What a Reporting Convention Hides: A Matched-Budget Audit of Quantum Natural Gradient with an Exactly Computed Metric

Lu Wei; Yufeng Wang; Haibin Ling

cs.LGOct 7, 2026

What a Reporting Convention Hides: A Matched-Budget Audit of Quantum Natural Gradient with an Exactly Computed Metric

Authors: Lu Wei, Yufeng Wang, Haibin Ling

Organizations: Stony Brook University · Westlake University

Abstract

Several published comparisons of variational quantum optimizers time only runs that reach a target loss, or read the verdict at a single target. Either convention could decide whether an optimizer's costlier steps pay off. We measure how much each convention changes verdicts among Adam, simultaneous perturbation stochastic approximation (SPSA) and quantum natural gradient (QNG), on initializations held out from the selection of settings. We compute the exact metric that preconditions QNG, price every step in circuit evaluations and give every method the same budget. In a median pooled over circuit widths, cost families and a sweep of settings with common misses, SPSA needs more than twice Adam's evaluations to reach a loose target. Dropping the censored runs that miss the target hides this gap. On the global-cost family we hold fixed the settings selected for a strict target. QNG then usually reaches the loose target after Adam but the strict target first. QNG's strict-target lead disappears when the metric's simulator price, linear in the number of parameters, is replaced by an assumed hardware count quadratic in that number. We recommend charging every miss the budget and reporting verdicts across targets.

Figures & tables

Figure 1: The target

\tau

defines the crossing times

T

, and

\times

marks a run that does not reach

\tau

within the budget

B

. The choice of which runs enter then acts on these times in both branches (Figure 3 ). Medians are conditional (reached runs only) or budget-capped (misses charged the budget). The paired branch counts which method needs fewer evaluations to the target, a miss counting as the budget, and takes the median of the paired differences.

Figure 2: (a) The testbed. Each of the

L

layers applies

R_{Y}

and then

R_{Z}

to every qubit and then the ring

E

of controlled-

Z

gates, and the loss

C(\theta)

follows from the probabilities of measuring 0 (Section 3 ). Each optimizer step adds its price in circuit evaluations to the run’s count,

2p

for Adam,

2p+(p+1)

for QNG and

2

for SPSA, up to the budget

B=54p

. (b) Adam (

\eta=0.3

) and QNG (

\eta=1

\lambda=10^{-1}

), with settings selected for

\tau=0.01

, from one of the 20 new global-cost initializations at

n=6

(

B=3888

) whose

\tau=0.01

paired difference is nearest the median (rule in Appendix D ). Adam is first at

\tau=0.20

and QNG at

\tau=0.01

, which Adam misses within

B

. A black-edged marker is a run’s first step at or below a target.

Choice	Comparison; held fixed	Reading 1	Reading 2	Verdict change absent when
Which runs enter (Sec. 4 )	SPSA vs. Adam, ratio of median times pooled over eight configurations; original grid and initializations, loose target, SPSA’s shared perturbation sequence	Reached runs only 1.11 (0.84–1.40)	Misses charged the budget 2.24 (1.60–2.69)	Settings selected for the loose target (every Adam and SPSA run reaches it on both initialization sets, so the readings coincide) or one configuration taken alone (Reading 1 still shrinks the ratio but leaves SPSA slower in 7 of 8)
At which target (Sec. 5 )	QNG vs. Adam, number of initializations with QNG first and median paired difference; global-cost family, 80 new initializations, strict-target settings, simulator price	Loose target, $\tau=0.20$ 12 of 80 $+172$ (121.5–255)	Strict target, $\tau=0.01$ 77 of 80 $-958.5$ ( $-1099.5$ to $-842.5$ )	Settings selected for the loose target, evaluated at the strict target (40–29–11, sign-test p-value 0.23; Adam first at 5 and 6 qubits, 2–14–4 and 1–12–7), or the metric charged $p(p+1)/2$ evaluations per step (QNG first from 0 of 80 at the loose and 3 of 59 non-tied at the strict target)

Table 1: Each row gives one comparison under two readings on its own fixed set of runs. Pooled medians and median paired differences (QNG minus Adam) mix budgets. Bootstrap intervals, with initializations resampled within configurations, are Bonferroni-adjusted 99.5% (ten ratios per initialization set) in row 1 and 95% in row 2. In row 2 first means fewer evaluations to

\tau

, a miss counting as the budget, and counts are QNG first–Adam first–tied. With each target’s own settings, on different runs at the two targets, Adam is first from 66 of 80 at the loose and QNG from 77 at the strict target. Section 3 gives the definitions.

