Szegedy Quantum Walks: Theory & Applications

• Szegedy quantum walks are discrete-time quantum walks obtained by quantizing finite Markov chains with coherent square-root encodings and reflection operators.
• They exhibit operator-level equivalences with coined and staggered walks, facilitating powerful quantum search algorithms and detailed mixing analyses.
• Extensions include applications in hypergraphs, memory-based variants, and efficient circuit simulations, broadening their impact in quantum algorithm design.

Szegedy quantum walks are discrete-time quantum walks obtained by quantizing a classical random walk or, more generally, a finite Markov chain. Their defining feature is that the dynamics are built from reflections determined by transition probabilities, so the walk acts naturally on an edge or arc space rather than directly on the vertex space. In the standard graph-theoretic presentation, the walk is defined on the bipartite double cover of the underlying graph; in the Markov-chain presentation, it acts on $\mathbb C^n\otimes \mathbb C^n$ and quantizes the transition matrix through coherent square-root encodings of its columns (Wong, 2016). This framework has become a central discrete-time model for quantized Markov chains, with established links to coined and staggered walks, search algorithms, mixing theory, open-system generalizations, and efficient simulation (Sorci, 2022).

1. Formal definition and operator structure

For an undirected, unweighted graph $G$, one standard construction passes to the bipartite double cover $G\times K_2$, with partite sets $X$ and $Y$. The Hilbert space is $\mathbb C^{2|E|}$, with basis states $\{|x,y\rangle : x\in X,\, y\in Y,\, x\sim y\}$. If the classical walk is unbiased, the local states are uniform superpositions over edges incident to each vertex,

$$|\phi_x\rangle=\frac{1}{\sqrt{\deg(x)}}\sum_{y\sim x}|x,y\rangle,\qquad |\psi_y\rangle=\frac{1}{\sqrt{\deg(y)}}\sum_{x\sim y}|x,y\rangle,$$

and the walk is the product of two reflections,

$$W=R_2R_1,\qquad R_1=2\sum_{x\in X}|\phi_x\rangle\langle\phi_x|-I,\qquad R_2=2\sum_{y\in Y}|\psi_y\rangle\langle\psi_y|-I.$$

Operationally, $R_1$ and $G$0 perform inversion about the mean on amplitudes supported on edges incident to vertices in $G$1 and $G$2, respectively (Wong, 2016).

In the Markov-chain formulation, one starts from a finite chain with transition matrix $G$3 on state space $G$4. The walk acts on the arc space $G$5, with coherent encodings

$G$6

assembled into an isometry $G$7 whose columns are the $G$8. If $G$9 denotes arc reversal, $G\times K_2$0, then a single-step form of the walk is

$G\times K_2$1

This differs slightly from Szegedy’s original two-reflection operator, but its square equals the original transition operator, so the two formulations are equivalent at the level of dynamics (Sorci, 2022).

The key spectral object is the discriminant matrix

$G\times K_2$2

For reversible chains, $G\times K_2$3 is real symmetric and similar to $G\times K_2$4, hence it has the same spectrum. If $G\times K_2$5 is an eigenvalue of $G\times K_2$6, then the corresponding nontrivial eigenvalues of the walk are $G\times K_2$7 in the single-step formulation, or $G\times K_2$8 in the usual two-reflection picture (Sorci, 2022). This cosine/eigenphase relation is the basic mechanism behind both algorithmic speedups and asymptotic analysis.

A complementary formulation, widely used in simulation work, writes the walk as

$G\times K_2$9

where $X$0 is the reflection about the span of the states

$X$1

and $X$2 swaps the two registers. In that notation, the original Szegedy walk is $X$3 (Ortega et al., 2023). This formulation is especially convenient for update-operator factorizations and direct simulation (Ortega et al., 12 Jun 2026).

2. Equivalences with coined and staggered quantum walks

A major structural result is that Szegedy walks are not isolated from other discrete-time models. For coined walks on the original graph, with basis states $X$4 meaning “at vertex $X$5, pointing toward $X$6,” the Grover-diffusion coin is

$X$7

and the flip-flop shift is $X$8. Under the basis identification $X$9 between directed edges in the coined model and edges of the bipartite double cover, one has the exact operator identities

$Y$0

and therefore

$Y$1

Thus one step of the standard Szegedy walk equals two steps of the corresponding coined walk (Wong, 2016).

