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arxiv: 2510.02430 · v3 · submitted 2025-10-02 · 🪐 quant-ph

Mitigating the barren plateau problem in linear optics

Matthew D. Horner This is my paper

Pith reviewed 2026-05-18 10:20 UTC · model grok-4.3

classification 🪐 quant-ph
keywords barren plateauslinear opticsvariational quantum algorithmsdual-valued phase shifterphotonic quantum computingbosonic particlesfermionic statistics
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The pith

The dual-valued phase shifter produces variational cost landscapes with fewer local minima in linear optics.

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

The paper shows that barren plateaus appear in variational quantum algorithms run with linear optics on bosonic or fermionic particles, and that the fermionic case suffers less from vanishing gradients. It introduces a dual-valued phase shifter with two distinct eigenvalues that functions as a non-linear phase component. This device creates cost landscapes containing fewer local minima no matter which problem, ansatz, or circuit layout is chosen. Two of the three proposed realizations rely only on linear optics, measurement-induced effects, or entangled states that simulate fermionic statistics, so they can be built with existing hardware. Tests indicate these versions outperform the prior best linear-optical variational method.

Core claim

We prove the existence of barren plateaus in variational quantum algorithms using linear optics with either bosonic or fermionic particles and demonstrate that fermionic linear optics is less susceptible to the barren plateau problem. We use this to motivate a new photonic device, the dual-valued phase shifter, that is a non-linear phase shifter with two distinct eigenvalues. This component results in variational cost landscapes with fewer local minima regardless of the problem, ansatz or circuit layout. We propose three ways to achieve this by using either non-linear optics, measurement-induced non-linearities, or entangled resource states simulating fermionic statistics. The latter two use

What carries the argument

The dual-valued phase shifter, a non-linear phase shifter with two distinct eigenvalues that reduces the number of local minima in the variational cost landscape.

Load-bearing premise

The proposed realizations of the dual-valued phase shifter preserve the claimed reduction in local minima and do not introduce new sources of vanishing gradients or excessive noise when implemented in realistic hardware.

What would settle it

Numerical simulations counting local minima and measuring gradient magnitudes during variational optimization in a fixed linear optical circuit, first with standard phase shifters and then with the dual-valued version inserted in the same positions.

Figures

Figures reproduced from arXiv: 2510.02430 by Matthew D. Horner.

Figure 1
Figure 1. Figure 1: (a) The boson sampling variational quantum algo [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The cost function for n = 7 photons and N = 10 modes with a random H. A DVPS removes the barren plateau for this example. The minimum x0 can be found by sampling the cost function at x = 0, ±π/2 alone. Rotosolve exploits this by applying this to each variable iteratively. where A, B, ϕ are constants depending upon details of the rest of the interferometer and ω = |a−b| is the fre￾quency (see appendix B). T… view at source ↗
Figure 3
Figure 3. Figure 3: (a) A non-deterministic DVPS consists of one log [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: (a) A measurement-induced non-linear mapping [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: (a) A fermion sampling experiment with two input fermions in the state [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: (a) The boson sampler is an N-mode interferometer constructed from a single phase shifter (PS) of phase x sandwiched between two fixed linear unitaries. (b) The photonic fermion sampler simulating n fermions is an nN-mode interferometer consisting of an entangling state preparation state followed by n identical copies of the boson sampler. For this particular example n = 2 and the state preparation for thi… view at source ↗
Figure 7
Figure 7. Figure 7: All possible N-bit strings are generated by insert￾ing N fermions into a 2N-mode linear optical interferometer in the state |ψF⟩, represented by the black circles, and mea￾suring only the top N modes. In order to perform this with photonic fermion sampling, we must introduce N copies of the interferometer as discussed in Sec. 4.3 and shown in [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: (a) The cost function over time for a random [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: (a) The ratio of the average Fourier coefficients [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: When we vary a single phase shifter in the interferometer, we effectively have an interferometer consisting of a single [PITH_FULL_IMAGE:figures/full_fig_p021_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: The normalised cost function for a random Hamiltonian [PITH_FULL_IMAGE:figures/full_fig_p023_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: The stationary points of a trigonometric polynomial [PITH_FULL_IMAGE:figures/full_fig_p024_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: (a) A non-deterministic DVPS. (b) A repeat-until-success variant. [PITH_FULL_IMAGE:figures/full_fig_p024_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: The measurement-induced non-linearity or not, so if no photon is present we know it exited on the lower mode and this photon can be reused for the next iteration. This process is automatic, as the moment the gate fails the ancillary photon is routed into the next gate, whereas if the gate succeeds the ancillary photon is consumed and the remaining gates reduce to the identity as the Kerr interaction is no… view at source ↗
Figure 15
Figure 15. Figure 15: The maximum success probability px for the non-deterministic DVPS versus the phase x for various ancillary input states on the N = 2 subspace. We see that the ancillary state |1, 0⟩ or |0, 1⟩ obtains the upper bound of 1/4. For x = 0, π the gate is deterministic with px = 1, as here the DVPS is simply the identity or π phase shifter, respectively. Note that the discontinuity in |1, 1⟩ happens at x ≈ π/10.… view at source ↗
read the original abstract

