Staggered Quantum Walks

• Staggered quantum walks are discrete-time models that partition graph vertices into disjoint cliques, using orthogonal reflections to drive the dynamics.
• The framework unifies several quantum walk formulations, including Szegedy’s and coined walks, and extends to Hamiltonian evolutions for continuous-time analogs.
• Applications range from optimized search algorithms to experimental implementations in superconducting circuits and Rydberg-atom architectures.

Staggered quantum walks (SQWs) are discrete-time quantum walks on graphs in which the local dynamics are defined by a tessellation cover: a collection of partitions of the vertex set into disjoint cliques, called polygons, whose union covers the graph edges. Each tessellation induces an orthogonal reflection, and one walk step is the ordered product of these reflections; in the Hamiltonian variant, the reflections are used as time-independent Hamiltonians through exponentials of the form $e^{i\theta_j H_j}$ (Portugal et al., 2015, Portugal et al., 2016). The model was formalized as a unifying framework containing every instance of Szegedy’s walk and a large class of coined walks, while also admitting genuinely staggered constructions that cannot be cast into Szegedy’s framework (Portugal et al., 2015, Portugal, 2016).

1. Core definition and local operators

Let $\Gamma=(V,E)$ be a finite graph and let the walker live in the Hilbert space

$\mathcal H=\mathrm{span}\{|v\rangle: v\in V\}.$

A tessellation $\mathcal T_i$ is a partition of $V$ into disjoint cliques. If

$$\mathcal T_i=\{C^{(i)}_0,C^{(i)}_1,\dots,C^{(i)}_{m_i-1}\},$$

then each polygon $C^{(i)}_k$ is assigned a normalized vector

$$|\alpha^{(i)}_k\rangle=\sum_{v\in C^{(i)}_k} a^{(i)}_{k,v}|v\rangle, \qquad \sum_v |a^{(i)}_{k,v}|^2=1.$$

Because polygons in the same tessellation do not overlap, these vectors are mutually orthogonal. The associated local reflection is

$$R_i=2\sum_{k=0}^{m_i-1}|\alpha^{(i)}_k\rangle\langle \alpha^{(i)}_k|-I,$$

which is unitary and Hermitian, with $R_i^2=I$. For a tessellation cover with $\Gamma=(V,E)$0 tessellations, one step of the standard staggered walk is

$\Gamma=(V,E)$1

In the frequently studied 2-tessellable case, this reduces to $\Gamma=(V,E)$2 or $\Gamma=(V,E)$3 (Portugal et al., 2015).

The search version is obtained either by partial tessellation or by an additional reflection on marked vertices. For a marked set $\Gamma=(V,E)$4, one may insert

$\Gamma=(V,E)$5

so that, for two tessellations, the search evolution can be written as

$\Gamma=(V,E)$6

The time evolution is $\Gamma=(V,E)$7, and the success probability is the total probability on the marked set (Portugal et al., 2015).

The Hamiltonian extension keeps the same reflections but replaces them by continuous rotations generated by those reflections:

$\Gamma=(V,E)$8

where each $\Gamma=(V,E)$9 is itself Hermitian and unitary. When $\mathcal H=\mathrm{span}\{|v\rangle: v\in V\}.$0 and $\mathcal H=\mathrm{span}\{|v\rangle: v\in V\}.$1, the standard SQW step $\mathcal H=\mathrm{span}\{|v\rangle: v\in V\}.$2 is recovered up to a global phase. The same framework also admits Trotter–Suzuki approximations to evolution under $\mathcal H=\mathrm{span}\{|v\rangle: v\in V\}.$3 (Portugal et al., 2016).

2. Tessellations, clique graphs, and graph-theoretic constraints

The graph-theoretic structure of SQWs is unusually explicit. A connected graph $\mathcal H=\mathrm{span}\{|v\rangle: v\in V\}.$4 is 2-tessellable if and only if its clique graph $\mathcal H=\mathrm{span}\{|v\rangle: v\in V\}.$5 is 2-colorable (Portugal, 2016). More generally, the tessellation number $\mathcal H=\mathrm{span}\{|v\rangle: v\in V\}.$6 is the minimum number of tessellations in a tessellation cover. If $\mathcal H=\mathrm{span}\{|v\rangle: v\in V\}.$7, then

$\mathcal H=\mathrm{span}\{|v\rangle: v\in V\}.$8

where $\mathcal H=\mathrm{span}\{|v\rangle: v\in V\}.$9 is the chromatic number of the clique graph. The upper bound comes from coloring maximal cliques so that intersecting cliques receive different colors, each color class then defining a tessellation (Abreu et al., 2017).

