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arxiv: 2604.14277 · v1 · submitted 2026-04-15 · 🪐 quant-ph · math-ph· math.MP· math.PR

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Entanglement and circuit complexity in finite-depth random linear optical networks

Alexey V. Gorshkov, Joseph T. Iosue, Laura Shou, Victor Galitski, Yu-Xin Wang

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Pith reviewed 2026-05-10 13:20 UTC · model grok-4.3

classification 🪐 quant-ph math-phmath.MPmath.PR
keywords entanglement dynamicscircuit complexitylinear optical networksrandom circuitsRényi entropybrickwall circuitsGaussian statesHaar random unitaries
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The pith

In random brickwall linear optical circuits, Rényi-2 entanglement entropy grows at most diffusively with depth, and robust circuit complexity does the same.

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

This paper analyzes the evolution of entanglement and circuit complexity in finite-depth random passive linear optical networks. Beginning with a Gaussian state of squeezed modes, it establishes upper bounds on the growth rate of Rényi-2 entropy for brickwall geometries, showing it increases no faster than diffusively. For arbitrary geometries, it derives depth requirements for near-maximal subsystem entanglement and proximity to Haar-random unitaries. It further shows that the minimal number of gates for approximate implementations, termed robust circuit complexity, scales diffusively with depth at high probability. These findings clarify the minimal resources needed to produce highly entangled states in optical quantum systems.

Core claim

For random brickwall circuits, entanglement as measured by the Rényi-2 entropy grows at most diffusively as a function of the depth. In the other direction, for arbitrary circuit geometries we prove bounds on depths which ensure the average subsystem entanglement reaches within a constant factor of the maximum value in all subsystems, and bounds which ensure closeness of the random linear optical unitary to a Haar random unitary in L2 Wasserstein distance. The robust circuit complexity for random one-dimensional brickwall circuits scales at most diffusively in the depth with high probability, where the corresponding approximate Gaussian unitary retains high output fidelity for pure states |ψ

What carries the argument

The robust circuit complexity, defined as the minimum number of gates in any circuit that approximately implements the linear optical unitary while retaining high output fidelity for pure states with constrained expected photon number.

If this is right

  • Subsystem entanglement approaches a constant fraction of its maximum value once circuit depth exceeds geometry-dependent thresholds.
  • The random linear optical unitary becomes close to a Haar-random unitary in L2 Wasserstein distance after sufficient depth.
  • Robust circuit complexity remains at most a diffusive function of depth with high probability, so shallow circuits suffice for complex transformations.
  • Approximate implementations preserve high fidelity on photon-number constrained states.

Where Pith is reading between the lines

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

  • The diffusive bound may extend to other entanglement measures or initial states beyond Gaussians, though the paper restricts to the given setting.
  • These scaling results could guide resource estimates for optical implementations of quantum algorithms that rely on entanglement generation.
  • Analogous diffusive limits might appear in related random-circuit models from condensed-matter physics.

Load-bearing premise

The linear optical unitaries are passive and drawn randomly from brickwall or general ensembles, with the system beginning in a Gaussian state where all modes are squeezed.

What would settle it

A numerical simulation of a brickwall linear optical circuit in which the Rényi-2 entropy grows faster than diffusively, for example linearly with depth, would disprove the upper bound.

Figures

Figures reproduced from arXiv: 2604.14277 by Alexey V. Gorshkov, Joseph T. Iosue, Laura Shou, Victor Galitski, Yu-Xin Wang.

