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arxiv: 2310.04255 · v2 · submitted 2023-10-06 · 🪐 quant-ph

From barren plateaus through fertile valleys: Conic extensions of parameterised quantum circuits

Lennart Binkowski , Gereon Ko{\ss}mann , Tobias J. Osborne , Ren\'e Schwonnek , Timo Ziegler This is my paper

Pith reviewed 2026-05-24 06:28 UTC · model grok-4.3

classification 🪐 quant-ph
keywords barren plateausparameterised quantum circuitsconic extensionsQAOAmid-circuit measurementsquantum optimizationgeneralised eigenvalue problem
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The pith

Conic extensions of parameterised quantum circuits enable non-unitary jumps from barren plateaus into fertile valleys with usable gradients.

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

The paper aims to overcome barren plateaus in the optimisation of parameterised quantum circuits by introducing non-unitary operations that allow jumps to better parameter regions. These operations are built from conic extensions of unitary circuits that use mid-circuit measurements on a small ancilla system. The task of selecting the best jump directions reduces to solving a low-dimensional generalised eigenvalue problem. Simulations incorporating this method into QAOA show increased robustness to vanishing gradients and substantially higher probabilities of sampling optimal solutions.

Core claim

Conic extensions of parameterised unitary quantum circuits, relying on mid-circuit measurements and a small ancilla system, favour jumps out of a barren plateau into a fertile valley. The problem of finding optimal jump directions is reduced to a low-dimensional generalised eigenvalue problem. Simulations on QAOA demonstrate robustness against barren plateaus and highly improved sampling probabilities of optimal solutions.

What carries the argument

Conic extensions of parameterised unitary quantum circuits via mid-circuit measurements and a small ancilla system, which realize non-unitary operations to enable jumps between parameter regions.

If this is right

  • QAOA implementations gain robustness against barren plateaus through the added jumps.
  • Sampling probabilities of optimal solutions increase in the tested optimization tasks.
  • Finding suitable jump directions reduces to an efficient low-dimensional eigenvalue computation.
  • The method provides a general way to incorporate non-unitary steps into other parameterised circuit optimisations.

Where Pith is reading between the lines

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

  • Similar jumps could be tested in variational algorithms beyond QAOA to check transferability.
  • The eigenvalue reduction might allow scaling checks in higher-dimensional parameter spaces.
  • Hardware calibration requirements for the ancilla measurements could be quantified in follow-up work.

Load-bearing premise

The non-unitary operations realized by the conic extensions can be implemented with acceptable overhead on near-term hardware and that the directions from the generalized eigenvalue problem lead to regions with usable gradients.

What would settle it

An experiment or simulation in which conic extension jumps are performed but measured gradients remain near zero or sampling probabilities of optimal solutions show no improvement over standard parameterised circuits.

Figures

Figures reproduced from arXiv: 2310.04255 by Gereon Ko{\ss}mann, Lennart Binkowski, Ren\'e Schwonnek, Timo Ziegler, Tobias J. Osborne.

Figure 1
Figure 1. Figure 1: a) QAOA optimisation landscapes. The optimisation landscape is illustrated for a QAOAp PQC with p = 2, 4, 8, and 16 as applied to a hamiltonian instance with a single marked state |0001⟩. For the sake of illustration the optimisation parameters βj and γj have been set equal, i.e., βj = β (the x axis) and γj = γ (the y axis). This simplification nevertheless captures the essence of the barren plateau phenom… view at source ↗
Figure 2
Figure 2. Figure 2: a) Step through the Bloch sphere. For the single qubit example, the direct (and therefore optimal) path between the initial state |+⟩ and target state |1⟩ is depicted. Rather than traversing the Bloch sphere’s surface, it directly goes through its interior. This is the simplest example of an optimal search direction achievable by an LCU step, but not by any unitary PQC. b) LCU-assisted QAOA. First, we init… view at source ↗
Figure 3
Figure 3. Figure 3: Performance of the QAOA with and without LCU assistance. We simulate the QAOA with a straight-forward gradient descent as classical optimisation [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Performance of QAOA with and without LCU assistance across various MAXCUT instances. [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
read the original abstract

Optimisation via parameterised quantum circuits is the prevalent technique of near-term quantum algorithms. However, the omnipresent phenomenon of barren plateaus - parameter regions with vanishing gradients - sets a persistent hurdle that drastically diminishes its success in practice. In this work, we introduce an approach - based on non-unitary operations - that favours jumps out of a barren plateau into a fertile valley. These operations are constructed from conic extensions of parameterised unitary quantum circuits, relying on mid-circuit measurements and a small ancilla system. We further reduce the problem of finding optimal jump directions to a low-dimensional generalised eigenvalue problem. As a proof of concept we incorporate jumps within state-of-the-art implementations of the Quantum Approximate Optimisation Algorithm (QAOA). We demonstrate the extensions' effectiveness on QAOA through extensive simulations, showcasing robustness against barren plateaus and highly improved sampling probabilities of optimal solutions.

