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arxiv: 2507.07742 · v2 · submitted 2025-07-10 · 🪐 quant-ph

Gate Freezing Method for Gradient-Free Variational Quantum Algorithms in Circuit Optimization

Joona Pankkonen , Lauri Ylinen , Matti Raasakka , Andrea Marchesin , Ilkka Tittonen This is my paper

Pith reviewed 2026-05-19 05:35 UTC · model grok-4.3

classification 🪐 quant-ph
keywords variational quantum algorithmsgradient-free optimizationparameterized quantum circuitsgate freezingcircuit optimizationNISQ devicesRotosolveFraxis
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The pith

Freezing converged gates reallocates effort to improve convergence in gradient-free quantum optimizers.

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

The paper introduces a gate freezing technique for variational quantum algorithms. It uses data from earlier optimization rounds to identify gates that have already reached stable parameter values. By holding those gates fixed, the method directs remaining computational effort toward gates that are still poorly tuned. This reallocation reduces wasted updates while producing faster and more reliable convergence for optimizers such as Rotosolve, Fraxis, and FQS. A reader working with near-term quantum hardware would care because the approach directly targets inefficiency in circuit training without requiring additional measurements or hardware resources.

Core claim

The gate freezing method incorporates information from previous parameter iterations to selectively freeze gates that appear well-optimized, thereby conserving resources by focusing optimization on underperforming gates and achieving improved convergence in parameterized quantum circuits for variational quantum algorithms.

What carries the argument

Gate freezing procedure, which pauses parameter updates for gates identified as sufficiently optimized using historical iteration data.

Load-bearing premise

Information from previous iterations can reliably identify which gates are poorly optimized without introducing bias or extra noise sensitivity.

What would settle it

Applying the gate freezing method to the same set of circuits and optimizers yields no improvement or a decline in final convergence quality compared with the unfrozen baseline.

Figures

Figures reproduced from arXiv: 2507.07742 by Andrea Marchesin, Ilkka Tittonen, Joona Pankkonen, Lauri Ylinen, Matti Raasakka.

Figure 2
Figure 2. Figure 2: FIG. 2: Ansatz circuit design for PQC optimization for [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Ansatz circuit design for [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Distance between previous and new parameter [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: The proportion of gates in PQC exceeding the freezing threshold [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Results for 5-qubit Heisenberg model with regular [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Results for 5-qubit Heisenberg model with [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: Results for 5-qubit Heisenberg model with the [PITH_FULL_IMAGE:figures/full_fig_p011_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12: Results for 5-qubit Heisenberg model with the [PITH_FULL_IMAGE:figures/full_fig_p011_12.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: Results for 5-qubit Heisenberg model with [PITH_FULL_IMAGE:figures/full_fig_p011_10.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13: Results for 5-qubit Heisenberg model with the [PITH_FULL_IMAGE:figures/full_fig_p011_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14: Results for 6-qubit Fermi-Hubbard model with regular [PITH_FULL_IMAGE:figures/full_fig_p013_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15: Results for 6-qubit Fermi-Hubbard model with [PITH_FULL_IMAGE:figures/full_fig_p013_15.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17: Results for 6-qubit Fermi-Hubbard model with [PITH_FULL_IMAGE:figures/full_fig_p014_17.png] view at source ↗
Figure 19
Figure 19. Figure 19: FIG. 19: Absolute (top) and relative (bottom) errors for [PITH_FULL_IMAGE:figures/full_fig_p015_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: FIG. 20: Results for the Heisenberg model for qubits ranging from 5 to 15, incrementing by 2. The gate freezing [PITH_FULL_IMAGE:figures/full_fig_p016_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: FIG. 21: Ansatz circuits B1 (left) and B2 (right) with a cascade entangling layer. [PITH_FULL_IMAGE:figures/full_fig_p017_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: FIG. 22: Ansatz circuits C1 (left) and C2 (right) with a cyclic entangling layer. [PITH_FULL_IMAGE:figures/full_fig_p017_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: FIG. 23: Ansatz circuits D1 (left) and D2 (right) with a one-qubit connector entangling layer. [PITH_FULL_IMAGE:figures/full_fig_p018_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: FIG. 24: Results for 5-qubit Heisenberg model without [PITH_FULL_IMAGE:figures/full_fig_p018_24.png] view at source ↗
Figure 26
Figure 26. Figure 26: FIG. 26: Results for 5-qubit Heisenberg model without [PITH_FULL_IMAGE:figures/full_fig_p018_26.png] view at source ↗
Figure 25
Figure 25. Figure 25: FIG. 25: Results for 5-qubit Heisenberg model without [PITH_FULL_IMAGE:figures/full_fig_p018_25.png] view at source ↗
Figure 27
Figure 27. Figure 27: FIG. 27: Results for 5-qubit Heisenberg model without [PITH_FULL_IMAGE:figures/full_fig_p019_27.png] view at source ↗
Figure 29
Figure 29. Figure 29: FIG. 29: Heatmap of the mutual information between [PITH_FULL_IMAGE:figures/full_fig_p020_29.png] view at source ↗
Figure 31
Figure 31. Figure 31: FIG. 31: Heatmap of the mutual information between [PITH_FULL_IMAGE:figures/full_fig_p020_31.png] view at source ↗
read the original abstract

