these observations with walks on Cayley graphs; we show that walks on fibre-constant initial states for Cayley graphs are equivalent to walks on the induced Schreier graph. We also consider the problem of constructing efficient gate sequences simulating the time evolution of a continuous-time quantum walk. For the case of the walk on the m-torus graph T^m on 2^n vertices we construct a gate sequence which uses O(\poly(n)) gates which is independent of the time t the walk is simulated for (and so the sequence can simulate the walk for exponential times). We argue that there exists a wide class of nontrivial operators based on quantum walks on graphs which can be measured efficiently. We introduce a new general class of computational problems, HiddenCover, which includes a variant of the general hidden subgroup problem as a subclass. We argue that quantum computers ought to be able to utilise covering space structures to efficiently solve HiddenCover problems.