Figure 3: Two reporting choices, each applied to its own fixed set of runs. Only Adam’s

\eta=0.3

runs on the new global-cost initializations enter both panels (Appendix D ). (a) Original grid at

\tau=0.20

, original and new initializations in separate rows, all settings and the eight configurations of both cost families pooled, so medians mix budgets. Open markers use reached runs only, and filled ones charge every miss

B

. The line at 1 is Adam, whose runs all reach the target. SPSA’s shared sequence is defined in Section 3 . Bars are Bonferroni-adjusted 99.5% bootstrap intervals, ten ratios per initialization set. (b) Global-cost family, 80 new initializations, settings selected for

\tau=0.01

, held fixed. A grey marker merges initializations with the same Adam and the same QNG step count, each displaced by at most 1%, area proportional to their number. At Adam’s value 1, Adam reached the target at its last step. A cross in a strip is a miss, placed at the other run’s value, and position across strips carries no value. First means fewer evaluations to the target, a miss counting as

B

, and counts include the strips. The metric is charged its simulator price. Settings selected for

\tau=0.20

give other counts (Section 5 ). The circled marker is the pair of Figure 2 (b).

Depth $L$	1	2	3	4	5	6
$p={}$ true rank	12	24	36	48	60	72
$\log_{10}\kappa(g)$	1.87	3.64	4.05	4.89	5.44	5.22
Effective rank	8.9	8.0	9.2	9.5	9.6	10.5
Alignment	1.000	0.434	0.436	0.330	0.301	0.294

Table 2: The exact metric

g

for the global cost at

n=6

is measured apart from the optimizer runs, not per run, at 20 random initial points per depth; entries are means, without an uncertainty estimate. The true rank counts the eigenvalues

{\lambda_{\max}\geq\dots\geq\lambda_{\min}}

g

above

10^{-10}\lambda_{\max}

and equals

p

, so every point is full rank and

{\kappa(g)=\lambda_{\max}/\lambda_{\min}}

. Effective rank is

\operatorname{tr}(g)/\lambda_{\max}

; alignment is

{\cos(\nabla C,(g+\lambda I)^{-1}\nabla C)}

at damping

{\lambda=10^{-3}}

Appendix figures & tables13 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 4: Gradient variance against width for (a) the global-cost family and (b) the local-cost family of Section 3 , with a shared vertical axis. Here

\partial_{1}C

is the derivative of the loss

C

with respect to the

R_{Y}

angle of the first qubit in the first layer. In (a) and (b), each marker is the variance of

\partial_{1}C

over 200 initial points with independent, uniformly random angles. Each line is the least-squares fit of the natural logarithm of this variance against

n

over one set of random initial points, and each set draws its own points. (c) The fitted slope of each set for each family with its unadjusted percentile bootstrap interval from 400 replicates that resample the derivatives at each

n

. Marker shape and grey level identify the set, and the dashed line in (c) marks zero slope.

Run set	Settings	Initializations	Targets	Used for
Original grid	Adam and SPSA $\eta\in\{0.1,0.3\}$ ; QNG also $\lambda\in\{10^{-3},10^{-1}\}$	10 original	six, from 0.5 to 0.01	Section 4 ; best original setting; fixing the strict target
Extended grid	six $\eta$ from 0.03 to 10 for Adam and SPSA; 27 settings for QNG, $\lambda$ from $10^{-5}$ to 1	10 original	0.20, 0.01	The 48 selections (Table 4 )
New initializations	every original and every selected setting	20 new, held out from selection	0.20, 0.01; six at the strict-target selection	Replication of Section 4 ; main comparison and out-of-sample tests of Section 5
Five SPSA sequences	every SPSA setting in the rows above	10 original; 20 new	0.20, 0.01	Per-initialization median of five sequences (Appendix E )
Budgets $2B$ and $4B$	strict-target selections; global-cost loose-target selections	10 original and 20 new; 10 original	0.20, 0.01 and final loss; final loss	Counts and final loss at longer budgets (Appendices J and L )

Appendix

Table 3: The sets of runs of Section 3 . Initialization counts are per configuration, and the three methods share the initializations; every run is charged the per-step prices of Section 3 within its budget.