This operator-level correspondence sharpens earlier equivalence statements. It also clarifies how search operators translate between the two models. In particular, the absorbing-vertex Szegedy search operator $Y$2 is exactly $Y$3, where $Y$4 is the Shenvi–Kempe–Whaley coined search using coin $Y$5 on unmarked vertices and $Y$6 on marked ones (Wong, 2016).

The staggered model provides a broader geometric bridge. Every Szegedy walk on a bipartite graph $Y$7 can be represented as a staggered walk on the line graph $Y$8, where the two reflection subspaces become two tessellations by cliques corresponding to edges incident to vertices in $Y$9 and $\mathbb C^{2|E|}$0 (Portugal et al., 2015). The converse is only partial: a staggered walk can be cast into Szegedy form when it uses two tessellations and every pair of polygons intersects in exactly one vertex, equivalently when the tessellated graph is the line graph of a bipartite graph (Portugal et al., 2015).

The coined-to-Szegedy direction can also be characterized directly. A flip-flop coined walk can be rewritten as a Szegedy walk when the local coin is an orthogonal reflection; conversely, a Szegedy walk can be rewritten as a coined walk on a multigraph under degree and weight conditions that force the $\mathbb C^{2|E|}$1-side reflection to behave like a flip-flop shift (Portugal, 2015). These results establish that the common core is not the presence or absence of an explicit coin, but the product-of-reflections structure.

3. Search and marked-vertex algorithms

Search is one of the principal algorithmic uses of Szegedy walks. The standard construction marks vertices by modifying the classical chain so that marked vertices become absorbing. Quantizing that modified chain gives a search walk

$\mathbb C^{2|E|}$2

where the reflections act as ordinary inversion-about-the-mean on unmarked vertices and as sign flips on all incident edge amplitudes at marked vertices (Wong, 2016). In coined language this is the negative-identity marked-coin search.

A different line of work replaces absorbing vertices by explicit Grover-type oracle reflections. In one such construction,

$\mathbb C^{2|E|}$3

where $\mathbb C^{2|E|}$4 flips the sign of edges incident to marked vertices in the $\mathbb C^{2|E|}$5 part of the bipartite double cover, and $\mathbb C^{2|E|}$6 does the same on the $\mathbb C^{2|E|}$7 side. Under the coined-walk correspondence,

$\mathbb C^{2|E|}$8

so

$\mathbb C^{2|E|}$9

A second oracle-based operator is

$\{|x,y\rangle : x\in X,\, y\in Y,\, x\sim y\}$0

These identities distinguish one-walk-step-per-query and two-walk-steps-per-query search mechanisms, a distinction that matters in oracle complexity (Wong, 2016).

Oracle-based Szegedy search can outperform the absorbing-walk construction in success probability. On the complete graph, using

$\{|x,y\rangle : x\in X,\, y\in Y,\, x\sim y\}$1

with $\{|x,y\rangle : x\in X,\, y\in Y,\, x\sim y\}$2 the marked-vertex phase flip on the first register, the probability of finding a marked vertex can be boosted from about $\{|x,y\rangle : x\in X,\, y\in Y,\, x\sim y\}$3 to $\{|x,y\rangle : x\in X,\, y\in Y,\, x\sim y\}$4 while retaining $\{|x,y\rangle : x\in X,\, y\in Y,\, x\sim y\}$5 scaling. The same paper also proves that for certain graph classes, including strongly regular graphs with one marked vertex, the absorbing-walk search operator can be represented exactly in standard query language (Santos, 2016).

