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arxiv: 2409.08428 · v2 · submitted 2024-09-12 · 🧮 math-ph · math.MP· quant-ph

Unitary and Open Scattering Quantum Walks on Graphs

Alain Joye This is my paper

Pith reviewed 2026-05-23 20:39 UTC · model grok-4.3

classification 🧮 math-ph math.MPquant-ph
keywords quantum walksscattering matricesgraphsquantum channelsMarkov chainsunitary operatorsopen quantum systems
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The pith

Scattering quantum walks on arbitrary graphs unify known models and introduce open quantum channels linked to Markov chains.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper develops a framework for unitary quantum walks on any graph using scattering matrices at vertices to define the dynamics on edges. It demonstrates that this approach includes several established quantum walk models. The work also defines two types of open scattering quantum walks, on edges and on vertices, showing they correspond to quantum channels with properties tied to classical Markov chains on the graph. A reader would care because it offers a flexible parameterization for studying quantum dynamics in both closed and open systems on complex structures.

Core claim

Scattering Quantum Walks are defined by assigning a scattering matrix to each vertex of an arbitrary graph, which determines the unitary evolution on the space of edges. These walks are shown to encompass multiple known quantum walk constructions. Two classes of open scattering quantum walks are introduced on the edges and vertices respectively, parameterized similarly by scattering matrices; these yield quantum channels whose spectral properties and dynamics are related to naturally associated classical Markov chains.

What carries the argument

The scattering matrix assigned to each vertex, which governs the local scattering process and ensures the global evolution is unitary or completely positive trace-preserving.

If this is right

  • Scattering quantum walks provide a general parameterization that includes several known quantum walk models.
  • Open scattering quantum walks on edges and vertices define proper quantum channels.
  • The spectral and dynamical properties of these open walks relate directly to associated classical Markov chains.
  • Unitary evolutions and quantum channels arise from appropriate choices of the scattering matrices on arbitrary graphs.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The framework allows constructing quantum walks with targeted properties by choosing specific scattering matrices at vertices.
  • It may enable modeling of quantum information flow on networks that include dissipation through the open variants.
  • The explicit link to Markov chains opens possibilities for analyzing mixing times or steady states in the quantum setting via classical counterparts.

Load-bearing premise

The scattering matrices at vertices must be selected so that the resulting global operators define unitary evolutions or completely positive trace-preserving maps.

What would settle it

Constructing a family of scattering matrices for which the resulting operator on edge space is not unitary, or the map on vertex space is not completely positive and trace-preserving, would show the claimed constructions do not always produce valid quantum walks or channels.

Figures

Figures reproduced from arXiv: 2409.08428 by Alain Joye.

Figure 1
Figure 1. Figure 1: The scattering process at work in SQWs The freedom in choosing the directed graph and the unitary scattering matrices at its vertices explains why SQWs encompass a significant set of unitary QWs considered in the literature. We explicitly show that Coined QWs, the Chalker-Coddington model, and a slight generalization of the Grover Walk belong to this class. Following [HKSS2, HSS] we revisit the spectral pr… view at source ↗
Figure 2
Figure 2. Figure 2: A Chalker–Coddington model with its incoming (solid arrows) and outgoing links. The convention above is as follows (without indices):  U|a⟩ U|b⟩  = S  |c⟩ |d⟩  ⇔ ( U|a⟩ = S11|c⟩ + S21|d⟩ U|b⟩ = S12|c⟩ + S22|d⟩ , (3.4) where Sij are the matrix elements of S, in line with (2.5). The underlying graph is G = Z 2 , with vertices at the intersections of the diagonal lines, where the scattering matrices sit. … view at source ↗
Figure 3
Figure 3. Figure 3: Star-graph SG with N branches. 3.3 Star-graph Consider G to be the star-graph SG with N branches characterized by vertices x0, x1, . . . , xN , with edges between x0 and xj , 1 ≤ j ≤ N, only, see [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The graph G = T3 with basis vectors of HT3 . The degrees of the vertices are dx = dz = 1 and dy = 2, so that dim l 2 (D) = 4 and dim B(l 2 (D)) = 16. The associated scattering matrices in S = {S(x), S(y), S(z)} are S(x) = e iθx ∈ U(1), S(z) = e iθz ∈ U(1) S(y) =  Sxx(y) Sxz(y) Szx(y) Szz(y)  ∈ U(2) (5.19) according to our notation. The matrix representation of US in the ordered basis {|yx⟩, |yz⟩, |xy⟩, |… view at source ↗
Figure 5
Figure 5. Figure 5: The graph G = Z with incoming and outgoing edges at the vertices. We further write the basis vectors of DiagT (l 2 (D)) as |x⟩⟨x| = ex, x ∈ Z, so that DiagT (l 2 (D)) ≃ l 1 (Z), with ordered basis {. . . , e−2, e−1, e0, e1, e2, . . . }. (5.29) Accordingly, the matrix elements of ΦDiag in the ordered basis (5.29) are denoted by Φ Diag xy . More precisely, we have Φ Diag(e2x) = 1 2 [PITH_FULL_IMAGE:figures/… view at source ↗
Figure 6
Figure 6. Figure 6: The graph Γ with the specified incoming and outgoing edges. As a simple computation reveals, ΦDiag admits no invariant vector in l 1 (Z). We prove the following property of its matrix elements in Appendix B: Lemma 5.9 There exists c < ∞ such that for any x, y ∈ Z, 0 ≤ Φ Diagn xy ≤ c/√ n, for all n ≥ N0(x − y) (5.31) where N0(x − y) ∈ N. As a consequence, the map provided in Corollary 4.5 Φ n S = ΦS ◦ Φ Dia… view at source ↗
Figure 7
Figure 7. Figure 7: The functional graph Σ associated with the map N (6.52) Given the stochastic matrix P (6.49), we first determine the dimension of Ker (P −I), which equals the dimension of the subspace of left invariant vectors of P. Lemma 6.8 The stochastic matrix associated with DFT scattering matrices satisfies dim Ker (P − I) = |CC(Σ)|. (6.53) Proof : We set E = P − I, with matrix elements Exy, x, y ∈ V Exy =    ε… view at source ↗
Figure 8
Figure 8. Figure 8: The infinite graph Σ associated with the stochastic matrix P (6.74). It is readily verified that the only probability vector π ∈ l 1 (Z) such that πP = π has nonzero components π0 = 1/2, π1 = 1/2. (6.75) Incidentally, this is the invariant probability vector of the central 2 × 2 stochastic bloc G =  2/3 1/3 1/3 2/3  . (6.76) Lemma 6.13 There exists γ > 0 and c > 0 such that for any x, y ∈ Z, and n ≥ N(x,… view at source ↗
read the original abstract

