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:1510.08657 · Nature Communications

Efficient quantum walk on a quantum processor

Xiaogang Qiang, T. Loke, Ashley Montanaro, Kanin Aungskunsiri, Xiaoqi Zhou, Jeremy L. O’Brien, Jingbo B. Wang, Jonathan C. F. Matthews

Published May 5, 2016• 108 citations
arXivHTMLPDF

Abstract

The random walk formalism is used across a wide range of applications, from modelling share prices to predicting population genetics. Likewise, quantum walks have shown much potential as a framework for developing new quantum algorithms. Here we present explicit efficient quantum circuits for implementing continuous-time quantum walks on the circulant class of graphs. These circuits allow us to sample from the output probability distributions of quantum walks on circulant graphs efficiently. We also show that solving the same sampling problem for arbitrary circulant quantum circuits is intractable for a classical computer, assuming conjectures from computational complexity theory. This is a new link between continuous-time quantum walks and computational complexity theory and it indicates a family of tasks that could ultimately demonstrate quantum supremacy over classical computers. As a proof of principle, we experimentally implement the proposed quantum circuit on an example circulant graph using a two-qubit photonics quantum processor.

DOI
10.1038/ncomms11511

Related papers

Efficient quantum circuits for continuous-time quantum walks on composite graphs

T Loke, J B Wang

Abstract In this paper, we investigate the simulation of continuous-time quantum walks on specific classes of graphs, for which it is possible to fast-forward the time-evolution operator to achieve constant-time simulation complexity and to perform the simulation exactly, i.e. ϵ = 0 , while maintaining p o l y ( l o g ( n ) ) efficiency. In particular, we discuss two classes of composite graphs, commuting graphs and Cartesian product of graphs, that contain classes of graphs which can be simulated in this fashion. This allows us to identify new families of graphs that we can efficiently simulate in a quantum circuit framework, providing practical and explicit means to explore quantum-walk based algorithms in laboratories.

arXiv:1610.07107•Journal of Physics A Mathematical and Theoretical•Jan 6, 2017•19 citations

Quantum Walks in a Superconducting Quantum Computer

Jaime E. Santos, Bruno Chagas, Rodrigo Chaves

Quantum Walks are among the most widely used techniques with which we can construct new quantum algorithms. This paper aims to outline how to construct a circuit for the continuous-time quantum walk (CTQW) over circulant graphs using the Quantum Fourier Transform (QFT) due to the spectral properties of those graphs. Furthermore, we examine how the Approximate Quantum Fourier Transform (AQFT) allows us to shorten the size of the circuit by reducing the number of controlled rotation gates. The contributions of this paper consist of the development of a general circuit implementation of the CTQW for an important class of graphs that does not scale up with time, and the study of the cases where the AQFT decreases the error by controlling the approximation. Finally, we provide a statistical analysis for several circulant graphs, running experiments in a IBM's superconducting quantum computer, and we also explore the pretty good state transfer (PGST) for some graphs.

•Aug 16, 2021•2 citations

Quantum circuit model for continuous-time quantum walks on random graphs

Sabyasachi Chakraborty, Rohit Sarma Sarkar, Sonjoy Majumder +1 more

Quantum-circuit implementations of continuous-time quantum walks (CTQWs) can provide an efficient route to model graph-based algorithms. However, constructing circuits that faithfully reproduce CTQW dynamics across arbitrary graphs remains a major challenge. In this work, we introduce a Laplacian partitioning algorithm (LPA) that enables an efficient and scalable quantum-circuit realization of CTQWs on random graphs. A common algorithm to simulate a general graph (of size $N = 2^n$ for $n$ qubits) on a quantum circuit is based on Pauli decomposition of the graph Hamiltonian, which can yield $O(4^n)$ terms, and require $O(N^2\log N)$ time for coefficient computation. In contrast, our LPA uses $O(2^n)$ terms, in $O(N^2)$ time. Our circuit provides a graph-agnostic framework for CTQWs, implemented via a Trotter-Suzuki product formula and confirming error scaling consistent with theoretical Trotter error bounds. To further test the circuit performance, we study the localization behavior of the CTQW. In our case, localization originates from Laplacian spectral degeneracies rather than disorder (Anderson-type), and our circuit faithfully reproduces these localization phenomena and spectral structure for a random graphs with high accuracy.