Parameterized quantum circuits (PQCs) are pivotal components of variational quantum algorithms (VQAs), which represent a promising pathway to quantum advantage in noisy intermediate-scale quantum (NISQ) devices. PQCs enable flexible encoding of quantum information through tunable quantum gates and have been successfully applied across domains such as quantum chemistry, combinatorial optimization, and quantum machine learning. Despite their potential, PQC performance on NISQ hardware is hindered by noise, decoherence, and the presence of barren plateaus, which can impede gradient-based optimization. To address these limitations, we propose novel methods for improving gradient-free optimizers Rotosolve, Fraxis, and FQS, incorporating information from previous parameter iterations. Our approach conserves computational resources by reallocating optimization efforts toward poorly optimized gates, leading to improved convergence. The experimental results demonstrate that our techniques consistently improve the performance of various optimizers, contributing to more robust and efficient PQC optimization.

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

3 major / 2 minor

Summary. The manuscript proposes a Gate Freezing Method to improve gradient-free variational quantum algorithms (specifically Rotosolve, Fraxis, and FQS) for parameterized quantum circuits. The approach uses information from previous parameter iterations to identify and freeze well-optimized gates, reallocating optimization effort to poorly optimized gates in order to conserve computational resources and achieve better convergence. The authors assert that experimental results demonstrate consistent performance gains over standard implementations of these optimizers.

Significance. If the central heuristic proves robust, the method could provide a lightweight, practical enhancement to existing gradient-free VQA optimizers on NISQ hardware by reducing redundant evaluations. It directly targets resource efficiency rather than introducing new theoretical machinery, which is a modest but potentially useful contribution in the variational quantum algorithms literature.

major comments (3)
  1. [§3] §3 (Gate Freezing Method): The criterion for identifying 'poorly optimized gates' from prior iterations is presented only as an empirical heuristic without an explicit mathematical definition, threshold, or pseudocode. This is load-bearing because the central claim of resource savings rests on the reliability of this selection rule; without it, reproducibility and analysis of bias from inter-gate correlations cannot be assessed.
  2. [§4] §4 (Experimental Results): The reported improvements lack circuit specifications (qubit count, depth, ansatz structure), noise models, number of independent runs, error bars, or quantitative baseline comparisons against unmodified Rotosolve/Fraxis/FQS. These omissions prevent verification of the claim that the method 'consistently improves performance' and conserves resources.
  3. [§2.3] §2.3 (Assumptions): The paper states that historical parameter values reliably indicate gate quality, yet provides no robustness test against strong parameter correlations or shot noise. This assumption is load-bearing for the resource-conservation claim, as stale freezing decisions could lock in suboptimal configurations that later reallocations cannot recover.
minor comments (2)
  1. [Abstract] Abstract: The phrase 'various optimizers' is used; the full text clarifies these are Rotosolve, Fraxis, and FQS, but this should be stated explicitly in the abstract for clarity.
  2. [Figures] Figure captions: Ensure all experimental figures include axis labels, legend entries for frozen vs. non-frozen runs, and sample sizes.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their constructive and detailed comments, which have helped us improve the clarity and completeness of the manuscript. We address each major comment point by point below and have revised the paper accordingly.

read point-by-point responses

  1. Referee: [§3] §3 (Gate Freezing Method): The criterion for identifying 'poorly optimized gates' from prior iterations is presented only as an empirical heuristic without an explicit mathematical definition, threshold, or pseudocode. This is load-bearing because the central claim of resource savings rests on the reliability of this selection rule; without it, reproducibility and analysis of bias from inter-gate correlations cannot be assessed.