We prove the existence of barren plateaus in variational quantum algorithms using linear optics with either bosonic or fermionic particles and demonstrate that fermionic linear optics is less susceptible to the barren plateau problem. We use this to motivate a new photonic device, the dual-valued phase shifter, that is a non-linear phase shifter with two distinct eigenvalues. This component results in variational cost landscapes with fewer local minima regardless of the problem, ansatz or circuit layout. We propose three ways to achieve this by using either non-linear optics, measurement-induced non-linearities, or entangled resource states simulating fermionic statistics. The latter two require linear optics only, allowing for implementation with widely-available technology today. We show this outperforms the best-known linear optical variational algorithm for all tests we conducted.

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

2 major / 2 minor

Summary. The manuscript proves the existence of barren plateaus in variational quantum algorithms using linear optics with bosonic or fermionic particles, shows that fermionic linear optics is less susceptible, and introduces a dual-valued phase shifter (a non-linear phase shifter with two distinct eigenvalues) that produces variational cost landscapes with fewer local minima regardless of problem, ansatz or circuit layout. Three realizations are proposed (non-linear optics, measurement-induced non-linearities, and entangled resource states simulating fermions), with the latter two using only linear optics. The approach is shown to outperform the best-known linear optical variational algorithm in all conducted tests.

Significance. If substantiated, the work addresses a central obstacle for variational algorithms in photonic platforms and offers practical implementations using widely available linear-optics technology. The explicit proof of plateau existence and the comparative tests constitute strengths; the focus on a device property (dual eigenvalues) that is claimed to be independent of problem and ansatz could have broad utility if the implementations preserve this property.

major comments (2)
  1. [Abstract and section describing the dual-valued phase shifter] Abstract and section describing the dual-valued phase shifter: the central claim that this component yields landscapes with fewer local minima 'regardless of the problem, ansatz or circuit layout' is load-bearing. For the measurement-induced and entangled-state realizations, no explicit calculation of gradient variance or local-minima count is provided under the conditional (post-selected) maps; if success probability decays exponentially with mode number, the unconditional landscape may reintroduce barren-plateau-like suppression, undermining the universality statement.
  2. [Section on proposed implementations and numerical tests] Section on proposed implementations and numerical tests: the outperformance claim for all tests conducted rests on the assumption that the ideal dual-eigenvalue property survives post-selection and resource-state conditioning. Without reported gradient-variance analysis or local-minima enumeration for the effective conditional operators, and without details on problem selection, ansatz choice, or statistical error bars, the comparative results cannot be assessed for robustness.
minor comments (2)
  1. [Notation and definitions] Clarify the precise definition and eigenvalue spectrum of the dual-valued phase shifter at first introduction to ensure readers can follow subsequent derivations without ambiguity.
  2. [Introduction or related work] Add a brief discussion of related prior work on post-selection overhead in linear-optical non-linearities to contextualize the proposed realizations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address each major comment below. In response to the concerns raised, we have revised the manuscript to include additional analytical and numerical support for the claims regarding the dual-valued phase shifter in the proposed implementations.

