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arxiv: 2606.09734 · v1 · pith:JCSONW3Tnew · submitted 2026-06-08 · 🪐 quant-ph · cs.LG

Adaptive directional gradients for parameterised quantum circuits

Brian Coyle , Snehal Raj , Virag Umathe , El Amine Cherrat , Elham Kashefi This is my paper

Pith reviewed 2026-06-27 16:29 UTC · model grok-4.3

classification 🪐 quant-ph cs.LG
keywords parameterised quantum circuitsforward gradientsgradient estimationvariational quantum algorithmsmeasurement costSPSAparameter-shift ruleadaptive optimization
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The pith

Forward-mode directional derivatives yield unbiased gradient estimates for parameterised quantum circuits at tunable measurement cost.

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

The paper sets out to establish that an unbiased estimator of the gradient for parameterised quantum circuits can be obtained by averaging any number of random directional derivatives computed via the forward mode. This construction requires no ancilla qubits or controlled gates and recovers the parameter-shift rule, SPSA, and random coordinate descent as special cases when the number of directions is varied. A sympathetic reader would care because the dominant cost in hardware training of such circuits is the number of measurements needed for gradients, and the new estimator allows that cost to be traded against variance in a controlled way. From the same second-moment analysis the authors derive an adaptive optimiser, QUIVER, whose shot allocation follows from a closed-form minimum-cost rule. Numerical experiments then show that circuits with up to 1770 parameters on 60 qubits can be trained orders of magnitude more efficiently than with the parameter-shift rule alone.

Core claim

A framework of forward gradient estimators for PQCs, based on the forward mode of automatic differentiation, yields an unbiased estimator of the gradient by averaging a freely tunable number of random directional derivatives and recovers SPSA, random coordinate descent, and the parameter-shift rule as limiting cases, with no ancilla qubits or controlled-gate overhead. Stochastic quantum forward gradient descent converges under standard assumptions, with an explicit second-moment expansion that interpolates between the single-direction extreme of SPSA and the full-gradient extreme of parameter-shift. Within this framework the authors derive QUIVER, an adaptive optimiser whose update rule foll

What carries the argument

The stochastic forward gradient estimator obtained by averaging a tunable number of random directional derivatives of the circuit output expectation value.

If this is right

  • Stochastic forward gradient descent converges under the same assumptions used for classical SGD.
  • The variance of the estimator interpolates continuously between the SPSA and parameter-shift extremes.
  • QUIVER's closed-form shot allocation minimises total measurement cost for a target variance.
  • Circuits with 60 qubits and 1770 parameters train orders of magnitude faster than with the parameter-shift rule.
  • QUIVER outperforms iCANS and gCANS on QAOA and VQE benchmark problems.

Where Pith is reading between the lines

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

  • The same directional-derivative construction could be applied to any variational quantum algorithm whose cost function is an expectation value.
  • Because the method is ancilla-free it may combine directly with existing error-mitigation protocols without increasing circuit depth.
  • At large parameter counts the optimal number of directions may become a hyper-parameter that itself needs adaptive tuning.
  • If the variance model holds, similar cost-optimal allocation rules could be derived for other stochastic estimators used in quantum machine learning.

Load-bearing premise

That the second-moment expansion of the directional-derivative estimator correctly predicts variance under the measurement-cost model used to derive QUIVER's allocation rule.

What would settle it

Compute the empirical bias of the averaged directional derivative estimator on a single-parameter circuit whose analytic gradient is known; the bias must remain zero for any finite number of directions.

Figures

Figures reproduced from arXiv: 2606.09734 by Brian Coyle, El Amine Cherrat, Elham Kashefi, Snehal Raj, Virag Umathe.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
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Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
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Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
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Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
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Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
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Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
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Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
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Figure 10. Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p017_10.png] view at source ↗
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Figure 11. Figure 11: FIG. 11 [PITH_FULL_IMAGE:figures/full_fig_p022_11.png] view at source ↗
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Figure 13. Figure 13: FIG. 13 [PITH_FULL_IMAGE:figures/full_fig_p035_13.png] view at source ↗
read the original abstract

Training parameterised quantum circuits (PQCs) on quantum hardware is bottlenecked by the measurement cost of gradient estimation, which under the parameter-shift rule scales linearly in the number of trainable parameters and dominates the total shot budget of training at scale. In this work, we propose a framework of forward gradient estimators for PQCs, based on the forward mode of automatic differentiation, that yields an unbiased estimator of the gradient by averaging a freely tunable number of random directional derivatives and recovers SPSA, random coordinate descent, and the parameter-shift rule as limiting cases, with no ancilla qubits or controlled-gate overhead. We prove that stochastic quantum forward gradient descent converges under standard assumptions, with an explicit second-moment expansion that interpolates between the single-direction extreme of SPSA and the full-gradient extreme of parameter-shift. Within this framework we derive QUIVER (Quantum Iterative V-adaptive Estimator Rule), an adaptive optimiser for parameterised circuits whose update rule follows from a closed-form minimum measurement-cost allocation. We show numerically that forward gradients train Hamming-weight-preserving orthogonal quantum neural networks with up to 60 qubits and 1770 parameters on the ECG5000 and MNIST datasets orders of magnitude more efficiently than the parameter-shift rule. We also demonstrate that our proposed QUIVER optimiser can outperform iCANS and gCANS measurement-frugal optimisers on optimisation problems using the quantum approximate optimisation algorithm and quantum simulation with the variational quantum eigensolver.