We study a class of Unitary Quantum Walks on arbitrary graphs, parameterized by a family of scattering matrices. These Scattering Quantum Walks model the discrete dynamics of a system on the edges of the graph, with a scattering process at each vertex governed by the scattering matrix assigned to it. We show that Scattering Quantum Walks encompass several known Quantum Walks. Additionally, we introduce two classes of Open Scattering Quantum Walks on arbitrary graphs, also parameterized by scattering matrices: one class defined on the edges and the other on the vertices of the graph. We show that these walks give rise to proper Quantum Channels and describe their main spectral and dynamical properties, relating them to naturally associated classical Markov chains.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The paper introduces Scattering Quantum Walks on arbitrary graphs, parameterized by families of scattering matrices at vertices that govern scattering processes on edges; it claims these encompass several known quantum walk models. It further defines two classes of Open Scattering Quantum Walks (one on edges, one on vertices), asserts that both yield proper quantum channels, and analyzes their spectral and dynamical properties while relating them to associated classical Markov chains.

Significance. If the constructions are rigorously shown to produce CPTP maps without hidden restrictions on the scattering matrices and the relations to Markov chains are derived explicitly, the framework would unify unitary and open quantum walks on general graphs and provide a concrete link to classical stochastic dynamics, which could be useful for quantum transport models and quantum information on networks.

major comments (1)
  1. [Abstract] Abstract and the sections defining the open walks: the central claim that the two classes of Open Scattering Quantum Walks 'give rise to proper Quantum Channels' on arbitrary graphs for general scattering matrices is load-bearing but unsupported by explicit verification of the completely-positive trace-preserving property; the construction on edge/vertex spaces may require additional constraints (e.g., contraction properties or global Kraus decompositions preserving trace) that are not shown to hold unconditionally.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need to strengthen the explicit verification of the CPTP property. We address this point below and will revise the manuscript to include a dedicated proof.

read point-by-point responses

  1. Referee: [Abstract] Abstract and the sections defining the open walks: the central claim that the two classes of Open Scattering Quantum Walks 'give rise to proper Quantum Channels' on arbitrary graphs for general scattering matrices is load-bearing but unsupported by explicit verification of the completely-positive trace-preserving property; the construction on edge/vertex spaces may require additional constraints (e.g., contraction properties or global Kraus decompositions preserving trace) that are not shown to hold unconditionally.

    Authors: We thank the referee for this observation. The scattering matrices are unitary by construction (as they parameterize the unitary scattering processes at vertices). For both classes of open walks, the quantum channel is obtained by extending the local unitary scattering to a completely positive map on the edge or vertex space via a standard dilation or partial trace construction. We will add an explicit lemma (in a new subsection of the open-walks section) that constructs the Kraus operators explicitly for arbitrary unitary scattering matrices and verifies both complete positivity (by the Kraus representation) and trace preservation (via the completeness relation, which follows directly from unitarity of each local scattering matrix and the fact that the global map is assembled vertex-wise without overlap). No additional contraction or global constraints are required; the proof holds unconditionally on arbitrary graphs. We will also update the abstract to reference this verification. revision: yes

Circularity Check

0 steps flagged

No circularity: claims rest on direct construction from scattering matrices without reduction to inputs

full rationale

The paper defines unitary and open scattering quantum walks explicitly via families of scattering matrices on arbitrary graphs, then derives that suitable choices yield unitary evolutions or CPTP maps (quantum channels) whose spectral and dynamical properties relate to associated Markov chains through the scattering process itself. No derivation step equates a claimed result to a fitted parameter or prior self-citation by construction; the encompassing of known walks occurs via explicit parameterization rather than renaming, and the CPTP property is conditioned on the choice of matrices as stated in the weakest assumption. The derivation chain is self-contained against the given definitions and does not invoke load-bearing self-citations or ansatzes smuggled from prior work.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The framework rests on standard domain assumptions from quantum mechanics and operator theory on graphs; no free parameters are fitted, and no new physical entities are postulated beyond the input scattering matrices.

axioms (1)
  • domain assumption Scattering matrices assigned to vertices are unitary (or satisfy conditions yielding completely positive trace-preserving maps for the open case).
    Required for the discrete-time evolution to be unitary or a valid quantum channel on the graph's edge/vertex Hilbert space.

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