Variance-Based Shot Allocation

• Variance-based shot allocation is a statistical framework that optimally distributes measurement shots across quantum devices to reduce estimator variance and mean-squared error.
• It leverages empirical variances, bias estimates, and reliability scores to dynamically calibrate and reallocate shots based on online performance feedback.
• The method is applied in distributed quantum processing, variational quantum algorithms, and quasi-probability stratification to meet strict cost and hardware constraints.

Variance-based shot allocation is a family of measurement-budgeting methods that distributes a finite shot budget across heterogeneous QPUs, gradient components, Hamiltonian or commutator cliques, or quasi-probability strata in order to control estimator variance, mean-squared error (MSE), or the distributional analogue MISE under hardware and cost constraints. In the NISQ setting, the central premise is that shots need not be treated as a singular monolithic unit: they can be calibrated, split, merged, and updated online, with allocation decisions informed by empirical variances, bias estimates, Hellinger-distance reliability scores, or related distribution-level criteria (Bisicchia et al., 2024, Gu et al., 2021, Ikhtiarudin et al., 22 Jul 2025, Dai et al., 11 Feb 2026).

1. Statistical setting and estimator structure

The basic setting in distributed execution is a single circuit $U$ measured for a total of $N$ shots across $M$ QPUs, with allocation $n_i$ satisfying $\sum_i n_i=N$. Two observation models recur. In the scalar case, a QPU-specific $\pm 1$-valued observable has true mean $\mu_i$, bias $b_i=\mu_i-\mu^\*$ relative to an ideal target, sample mean $\bar y_i$, and variance $\mathrm{Var}(\bar y_i)=\sigma_i^2/n_i$ with $N$0. In the distributional case, counts are multinomial, empirical probabilities $N$1 are unbiased for $N$2, and per-outcome marginal variance is commonly approximated by $N$3 (Bisicchia et al., 2024).

In VQAs, the same logic is applied to gradient estimation rather than directly to output probabilities. With $N$4, Pauli decomposition of $N$5, weighted random operator sampling, and the parameter-shift rule, each gradient component $N$6 is an unbiased estimator of $N$7 with variance $N$8, where $N$9 is the shot count assigned to parameter $M$0 (Gu et al., 2021). In ADAPT-VQE, both Hamiltonian energy estimation and operator-gradient estimation are expressed as weighted sums of Pauli expectations, usually grouped into commuting cliques so that empirical clique variances become the allocation primitives (Ikhtiarudin et al., 22 Jul 2025).

A distinct but related setting arises in quasi-probability decompositions (QPDs). There the estimator variance decomposes into Born-rule shot noise and configuration variance introduced by randomizing over circuit variants and reweighting by signed quasi-probabilities. Variance-based allocation then operates over strata of classical configurations rather than only over physical measurements (Dai et al., 11 Feb 2026).

Context	Allocation variable	Representative criterion
Distributed circuit across QPUs	$M$1, $M$2	MISE, Hellinger unreliability
VQA gradient estimation	$M$3	maximize $M$4
ADAPT-VQE grouped measurements	$M$5	minimize $M$6 or satisfy a target variance threshold
QPD configuration sampling	$M$7	proportional or Neyman stratified variance reduction

2. Objective functions and canonical allocation rules

At the scalar level, the central quantity is usually

$M$8

For full output distributions, the shot-wise framework formalizes the analogue

$M$9

where a merged estimate is formed as $n_i$0. The corresponding optimum is obtained from a linear system

$n_i$1

with $n_i$2 and $n_i$3 (Bisicchia et al., 2024).

When bias is negligible and independent scalar estimators are being merged, the classical variance-minimizing weights are recovered: $n_i$4 which yields

$n_i$5

Under a total cost constraint $n_i$6, a Neyman-type split gives

$n_i$7

This is the standard proportional-to-standard-deviation and inverse-$n_i$8 allocation (Bisicchia et al., 2024).

For a general weighted sum $n_i$9 with independent estimators $\sum_i n_i=N$0, minimizing

$\sum_i n_i=N$1

subject to $\sum_i n_i=N$2 gives the Lagrangian optimum

$\sum_i n_i=N$3

In ADAPT-VQE this is the baseline derivation behind variance-based assignment for Hamiltonian terms and commutator-gradient terms (Ikhtiarudin et al., 22 Jul 2025).

QPD stratification yields a complementary set of rules. If strata $\sum_i n_i=N$4 have weights $\sum_i n_i=N$5, conditional means $\sum_i n_i=N$6, and conditional variances $\sum_i n_i=N$7, the stratified estimator has variance

$\sum_i n_i=N$8

Under proportional allocation, $\sum_i n_i=N$9, the variance is never worse than naïve sampling: $\pm 1$0 Under equal costs, Neyman allocation becomes

$\pm 1$1

and the cost-aware version is

$\pm 1$2

These formulas move variance-based shot allocation from direct quantum measurements to stratified configuration budgets (Dai et al., 11 Feb 2026).