		Adam $\eta$		QNG				SPSA $\eta$
		$\tau=0.20$	$\tau=0.01$	$\tau=0.20$		$\tau=0.01$		$\tau=0.20$	$\tau=0.01$
Family	$n$			$\eta$	$\lambda$	$\eta$	$\lambda$
global-cost	3	0.3	0.3	1	$10^{-1}$ $\dagger$	1	$10^{-1}$	1	1
	4	0.3	0.3	1	$10^{-1}$ $\dagger$	1	$10^{-1}$	1	1
	5	0.3	0.3	1	$10^{-2}$	1	$10^{-1}$	1	1
	6	0.3	0.3	1	$10^{-2}$	1	$10^{-1}$	1	1

Appendix

Table 4: Selected learning rate

\eta

, and for QNG damping

\lambda

, for each method, configuration and target on the extended grid. The selected setting has the lowest budget-capped median over its 10 runs at the budget

B

, and ties go to the lower median final loss, then to the smaller

\eta

, then to the smaller

\lambda

. SPSA uses its shared perturbation sequence, and QNG’s metric is charged at its simulator price. Section 3 defines the budget-capped median and the per-step prices. Selection used only the 10 original initializations of each configuration and none of the 20 new ones. The marks identify the 8 selections that the tie-break decided.

Perturbation sequence	Conditional	Budget- capped	Reached of 160	Conditional	Conditional	Budget- capped	Reached of 320	Conditional
	Original initializations				New initializations
	SPSA/Adam, $\tau=0.20$		SPSA, $\tau=0.01$		SPSA/Adam, $\tau=0.20$		SPSA, $\tau=0.01$
Shared sequence	1.114	2.243	45	536	0.989	2.289	98	637
Sequence 1	1.114	2.229	46	532	1.014	2.304	89	674
Sequence 2	0.979	2.318	43	544	0.971	2.239	95	694
Sequence 3	1.004	2.211	46	653	0.936	2.221	94	657

Appendix

Table 5: SPSA under its shared perturbation sequence and five independent sequences, on the original grid with learning rates

\eta\in\{0.1,0.3\}

for Adam and SPSA. Sequences 1–5 are drawn per initialization. Each entry covers 160 runs on the original initializations or 320 on the new ones (10 or 20 per configuration and learning rate), pooled over configurations with different budgets (the eight of both cost families, 648 to 3888 evaluations), so these medians mix budgets and need not equal Kaplan–Meier medians. At

\tau=0.20

the entries are SPSA/Adam ratios of the conditional and the budget-capped median; at

\tau=0.01

they are SPSA’s reached runs and its conditional median in circuit evaluations, a point estimate over reached runs only. In the row for the per-initialization median of five sequences, the 160 or 320 are units; a unit is one initialization at one learning rate and reaches the target when at least three of its five sequences do. Adam’s runs are the same in every row. All of them reach

\tau=0.20

, and Adam’s two medians are 280 on each initialization set; at

\tau=0.01

its conditional medians are 864 and 880, with 113 of 160 and 234 of 320 runs reaching the target. Section 3 defines both medians, the shared sequence and the per-initialization median of five sequences.

		$\tau=0.20$			$\tau=0.01$
			Median			Median
Method	Runs	Reached	Conditional	Budget- capped	Reached	Conditional	Budget- capped
Adam	160	160	280	280	113	864	1000
QNG	320	188	584	935	85	776	1080 ≥B
SPSA (shared sequence)	160	109	312	628	45	536	1080 ≥B

Appendix

Table 6: Reach and both medians at

\tau=0.20

and

\tau=0.01

on the original grid and initializations. Each row pools all settings of one method over eight configurations with different budgets, so its medians mix budgets and need not equal Kaplan–Meier medians. QNG’s two damping values double its number of runs; SPSA uses its shared perturbation sequence. The Reached columns count runs that reach

\tau

. Medians are in circuit evaluations, with the metric charged its simulator price; Section 3 defines them, the price and the budget

B

. A superscript

{\geq}B

marks a budget-capped median at a target that at most half the method’s runs reach; both marked values equal the budget of the local-cost configuration at

n=5

and are not times to target. No entry carries an interval; Figure 3 (a) gives adjusted intervals for these medians’ ratios to Adam’s at

\tau=0.20

only.