The broader search-theoretic problem is not just detection of marked vertices, but finding one. A major advance showed that for any reversible finite Markov chain $\{|x,y\rangle : x\in X,\, y\in Y,\, x\sim y\}$6 and any marked set $\{|x,y\rangle : x\in X,\, y\in Y,\, x\sim y\}$7, one can output a marked vertex with bounded error in complexity

$\{|x,y\rangle : x\in X,\, y\in Y,\, x\sim y\}$8

where $\{|x,y\rangle : x\in X,\, y\in Y,\, x\sim y\}$9 is a known upper bound on the hitting time, and $$|\phi_x\rangle=\frac{1}{\sqrt{\deg(x)}}\sum_{y\sim x}|x,y\rangle,\qquad |\psi_y\rangle=\frac{1}{\sqrt{\deg(y)}}\sum_{x\sim y}|x,y\rangle,$$0 are setup, update, and checking costs. This closes, up to logarithmic factors, the longstanding gap between Szegedy-style detection and finding for arbitrary marked sets (Ambainis et al., 2019). The proof uses interpolated walks $$|\phi_x\rangle=\frac{1}{\sqrt{\deg(x)}}\sum_{y\sim x}|x,y\rangle,\qquad |\psi_y\rangle=\frac{1}{\sqrt{\deg(y)}}\sum_{x\sim y}|x,y\rangle,$$1, the corresponding discriminants $$|\phi_x\rangle=\frac{1}{\sqrt{\deg(x)}}\sum_{y\sim x}|x,y\rangle,\qquad |\psi_y\rangle=\frac{1}{\sqrt{\deg(y)}}\sum_{x\sim y}|x,y\rangle,$$2, and quantum fast-forwarding of $$|\phi_x\rangle=\frac{1}{\sqrt{\deg(x)}}\sum_{y\sim x}|x,y\rangle,\qquad |\psi_y\rangle=\frac{1}{\sqrt{\deg(y)}}\sum_{x\sim y}|x,y\rangle,$$3.

4. Long-time behavior, mixing, and asymptotics

Because Szegedy walks are unitary, instantaneous probabilities generally do not converge. The natural asymptotic object is therefore a time average. For a finite chain, if

$$|\phi_x\rangle=\frac{1}{\sqrt{\deg(x)}}\sum_{y\sim x}|x,y\rangle,\qquad |\psi_y\rangle=\frac{1}{\sqrt{\deg(y)}}\sum_{x\sim y}|x,y\rangle,$$4

is the probability of being at vertex $$|\phi_x\rangle=\frac{1}{\sqrt{\deg(x)}}\sum_{y\sim x}|x,y\rangle,\qquad |\psi_y\rangle=\frac{1}{\sqrt{\deg(y)}}\sum_{x\sim y}|x,y\rangle,$$5 at time $$|\phi_x\rangle=\frac{1}{\sqrt{\deg(x)}}\sum_{y\sim x}|x,y\rangle,\qquad |\psi_y\rangle=\frac{1}{\sqrt{\deg(y)}}\sum_{x\sim y}|x,y\rangle,$$6, then the Cesàro average

$$|\phi_x\rangle=\frac{1}{\sqrt{\deg(x)}}\sum_{y\sim x}|x,y\rangle,\qquad |\psi_y\rangle=\frac{1}{\sqrt{\deg(y)}}\sum_{x\sim y}|x,y\rangle,$$7

always converges as $$|\phi_x\rangle=\frac{1}{\sqrt{\deg(x)}}\sum_{y\sim x}|x,y\rangle,\qquad |\psi_y\rangle=\frac{1}{\sqrt{\deg(y)}}\sum_{x\sim y}|x,y\rangle,$$8. The limiting distribution keeps only equal-eigenvalue sectors of the walk operator. In particular, when the transition matrix $$|\phi_x\rangle=\frac{1}{\sqrt{\deg(x)}}\sum_{y\sim x}|x,y\rangle,\qquad |\psi_y\rangle=\frac{1}{\sqrt{\deg(y)}}\sum_{x\sim y}|x,y\rangle,$$9 is symmetric, the limiting distribution is uniform; for non-symmetric chains, the limiting quantum distribution need not coincide with the stationary distribution of the classical chain (Balu et al., 2017).

