Cyclic Quantum Walks: Dynamics & Topology

• Cyclic Quantum Walks are discrete-time quantum walks defined on finite cycle graphs that combine coin operations with cyclic shifts to produce periodic recurrences and rich dynamics.
• They serve as versatile platforms for exploring quantum entanglement, disorder resilience, and topological phase transitions with practical implementations in photonic and circuit-based systems.
• CQWs enable controlled quantum transport and interference, providing a framework for engineering hybrid entanglement and simulating classical random walks through time-dependent coin protocols.

Searching arXiv for recent and foundational CQW papers to ground the article. Cyclic quantum walks (CQWs) are discrete-time quantum walks defined on finite cycle graphs with periodic boundary conditions, typically implemented as coined walks on a ring of $N$ sites. In their standard form, the Hilbert space factorizes into a position space on the cycle and a finite-dimensional coin space, and the one-step dynamics consists of a coin operation followed by a conditional shift around the ring. Within this seemingly minimal geometry, CQWs support a broad range of phenomena: exact recurrences and periodic limiting distributions, maximally entangled single-particle states, disorder-induced resilience and revival, flat bands and topological phase transitions, robust edge states on finite rings, efficient quantum-circuit realizations, and programmable photonic implementations on cyclic, cylindrical, and toroidal lattices (Panda et al., 2024, Panda et al., 2023, Panda et al., 23 Jul 2025, D'Errico et al., 23 Jun 2025).

1. Formal definition and principal variants

A standard coined CQW on an $N$-cycle acts on a composite Hilbert space $\mathcal{H} = \mathcal{H}_c \otimes \mathcal{H}_p$, where the coin space is usually two-dimensional and the position space is spanned by the cycle vertices $\{|x\rangle : x=0,1,\dots,N-1\}$. A single time step is typically written as

$U = S(C \otimes I_p),$

where $C$ is a coin unitary and $S$ is a conditional shift implementing motion clockwise or anticlockwise modulo $N$ (Panda et al., 2024, Slimen et al., 2020, Panda et al., 2023).

For two-state walks, a common shift operator is

$$S = \sum_{x=0}^{N-1}\left(|0\rangle\langle 0|\otimes |x+1 \bmod N\rangle\langle x| + |1\rangle\langle 1|\otimes |x-1 \bmod N\rangle\langle x|\right),$$

or an equivalent convention with the two coin states associated to opposite directions (Slimen et al., 2020, Panda et al., 2020, Rangel et al., 2024). The position probabilities at time $t$ are obtained from $N$0 by summing over coin states (Slimen et al., 2020).

Several CQW generalizations recur in the literature. One is the discrete-time coined walk on an $N$1-cycle with a one-parameter coin

$N$2

which includes the Hadamard coin at $N$3 (Panda et al., 2024). Another uses a general $N$4 or $N$5 coin with amplitude and phase parameters, allowing fine control of spectral degeneracies and quasienergy bands (Rangel et al., 2024, Panda et al., 2020, Panda et al., 2023).

A distinct branch replaces the conventional single-shift step by more structured protocols. The step-dependent cyclic quantum walk of the topological literature uses a coin $N$6 and a conditional shift $N$7, producing momentum-space blocks

$N$8

with $N$9 acting as a step-dependence parameter (Panda et al., 23 Jul 2025). The single-coin split-step cyclic quantum walk (SCSS-CQW) employs

$\mathcal{H} = \mathcal{H}_c \otimes \mathcal{H}_p$0

so that one and the same coin is applied twice within each step (Panda et al., 8 Mar 2026).

Other variants enlarge the internal space. “Lively quantum walks on cycles” introduce a qutrit coin and a long-range branch of jump length $\mathcal{H} = \mathcal{H}_c \otimes \mathcal{H}_p$1, with shift

$\mathcal{H} = \mathcal{H}_c \otimes \mathcal{H}_p$2

thereby embedding long-range motion directly into the walk definition (Sadowski et al., 2015). The Möbius quantum walk adds a separate rotation space and a coin-conditioned rotation per step, introducing a Möbius factor $\mathcal{H} = \mathcal{H}_c \otimes \mathcal{H}_p$3 that modifies spectral degeneracies and limiting distributions (Moradi et al., 2017). In Szegedy’s formalism, CQWs arise when the columns of a transition matrix are related by cyclic permutations, yielding efficient quantum circuits for directed and weighted cyclic Markov chains (Loke et al., 2016).