			$\tau=0.20$ , settings pooled					$\tau=0.20$ , best setting				$\tau=0.01$ , best setting
Family	$n$	$B$	Adam	QNG		SPSA		Adam	QNG		SPSA	Adam	QNG	SPSA
global-cost	3	972	198	522	.5	326		144	275		127	702	385	304
	4	1728	320	970		848		288	533	.5	338	1504	776	1009
	5	2700	650	2633	.5 ≥B	2700	=B	450	1434	.5	844	2400	1812	2700	=B
	6	3888	1008	3888	=B	3888	=B	720	2821		3505	3672	3255	3888	=B
local-cost	3	648	84	222		98		60	129	.5	28	312	370	302

Appendix

Table 7: Budget-capped median circuit evaluations to the target per configuration, on the original grid and initializations, with SPSA under its shared perturbation sequence and QNG’s metric at its simulator price. Section 3 and Appendix B define this median and relate it to the Kaplan–Meier median.

{=}B

marks an entry equal to the budget

B

, and

{\geq}B

any other entry from a group in which at most half of the runs reach the target. A marked entry gives no time estimate. The left block is at

\tau=0.20

with each method’s settings pooled, 20 runs per entry and 40 for QNG. The right blocks take each method’s best original setting (Section 3 ), 10 runs per entry.

Family	$n$	$B$	Reached of 20	Conditional	Budget- capped	Reached of 40	Conditional	Budget- capped
			Original initializations			New initializations
			SPSA	SPSA/Adam		SPSA	SPSA/Adam
global-cost	3	972	18	1.348	1.646	35	0.972	1.009
	4	1728	15	1.269	2.650	27	1.057	2.659
	5	2700	9	1.148	4.154 =B	23	1.333	3.070
	6	3888	6	2.245	3.857 =B	15	2.366	4.500 =B

Appendix

Table 8: SPSA against Adam in each configuration on the original grid at the loose target

\tau=0.20

, with SPSA under its shared perturbation sequence and each method’s two learning rates pooled, 20 runs per method and configuration on the original initializations and 40 on the new ones. Each block gives SPSA’s reached runs and the ratio of SPSA’s median to Adam’s for the conditional median over reached runs only and the budget-capped median with misses charged the budget

B

(Section 3 ). Adam reaches the target in all its runs, so its two medians coincide, and the two ratios differ only through SPSA’s misses. A ratio marked

{=}B

has SPSA’s budget-capped median equal to

B

because fewer than half of SPSA’s runs reach the target. Such a ratio is

B

over Adam’s median and gives no time estimate.

Figure 5: SPSA against Adam on the original grid in the local-cost configuration at

n=3

, at the loose target

\tau=0.20

with budget

B=648

circuit evaluations and the learning rates

\eta=0.1

and

0.3

pooled. Adam and each SPSA sequence have 20 runs from the 10 original initializations (definitions in Section 3 ). SPSA uses its shared perturbation sequence or one of five independent sequences (1–5), drawn per initialization. Open circles mark the conditional median, filled circles the budget-capped median and diamonds the lower-endpoint Kaplan–Meier median, here the lower of the two central values of

\min(T,B)

(Appendix B ). Each marker is compared with Adam’s line for the same convention. Adam reaches

\tau

in every run, so its conditional and budget-capped medians are equal and share the solid line. The dashed line is its lower-endpoint Kaplan–Meier median. Runs from the two initializations that start below

\tau

record a crossing at zero evaluations and are kept in every median (Appendix C ).