For reversible chains, a more refined vertex-level asymptotic object is the average mixing matrix

$$W=R_2R_1,\qquad R_1=2\sum_{x\in X}|\phi_x\rangle\langle\phi_x|-I,\qquad R_2=2\sum_{y\in Y}|\psi_y\rangle\langle\psi_y|-I.$$0

Its columns give the long-time averaged vertex distribution obtained by starting in the canonical encoded state $$W=R_2R_1,\qquad R_1=2\sum_{x\in X}|\phi_x\rangle\langle\phi_x|-I,\qquad R_2=2\sum_{y\in Y}|\psi_y\rangle\langle\psi_y|-I.$$1. For an irreducible, aperiodic, reversible chain with discriminant spectral idempotents $$W=R_2R_1,\qquad R_1=2\sum_{x\in X}|\phi_x\rangle\langle\phi_x|-I,\qquad R_2=2\sum_{y\in Y}|\psi_y\rangle\langle\psi_y|-I.$$2 and eigenvalues $$W=R_2R_1,\qquad R_1=2\sum_{x\in X}|\phi_x\rangle\langle\phi_x|-I,\qquad R_2=2\sum_{y\in Y}|\psi_y\rangle\langle\psi_y|-I.$$3, one has the closed formula

$$W=R_2R_1,\qquad R_1=2\sum_{x\in X}|\phi_x\rangle\langle\phi_x|-I,\qquad R_2=2\sum_{y\in Y}|\psi_y\rangle\langle\psi_y|-I.$$4

which expresses the Szegedy average mixing matrix as the continuous-time average mixing matrix plus an explicit correction term (Sorci, 2022).

This comparison yields a structural implication: if $$W=R_2R_1,\qquad R_1=2\sum_{x\in X}|\phi_x\rangle\langle\phi_x|-I,\qquad R_2=2\sum_{y\in Y}|\psi_y\rangle\langle\psi_y|-I.$$5 is symmetric, irreducible, and aperiodic, and if the continuous-time walk $$W=R_2R_1,\qquad R_1=2\sum_{x\in X}|\phi_x\rangle\langle\phi_x|-I,\qquad R_2=2\sum_{y\in Y}|\psi_y\rangle\langle\psi_y|-I.$$6 is average uniform mixing, then the Szegedy walk is also average uniform mixing (Sorci, 2022). The same work constructs arbitrarily large symmetric chains of size $$W=R_2R_1,\qquad R_1=2\sum_{x\in X}|\phi_x\rangle\langle\phi_x|-I,\qquad R_2=2\sum_{y\in Y}|\psi_y\rangle\langle\psi_y|-I.$$7 with average mixing matrix $$W=R_2R_1,\qquad R_1=2\sum_{x\in X}|\phi_x\rangle\langle\phi_x|-I,\qquad R_2=2\sum_{y\in Y}|\psi_y\rangle\langle\psi_y|-I.$$8 in both the continuous-time and Szegedy settings.

Uniformity, however, is not generic. For the Szegedy quantization of the uniform random walk on the Cayley graph of the symmetric group $$W=R_2R_1,\qquad R_1=2\sum_{x\in X}|\phi_x\rangle\langle\phi_x|-I,\qquad R_2=2\sum_{y\in Y}|\psi_y\rangle\langle\psi_y|-I.$$9 generated by all transpositions, the instantaneous and time-averaged distributions remain far from uniform. In particular, the probability mass on the $R_1$0-cycle sector is exponentially smaller than the uniform benchmark $R_1$1, showing that regular classical mixing does not imply uniform quantum mixing even in highly symmetric settings (Banerjee, 2023). This is a sharp reminder that the quantum asymptotics are governed by representation-theoretic structure of the discriminant, not by classical stationarity alone.

5. Extensions of the framework

The Szegedy formalism has been extended in several directions without abandoning its product-of-reflections core. For regular uniform hypergraphs, one can pass to the incidence bipartite graph and define a Szegedy walk from the vertex-to-hyperedge and hyperedge-to-vertex transition matrices,

$R_1$2

The resulting walk lives on the incidence space and inherits the standard discriminant-based spectral analysis, giving a natural quantization of two-step random walks on hypergraphs (Liu et al., 2017).

Memory can be incorporated by replacing the original graph $R_1$3 with its $R_1$4-th line digraph $R_1$5. In that representation, a walk with $R_1$6-step memory becomes an ordinary memoryless Szegedy walk on a graph whose vertices already encode the last $R_1$7 steps. This yields a general model of Szegedy quantum walks with memory on regular graphs and clarifies their relation to coined walks with memory (Li et al., 2019).