This graph-theoretic viewpoint separates several classes relevant to model reduction. Graphs that are not line graphs cannot embed into Szegedy or coined models. Line graphs of non-bipartite graphs still cannot become Szegedy walks, while line graphs of bipartite graphs may become Szegedy walks precisely in the 2-tessellable case with no edge in the intersection of the two tessellations. A further subclass admitting a perfect matching partition yields coined walks when uniform polygon states are used (Portugal, 2016).

The minimum tessellation problem is algorithmically nontrivial. For any fixed $\mathcal T_i$0, deciding whether a graph is $\mathcal T_i$1-tessellable is NP-complete. In triangle-free graphs, every clique has size at most $\mathcal T_i$2, so each tessellation is exactly a matching; the reduction from $\mathcal T_i$3-edge-coloring makes the hardness statement particularly transparent (Abreu et al., 2017).

Polygon intersections are structurally important. If the intersection $\mathcal T_i$4 of polygons has size $\mathcal T_i$5, new eigenvectors supported entirely on that intersection can appear. In the framework of intersection expansion and reduction, replacing a single vertex by a $\mathcal T_i$6-clique or collapsing a $\mathcal T_i$7-vertex intersection back to a single vertex preserves the coarse-grained dynamics outside the internal intersection modes. In particular, a $\mathcal T_i$8-vertex intersection contributes $\mathcal T_i$9 additional eigenvectors with eigenvalue $V$0, where $V$1 is the number of tessellations in which the intersection participates (Santos, 2018). This enables equivalences between SQWs on different graphs and can convert an SQW not directly included in Szegedy’s model into one that is equivalent after reduction (Santos, 2018).

3. Relations to Szegedy, coined, partition-based, hypergraph, and continuous-time models

The best-known structural result is that any instance of Szegedy’s model is equivalent to an instance of the staggered model. The construction passes through the line graph of a bipartite graph: the $V$2- and $V$3-tessellations are induced by the Szegedy reflection states, and the staggered evolution $V$4 reproduces the Szegedy walk up to relabeling of basis vectors (Portugal et al., 2015). The converse requires a restriction: a two-tessellation SQW is of Szegedy type when each intersection between one polygon from the $V$5-tessellation and one polygon from the $V$6-tessellation consists of exactly one vertex (Portugal et al., 2015).

The same staggered formalism also mediates the relation between coined and Szegedy walks. For flip-flop coined walks, the coin can be represented as an orthogonal reflection on a clique replacing each original vertex, while the shift becomes a reflection on 2-vertex polygons corresponding to matched edge states. Under this construction, coined search, staggered partial-tessellation search, and Szegedy search with sinks are exactly equivalent under the associated graph transformations (Portugal, 2015).

The Hamiltonian formulation broadens these inclusions. The staggered walk with Hamiltonians contains the standard SQW, Szegedy’s model, and an important subset of the coined model. Any flip-flop coined walk $V$7 with a coin $V$8 and a shift $V$9 can be represented on an enlarged graph by

$$\mathcal T_i=\{C^{(i)}_0,C^{(i)}_1,\dots,C^{(i)}_{m_i-1}\},$$0

which is precisely of staggered-Hamiltonian form (Portugal et al., 2016).

The unification extends beyond named walk models. The two-partition model, based on two equivalence-class partitions of the computational basis, is unitarily equivalent to the 2-tessellable staggered model, to the square of the coined walk, and to the corresponding bipartite walk. In that setting, changing formalism does not alter the spectrum or dynamics (Konno et al., 2017). At a still higher level of abstraction, every staggered quantum walk is a strong instance of a generalized hyperwalk on the hypergraph $$\mathcal T_i=\{C^{(i)}_0,C^{(i)}_1,\dots,C^{(i)}_{m_i-1}\},$$1, with identity coin and time-dependent shift operators chosen to match the staggered reflections (Sadowski et al., 2018).

A more selective connection links staggered walks with Hamiltonians to continuous-time quantum walks. On certain Cayley graphs, when the tessellation cover is a factorization and the local Hamiltonians commute, the staggered step can be written as

$$\mathcal T_i=\{C^{(i)}_0,C^{(i)}_1,\dots,C^{(i)}_{m_i-1}\},$$2

so that after $$\mathcal T_i=\{C^{(i)}_0,C^{(i)}_1,\dots,C^{(i)}_{m_i-1}\},$$3 steps one obtains the continuous-time propagator $$\mathcal T_i=\{C^{(i)}_0,C^{(i)}_1,\dots,C^{(i)}_{m_i-1}\},$$4 sampled at $$\mathcal T_i=\{C^{(i)}_0,C^{(i)}_1,\dots,C^{(i)}_{m_i-1}\},$$5, up to a global phase. In this sense the staggered model acts as a discretization of the continuous-time walk on that class of graphs (Coutinho et al., 2017).