arXiv:2510.14905•Physical Review A•Feb 11, 2026•1 citation

Efficient Circuit Implementations of Continuous-Time Quantum Walks for Quantum Search

Renato Portugal, Jalil Khatibi Moqadam

Quantum walks are a powerful framework for simulating complex quantum systems and designing quantum algorithms, particularly for spatial search on graphs, where the goal is to find a marked vertex efficiently. In this work, we present efficient quantum circuits that implement the evolution operator of continuous-time quantum-walk-based search algorithms for three graph families: complete graphs, complete bipartite graphs, and hypercubes. For complete and complete bipartite graphs, our circuits exactly implement the evolution operator. For hypercubes, we propose an approximate implementation that closely matches the exact evolution operator as the number of vertices increases. Our Qiskit simulations demonstrate that even for low-dimensional hypercubes, the algorithm effectively identifies the marked vertex. Furthermore, the approximate implementation developed for hypercubes can be extended to a broad class of graphs, enabling efficient quantum search in scenarios where exact implementations are impractical.

arXiv:2212.08889•Entropy•Apr 23, 2025•1 citation

A matching decomposition algorithm for simulating quantum walk Hamiltonians

Mostafa Atallah, Alvin Gonzales, Daniel Dilley +3 more

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.

quant-pharXiv:2601.11418•Jan 19, 2026•0 citations

Experimental implementation of a continuous-time quantum random walk on a solid-state quantum information processor*

Maimaitiyiming Tusun, Yang Wu, Wenquan Liu +2 more

There are some problems that quantum computers seem to be exponentially faster than classical computers, like factoring large numbers, machine learning, and simulation of quantum systems. Constructing an appropriate quantum algorithm becomes more important for solving these specific problems. In principle, any quantum algorithm can recast by a quantum random walk algorithm. Although quantum random walk with a few qubits has been implemented in a variety of systems, the experimental demonstration of solid-state quantum random walk remains elusive. Here we report the experimental implementation of the quantum continuous-time random walk algorithm by a two-qubit quantum processor in a nitrogen–vacancy center in diamond. We found that quantum random walk on a circle does not converge to any stationary distribution and exhibit a reversible property. Our results represent a further investigation of quantum walking dynamics in solid spin platforms, may also lead to other practical applications by the use of quantum continuous-time random walk for quantum algorithm design and quantum coherence transport.