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 / 1 minor

Summary. The manuscript proposes a framework of forward gradient estimators for parameterised quantum circuits based on forward-mode automatic differentiation. It constructs an unbiased gradient estimator by averaging a tunable number of random directional derivatives, recovering SPSA, random coordinate descent, and the parameter-shift rule as limiting cases without ancilla qubits or controlled gates. The authors prove convergence of stochastic forward gradient descent under standard assumptions, supply an explicit second-moment expansion of the estimator, and derive the QUIVER adaptive optimizer from a closed-form minimum-measurement-cost allocation rule. Large-scale numerical results are presented for training up to 60-qubit, 1770-parameter Hamming-weight-preserving orthogonal quantum neural networks on ECG5000 and MNIST, as well as comparisons on QAOA and VQE problems against iCANS and gCANS.

Significance. If the central claims hold, the work provides a tunable, ancilla-free alternative to the parameter-shift rule that can substantially reduce measurement overhead for large PQCs. The explicit convergence proof for stochastic quantum forward gradient descent and the large-scale numerical demonstrations on circuits with 1770 parameters constitute clear strengths. The QUIVER rule offers a principled adaptive strategy whose practical advantage, however, is tied to the validity of the underlying variance model.

major comments (1)
  1. [section deriving the QUIVER allocation rule and second-moment expansion] The second-moment expansion used to derive the closed-form QUIVER allocation rule assumes a specific measurement-cost model under which the variance interpolates between the SPSA (single-direction) and parameter-shift (full-basis) extremes. For general PQCs this scaling may be violated by circuit-specific correlations, non-independent shot noise, or the multi-frequency dependence of f(θ + t v) when v is non-coordinate; in that case the derived allocation ceases to be optimal and the headline measurement-efficiency claims for QUIVER no longer follow. This assumption is load-bearing for the adaptive optimizer and the numerical advantage reported in the experiments.
minor comments (1)
  1. [Abstract and numerical experiments] The abstract and experimental sections report large efficiency gains but omit error bars, dataset splits, and explicit variance-model parameters; adding these would strengthen verifiability of the comparisons.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the key assumptions in the QUIVER derivation. We respond to the major comment below.

read point-by-point responses

  1. Referee: The second-moment expansion used to derive the closed-form QUIVER allocation rule assumes a specific measurement-cost model under which the variance interpolates between the SPSA (single-direction) and parameter-shift (full-basis) extremes. For general PQCs this scaling may be violated by circuit-specific correlations, non-independent shot noise, or the multi-frequency dependence of f(θ + t v) when v is non-coordinate; in that case the derived allocation ceases to be optimal and the headline measurement-efficiency claims for QUIVER no longer follow. This assumption is load-bearing for the adaptive optimizer and the numerical advantage reported in the experiments.

    Authors: The second-moment expansion is derived under the explicit assumption of independent additive shot noise with variance scaling as 1/M per direction. This produces the stated interpolation and the closed-form allocation. We agree that circuit-specific correlations, non-independent noise, or multi-frequency effects in non-coordinate directions can violate the model, rendering the allocation suboptimal in those cases. The estimator itself remains unbiased for any choice of directions. The reported numerical advantages are observed on the specific circuits tested (Hamming-weight-preserving QNNs, QAOA, VQE). We will revise the manuscript to state the variance-model assumptions more prominently, add a limitations paragraph discussing potential violations, and qualify the optimality claims for general PQCs. This is a partial revision. revision: partial

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The core estimator is obtained directly from forward-mode automatic differentiation and is unbiased by construction. The second-moment expansion is stated to be explicit and derived from the estimator itself, interpolating between known limits. QUIVER follows from a closed-form allocation rule under an explicitly stated measurement-cost model; this is a derivation under assumptions rather than a reduction of the result to its inputs by definition or by fitting. No self-citations are invoked as load-bearing uniqueness theorems, no ansatzes are smuggled, and no known empirical patterns are merely renamed. The derivation chain is therefore self-contained.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The framework relies on standard stochastic-gradient convergence assumptions and an explicit measurement-cost model whose parameters enter the QUIVER allocation; no new physical entities are introduced.

free parameters (1)
  • number of random directions
    Tunable integer that controls the bias-variance tradeoff of the gradient estimator and directly affects total measurement cost.
axioms (2)
  • domain assumption Standard assumptions for convergence of stochastic gradient descent
    Invoked to prove convergence of stochastic quantum forward gradient descent with the stated second-moment expansion.
  • domain assumption Measurement cost is linear in the number of directional derivative estimates and independent of circuit depth
    Underlies the closed-form minimum-cost allocation rule for QUIVER.

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    IfE[∥g (t)(θ)∥2]≤β 2∥∇f(θ)∥ 2 for allθ, tandη= 1/(Lβ 2), then E[f(θ (T) )]−f(θ ⋆)≤ 1− µ Lβ2 T f(θ (0))−f(θ ⋆) .(D6) Proof of Proposition 4.Part (i).By App. A,E[ egF(θ)] =∇f(θ), so the estimator is unbiased. By Lemma 4 with κ= 1(Rademacher), E ∥egF∥2 = N+V−1 V ∥∇f∥2 =:β 2 ∥∇f∥2. This is the bounded relative second moment condition of part 2 of Lemma 5. Set...

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    Proof of Corollary 1: CRB-level optimality The proof has two parts. First we establish the Cramér–Rao lower bound eq. (40) on the MSE of any unbiased estimator ofgthat queries the shot-noise oracle a total ofBtimes. Second we show that the forward-gradient estimator at the optimal allocation of Theorem 1 attains this bound up to a constant that vanishes a...