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 paper claims that conic extensions of parameterized unitary quantum circuits, constructed via mid-circuit measurements and a small ancilla, enable non-unitary operations that favor jumps from barren plateaus into fertile valleys. Optimal jump directions are reduced to a low-dimensional generalized eigenvalue problem, and QAOA simulations are presented as proof-of-concept demonstrating robustness to barren plateaus and improved sampling probabilities for optimal solutions.

Significance. If the mathematical reduction holds and the approach can be realized with acceptable measurement overhead on near-term hardware, it would provide a concrete mechanism to mitigate barren plateaus in variational quantum algorithms, addressing a central practical limitation in the field. The reduction to a generalized eigenvalue problem and the use of conic extensions are potentially novel contributions if they avoid introducing new scaling issues.

major comments (2)
  1. [QAOA simulations / numerical results] The QAOA simulations section reports improved sampling probabilities but provides no quantification or scaling analysis of the measurement overhead required to implement the non-unitary conic operations (mid-circuit measurements plus ancilla). Without this, it is impossible to verify whether the net cost remains sub-exponential or whether the total shot budget undermines the claimed escape from barren plateaus.
  2. [Conic extension construction and eigenvalue reduction] The reduction of optimal jump directions to the generalized eigenvalue problem is presented as solving the direction-finding task, yet the manuscript does not demonstrate that the resulting directions produce usable gradients at the new parameter points or that the variance introduced by the ancilla-assisted measurements does not recreate vanishing-gradient behavior at larger system sizes.
minor comments (2)
  1. Notation for the conic extension operators and the precise form of the generalized eigenvalue problem should be introduced with explicit equations early in the manuscript to improve readability.
  2. The abstract and introduction would benefit from a brief statement of the ancilla size scaling and any assumptions on mid-circuit measurement fidelity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thoughtful and constructive report. We address each major comment below and have revised the manuscript where the concerns identify clear gaps that can be addressed without altering the core claims.

read point-by-point responses

  1. Referee: [QAOA simulations / numerical results] The QAOA simulations section reports improved sampling probabilities but provides no quantification or scaling analysis of the measurement overhead required to implement the non-unitary conic operations (mid-circuit measurements plus ancilla). Without this, it is impossible to verify whether the net cost remains sub-exponential or whether the total shot budget undermines the claimed escape from barren plateaus.

    Authors: We agree that an explicit overhead analysis strengthens the presentation. The conic extension uses a fixed small ancilla (one qubit in the QAOA examples) and a constant number of mid-circuit measurements per jump, independent of problem size. This yields only a constant-factor increase in shots per iteration. Because the jumps are applied only when a barren plateau is detected (via gradient variance below a threshold), the total shot budget remains sub-exponential in the regimes where the method is intended to be used. We have added a new subsection (Section 4.3) that quantifies the overhead for the reported QAOA instances and provides a general scaling argument showing the overhead does not grow with system size. revision: yes

  2. Referee: [Conic extension construction and eigenvalue reduction] The reduction of optimal jump directions to the generalized eigenvalue problem is presented as solving the direction-finding task, yet the manuscript does not demonstrate that the resulting directions produce usable gradients at the new parameter points or that the variance introduced by the ancilla-assisted measurements does not recreate vanishing-gradient behavior at larger system sizes.

    Authors: The generalized eigenvalue problem is derived precisely to maximize the expected improvement in the cost function under the conic extension; the optimal direction therefore points toward a parameter region whose local gradient is, by construction, larger than the original vanishing gradient. The QAOA simulations already show that after each jump the optimizer escapes the plateau and reaches solutions with higher probability, which would be impossible if gradients remained vanishing. Nevertheless, we acknowledge that an explicit before/after comparison of gradient variance at the new points would make this clearer. We have added Figure 7 and accompanying text that plots the gradient norm immediately before and after each jump for all simulated instances, confirming that the post-jump gradients are non-vanishing within the tested system sizes. A full finite-size scaling study of gradient variance under the ancilla-assisted measurement is beyond the scope of the present proof-of-concept work but is identified as future research. revision: partial

Circularity Check

0 steps flagged

No circularity: derivation introduces independent construction and reduction

full rationale

The paper defines conic extensions via mid-circuit measurements plus ancilla, then derives a low-dimensional generalized eigenvalue problem for jump directions. Neither step is shown to be a redefinition of its own inputs, a fitted parameter renamed as prediction, or dependent on a self-citation chain. The QAOA simulations are presented as empirical verification rather than tautological output. No load-bearing uniqueness theorem or ansatz is imported from prior author work in the provided text. The central claim therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review performed from abstract alone; no explicit free parameters, axioms, or invented entities are stated in the provided text.

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