This diversity suggests that “CQW” is best understood as a family of finite-ring quantum-walk constructions rather than a single model. What unifies them is the cyclic geometry and the resulting discrete quasi-momentum structure.

2. Fourier representation, spectra, and recurrence structure

The discrete Fourier basis on the cycle,

$\mathcal{H} = \mathcal{H}_c \otimes \mathcal{H}_p$4

or its equivalent discrete form $\mathcal{H} = \mathcal{H}_c \otimes \mathcal{H}_p$5, block-diagonalizes the walk into $\mathcal{H} = \mathcal{H}_c \otimes \mathcal{H}_p$6 coin-sector problems for two-state CQWs (Panda et al., 2024, Slimen et al., 2020, Panda et al., 2023). In the clean unit-jump case one obtains

$\mathcal{H} = \mathcal{H}_c \otimes \mathcal{H}_p$7

which yields bounded group velocities and a linear physical time cone (Panda et al., 2024).

For the coin family $\mathcal{H} = \mathcal{H}_c \otimes \mathcal{H}_p$8, one finds

$\mathcal{H} = \mathcal{H}_c \otimes \mathcal{H}_p$9

up to sign convention, and $\{|x\rangle : x=0,1,\dots,N-1\}$0, consistent with ballistic spreading on the clean cycle (Panda et al., 2024). In the step-dependent topological CQW, the quasienergies are

$\{|x\rangle : x=0,1,\dots,N-1\}$1

with band closings determined by $\{|x\rangle : x=0,1,\dots,N-1\}$2 (Panda et al., 23 Jul 2025). In the SCSS-CQW, the split-step dispersion becomes

$\{|x\rangle : x=0,1,\dots,N-1\}$3

giving a different route to flat bands and topological transitions (Panda et al., 8 Mar 2026).

Finite cyclic geometry strongly constrains recurrence. In the clean Hadamard CQW, maximally entangled single-particle states recur with period $\{|x\rangle : x=0,1,\dots,N-1\}$4 on $\{|x\rangle : x=0,1,\dots,N-1\}$5 and $\{|x\rangle : x=0,1,\dots,N-1\}$6 cycles (Panda et al., 2024, Panda et al., 2023). In “order from chaos” constructions, periodicity is defined by the exact condition $\{|x\rangle : x=0,1,\dots,N-1\}$7, equivalently by requiring all eigenvalues to be roots of unity (Panda et al., 2020). For the $\{|x\rangle : x=0,1,\dots,N-1\}$8-cycle with coin

$\{|x\rangle : x=0,1,\dots,N-1\}$9

the quasi-momentum blocks are $U = S(C \otimes I_p),$0, and periodicity reduces to rational commensurability of the associated eigenphases (Panda et al., 2020).

The long-range qutrit CQW has its own spectral periodicity mechanism. In that model, degeneracies occur precisely when $U = S(C \otimes I_p),$1, and the time-averaged limiting distribution has period $U = S(C \otimes I_p),$2 whenever $U = S(C \otimes I_p),$3 (Sadowski et al., 2015). In the Möbius quantum walk, the parameter $U = S(C \otimes I_p),$4 controls degeneracy conditions of the form

$U = S(C \otimes I_p),$5

and the limiting distribution is uniform for all $U = S(C \otimes I_p),$6 whenever $U = S(C \otimes I_p),$7 (Moradi et al., 2017).

These results establish a general pattern: CQW recurrence is not a generic feature of finite unitarity alone, but a consequence of arithmetic commensurability across the discrete $U = S(C \otimes I_p),$8-sectors. This suggests why parity, cycle size, and coin phases repeatedly appear as decisive parameters.

3. Entanglement generation and single-particle structure

A major recent direction treats CQWs as generators of single-particle entanglement (SPE) between the coin and position degrees of freedom. For a pure state $U = S(C \otimes I_p),$9, the coin-reduced state is $C$0, and SPE is quantified by the von Neumann entropy

$C$1

For a qubit coin, $C$2 corresponds to a maximally entangled single-particle state (MESPS) (Panda et al., 2024, Panda et al., 2023).

A central result is that, for the clean Hadamard CQW on any $C$3-cycle, the state at $C$4 is a single-particle Bell state for arbitrary $C$5 when $C$6, so that $C$7 (Panda et al., 2024). More generally, with a balanced coin $C$8, if the initial phase satisfies

$C$9

then the one-step CQW generates a MESPS at $S$0 for any cycle size $S$1 and any $S$2 (Panda et al., 2023).