3. Distributed execution across heterogeneous QPUs

The shot-wise framework for distributing a single circuit across multiple QPUs is organized as calibration, ranking, and production. Calibration begins with a pilot allocation $\pm 1$3, typically uniform in the reported experiments, and a set of calibration circuits whose ideal distributions are computed classically. Each calibration circuit is executed on each QPU, empirical distributions are formed, and a reliability score $\pm 1$4 is computed from the mean squared Hellinger distance to ideal. The Hellinger distance is

$\pm 1$5

and the framework uses a jackknife correction for bias reduction in the Hellinger estimator (Bisicchia et al., 2024).

Production then separates two decisions. The split policy determines the next shot allocation $\pm 1$6 or batch sizes, and the merge policy aggregates the per-QPU outputs. The paper proposes uniform, Hellinger-weighted, and MISE-based strategies. In the Hellinger formulation, smaller unreliability $\pm 1$7 implies higher weight. For distribution merging under weighted squared Hellinger distance, the minimizing distribution is

$\pm 1$8

again with jackknife bias correction for the square-root terms.

A defining feature of the framework is incremental execution. After each batch, unreliability scores, variance surrogates, and bias estimates can be updated; the next split and merge are then recomputed. The paper explicitly recommends frequent recalibration on hours-scale because empirical $\pm 1$9 values vary over time. It also allows cost-aware and availability-aware policies, fairness floors, minimum exploration shots, shrinkage of $\mu_i$0 and $\mu_i$1, weight clipping, and confidence bounds to prevent over-commitment to a single device.

Empirically, shot-wise distribution across QPUs improves stability, lowers worst-case error, and typically aligns with average single-QPU outcomes. It does not systematically beat the best individual QPU. MISE and Hellinger strategies generally outperform purely uniform split and merge, but circuit dependence persists, and calibration on random circuits may not perfectly predict performance on specific target circuits (Bisicchia et al., 2024).

4. Variational algorithms and grouped-measurement allocation

In variational optimization, variance-based shot allocation operates at the level of gradient components. The gCANS method defines the expected one-step gain

$\mu_i$2

and maximizes the global efficiency

$\mu_i$3

The resulting global, coupled rule is

$\mu_i$4

This contrasts with iCANS, which uses $\mu_i$5, and with CABS, which chooses a single uniform batch size. Under unbiasedness, variance scaling $\mu_i$6, $\mu_i$7-strong convexity, $\mu_i$8-Lipschitz gradients, and the idealized gCANS rule, the method achieves geometric convergence, $\mu_i$9, and therefore $b_i=\mu_i-\mu^\*$0 iterations in the convex setting. For He2$b_i=\mu_i-\mu^\*$1, chemical accuracy was reached with $b_i=\mu_i-\mu^\*$2 iterations and $b_i=\mu_i-\mu^\*$3 shots for gCANS, versus $b_i=\mu_i-\mu^\*$4, $b_i=\mu_i-\mu^\*$5 for iCANS, $b_i=\mu_i-\mu^\*$6, $b_i=\mu_i-\mu^\*$7 for SGD-DS, and $b_i=\mu_i-\mu^\*$8, $b_i=\mu_i-\mu^\*$9 for ADAM; analogous gains were reported for NH$\bar y_i$0 (Gu et al., 2021).

ADAPT-VQE applies the same principle to commuting groups rather than gradient coordinates. With Hamiltonian

$\bar y_i$1

clique-level per-shot statistics are

$\bar y_i$2

and the total estimator variance is minimized by allocating more shots to higher-variance cliques. The paper implements two clique-level policies after a uniform pilot $\bar y_i$3. Variance-Minimized Shot Assignment is

$\bar y_i$4

and Variance-Preserved Shot Reduction is

$\bar y_i$5

The same framework is used for commutator-based ADAPT gradients, after qubit-wise commutativity grouping. The method is integrated with Pauli-measurement reuse across successive ADAPT iterations: grouped measurement alone used, on average, approximately $\bar y_i$6 of naive full measurement, while grouping plus reuse used approximately $\bar y_i$7, with reuse consistently saving an additional approximately $\bar y_i$8. For H$\bar y_i$9, VMSA saved $\mathrm{Var}(\bar y_i)=\sigma_i^2/n_i$0 shots and VPSR $\mathrm{Var}(\bar y_i)=\sigma_i^2/n_i$1; for LiH, VMSA saved $\mathrm{Var}(\bar y_i)=\sigma_i^2/n_i$2 and VPSR $\mathrm{Var}(\bar y_i)=\sigma_i^2/n_i$3; both achieved chemical accuracy in the reported sampler simulations (Ikhtiarudin et al., 22 Jul 2025).