4. Spectral theory and eigenstructure

For 2-tessellable SQWs, the spectral problem reduces to a smaller matrix built from tessellation overlaps. If the two reflections are written as

$$\mathcal T_i=\{C^{(i)}_0,C^{(i)}_1,\dots,C^{(i)}_{m_i-1}\},$$6

with $$\mathcal T_i=\{C^{(i)}_0,C^{(i)}_1,\dots,C^{(i)}_{m_i-1}\},$$7 and $$\mathcal T_i=\{C^{(i)}_0,C^{(i)}_1,\dots,C^{(i)}_{m_i-1}\},$$8 isometries whose columns are polygon states, then

$$\mathcal T_i=\{C^{(i)}_0,C^{(i)}_1,\dots,C^{(i)}_{m_i-1}\},$$9

A determinant formula gives

$C^{(i)}_k$0

and equivalently with the $C^{(i)}_k$1 block $C^{(i)}_k$2. Hence the nontrivial spectrum is determined by the eigenvalues $C^{(i)}_k$3 of

$C^{(i)}_k$4

If $C^{(i)}_k$5, then the corresponding eigenvalues of $C^{(i)}_k$6 are $C^{(i)}_k$7, together with $C^{(i)}_k$8 of multiplicity $C^{(i)}_k$9 and $$|\alpha^{(i)}_k\rangle=\sum_{v\in C^{(i)}_k} a^{(i)}_{k,v}|v\rangle, \qquad \sum_v |a^{(i)}_{k,v}|^2=1.$$0 of multiplicity $$|\alpha^{(i)}_k\rangle=\sum_{v\in C^{(i)}_k} a^{(i)}_{k,v}|v\rangle, \qquad \sum_v |a^{(i)}_{k,v}|^2=1.$$1 (Konno et al., 2017).

For staggered search, the same reduction appears in singular-value form. If $$|\alpha^{(i)}_k\rangle=\sum_{v\in C^{(i)}_k} a^{(i)}_{k,v}|v\rangle, \qquad \sum_v |a^{(i)}_{k,v}|^2=1.$$2 and $$|\alpha^{(i)}_k\rangle=\sum_{v\in C^{(i)}_k} a^{(i)}_{k,v}|v\rangle, \qquad \sum_v |a^{(i)}_{k,v}|^2=1.$$3 are the polygon subspaces after marking, then the discriminant matrix

$$|\alpha^{(i)}_k\rangle=\sum_{v\in C^{(i)}_k} a^{(i)}_{k,v}|v\rangle, \qquad \sum_v |a^{(i)}_{k,v}|^2=1.$$4

has singular values $$|\alpha^{(i)}_k\rangle=\sum_{v\in C^{(i)}_k} a^{(i)}_{k,v}|v\rangle, \qquad \sum_v |a^{(i)}_{k,v}|^2=1.$$5, and the search operator has eigenvalues $$|\alpha^{(i)}_k\rangle=\sum_{v\in C^{(i)}_k} a^{(i)}_{k,v}|v\rangle, \qquad \sum_v |a^{(i)}_{k,v}|^2=1.$$6. The $$|\alpha^{(i)}_k\rangle=\sum_{v\in C^{(i)}_k} a^{(i)}_{k,v}|v\rangle, \qquad \sum_v |a^{(i)}_{k,v}|^2=1.$$7 and $$|\alpha^{(i)}_k\rangle=\sum_{v\in C^{(i)}_k} a^{(i)}_{k,v}|v\rangle, \qquad \sum_v |a^{(i)}_{k,v}|^2=1.$$8 sectors are then described by intersections such as $$|\alpha^{(i)}_k\rangle=\sum_{v\in C^{(i)}_k} a^{(i)}_{k,v}|v\rangle, \qquad \sum_v |a^{(i)}_{k,v}|^2=1.$$9, $$R_i=2\sum_{k=0}^{m_i-1}|\alpha^{(i)}_k\rangle\langle \alpha^{(i)}_k|-I,$$0, $$R_i=2\sum_{k=0}^{m_i-1}|\alpha^{(i)}_k\rangle\langle \alpha^{(i)}_k|-I,$$1, and $$R_i=2\sum_{k=0}^{m_i-1}|\alpha^{(i)}_k\rangle\langle \alpha^{(i)}_k|-I,$$2 (Portugal et al., 2015).

