---
title: Szegedy's Quantum Walk
url: https://www.emergentmind.com/topics/szegedy-s-quantum-walk
type: topic
---

# Szegedy's Quantum Walk

Szegedy’s quantum walk is a discrete-time construction that quantizes a classical Markov chain via two successive reflections, yielding a unitary process defined on the bipartite double cover of the underlying graph. It generalizes coined quantum walks but is formulated natively on the edge (or vertex-pair) Hilbert space, enabling rigorous analysis and quadratic speedup in mixing, search, and related algorithmic problems.

## 1. Formal Structure and Operator Definition

Let $G=(V,E)$ be an undirected (possibly weighted) graph; $N=|V|$, $|E|$ edges. Szegedy’s walk is defined on the bipartite double cover $B=G \times K_2$, whose vertex set splits into two labeled copies $X$ and $Y$ of $V$. The Hilbert space is
$$
H = \operatorname{span}\Big\{\,|x, y\rangle : x \in X, y \in Y,\, \{x, y\} \in E\,\Big\}, \qquad \dim H = 2|E|
$$
For a classical random walk with transition matrix $P$,
$$
P_{uv} = \begin{cases}
1/\deg(u) & \text{if } u \sim v \\
0 & \text{otherwise}
\end{cases}
$$
define “coinless” superpositions
$$
|\psi_u\rangle = \sum_{v \sim u} \sqrt{P_{uv}}\,|u, v\rangle = \frac{1}{\sqrt{\deg(u)}} \sum_{v \sim u} |u, v\rangle
$$
The Szegedy reflections are
$$
R_1 = 2\sum_{u \in X} |\psi_u\rangle\langle \psi_u| - I, \qquad R_2 = 2\sum_{v \in Y} |\psi_v\rangle\langle \psi_v| - I
$$
and the walk operator is
$$
W = R_2 R_1
$$
This operator is unitary and, for reversible $P$, acts as a direct sum of two-dimensional rotation blocks whose phases are determined by the singular values of a discriminant matrix, ensuring rich spectral structure [1611.02238].

## 2. Spectral Properties and Quantum Speedup

The discriminant matrix $D = \operatorname{diag}(\sqrt{\pi})\,P\,\operatorname{diag}(\pi^{-1/2})$ is real symmetric (for reversible $P$), with eigenvalues $\lambda_j \in [-1,1]$. The corresponding quantum walk has eigenphases $\theta_j = \arccos(\lambda_j)$, yielding spectral gaps that scale quadratically with the classical Markov gap,
$$
\Delta_Q = \min_{j\neq 0}\, |\theta_j| = \arccos(\lambda_1) \in \Omega(\sqrt{\delta})
$$
for small classical gap $\delta = 1 - \max\{|\lambda_1|,|\lambda_{n-1}|\}$ [2506.11576]. The quantum mixing time is thus
$$
\tau_{\text{quantum}} = O(\Delta_Q^{-1}\log(1/\epsilon)) = O(\sqrt{\delta^{-1}}\log(1/\epsilon))
$$
realizing a quadratic reduction compared to classical mixing, with analogous results in search and hitting algorithms [0808.0059, 1903.07493].

## 3. Search Algorithms: Absorbing vs. Oracle Models

For quantum search, one designates a marked set $M \subset V$ and modifies $P$ to an absorbing walk:
$$
P'_{mv} = \delta_{mv},\quad P'_{um}=P_{um}\;\text{for } u \notin M
$$
The corresponding quantum walk uses marked-vertex versions of the reflections:
- On $u\in M$, $R_1'$ acts as $-I$ on $|u,v\rangle$ ($v \sim u$); similarly for $v\in M$ in $R_2'$.
- The walk operator is $W' = R_2' R_1'$.

The spectral analysis shows that, e.g., on the complete graph with $k$ marked vertices,
- $W'$ has eigenphases $\pm 2\theta$ where $\cos\theta = 1 - k/(N-1)$
- The success amplitude evolves as $\sin((2t+1)\theta)$, maximizing at $(2t+1)\theta \approx \pi/2$
- Thus optimal quantum search complexity is $O(1/\theta) = O(\sqrt{N/k})$, with constant success probability [1611.02238, 2103.12878].