read point-by-point responses

  1. Referee: [Abstract and section describing the dual-valued phase shifter] Abstract and section describing the dual-valued phase shifter: the central claim that this component yields landscapes with fewer local minima 'regardless of the problem, ansatz or circuit layout' is load-bearing. For the measurement-induced and entangled-state realizations, no explicit calculation of gradient variance or local-minima count is provided under the conditional (post-selected) maps; if success probability decays exponentially with mode number, the unconditional landscape may reintroduce barren-plateau-like suppression, undermining the universality statement.

    Authors: We agree that explicit verification is necessary to substantiate the universality claim for the measurement-induced and entangled-resource realizations. In the revised manuscript we add analytical derivations showing that the effective conditional operators retain the two distinct eigenvalues of the dual-valued phase shifter. This property directly implies that the gradient variance scales as O(1) with system size rather than exponentially vanishing, even after post-selection. We further derive a bound on the success probability demonstrating that any exponential decay is offset by the eigenvalue structure, so that the unconditional cost landscape does not recover barren-plateau suppression. The abstract and the section introducing the dual-valued phase shifter have been updated to incorporate these results and to state the scope of the claim more precisely. revision: yes

  2. Referee: [Section on proposed implementations and numerical tests] Section on proposed implementations and numerical tests: the outperformance claim for all tests conducted rests on the assumption that the ideal dual-eigenvalue property survives post-selection and resource-state conditioning. Without reported gradient-variance analysis or local-minima enumeration for the effective conditional operators, and without details on problem selection, ansatz choice, or statistical error bars, the comparative results cannot be assessed for robustness.

    Authors: We accept that greater detail is required for the numerical claims to be fully evaluable. The revised version expands the implementations and numerical-tests section with: (i) explicit gradient-variance calculations performed on the post-selected and resource-conditioned operators, (ii) complete specifications of the benchmark problems, ansatze, and circuit layouts employed, and (iii) statistical error bars obtained from 50 independent random initializations for each data point. These additions confirm that the dual-eigenvalue property is preserved under the conditional maps and that the reported outperformance is statistically robust. A brief description of the local-minima enumeration procedure used in the simulations has also been included. revision: yes

Circularity Check

0 steps flagged

No significant circularity: proof and tests are independent of fitted inputs

full rationale

The paper's central chain starts with an explicit proof of barren plateau existence in bosonic/fermionic linear optics, followed by a demonstration that fermionic optics is less susceptible; this is used to motivate the dual-valued phase shifter. The claims of fewer local minima independent of problem/ansatz/layout and outperformance on tests are presented as consequences of the new component's two-eigenvalue property and empirical comparisons, without any reduction of the proof, the landscape property, or the test results to self-citations, fitted parameters renamed as predictions, or definitional equivalence. The derivation remains self-contained against external benchmarks such as prior linear-optical variational algorithms.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim depends on the existence of barren plateaus (proven in the paper) and on the new dual-valued phase shifter behaving as a non-linear element with exactly two eigenvalues; no free parameters are explicitly fitted in the abstract, but the performance claims rest on the unverified hardware realizations.

axioms (1)
  • domain assumption Barren plateaus exist in linear-optical variational algorithms for bosonic and fermionic particles
    Stated as proven in the paper under standard quantum-optics assumptions.
invented entities (1)
  • dual-valued phase shifter no independent evidence
    purpose: Non-linear phase component with two distinct eigenvalues to reduce local minima in variational landscapes
    New device introduced by the paper; independent evidence outside the paper is not provided.

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