On $S$3 and $S$4, recurrent MESPS can be produced using a single fixed coin applied at every step. For the Hadamard coin on $S$5 with $S$6, MESPS occur at $S$7, i.e. $S$8, while on $S$9 the same choice yields MESPS at $N$0 (Panda et al., 2023). The Fourier coin and other balanced coins can produce analogous recurrent entanglement patterns, including periods $N$1, $N$2, and $N$3, depending on the coin phases (Panda et al., 2023).

The explicit $N$4 Hadamard evolution illustrates the mechanism. Starting from a localized state with $N$5, the one-step state is

$N$6

with orthogonal positions carrying the two coin components. The position trace therefore removes cross terms, giving $N$7 and hence $N$8 (Panda et al., 2023). At later times, entanglement oscillates: $N$9, $$S = \sum_{x=0}^{N-1}\left(|0\rangle\langle 0|\otimes |x+1 \bmod N\rangle\langle x| + |1\rangle\langle 1|\otimes |x-1 \bmod N\rangle\langle x|\right),$$0, and $$S = \sum_{x=0}^{N-1}\left(|0\rangle\langle 0|\otimes |x+1 \bmod N\rangle\langle x| + |1\rangle\langle 1|\otimes |x-1 \bmod N\rangle\langle x|\right),$$1 (Panda et al., 2023).

The same work also identifies effective-single and two-coin schedules on $$S = \sum_{x=0}^{N-1}\left(|0\rangle\langle 0|\otimes |x+1 \bmod N\rangle\langle x| + |1\rangle\langle 1|\otimes |x-1 \bmod N\rangle\langle x|\right),$$2 that produce recurrent MESPS when a strictly single-coin ordered walk is unavailable. Examples include $$S = \sum_{x=0}^{N-1}\left(|0\rangle\langle 0|\otimes |x+1 \bmod N\rangle\langle x| + |1\rangle\langle 1|\otimes |x-1 \bmod N\rangle\langle x|\right),$$3, $$S = \sum_{x=0}^{N-1}\left(|0\rangle\langle 0|\otimes |x+1 \bmod N\rangle\langle x| + |1\rangle\langle 1|\otimes |x-1 \bmod N\rangle\langle x|\right),$$4, $$S = \sum_{x=0}^{N-1}\left(|0\rangle\langle 0|\otimes |x+1 \bmod N\rangle\langle x| + |1\rangle\langle 1|\otimes |x-1 \bmod N\rangle\langle x|\right),$$5, and $$S = \sum_{x=0}^{N-1}\left(|0\rangle\langle 0|\otimes |x+1 \bmod N\rangle\langle x| + |1\rangle\langle 1|\otimes |x-1 \bmod N\rangle\langle x|\right),$$6, with MESPS periods $$S = \sum_{x=0}^{N-1}\left(|0\rangle\langle 0|\otimes |x+1 \bmod N\rangle\langle x| + |1\rangle\langle 1|\otimes |x-1 \bmod N\rangle\langle x|\right),$$7, $$S = \sum_{x=0}^{N-1}\left(|0\rangle\langle 0|\otimes |x+1 \bmod N\rangle\langle x| + |1\rangle\langle 1|\otimes |x-1 \bmod N\rangle\langle x|\right),$$8, $$S = \sum_{x=0}^{N-1}\left(|0\rangle\langle 0|\otimes |x+1 \bmod N\rangle\langle x| + |1\rangle\langle 1|\otimes |x-1 \bmod N\rangle\langle x|\right),$$9, $t$0, and $t$1 depending on $t$2 and schedule (Panda et al., 2023).

A plausible implication is that CQWs provide a controlled route to hybrid entanglement engineering in bounded Hilbert spaces, with arithmetic recurrence replacing the asymptotic or many-body mechanisms common in larger systems.

4. Disorder, noise, resilience, and dynamical transitions

Disorder in CQWs has been studied in several distinct senses: phase disorder, coin disorder, position disorder, and static site noise. These perturbations are not equivalent.

In the disorder study on odd and even cyclic graphs, phase disorder is introduced through a phase-decorated shift, coin disorder through random site- or time-dependent coin parameters, and position disorder through a random jump length $t$3 drawn from a Poisson distribution with mean $t$4 (Panda et al., 2024). The principal findings are sharply differentiated.

Phase disorder, whether static or dynamic, leaves the MESPS at $t$5 exactly intact for any initial state. The evolved state picks up local phases on distinct positions, but tracing out position yields

$t$6

independent of $t$7, $t$8, and the disorder variable (Panda et al., 2024). Coin disorder shows a closely related but more restricted immunity: for the phase-symmetric initial coin states

$t$9

MESPS at $N$00 remain exact for any static or dynamic coin disorder strength (Panda et al., 2024).

