Quantum Random Walks on Graphs

Updated 6 February 2026

Quantum random walks on graphs are quantum analogs of classical walks featuring unitary evolution, interference, and potential entanglement.
They encompass models like discrete-time coined, continuous-time, open, and noisy walks to study transport, localization, and network growth.
These walks enable algorithmic speedups in quantum search, simulation on photonic and circuit devices, and novel approaches to graph isomorphism.

Quantum random walks on graphs are quantum analogs of classical random walks, exhibiting fundamentally different transport and spreading features due to quantum interference, unitary evolution, and the possibility of entanglement. These walks underpin algorithmic speedups, quantum simulations, device architectures, and new perspectives on graph structures.

1. Foundational Models of Quantum Random Walks

Quantum random walks (QRWs) on graphs exist in several principal models characterized by the evolution rule, system-environment coupling, and the Hilbert space structure:

Unitary (Closed) Quantum Walks: Evolutions are fully unitary, involving a coin space (locally at each vertex) and a shift operator on the graph’s edge structure. The walker's state is a superposition over vertex–coin basis states. The discrete-time coined quantum walk is described by

$|\Psi(t+1)\rangle = U|\Psi(t)\rangle, \quad U = S W, \quad W = \bigoplus_v w^{(v)} \in U(d(v)),$

with $S$ mapping coin states to neighbor vertices via local permutations (Andrade et al., 2019). The probability of finding the walker at vertex $v$ is given by summing the amplitudes squared over all coin states (Andrade et al., 2021, Andrade et al., 2019).

Continuous-Time Quantum Walks (CTQW): Generated by a Hermitian graph operator, often the adjacency matrix $A$ or graph Laplacian $L$ , with unitary evolution

$U(t) = e^{-iHt},\ \quad H = -\gamma A\ \text{or}\ -\gamma L$

(Frigerio et al., 2021, Chakraborty et al., 16 Oct 2025, Jnane et al., 2020). The choice of $H$ admits further degrees of freedom including on-site energies and edge-dependent Peierls phases.

Open Quantum Random Walks (OQRW): Evolution is defined by completely positive trace-preserving (CPTP) maps over a "coin+vertex" Hilbert space, realized via sets of Kraus operators $B^i_j$ per directed edge with

$\rho^{(n+1)}_i = \sum_j B^i_j \rho^{(n)}_j (B^i_j)^\dagger,$

and $\rho^{(n)} = \sum_i \rho^{(n)}_i \otimes |i\rangle\langle i|$ (Attal et al., 2014). OQRWs exactly generalize Markovian random walks to the quantum setting and admit a measurement-based unraveling.

Non-Markovian/Noisy QRWs: Discrete-time open quantum walks using environment-induced noise channels (amplitude damping, dephasing, depolarizing) are implemented by Kraus-coin operators, constructed from Weyl operator bases acting on the coin space (Rani et al., 2024). These allow detailed analysis of coherence, fidelity, and non-Markovian memory on arbitrary graphs.

Alternate frameworks include the state diffusion model—every vertex hosts a walker at all times and only swaps occur between neighboring vertices, effecting mixing and entanglement purely by local operations and measurements (III et al., 2024).

2. Quantum-Classical Correspondence and Simulation

A cornerstone of QRW theory is the formal equivalence between the statistics of quantum and classical random walks when time-dependent and inhomogeneous classical dynamics are allowed:

For any discrete-time coined quantum walk on a finite graph, there exists a time-dependent classical random walk with a sequence of stochastic matrices $P(t)$ such that the vertex (marginal) probabilities match those of the quantum walk at every step (Andrade et al., 2019). The explicit construction of $P(t)$ exploits the unitary walker's amplitudes and coin operator:

$P(t)_{v,u} = \frac{\left| \sum_{j} \Psi(u,j,t) w^{(u)}_{j,\sigma(u,v)} \right|^2}{p_Q(u,t)},$

for $p_Q(u,t) > 0$ and $(u,v)\in E$ .

Conversely, for any arbitrary (possibly time-dependent) classical random walk sequence $\pi(t+1) = P(t)\pi(t)$ on a finite graph, a one-parameter family of unitary coined QRWs $(U_t)$ can be constructed so that vertex measurement at time $t$ yields $\pi_v(t)$ at every $t$ (Andrade et al., 2021). This generalizes to non-homogeneous and aperiodic processes.

This correspondence extends to multi-particle and interacting cases (Andrade et al., 2019), with the evolution of composite vertex distributions governed by time-dependent Markov chains on graph products.

3. Quantum Transport, Speedup, and Algorithmic Features

Quantum interference enables transport phenomena and algorithmic speedups unobtainable classically.