Figure 6: Attainment over the budget with each target’s own selection. Rows are the two cost families, each pooled over its four configurations (

n=3

–

6

qubits) on the 20 new initializations per configuration. Columns are the two targets; in each column every method runs, in each configuration, at the setting selected for that target on the original initializations (Table 4 ). SPSA contributes five runs per initialization, one under each of five independent perturbation sequences. The curve at

f\in[0,1]

is the share of a method’s runs with

T\leq fB

, where

T

and the budget

B

of each run’s configuration are those of Section 3 and QNG’s metric is charged its simulator price. The curve at

f=1

is the method’s reach rate. The value printed for each method is its restricted mean time to target (RMST), the area between its curve and 1 over

0\leq f\leq 1

, with misses charged the budget.

(a) Each family pooled over $n$			Evaluated at $\tau=0.20$		Evaluated at $\tau=0.01$		Final loss
Selected at	Family	QNG search	Count	$p$	Count	$p$	Count	$p$
$\tau=0.01$	global-cost	27 settings	12–68–0	$1.2\times 10^{-10}$	77–3–0	$1.4\times 10^{-19}$	79–1–0	$1.3\times 10^{-22}$
		6 at $\lambda=10^{-3}$	1–79–0	$1.3\times 10^{-22}$	73–7–0	$5.8\times 10^{-15}$	76–4–0	$2.8\times 10^{-18}$
	local-cost	27 settings	36–43–1	$0.50$	79–1–0	$1.3\times 10^{-22}$	80–0–0	$1.7\times 10^{-24}$
$\tau=0.20$	global-cost	27 settings	14–66–0	$3.1\times 10^{-9}$	40–29–11	$0.23$	46–34–0	$0.22$
	local-cost	27 settings	24–55–1	$6.4\times 10^{-4}$	78–2–0	$5.4\times 10^{-21}$	80–0–0	$1.7\times 10^{-24}$

Appendix

Table 9: QNG against Adam on the 20 new initializations per configuration, by the target that selected the settings and the target at which they are evaluated. Settings were selected on the original initializations at the strict target

\tau=0.01

or the loose target

\tau=0.20

; Adam searches six learning rates, QNG 27 settings of learning rate

\eta

and damping

\lambda

or, in the restricted row, Adam’s six learning rates at

\lambda=10^{-3}

. A count gives the pairs sharing an initialization in which QNG is first, Adam is first, or the two tie; at a target, first means reaching

\tau

with fewer circuit evaluations, a miss charged the budget

B

, and on final loss it means the lower loss after the full budget. Every row compares one setting per method in each configuration, so each initialization gives one pair and every count carries its exact two-sided sign-test

p

, ties dropped. QNG’s metric is charged at its simulator price; Section 3 defines the cost families, the initialization sets and the per-step prices. (a) Each family pooled over its four configurations,

n=3

–

6

qubits, 80 pairs per row. (b) The global-cost family under the loose-target selection, 20 pairs per configuration; Adam selects

\eta=0.3

in every global-cost configuration under both targets.

Global-cost family
Target $\tau$	0.50	0.20	0.10	0.05	0.02	0.01
QNG first–Adam first–tied	3–77–0	12–68–0	35–45–0	63–17–0	75–5–0	77–3–0
Sign-test $p$	$1.4\times 10^{-19}$	$1.2\times 10^{-10}$	$0.31$	$2.3\times 10^{-7}$	$4.3\times 10^{-17}$	$1.4\times 10^{-19}$
Adam reached	80	80	80	80	80	65
QNG reached	79	79	79	79	79	79
Local-cost family

Appendix

Table 10: QNG against Adam at six targets on the 80 new initializations of each cost family, 20 in each configuration with

n=3

–

6

qubits. Each method runs at the setting selected per configuration for the strict target

\tau=0.01

on the original initializations and held fixed at every target (the

\tau=0.01

columns of Table 4 ). A count gives the pairs sharing an initialization in which QNG reaches

\tau

with fewer circuit evaluations, in which Adam does, and in which the two tie, a miss charged the budget

B

. Every tie is a pair whose initial loss is already at or below

\tau

, and every count carries its exact two-sided sign-test

p

, ties dropped. The rows Adam reached and QNG reached count each method’s runs that reach

\tau

within

B

. QNG’s metric is charged at its simulator price; Section 3 defines the cost families, the initialization sets and the per-step prices. The columns at

\tau=0.20

and

0.01

repeat the rows of Table 9 (a) selected at

\tau=0.01

with QNG’s 27-setting search.