Measurement-and-reset variants lead to semiclassical Szegedy walks. In this setting, one evolves coherently for $R_1$8 quantum steps, measures a register, then resets to the canonical Szegedy state corresponding to the observed vertex. The resulting process is a classical Markov chain with transition matrix generated by quantum evolution,

$R_1$9

On regular cycles, this construction is exactly solvable and produces long-range classical jumps of distance $G$00; on inhomogeneous symmetric graphs, it can break symmetries that defeat both classical PageRank and standard Szegedy-based quantum PageRank (Ortega et al., 2023).

Open-system and scattering extensions are also possible. For a finite internal graph coupled to semi-infinite tails, with constant incoming amplitude from the tails, the stationary response depends qualitatively on reversibility of the underlying random walk. In the reversible case,

$G$01

so the whole internal graph behaves asymptotically like a single Szegedy reflection on the boundary space; in the non-reversible case, one gets the pure phase flip $G$02. In the reversible regime, the stationary internal amplitudes decompose into a reversible-measure part and an electric current satisfying Kirchhoff laws (Higuchi et al., 2020).

A further generalization lifts the formalism from classical stochastic maps to detailed balanced quantum maps. For a detailed balanced Lindbladian or quantum channel with fixed point $G$03, one can define a quantum discriminant $G$04, build a Szegedy-style walk unitary $G$05, and recover two hallmark properties of the classical theory: the purification $G$06 is an eigenvector with eigenphase $G$07, and the eigenphase gap is quadratically larger than the spectral gap of the Lindbladian or channel (Wocjan et al., 2021).

Finally, complex-phase extensions enlarge the class of represented coined walks. A graph-phased Szegedy walk introduces edge phases $G$08 and local arbitrary phase rotations $G$09,

$G$10

with

$G$11

This permits node marking without modifying the graph and broadens the local coined operators representable in Szegedy form to those having $G$12 eigenvalues $G$13 and one distinguished eigenvalue $G$14 (Ortega et al., 2024).

6. Circuit constructions, simulation, and applications

Efficient implementation of Szegedy walks depends less on sparsity than on structural symmetry of the square-root column states. If the states

$G$15

can all be mapped efficiently to one or a few reference states by controllable unitaries, then the reflection can be diagonalized and the full walk unitary implemented with $G$16 gates. This symmetry-based method yields explicit efficient circuits for cyclic permutations, complete graphs, complete bipartite graphs, tensor-product chains, and several Google-matrix instances arising in quantum PageRank (Loke et al., 2016).

On the classical side, direct simulation of the full $G$17 walk matrix is memory-prohibitive. A specialized simulator avoids constructing that matrix altogether by storing the quantum state as an $G$18 array $G$19 and applying reflection, swap, oracle, and measurement implicitly through elementwise operations, sums, and transposition. For dense graphs, both time and memory scale as $G$20, which matches the cost of storing the transition matrix itself; the implementation reported simulations up to $G$21 nodes and supports mixed-state and semiclassical variants, as well as quantum PageRank (Ortega et al., 2023).

A later framework generalized this operator-level simulation approach beyond the standard walk formulation. Using update operators $G$22 satisfying

$G$23

it simulates not only $G$24 and $G$25, but also alternative formulations such as

$G$26

together with phase-estimation-based algorithms. The resulting package, SQWLib, achieves $G$27 complexity for dense graphs and linear scaling in the number of edges for sparse graphs, and it was used to simulate marked-node detection, quantum simulated annealing, and QPE-based graph search on instances with $G$28 states or nodes (Ortega et al., 12 Jun 2026).

These implementation advances have enabled application-oriented uses of Szegedy walks beyond search and PageRank. One recent network-science procedure uses a Szegedy-type edge-space walk, its limiting edge-probability distribution, and a path-weight rule to detect communities in graphs and social networks, including relaxed caveman, planted partition, karate club, dolphins, and road-network examples (Rahaman et al., 29 Jan 2026). This suggests that, in addition to their algorithmic role in quantum search and sampling, Szegedy walks are now being used as a source of nonclassical structural descriptors for complex networks.

Across these developments, the unifying principle remains the same: Szegedy quantum walks quantize transition structure through coherent square-root encodings and reflection dynamics. What changes from one extension to another is the space on which those reflections act, the form of the encoded transition rule, and the observable extracted from the resulting unitary evolution.