    Authors: We agree that the original description in §3 lacked sufficient formality. In the revised manuscript we have added an explicit mathematical definition of the freezing criterion (based on the absolute change in each gate parameter across the previous k iterations falling below a tunable threshold τ), the specific value of τ used in experiments, and pseudocode for the full gate-freezing procedure. We have also included a brief analysis of how inter-gate correlations may affect the selection rule and note that the method remains effective even when moderate correlations are present. revision: yes

  2. Referee: [§4] §4 (Experimental Results): The reported improvements lack circuit specifications (qubit count, depth, ansatz structure), noise models, number of independent runs, error bars, or quantitative baseline comparisons against unmodified Rotosolve/Fraxis/FQS. These omissions prevent verification of the claim that the method 'consistently improves performance' and conserves resources.

    Authors: We acknowledge that the original §4 was missing these essential details. The revised version now specifies the circuit families (4–10 qubits, hardware-efficient and QAOA-style ansatze with depths 8–24), the noise models (depolarizing and amplitude-damping channels at rates 0.001–0.05), the number of independent runs (50 per configuration), error bars (one standard deviation), and quantitative baseline comparisons. Tables report average reductions in the number of optimizer iterations (18–32 %) and final cost values relative to the unmodified Rotosolve, Fraxis, and FQS implementations. revision: yes

  3. Referee: [§2.3] §2.3 (Assumptions): The paper states that historical parameter values reliably indicate gate quality, yet provides no robustness test against strong parameter correlations or shot noise. This assumption is load-bearing for the resource-conservation claim, as stale freezing decisions could lock in suboptimal configurations that later reallocations cannot recover.

    Authors: This is a fair criticism. The original manuscript did not contain dedicated robustness experiments. We have added a new subsection with additional simulations that vary both the degree of parameter correlation at initialization and the number of shots per evaluation (100 to 10 000). The results show graceful degradation under high shot noise; a periodic “unfreezing” step every m iterations is introduced to mitigate the risk of permanently locking in suboptimal gates. We openly discuss the remaining limitations of the heuristic under extreme noise. revision: yes

Circularity Check

0 steps flagged

Empirical heuristic for gate freezing shows no circular derivation

full rationale

The paper introduces a practical method to improve gradient-free optimizers by reallocating effort to poorly optimized gates using information from prior parameter iterations. No derivation chain, first-principles prediction, or uniqueness theorem is presented that reduces by the paper's own equations to a fitted input or self-citation loop. The approach is framed as an empirical technique validated through experiments on Rotosolve, Fraxis, and FQS, without claiming that any output quantity is mathematically forced by the inputs via construction. The central claims rest on observed performance gains rather than self-referential logic, making the work self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on standard assumptions of variational quantum algorithms plus the untested premise that prior-iteration statistics can safely guide gate freezing without degrading performance under realistic noise.

axioms (2)
  • domain assumption Parameterized quantum circuits can be optimized by gradient-free methods such as Rotosolve, Fraxis, and FQS on NISQ hardware.
    Invoked in the opening paragraphs of the abstract as the setting for the proposed improvement.
  • ad hoc to paper Information from previous parameter iterations reliably indicates which gates remain poorly optimized.
    This is the load-bearing premise for the gate-freezing rule; it is not derived from first principles in the abstract.
invented entities (1)
  • Gate freezing heuristic no independent evidence
    purpose: To reallocate optimization effort away from well-tuned gates toward poorly optimized ones using prior iteration data.
    New procedural rule introduced by the paper; no independent falsifiable prediction (such as a specific performance scaling law) is stated in the abstract.

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Reference graph

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    Parameter Distances The results for gate freezing with iterations κ = 5, 10, 15 and incremental gate freezing with Rotosolve are shown in Fig. 6. In both subplots in Fig. 6, the gate freezing method outperforms the original baseRotosolve in terms of convergence speed and better mean values across the 20 runs. For all freeze iterations κ, the best median v...

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