5. Stratification, configuration variance, and online stopping

In QPD-based estimators, variance-based allocation is not restricted to physical shots. The total variance decomposes as

$\mathrm{Var}(\bar y_i)=\sigma_i^2/n_i$4

separating Born-rule shot noise from configuration variance. Stratification targets the second term. For product-form QPDs, a universal statistic is the counts vector $\mathrm{Var}(\bar y_i)=\sigma_i^2/n_i$5 of local indices. The number of strata becomes polynomial in circuit length for fixed local width, and a forward dynamic programme computes stratum weights in $\mathrm{Var}(\bar y_i)=\sigma_i^2/n_i$6 time and $\mathrm{Var}(\bar y_i)=\sigma_i^2/n_i$7 memory, while backward conditional sampling costs $\mathrm{Var}(\bar y_i)=\sigma_i^2/n_i$8 per configuration. Empirically, on Probabilistic Angle Interpolation and Probabilistic Error Cancellation benchmarks, proportional stratification gave robust $\mathrm{Var}(\bar y_i)=\sigma_i^2/n_i$9 savings in the pessimistic single-shot regime and oracle-model variance reductions up to $N$00–$N$01; for PAI the oracle variance ratio was approximately $N$02–$N$03, and for PEC approximately $N$04–$N$05 (Dai et al., 11 Feb 2026).

StableShots addresses a different but adjacent problem: when to stop measuring a fixed static circuit. It executes a circuit in batches of size $N$06, compares cumulative empirical distributions using

$N$07

and stops when the TVD between $N$08 and $N$09 is at most $N$10 for $N$11 consecutive checks. The held-out evaluation selected $N$12, $N$13, $N$14, $N$15. Across 180 QSimBench traces, six circuit families, six sizes from 4 to 14 qubits, five noisy IBM simulated backends, and 100 backend-holdout splits, the selected configuration reached $N$16 on all held-out test evaluations with median 7,650 shots; fixed-shot baselines at 5k, 10k, and 18k either failed more often or spent substantially more shots. The paper is explicit that StableShots is an empirical, black-box stopping rule and does not certify closeness to the unknown backend-induced distribution (Bisicchia et al., 20 Jun 2026).

These two lines of work are complementary. Stratified QPD sampling reallocates effort across classical configuration strata to reduce configuration variance, whereas StableShots monitors the marginal change of the cumulative empirical output distribution for a single static circuit. The latter can be attached to multi-circuit execution by maintaining independent stopping states per circuit or measurement group, which the paper identifies as a future direction.

6. Constraints, limitations, and broader methodological relations

Across the quantum literature, variance-based shot allocation is consistently formulated under explicit operational constraints. The shot-wise framework allows cost-aware splits $N$17, device-availability constraints $N$18, fairness floors $N$19, and online reweighting under drift (Bisicchia et al., 2024). gCANS uses a total shot budget $N$20, minimum shots $N$21, and exponential moving averages to forecast gradient norms and variances (Gu et al., 2021). ADAPT-VQE imposes small pilot budgets $N$22 per clique and then redistributes the residual budget according to empirical clique standard deviations (Ikhtiarudin et al., 22 Jul 2025). QPD stratification adds the practical requirement of residual-aware integer apportionment to preserve unbiasedness at finite $N$23 (Dai et al., 11 Feb 2026).

Several recurrent limitations also delimit the scope of these methods. Shot-wise QPU distribution improves robustness but generally does not exceed the best individual QPU and remains sensitive to calibration mismatch and device drift (Bisicchia et al., 2024). The geometric convergence theorem for gCANS assumes strong convexity and Lipschitz gradients, while VQA landscapes are non-convex in general; the paper therefore states the theorem for the convex setting and expects the behavior to manifest once the iterate enters a convex basin (Gu et al., 2021). StableShots controls local TVD change between empirical prefixes, not the distance to the true distribution, and full-distribution concentration bounds such as Hoeffding and Weissman remain too conservative in the reported benchmarks, with Hoeffding requiring 82,707 shots at 4 qubits, 30.3 million at 8 qubits, and 179.8 billion at 14 qubits for $N$24, $N$25 (Bisicchia et al., 20 Jun 2026). QPD stratification reduces constant variance prefactors but does not change the $N$26 overhead induced by quasi-probability negativity (Dai et al., 11 Feb 2026).

A broader methodological analogue appears in joint rate allocation for HEVC. There, the minimum distortion variance objective is derived from the equal-distortion condition across streams, and the look-ahead-and-feedback allocation model combines complexity measures with encoder feedback on distortion and bitrate; integrated into HM16.0, it reports that an average of $N$27 variance of mean square error is saved with different complexity measures (Fan et al., 2018). This suggests that feedback terms analogous to reliability histories, recent variances, or past distortions can stabilize future shot-allocation decisions when purely look-ahead proxies are imperfect.

Taken together, these results define variance-based shot allocation not as a single algorithm but as a statistical design principle. Its implementations vary—MISE minimization over QPUs, globally coupled gradient budgeting, clique-level variance matching, proportional or Neyman stratification, and TVD-based stopping—but they share the same operational objective: spend measurements where they most reduce uncertainty, subject to the structure and failure modes of the underlying workload.