Layerwise Richardson Extrapolation (LRE)

• Layerwise Richardson Extrapolation (LRE) is a family of methods that cancels leading-order errors via layer-specific folding and multivariate interpolation.
• In quantum error mitigation, LRE independently scales noise across circuit layers, enhancing estimator accuracy and reducing bias in noisy quantum devices.
• In numerical finite-element solvers, LRE applies time-layer extrapolation to significantly improve convergence rates and achieve higher solution accuracy.

Layerwise Richardson Extrapolation (LRE) denotes a family of extrapolation procedures in which error amplification and extrapolation are performed with respect to individual layers rather than a single global control parameter. In contemporary quantum error mitigation, LRE generalizes standard zero-noise extrapolation by treating the noise associated with different circuit layers or chunks as independent variables and combining measurements from layerwise folded circuits through multivariate interpolation (Russo et al., 2024). In a distinct numerical-analysis usage, LRE refers to Richardson combinations formed at aligned time layers of solutions computed with different time steps for finite-element Crank–Nicolson schemes with discrete transparent boundary conditions for the 1D generalized Schrödinger equation (Zlotnik et al., 2014). A later hybrid quantum neural network study adopts the quantum meaning, implementing LRE through Mitiq on PennyLane circuits simulated under Qiskit Aer noise models and evaluating its robustness under five noise channels (Njiki et al., 19 Apr 2026).

1. Terminology and conceptual scope

The term “layerwise” is context dependent. In quantum error mitigation, a layer is a circuit block or chunk, and LRE amplifies noise locally by folding one layer at a time while leaving the ideal unitary unchanged (Russo et al., 2024). In the finite-element Schrödinger setting, “layerwise” refers instead to time layers: one computes numerical solutions on several temporal grids and combines their values at coincident physical times to cancel leading even-power time-discretization errors (Zlotnik et al., 2014).

In both usages, the underlying mechanism is Richardson extrapolation. A quantity measured or computed at nonzero error scale is assumed to admit an expansion in powers of that scale, and carefully chosen linear combinations cancel the lowest-order terms. In the univariate quantum setting this takes the form

$$E(\lambda)=E_0+a_1\lambda+a_2\lambda^2+a_3\lambda^3+\cdots,$$

with extrapolated estimator

$$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$

where the coefficients satisfy

$$\sum_{i=0}^m c_i=1,\qquad \sum_{i=0}^m c_i s_i^k=0,\quad k=1,\ldots,m.$$

The corresponding coefficients are obtained from a Vandermonde system $Vc=b$, with $V_{ki}=s_i^k$ and $b=(1,0,\ldots,0)^T$ (Njiki et al., 19 Apr 2026).

Classic examples already used in the LRE literature are

$E_0 \approx 2E(\lambda)-E(2\lambda),$

for $m=1$, $S=\{1,2\}$,

$$E_0 \approx \frac{3}{2}E(\lambda)-\frac{1}{2}E(3\lambda),$$

for $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$0, $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$1, and

$$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$2

for $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$3, $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$4 (Njiki et al., 19 Apr 2026).

2. Multivariate quantum formulation

The quantum formulation introduced by Russo and Mari decomposes a circuit into $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$5 layers,

$$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$6

associates an independent noise-scale variable $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$7 to each layer, and regards the measured observable expectation as a multivariate function $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$8 (Russo et al., 2024). The smoothness assumption is that $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$9 is analytic near the zero-noise point and admits a multivariate Taylor expansion,

$$\sum_{i=0}^m c_i=1,\qquad \sum_{i=0}^m c_i s_i^k=0,\quad k=1,\ldots,m.$$0

The paper’s main formulation uses total-degree multivariate interpolation. For polynomial degree at most $$\sum_{i=0}^m c_i=1,\qquad \sum_{i=0}^m c_i s_i^k=0,\quad k=1,\ldots,m.$$1,