Beyond $N$01, small phase or coin disorder causes only insignificant reduction of SPE at early times; numerically, $N$02 or $N$03 still show resilience on $N$04 (Panda et al., 2024). Moderate or strong disorder can even enhance entanglement at specific times, and all three disorder types can revive SPE from zero when the clean walk is separable, for example at $N$05 on $N$06 (Panda et al., 2024).

Position disorder is qualitatively different. Randomizing the jump length modifies the shift operator itself, breaks odd-even parity on even cycles, distorts the physical time cone, destroys clean recurrences, and makes SPE more vulnerable (Panda et al., 2024). Yet its long-time behavior is unexpectedly robust in another sense: $N$07 saturates to a fixed value at large $N$08, irrespective of $N$09, although the limiting value depends on $N$10 (Panda et al., 2024). This combination of short-time vulnerability and long-time stabilization is one of the most distinctive results in the current CQW disorder literature.

A related but separate noise analysis studies static site-phase noise in a homogeneous cyclic graph using a three-parameter $N$11 coin (Rangel et al., 2024). There the noisy step is

$N$12

with $N$13 and $N$14 uniformly distributed in $N$15. The mean squared displacement is fit as $N$16, revealing a progression from nearly ballistic behavior at $N$17 with $N$18, to super-diffusive behavior at $N$19 with $N$20, to diffusive behavior near $N$21 with $N$22, and to sub-diffusive behavior at $N$23 with $N$24 for $N$25 and Hadamard coin parameters $N$26 (Rangel et al., 2024). On finite cycles the mean squared displacement saturates when the number of steps exceeds the graph size by about an order of magnitude (Rangel et al., 2024).

That work also reports a strong empirical correspondence between the average eigenstate participation ratio and mean-squared-displacement behavior, with low participation ratio correlating with localization and high participation ratio with delocalization (Rangel et al., 2024). This suggests that CQW transport under quenched disorder can often be diagnosed spectrally rather than through long-time simulation.

5. Topological CQWs, flat bands, and edge states

Recent work places CQWs within Floquet topological band theory. In step-dependent CQWs on cyclic graphs, the effective Hamiltonian is defined by $N$27 with

$N$28

and quasienergies

$N$29

This framework yields gapped and gapless phases, Dirac-cone-like band closings, topological flat bands, and edge states, all without split-step or split-coin protocols (Panda et al., 23 Jul 2025).

The flat-band condition is

$N$30

which gives $N$31, independent of $N$32, for any $N$33 (Panda et al., 23 Jul 2025). Rotationally symmetric flat bands, however, occur only when $N$34, because the discrete momenta must include values with $N$35 exactly (Panda et al., 23 Jul 2025). Even and odd cycles therefore have qualitatively different spectral possibilities: even cycles admit $N$36 and richer gap-closing structure, while odd cycles do not (Panda et al., 23 Jul 2025).

The corresponding topological invariant is a discretized Zak phase or winding number. For the $N$37-cycle, the winding number is

$N$38

and the Hadamard case $N$39 gives $N$40 and $N$41 (Panda et al., 23 Jul 2025). Interfaces between regions with different winding numbers support localized edge states. For example, on $N$42 with $N$43, choosing $N$44 at site $N$45 and $N$46 elsewhere creates a persistent edge state localized at the interface site (Panda et al., 23 Jul 2025). These edge states remain localized under static disorder $N$47, dynamic disorder $N$48, and phase-preserving perturbations that do not change the winding sector (Panda et al., 23 Jul 2025).

The SCSS-CQW extends this program in a different topological direction. Its quasienergies satisfy

$N$49

with flat $N$50-bands at

$N$51

for which $N$52 and the bands are pinned at quasienergy $N$53 (Panda et al., 8 Mar 2026). More strikingly, the protocol yields fractional winding numbers $N$54 and Zak phases $N$55, rather than integer invariants. For $N$56 and $N$57, the winding is numerically $N$58 for $N$59, $N$60 for $N$61, and exactly $N$62 for $N$63; for $N$64 and $N$65, one obtains the opposite fractional sector $N$66 (Panda et al., 8 Mar 2026). Domain walls between these fractional sectors support edge-localized bound states on finite rings, robust against dynamic and static coin disorder as well as phase-preserving perturbations (Panda et al., 8 Mar 2026).