			Budget-capped time to $\tau=0.01$							Final loss
			New initializations			Original initializations			Best original setting	New initializations
$n$	$p{+}1$	$p(p{+}1)/2$	$m^{\ast}$ (95% interval)		$f$	$m^{\ast}$ (95% interval)		$f$	$m^{\ast}$	$m^{\ast}$ (95% interval)		$f$
Global-cost family ( $L=n$ )
3	19	171	161	(91–197)	0.13	104	(97–179)	0.06	64	158	(126–207)	0.14
4	33	528	223	(170–255)	0	236	(179–249)	0	123	224	(182–281)	0
5	51	1275	271	(188–333)	0	199	(155–249)	0	99	285	(200–350)	0

Appendix

Table 11: Break-even price of the metric. Each

m^{\ast}

is the largest per-step price

m

of the metric (Section 3 ) at which QNG’s budget-capped median at

\tau=0.01

, or in the final-loss columns its median final loss after the last step that fits within

B

, stays below Adam’s; “none” means that QNG’s median is not below Adam’s even at

m=p+1

. Every

m^{\ast}

is computed by re-pricing noiseless simulated runs and is not a hardware measurement. The simulator price is

m=p+1

, and the assumed hardware count

p(p{+}1)/2

charges one circuit for each independent entry of the symmetric

p\times p

metric. Outside the column for the best original setting, Adam runs at learning rate

\eta=0.3

and QNG at learning rate and damping

(\eta,\lambda)=(1,10^{-1})

on the global-cost family and

(1,10^{-3})

(3,10^{-1})

(3,10^{-4})

and

(3,10^{-4})

on the local-cost family at

n=3

–

6

; these settings were selected for

\tau=0.01

on the 10 original initializations per configuration and never on the 20 new ones. The best original setting of Section 3 , QNG at

(0.3,10^{-3})

and Adam at

\eta=0.3

, is evaluated on the original initializations. Intervals are 95% within-configuration percentile bootstrap intervals over 2000 replicates; a replicate in which QNG’s median is not below Adam’s at

m=p+1

counts as

m^{\ast}=p

, and

f

is the fraction of replicates with

m^{\ast}\geq p(p{+}1)/2

(a) Strict-target settings	Count			Pooled ratio $\rho$
Comparison	$B$	$2B$	$4B$	$B$	$2B$	$4B$
New initializations (20 per configuration), global-cost family
QNG–Adam, first to $\tau=0.20$	12–68–0			$1.541$	$1.541$	$1.541$
QNG–Adam, first to $\tau=0.01$	77–3–0			$0.504$	$0.481$	$0.481$
QNG–Adam, lower final loss	79–1	79–1	62–18	$8.0\times 10^{-3}$	$5.3\times 10^{-3}$	$1.2\times 10^{-2}$
SPSA–Adam, lower final loss	46–34 (39–41)	1–79	0–80	$0.925$	$7.96$	$631$

Appendix

Table 12: Longer and shorter budgets, QNG’s metric at its simulator price (definitions in Section 3 ). (a) Settings selected on the extended grid for

\tau=0.01

B

on the original initializations, held fixed at every budget. Counts give the initializations from which QNG or SPSA is first, Adam is first and, at a target, the two tie. At a target, first means fewer circuit evaluations, a miss counting as the budget, and on final loss the lower final loss. Each first-to-target count is printed once because it does not change with the budget.

\rho

is QNG’s or SPSA’s median over Adam’s (final loss, or budget-capped at a target), pooled over configurations with different budgets (972–3888 at

B

global-cost, 648–1296 local-cost), so these medians mix budgets and need not equal Kaplan–Meier medians. SPSA entries in (a) and (b) use the shared sequence except the parenthesized 39–41, which uses the per-initialization median of five sequences. The global-cost count at

B

switches from SPSA to Adam between the two, and from

2B

SPSA’s pooled global-cost final loss is above Adam’s. (b) Runs reaching the target on the original grid and initializations at the best original setting of Section 3 . Reach within

B/2

is reach at a budget of

27p

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Variational Quantum Circuits Adaptive Gradient Methods Variational Quantum Algorithms Quantum Neural Networks

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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.

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