Figure 1
Figure 1. Figure 1: A one-dimensional depth d brickwall circuit on modes {1, . . . , n}, n ∈ 2N, is constructed as the product of d steps: U = Q1 i=d U (i) = U (d) · · ·U (2)U (1). Each 2-mode gate corresponds to a 2 × 2 unitary matrix representing a beamsplitter with phaseshifters, and each 1-mode gate corresponds to a 1×1 unitary matrix representing a phase-shifter. Note that, in the circuit, U (1) is applied first, then U … view at source ↗
Figure 2
Figure 2. Figure 2: Matrix structure of U (i) . A single step U (i) is the product of two mis￾matching block matrices U R,(i) = u0 ⊕ Ln/2−1 j=1 uj  ⊕ u′ 0 and U L,(i) = Ln/2 j=1 uj , where all uj are 2 × 2 unitary blocks except for u0 and u′ 0 which are 1 × 1 unitary. The values of the blocks u (i) j can differ between different steps U (i) . When consid￾ering random brickwall circuits, we will always take all 2 × 2 and 1 ×… view at source ↗
Figure 3
Figure 3. Figure 3: Depiction of how single photons split into superpositions. The two halves of the superposition propagate in opposite directions, ballistically (left, worst-case) or diffusively like in a random walk (right, average-case). In the diffusive case, after depth d only ∼ O( √ d) trajectories have crossed the cut between subsystems. Corollary 1.2 (D-dimensional brickwork upper bounds). Consider the D-dimensional … view at source ↗
Figure 4
Figure 4. Figure 4: (Left) Growth of the R´enyi-2 entropy for the random brickwall geometry as a function of the depth, for n = 100, k = 50, s = 1, shown for a single random realization (orange), and averaged over 5000 trials (blue). The early time shows a square-root diffusive growth. (Right) Variance of the R´enyi-2 entropy for the setting in the left figure. The variance reaches a constant value nearly immediately (in cont… view at source ↗
Figure 5
Figure 5. Figure 5: Forming V˜ or V from the banded matrix UUT requires cutting out some entries from the last O(d) rows in the subsystem, making those rows no longer or￾thonormal. Applying (2.2) and (2.4), we then have S2(U) = k log cosh(2s) + 1 2 Tr log(1 − tanh2 (2s)W) = k log cosh(2s) + 1 2 Tr Ik−2wd log(1 − tanh2 (2s)) + 1 2 Tr log(1 − tanh2 (2s)RR† ) = 2wd log cosh(2s) + 1 2 Tr log(1 − tanh2 (2s)RR† ) ≤ 2wd log cosh(2s)… view at source ↗
Figure 6
Figure 6. Figure 6: Plots of the matrix entries of |UUT | for n = 100 and depth d = 15, for a single realization (left) and averaged over 1000 realizations (right). The color scale is truncated at 0.5 (all entries > 0.5 are plotted at the same color as 0.5). Entries near the edges of the band are much smaller than those near the center. This creates a smaller “effective” band width ≈ Θ(√ d) compared to the actual band width Θ… view at source ↗
Figure 7
Figure 7. Figure 7: Top k rows of UUT . Due to (2.11), we only need to upper bound the entries of UUT in the gray shaded region. We first prove the average case bound (1.7). There are only O(d 2 ) nonzero terms in the double sum in (2.11), and all of them satisfy |y − x| ≥ γ. Using the first part of Proposition 2.1 and taking [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: (Left) An example of a single step U (i) = u [i,4]u [i,3]u [i,2]u [i,1] consisting of M = 4 layers and containing non-local 2-mode beamsplitters. Beamsplitters are shown using connecting lines. (Right) The 4-layer step U (i) shown in the left can be associated with the octahedral graph shown in the right. More formally, let P12(n) denote the set of permutations π ∈ Sn which are a product of disjoint transp… view at source ↗
Figure 9
Figure 9. Figure 9: A brickwall random walk Zt as defined in Corollary 3.2. Modes are labeled from top (1) to bottom (n). Theorem 3.1 follows from the following rephrased version which makes writing the proof more convenient [PITH_FULL_IMAGE:figures/full_fig_p017_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Tree illustration of all possible paths, starting at x or y, in the brickwall circuit. The beamsplitter structure is superimposed on the tree to make indexing the path and π-meeting time easier. The two specific bolded color paths π-meet at time t = 2.5 where they have just passed through the same beamsplitter/phaseshifter. Example 4.2. For a sequence of graphs Gn on n vertices, let t ∗ mix := tmix(o(1/n)… view at source ↗
Figure 11
Figure 11. Figure 11: Illustration for brickwall iteration. We start by peeling off unitaries u [1], u[2] , . . . from the right side of U, which corresponds to advancing a pair of random walks starting at x and y in the left of the above circuit illustration. We need the resulting random walk paths σ1 and σ2 to π-meet in order to apply (4.28) and stop the iteration. With high probability over the choice of paths, this happens… view at source ↗
Figure 12
Figure 12. Figure 12: Illustration of the bounds and cancellation in (5.7) for j = 4. The squares represent matrix elements of D˜, with the diagonal entries of D˜ shown as shaded squares. The bound using column orthonormalization in the first line of (5.7) is represented in blue vertical lines, and the application of the diagonal estimate (5.6) on the k = i term is shown in red horizontal lines. After canceling terms, the rema… view at source ↗
read the original abstract

We study the growth of entanglement and circuit complexity in random passive linear optical networks as a function of the circuit depth. For entanglement dynamics, we start with an initial Gaussian state with all $n$ modes squeezed. For random brickwall circuits, we show that entanglement, as measured by the R\'enyi-2 entropy, grows at most diffusively as a function of the depth. In the other direction, for arbitrary circuit geometries we prove bounds on depths which ensure the average subsystem entanglement reaches within a constant factor of the maximum value in all subsystems, and bounds which ensure closeness of the random linear optical unitary to a Haar random unitary in $L^2$ Wasserstein distance. We also consider robust circuit complexity for random one-dimensional brickwall circuits, as measured by the minimum number of gates required in any circuit that approximately implements the linear optical unitary. Viewing this as a function of the number of modes and the circuit depth, we show the robust circuit complexity for random one-dimensional brickwall circuits scales at most diffusively in the depth with high probability. The corresponding Gaussian unitary $\tilde{\mathcal U}$ for the approximate implementation retains high output fidelity $|\langle\psi|\mathcal U^\dagger \tilde{\mathcal U}|\psi\rangle|^2$ for pure states $|\psi\rangle$ with constrained expected photon-number.