A more refined account interprets a 2-tessellable SQW as a quantum Markov chain on the underlying multigraph whose line graph is the original graph. In that language one defines the double discriminant matrix

$$R_i=2\sum_{k=0}^{m_i-1}|\alpha^{(i)}_k\rangle\langle \alpha^{(i)}_k|-I,$$3

and the notion of quantum detailed balance. If quantum detailed balance holds, there is an eigenvector for each fundamental cycle of the underlying multigraph; if it does not hold, two fundamental cycles linked by a path are required to construct the remaining eigenvectors (Higuchi et al., 2018). On the kagome lattice, this program yields explicit eigenvectors supported on 6-cycles in the $$R_i=2\sum_{k=0}^{m_i-1}|\alpha^{(i)}_k\rangle\langle \alpha^{(i)}_k|-I,$$4 sector and explains the associated localization behavior (Higuchi et al., 2018).

5. Search algorithms and asymptotic performance

The staggered model supports a native search framework. A marked set can be encoded either by omitting polygons from one tessellation or by inserting a marking reflection, and the natural analogue of Szegedy’s quantum hitting time is

$$R_i=2\sum_{k=0}^{m_i-1}|\alpha^{(i)}_k\rangle\langle \alpha^{(i)}_k|-I,$$5

This definition measures the time-averaged departure of the walk from the initial state once the marked set has been detected (Portugal et al., 2015).

A central result concerns the $$R_i=2\sum_{k=0}^{m_i-1}|\alpha^{(i)}_k\rangle\langle \alpha^{(i)}_k|-I,$$6 toroidal grid with one marked vertex and cyclic boundary conditions. In the staggered-with-Hamiltonians construction, four tessellations $$R_i=2\sum_{k=0}^{m_i-1}|\alpha^{(i)}_k\rangle\langle \alpha^{(i)}_k|-I,$$7 partition the edges into 2-cliques, each generating a reflection $$R_i=2\sum_{k=0}^{m_i-1}|\alpha^{(i)}_k\rangle\langle \alpha^{(i)}_k|-I,$$8. With $$R_i=2\sum_{k=0}^{m_i-1}|\alpha^{(i)}_k\rangle\langle \alpha^{(i)}_k|-I,$$9 one defines local unitaries $R_i^2=I$0 and the walk step

$R_i^2=I$1

Marking $R_i^2=I$2 by

$R_i^2=I$3

the search propagator is $R_i^2=I$4. Spectral analysis shows that $R_i^2=I$5 acquires two eigenvalues $R_i^2=I$6 with

$R_i^2=I$7

The first maximum occurs at

$R_i^2=I$8

and the peak success probability is $R_i^2=I$9. Amplitude amplification raises the cost to $\Gamma=(V,E)$00 for constant success, whereas Tulsi’s extra-qubit trick or Ambainis post-processing gives constant success in $\Gamma=(V,E)$01 (Portugal et al., 2017).

The angle choice is decisive. Numerical results show that the original coinless staggered walk with $\Gamma=(V,E)$02, or perturbations away from $\Gamma=(V,E)$03, do not produce the same spectral collapse: both the smallest gap of $\Gamma=(V,E)$04 and the special search gap of $\Gamma=(V,E)$05 tend to the nonzero constant $\Gamma=(V,E)$06 as $\Gamma=(V,E)$07, and the search remains as slow as a classical random walk, requiring no fewer than $\Gamma=(V,E)$08 steps (Portugal et al., 2017). This is one of the clearest examples in which the Hamiltonian generalization is not merely formal but algorithmically decisive.

The hexagonal lattice gives a complementary example in which more than two tessellations are essential. Because the edge-chromatic number of the hexagonal lattice is $\Gamma=(V,E)$09, the walk uses three tessellations and a step

$\Gamma=(V,E)$10

Fourier reduction yields $\Gamma=(V,E)$11 blocks with eigenvalues $\Gamma=(V,E)$12. The position standard deviation satisfies $\Gamma=(V,E)$13, and site amplitudes decay as $\Gamma=(V,E)$14, so there is no localization. For spatial search with one marked node and cyclic boundary conditions, choosing $\Gamma=(V,E)$15 makes the small-$\Gamma=(V,E)$16 spectral gap close to zero, giving

$\Gamma=(V,E)$17

with Tulsi’s modification boosting the success probability to $\Gamma=(V,E)$18 without changing the asymptotic runtime (Chagas et al., 2018).

The staggered framework also yields search constructions outside Szegedy’s class. On a toroidal array of $\Gamma=(V,E)$19 8-cliques linked by 4-cliques, marking the central 8-clique gives a success-probability peak

$\Gamma=(V,E)$20

an optimal step

$\Gamma=(V,E)$21

and, after amplitude amplification, total cost

$\Gamma=(V,E)$22

which is faster than the classical $\Gamma=(V,E)$23 random-walk hitting time (Portugal, 2016).