Hitting Times and Mixing: On hypercubes $Q_n$ , discrete-time QRWs achieve exponential speedup—reaching the antipode in $T \sim (\pi/2) n$ steps with probability $p \rightarrow 1$ as $n \rightarrow \infty$ (classical walks require $O(n2^n)$ steps) (Mulherkar et al., 2021). For all cubelike graphs, this linear relationship between degree and hitting time is numerically confirmed (Mulherkar et al., 2021).
Continuous–Time Quantum Walk Search: On hierarchical and supervertex graphs, zero-mode delocalization leads to superpolynomial or exponential speedup over classical hitting or traversal times (Balasubramanian et al., 2023). For example, the quantum time to reach the far end of a 1D hierarchical chain is $O(\exp(O(\sqrt{n})))$ , compared to classical $O(\exp(\Theta(n)))$ .
Grid Search and Hybrid Algorithms: On $N \times N$ grids, scattering QRW concentrates probability around marked targets in $O(N)$ steps; combining with classical BFS search in a radius around the observed position yields a polynomial speedup robust to random obstacles (Koch, 2018).
Localization and Spreading: Modifying graph structure (e.g., periodic attachment of stubs to $\mathbb Z$ ) induces localization of quantum walks via emergence of nontrivial eigenstates, dependent on the spectrum of the underlying random walk (Higuchi et al., 2015).
Extreme Event Statistics: Despite unitary dynamics, QRWs on scale-free and regular graphs exhibit power-law extreme-event probabilities in vertex degree, closely matching classical random walk behavior (Vyas et al., 26 Feb 2025).

4. Device Implementations and Photonic Quantum Walks

Quantum walks realize natural experimental platforms, notably in photonics:

Photonic Ring Resonators: Series-coupled PRRs map to graphs where nodes represent half-ring segments. The unitary transfer matrix is explicitly constructed; quantum interference at resonance boosts steady-state transport and reduces hitting times compared to classical analogs. Decoherence (phase averaging) recovers classical dynamics. Device feasibility depends critically on waveguide coupling, loss rates, and phase control (Adão et al., 2022).
Graph Convolution for High-Dimensional Walks: For graphs formed as Kronecker products (e.g., hypercycles, hypercubes), quantum walk dynamics can be exactly preserved on dramatically smaller "convolved" graphs, mapping high-dimensional walks onto weighted lines or lattices. This yields exponential resource savings in quantum/photonic hardware—e.g., simulating a $2^D$ -vertex hypercube with $D+1$ modes (Abramov et al., 21 Jul 2025).
Quantum Circuit Implementations: For Erdős-Rényi random graphs with $N=2^n$ vertices, CTQWs implemented via Trotterized time evolution of $H = -\gamma L$ allow scalable simulation on quantum hardware. Circuit resources scale as $O(n\cdot p 2^n)$ in the average-number of nonzero Laplacian partitions for random graphs with connection probability $p$ (Chakraborty et al., 16 Oct 2025).

5. Quantum Walks, Random Graphs, and Growth Processes

Continuous-time QRW-driven network growth models introduce quantum generalizations of random graph formation:

Starting from any seed, a quantum walker evolves via $U(t)$ , undergoes a random measurement at time $t$ (exponentially distributed), and attaches a new node at the measurement vertex. With two or more walkers, the process generates more clustered graphs. For large exploration times between measurements, the resulting degree distribution is scale-free with exponents $2.5 < \alpha < 3$ (Jnane et al., 2020).
Key parameters governing network structure are the measurement rate $\lambda=1/\tau$ and the number of walkers. For small $\tau$ , "star" structures emerge; for large $\tau$ , scale-free tails become prominent. Quantum interference and the choice of initial wavefunction offer control beyond classical preferential attachment.

6. Multiparticle Quantum Walks and Graph Theoretic Applications

The use of $k$ -particle quantum walks extends distinguishing power in graph isomorphism problems:

k-Particle XY Walks: Evolving in the $k$ -excitation subspace of $(\mathbb C^2)^{\otimes N}$ under the XY Hamiltonian, a $k$ -QW on graph $G$ explores the occupation graph $G^{(k)}$ . For certain hard graph families (CFI graphs), $k$ -QWs with uniform superposition or localized input states can distinguish cases that defeat $k$ -WL classical tests (Kasture et al., 7 Jan 2025).
Strongly Regular Graphs (SRGs): Non-interacting $p=2$ QWs cannot distinguish non-isomorphic SRGs of the same parameter family, whereas $p=3,4$ particles increase distinguishing power, but no fixed $p$ suffices universally (Rudinger et al., 2012). Interacting walks with hard-core repulsion further improve performance.

7. Spectral Theory, Convergence, and Open Quantum Walks

General Hamiltonian Correspondence: There exists an infinite family of quantum Hamiltonians on a given graph matching the off-diagonal transition rates via $|H_{ij}|^2 = -L_{ij}$ and arbitrary diagonal energies and edge phases, physically corresponding to scalar and gauge fields (Frigerio et al., 2021).
Mixing, Stationarity, and Convergence: For OQRWs on finite graphs, CPTP evolution yields a unique stationary state if the open map is irreducible; convergence to this state is exponentially fast, governed by the spectral gap of the channel (Attal et al., 2014). Reduced position distributions converge exactly to the steady-state law dictated by the stationary density operator.
Quantum Diffusion and Entanglement: In state-diffusion models, one walker per vertex is maintained and swaps drive diffusion; superpositions and projective measurements distribute entanglement throughout the network, with convergence and mixing time determined spectrally (III et al., 2024).