Chinese Physics B•Sep 18, 2019•1 citation
The random walk formalism is used across a wide range of applications, from modelling share prices to predicting population genetics. Likewise, quantum walks have shown much potential as a framework for developing new quantum algorithms. Here we present explicit efficient quantum circuits for implementing continuous-time quantum walks on the circulant class of graphs. These circuits allow us to sample from the output probability distributions of quantum walks on circulant graphs efficiently. We also show that solving the same sampling problem for arbitrary circulant quantum circuits is intractable for a classical computer, assuming conjectures from computational complexity theory. This is a new link between continuous-time quantum walks and computational complexity theory and it indicates a family of tasks that could ultimately demonstrate quantum supremacy over classical computers. As a proof of principle, we experimentally implement the proposed quantum circuit on an example circulant graph using a two-qubit photonics quantum processor.
Abstract In this paper, we investigate the simulation of continuous-time quantum walks on specific classes of graphs, for which it is possible to fast-forward the time-evolution operator to achieve constant-time simulation complexity and to perform the simulation exactly, i.e. ϵ = 0 , while maintaining p o l y ( l o g ( n ) ) efficiency. In particular, we discuss two classes of composite graphs, commuting graphs and Cartesian product of graphs, that contain classes of graphs which can be simulated in this fashion. This allows us to identify new families of graphs that we can efficiently simulate in a quantum circuit framework, providing practical and explicit means to explore quantum-walk based algorithms in laboratories.
Quantum Walks are among the most widely used techniques with which we can construct new quantum algorithms. This paper aims to outline how to construct a circuit for the continuous-time quantum walk (CTQW) over circulant graphs using the Quantum Fourier Transform (QFT) due to the spectral properties of those graphs. Furthermore, we examine how the Approximate Quantum Fourier Transform (AQFT) allows us to shorten the size of the circuit by reducing the number of controlled rotation gates. The contributions of this paper consist of the development of a general circuit implementation of the CTQW for an important class of graphs that does not scale up with time, and the study of the cases where the AQFT decreases the error by controlling the approximation. Finally, we provide a statistical analysis for several circulant graphs, running experiments in a IBM's superconducting quantum computer, and we also explore the pretty good state transfer (PGST) for some graphs.
Quantum-circuit implementations of continuous-time quantum walks (CTQWs) can provide an efficient route to model graph-based algorithms. However, constructing circuits that faithfully reproduce CTQW dynamics across arbitrary graphs remains a major challenge. In this work, we introduce a Laplacian partitioning algorithm (LPA) that enables an efficient and scalable quantum-circuit realization of CTQWs on random graphs. A common algorithm to simulate a general graph (of size N=2nN = 2^nN=2n for nnn qubits) on a quantum circuit is based on Pauli decomposition of the graph Hamiltonian, which can yield O(4n)O(4^n)O(4n) terms, and require O(N2log⁡N)O(N^2\log N)O(N2logN) time for coefficient computation. In contrast, our LPA uses O(2n)O(2^n)O(2n) terms, in O(N2)O(N^2)O(N2) time. Our circuit provides a graph-agnostic framework for CTQWs, implemented via a Trotter-Suzuki product formula and confirming error scaling consistent with theoretical Trotter error bounds. To further test the circuit performance, we study the localization behavior of the CTQW. In our case, localization originates from Laplacian spectral degeneracies rather than disorder (Anderson-type), and our circuit faithfully reproduces these localization phenomena and spectral structure for a random graphs with high accuracy.
Quantum walks are a powerful framework for simulating complex quantum systems and designing quantum algorithms, particularly for spatial search on graphs, where the goal is to find a marked vertex efficiently. In this work, we present efficient quantum circuits that implement the evolution operator of continuous-time quantum-walk-based search algorithms for three graph families: complete graphs, complete bipartite graphs, and hypercubes. For complete and complete bipartite graphs, our circuits exactly implement the evolution operator. For hypercubes, we propose an approximate implementation that closely matches the exact evolution operator as the number of vertices increases. Our Qiskit simulations demonstrate that even for low-dimensional hypercubes, the algorithm effectively identifies the marked vertex. Furthermore, the approximate implementation developed for hypercubes can be extended to a broad class of graphs, enabling efficient quantum search in scenarios where exact implementations are impractical.
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.
There are some problems that quantum computers seem to be exponentially faster than classical computers, like factoring large numbers, machine learning, and simulation of quantum systems. Constructing an appropriate quantum algorithm becomes more important for solving these specific problems. In principle, any quantum algorithm can recast by a quantum random walk algorithm. Although quantum random walk with a few qubits has been implemented in a variety of systems, the experimental demonstration of solid-state quantum random walk remains elusive. Here we report the experimental implementation of the quantum continuous-time random walk algorithm by a two-qubit quantum processor in a nitrogen–vacancy center in diamond. We found that quantum random walk on a circle does not converge to any stationary distribution and exhibit a reversible property. Our results represent a further investigation of quantum walking dynamics in solid spin platforms, may also lead to other practical applications by the use of quantum continuous-time random walk for quantum algorithm design and quantum coherence transport.