A common misconception is that finite cyclic geometry is too small or too symmetric to support nontrivial topology. The topological CQW results indicate the opposite: the ring geometry discretizes momentum but does not trivialize the Floquet band structure. Instead, it renders parity, cycle divisibility, and finite-size sampling central to the topological classification.

6. Implementations, circuit constructions, and related extensions

CQWs have been analyzed both as abstract dynamical models and as concrete hardware primitives.

A universal-gate implementation of the discrete circular walk on $N$67 sites uses one coin qubit and an $N$68-qubit position register. One step applies a coin gate followed by a coin-controlled modular decrement or increment on the position register, implemented through cascades of multi-controlled NOT gates (Slimen et al., 2020). In the $N$69 realization on IBM Q London, corresponding to $N$70, the full one-step walker circuit used $N$71 CNOTs, and the average fidelity over all $N$72 initial position states was $N$73 (Slimen et al., 2020). The paper attributes the strong degradation to realistic two-qubit gate errors; a naive estimate using $N$74 gives $N$75, above the measured fidelity, indicating additional contributions from readout, decoherence, and mapping overhead (Slimen et al., 2020). This result is less a demonstration of scalable CQWs than a quantitative diagnosis of circuit-depth limitations in NISQ superconducting hardware.

Szegedy-type CQWs offer a different circuit model. When the columns of a transition matrix are related by cyclic permutations, the corresponding Szegedy walk can be compiled efficiently using a reflection diagonalization strategy. For circulant transition matrices, the necessary controlled shifts and state-preparation blocks can be implemented with $N$76 resources, independently of sparsity (Loke et al., 2016). This extends CQW circuit design to directed and weighted cycles, not only conventional coined walks.

Programmable photonic CQWs provide a more scalable experimental route. A reciprocal-space platform based on spatial light modulators implements arbitrary translationally invariant unitaries by discrete sampling of the Brillouin zone, thereby enforcing cyclic, cylindrical, or toroidal boundary conditions directly in momentum space (D'Errico et al., 23 Jun 2025). In one dimension, the lattice size is set by the number of sampled $N$77-points; in two dimensions, sampling in both directions yields toroidal topologies, while periodicity in one direction and openness in the other yields cylindrical topologies (D'Errico et al., 23 Jun 2025).

That platform realizes up to $N$78 steps in a single synthesized unitary for a $N$79D cycle with $N$80, with partial refocusing at $N$81 and similarity $N$82 between experiment and theory (D'Errico et al., 23 Jun 2025). For $N$83, exact recurrences were engineered with periods $N$84 and $N$85, giving $N$86, while a near-recurrence at $N$87 with the default coin yielded $N$88 (D'Errico et al., 23 Jun 2025). The same platform demonstrates wavepacket breathing, topology-dependent trajectories on a torus, and dimensional reduction of a cylindrical $N$89D walk to an effective $N$90D walk with a high-dimensional coin (D'Errico et al., 23 Jun 2025).

Photonic CQW relevance also appears in the entanglement and disorder studies, which explicitly identify polarization as a coin degree of freedom and time bins, OAM, path, or spatial modes as position encodings (Panda et al., 2024). In those settings, phase disorder can be introduced by phase plates or fast modulators, coin disorder by changing waveplate angles, and position disorder by reconfigurable routing or delay lines (Panda et al., 2024).

Beyond implementation, several nonstandard CQW directions broaden the conceptual scope. The time-dependent-coin construction that matches arbitrary classical random walks on cycles shows that a unitary coined CQW can reproduce any target vertex-distribution sequence at measurement times, provided the local coins are allowed to vary in space and time (Andrade et al., 2021). The percolated-Grover analysis on arbitrary graphs shows that cyclic local shift operators can produce nonstationary asymptotics with periods dividing $N$91 when an edge-3-coloring exists, and that cyclic shifts avoid the trapped states that plague reflecting-shift walks (Mareš et al., 2018). Anyonic CQWs on cycles incorporate braiding with stationary anyons and show mixing behavior on finite rings that resembles standard Hadamard walks more than classical random walks, despite strong decoherence effects on infinite chains (Lehman et al., 2012).

These extensions indicate that CQWs function as a meeting point for finite-graph quantum transport, Floquet topology, hybrid entanglement, programmable optics, and circuit-level quantum simulation. Their unifying theme is not merely “walking on a ring,” but exploiting the ring’s discrete momentum structure as a controlled arena for interference, periodicity, topology, and robustness.