6. Hamiltonian realizations, implementations, and later extensions

The Hamiltonian extension was motivated by physical realizability. In many experimental platforms it is easier to alternate fixed, time-independent Hamiltonians than to synthesize arbitrary reflections directly, and the staggered-Hamiltonian step

$\Gamma=(V,E)$24

fits naturally with this constraint. The framework was proposed as suitable for implementation with cold atoms, superconducting resonators, and nano-electromechanical arrays, and it supports first-order Trotter and higher-order Suzuki decompositions when the target is an effective Hamiltonian $\Gamma=(V,E)$25 (Portugal et al., 2016).

A concrete superconducting implementation uses the single-photon subspace of an array of microwave resonators coupled by SQUIDs. In one dimension, alternating even-link and odd-link couplings realizes two Hamiltonians $\Gamma=(V,E)$26 and $\Gamma=(V,E)$27, with the full step $\Gamma=(V,E)$28. Reported parameters include resonator frequency $\Gamma=(V,E)$29, tunable coupling $\Gamma=(V,E)$30 up to $\Gamma=(V,E)$31, single-photon lifetime $\Gamma=(V,E)$32, and flux-pulse rise-time $\Gamma=(V,E)$33, implying that more than $\Gamma=(V,E)$34 steps are feasible (Moqadam et al., 2016). The same physical setting was later used to study boundary-induced coherence on different topologies. By modifying boundary couplings, the staggered dynamics was defined on the torus, Klein bottle, real projective plane, and sphere; for $\Gamma=(V,E)$35, $\Gamma=(V,E)$36, and $\Gamma=(V,E)$37, the sphere yielded the highest coherence and entropy, while the classical random walk analogue was insensitive to boundary twists (Moqadam et al., 2018).

Rydberg-atom architectures supply a more recent implementation route. In that proposal, each clique in a tessellation is processed by preparing a W-state, applying a native $\Gamma=(V,E)$38 gate, and undoing the preparation, thereby realizing the tessellation reflection

$\Gamma=(V,E)$39

A classical preprocessing algorithm assigns colors to edges so that edges of the same color form disjoint cliques; its overall complexity is $\Gamma=(V,E)$40, where $\Gamma=(V,E)$41 and $\Gamma=(V,E)$42 is the average degree. On random geometric graphs, the number of colors scales numerically as $\Gamma=(V,E)$43, and the total preprocessing time becomes $\Gamma=(V,E)$44. The same work reports numerically that spatial search on such graphs has query complexity $\Gamma=(V,E)$45, with an overall number of Rydberg-gate layers $\Gamma=(V,E)$46 (Almeida et al., 28 Jul 2025).

Optical networks provide another physical interpretation. A staggered array of 50/50 beam splitters arranged in alternating layers implements a coinless discrete-time walk in which a single beam splitter simultaneously splits amplitude and shifts it to the next layer. In that setting, onefold, twofold, and threefold coincidence functions reveal multiphoton interference effects beyond the single-particle walk, and the staggered geometry leads to exact destructive interference at specific detectors and higher-order correlation zeros (Gard et al., 2011).

Later extensions also addressed noise and generalization. Percolation-inspired decoherence in SQWs was modeled by breaking polygons or breaking vertices, producing random time-dependent reflection operators that map directly to edge and coin percolation in the coined walk picture. On the two-dimensional grid of 4-cliques, very small noise rates preserve the quantum speedup, while larger rates degrade it toward classical scaling; enlarging tessellation intersections from $\Gamma=(V,E)$47 to $\Gamma=(V,E)$48 improves robustness, with a heuristic degradation of order $\Gamma=(V,E)$49 (Santos et al., 2021). At the same time, the hypergraph perspective showed that every SQW can be embedded into a generalized hyperwalk, and the lattice literature extended staggered constructions to geometries such as the hexagonal and kagome lattices, where tessellation number, cycle structure, and localization become model-defining features (Sadowski et al., 2018, Chagas et al., 2018, Higuchi et al., 2018).

Staggered quantum walks therefore occupy a distinct position among discrete-time quantum walks: they are defined directly on graph vertices by tessellation-induced reflections, admit a sharp graph-theoretic characterization, possess a compact spectral theory in the 2-tessellable case, support search algorithms with lattice-dependent asymptotics, and admit multiple physically motivated Hamiltonian realizations (Portugal et al., 2015, Konno et al., 2017, Portugal et al., 2016).