Alternatively, standard reflection queries can be embedded:
- $Q_1$ flips the sign of edges $|x,y\rangle$ with $x\in X\cap M$
- $Q_2$ flips $|x,y\rangle$ with $y\in Y\cap M$
- One-step and two-step per-query algorithms correspond to $W_{q1}= R_2 Q_2 R_1 Q_1$ and $W_{q2}= R_2 R_1 Q_1$
Mapping to coined walks, $W_{q1}$ is two steps per query, $W_{q2}$ is two steps per Q [see correspondences in 1611.02238]. The one-step-per-query variant can achieve $p_{\max}=1$ on the complete graph, a notable improvement over absorbing-vertex Szegedy search which attains $p_{\max}=1/2$.

## 4. Relationship to Coined and Staggered Quantum Walks

Szegedy’s formalism is closely related to coined quantum walks:
- Coined model: Hilbert space on vertices with internal degree-of-freedom (“coin”) $\{|a,b\rangle : a\in V, b\sim a\}$
- Unit step $U = S C$ where $C$ is coin operator, $S$ is flip-flop shift
- Szegedy’s operators correspond: $R_1 \leftrightarrow C$, $R_2 \leftrightarrow S C S$

The mapping between Szegedy and coined walks is exact when coins are Hermitian reflections:
- Szegedy’s walk = two coined steps with specific coin choices [1509.08852]
- The staggered quantum walk model unifies both as particular cases on the line graph or bipartite cover [1505.04761]

Search algorithms (e.g., abstract coined search with $-I$ coin on marked vertices) are exactly Szegedy’s absorbing-vertex search on bipartite graphs with sinks, preserving both spectral and algorithmic characteristics.

## 5. Extensions: Complex Phases, Memory, and Circuit Implementations

Recent work generalizes Szegedy’s walk to more flexible settings:
- Local arbitrary phase rotations (APR) and link phases introduce new families of coins, broadening the spectrum and optimizing node marking [2410.22011]
- Quantum circuits for Szegedy walks, notably for Metropolis-Hastings kernels, can avoid expensive reversible arithmetic by working on edge-space with constant ancilla overhead [2506.11576, 1910.01659]
- Quantum walks with memory (QWM) embed $d$-step memory into Szegedy’s construction on line digraphs, establishing equivalence with coined QWM and facilitating spectral analysis for regular graphs [1905.00625]

Classical simulation algorithms for Szegedy walks, notably SQUWALS, achieve $\mathcal{O}(N^2)$ scaling by representing states as $N\times N$ matrices and compressing reflections to entry-wise operations [2307.14314].

## 6. Generalizations and Physical Interpretations

The Szegedy scheme encompasses directed and weighted graphs, quantum walk-based circuit interpretations, and can be extended to walks on simplicial complexes—where the spectral profile reflects topological invariants (homology) and governs localization and ballistic dynamics [1507.01194].

In models with open graph structure (e.g., infinite tails attached to finite graphs), the nature of reversibility in the underlying Markov chain determines scattering outcomes: reversible walks yield reflection matrices governed by classical measures, while non-reversible walks effect global $-1$ phase flips and altered Kirchhoff-like constraints [2002.05261].

Analogous constructions are available for detailed-balanced Lindbladians (open quantum system dynamics), with Szegedy walk unitaries providing purified fixed points at eigenphase zero and quadratic amplification of the spectral gap, enabling fast mixing in quantum semigroups [2107.07365].

## 7. Algorithmic and Practical Implications

Szegedy’s framework yields quadratic speedup in a range of computational problems:
- Search for marked vertices with complexity $O(1/\sqrt{\delta\varepsilon})$ for fraction $\varepsilon$ marked [0808.0059, 1903.07493]
- Efficient simulation and circuit design for PageRank and Metropolis–Hastings methods, avoiding the scaling bottlenecks of earlier schemes [1609.00173, 2303.18202]
- Extensions to mixed states, semiclassical walks, and multi-step memory enable broader class of algorithms, including robust node ranking in graphs where classical PageRank fails [2303.18202, 2307.14314, 2410.22011]

The walk operator’s fundamental equivalence to coined walks means standard result and intuition from the latter carry over, with the two-reflection construction endowing the Szegedy scheme with powerful spectral tools that underpin its algorithmic speedup. Open directions include characterizing families where multi-step-per-query strategies outperform and generalizing to nonregular or weighted kernels [1611.02238].

Source: https://www.emergentmind.com/topics/szegedy-s-quantum-walk