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 analyzes entanglement growth (via Rényi-2 entropy) and robust circuit complexity in finite-depth random passive linear optical networks. Starting from an all-mode squeezed Gaussian state, it proves that for random brickwall circuits entanglement grows at most diffusively with depth; for arbitrary geometries it gives depth thresholds ensuring near-maximal average subsystem entanglement and L2-Wasserstein closeness to Haar-random unitaries. For 1D brickwall circuits it shows that robust circuit complexity (minimum gates in an approximating circuit) scales at most diffusively with depth with high probability, where the approximating Gaussian unitary preserves high fidelity only for pure states with constrained expected photon number.

Significance. If the central bounds hold, the work supplies rigorous scaling results for entanglement and complexity in random linear-optical circuits, a setting where Gaussian states and covariance-matrix techniques permit exact analysis. The diffusive upper bounds and depth thresholds for Haar-like behavior are potentially useful for optical quantum information processing and scrambling studies. The manuscript does not include machine-checked proofs or fully reproducible code, but the parameter-free nature of the random-ensemble analysis is a strength.

major comments (1)
  1. [Abstract and robust circuit complexity section] The robust-complexity claim (diffusive scaling with high probability) is established only for approximating unitaries that retain high fidelity on states with constrained expected photon number. The entanglement results, however, apply the exact random unitary to an initial all-mode squeezed Gaussian state whose photon-number distribution has heavy tails for typical squeezing parameters r ≳ 1. No explicit transfer of the fidelity bound to the squeezed covariance matrix or to the Rényi-2 entropy calculation is provided, so the complexity statement does not yet control the entanglement dynamics studied in the paper.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for highlighting the distinction between the entanglement analysis and the robust complexity result. We address the major comment below and propose a clarifying revision.

read point-by-point responses

  1. Referee: [Abstract and robust circuit complexity section] The robust-complexity claim (diffusive scaling with high probability) is established only for approximating unitaries that retain high fidelity on states with constrained expected photon number. The entanglement results, however, apply the exact random unitary to an initial all-mode squeezed Gaussian state whose photon-number distribution has heavy tails for typical squeezing parameters r ≳ 1. No explicit transfer of the fidelity bound to the squeezed covariance matrix or to the Rényi-2 entropy calculation is provided, so the complexity statement does not yet control the entanglement dynamics studied in the paper.

    Authors: The entanglement growth bounds are obtained by direct, exact calculation: the random brickwall unitary is applied to the all-mode squeezed Gaussian state, and the Rényi-2 entropy is computed from the resulting covariance matrix via the standard symplectic formalism. This derivation never invokes an approximating circuit or any fidelity guarantee. The robust circuit complexity result is introduced separately in the manuscript as an additional property of the same random ensemble; it is explicitly restricted to approximating unitaries that preserve high fidelity only for pure states whose expected photon number is bounded (as stated in the abstract). We do not assert that the complexity bound controls, implies, or is required for the entanglement statements; the two are independent results. We agree that the all-mode squeezed state for r ≳ 1 has heavy-tailed photon statistics, so the existing fidelity guarantee does not automatically extend to it, and no such transfer is performed or claimed in the current text. To eliminate any possible misreading, we will insert a brief clarifying paragraph in the robust-complexity section stating that the complexity result applies to a different class of states and is not used to bound the entanglement dynamics of the squeezed initial state. This is a partial revision. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivations are self-contained mathematical bounds

full rationale

The paper derives bounds on Rényi-2 entanglement growth (at most diffusive for brickwall circuits) and robust circuit complexity (also at most diffusive) directly from properties of random linear optical unitaries acting on Gaussian states. These follow from explicit analysis of circuit ensembles, Wasserstein distances to Haar measure, and fidelity bounds for approximate implementations, without reducing any central claim to a fitted parameter, self-definition, or load-bearing self-citation. The distinction between constrained-photon-number states for complexity and all-mode squeezed states for entanglement is handled by separate statements in the abstract and does not create a definitional loop. The work is self-contained against external benchmarks such as random matrix theory and Gaussian optics.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard properties of Gaussian states, passive linear optics, and random matrix ensembles without introducing new free parameters or postulated entities.

axioms (2)
  • domain assumption Passive linear optical transformations are Gaussian unitaries preserving photon number statistics in the appropriate basis
    Invoked when defining the circuit action on squeezed states.
  • domain assumption Random brickwall and arbitrary-geometry circuits are drawn from ensembles allowing averaging over unitary distributions
    Required for the probabilistic statements on entanglement growth and Wasserstein distance.

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Reference graph

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