$$\sum_{i=0}^m c_i=1,\qquad \sum_{i=0}^m c_i s_i^k=0,\quad k=1,\ldots,m.$$2

where $$\sum_{i=0}^m c_i=1,\qquad \sum_{i=0}^m c_i s_i^k=0,\quad k=1,\ldots,m.$$3 enumerates all monomials of total degree $$\sum_{i=0}^m c_i=1,\qquad \sum_{i=0}^m c_i s_i^k=0,\quad k=1,\ldots,m.$$4, and

$$\sum_{i=0}^m c_i=1,\qquad \sum_{i=0}^m c_i s_i^k=0,\quad k=1,\ldots,m.$$5

The interpolation nodes are chosen as

$$\sum_{i=0}^m c_i=1,\qquad \sum_{i=0}^m c_i s_i^k=0,\quad k=1,\ldots,m.$$6

where $$\sum_{i=0}^m c_i=1,\qquad \sum_{i=0}^m c_i s_i^k=0,\quad k=1,\ldots,m.$$7, $$\sum_{i=0}^m c_i=1,\qquad \sum_{i=0}^m c_i s_i^k=0,\quad k=1,\ldots,m.$$8, and the $$\sum_{i=0}^m c_i=1,\qquad \sum_{i=0}^m c_i s_i^k=0,\quad k=1,\ldots,m.$$9 are all nonnegative integer vectors with $Vc=b$0-norm $Vc=b$1. Writing the sample matrix as

$Vc=b$2

and the measured data as $Vc=b$3, one has $Vc=b$4. The zero-noise estimator can then be written directly as

$Vc=b$5

with

$Vc=b$6

where $Vc=b$7 is obtained by replacing the $Vc=b$8-th row of $Vc=b$9 with $V_{ki}=s_i^k$0 (Russo et al., 2024).

An optional tensor-grid variant is also described. If one chooses per-layer node sets $V_{ki}=s_i^k$1, then the multivariate Lagrange basis factorizes:

$V_{ki}=s_i^k$2

and

$V_{ki}=s_i^k$3

with

$V_{ki}=s_i^k$4

The paper adopts the total-degree scheme because it keeps the number of nodes polynomial in $V_{ki}=s_i^k$5 at fixed $V_{ki}=s_i^k$6, whereas the tensor-grid construction scales multiplicatively in $V_{ki}=s_i^k$7 (Russo et al., 2024).

LRE reduces to standard Richardson extrapolation when $V_{ki}=s_i^k$8. This reduction is exact and holds for both the total-degree and tensor-grid viewpoints (Russo et al., 2024).

3. Layerwise folding, sampling overhead, and quantum performance

The mechanism used to realize layer-resolved noise amplification is layerwise unitary folding. For a layer $V_{ki}=s_i^k$9, global folding replaces it by

$b=(1,0,\ldots,0)^T$0

which yields odd integer scale factors

$b=(1,0,\ldots,0)^T$1

The paper also describes local folding within a layer chunk: if $b=(1,0,\ldots,0)^T$2, one may fold at the sublayer level while preserving the same chunk-level odd scaling rule (Russo et al., 2024).

The basic algorithm is: partition the circuit into $b=(1,0,\ldots,0)^T$3 chunks, choose the multi-layer scale vectors $b=(1,0,\ldots,0)^T$4, generate the corresponding folded circuits, measure $b=(1,0,\ldots,0)^T$5, compute the interpolation weights $b=(1,0,\ldots,0)^T$6, and return $b=(1,0,\ldots,0)^T$7 (Russo et al., 2024). A worked micro-example with $b=(1,0,\ldots,0)^T$8, $b=(1,0,\ldots,0)^T$9, and $E_0 \approx 2E(\lambda)-E(2\lambda),$0 uses the nodes

$E_0 \approx 2E(\lambda)-E(2\lambda),$1

and monomials $E_0 \approx 2E(\lambda)-E(2\lambda),$2, yielding $E_0 \approx 2E(\lambda)-E(2\lambda),$3, $E_0 \approx 2E(\lambda)-E(2\lambda),$4, and $E_0 \approx 2E(\lambda)-E(2\lambda),$5 (Russo et al., 2024).

Sampling overhead is a central feature rather than a secondary implementation detail. For the total-degree construction, the number of circuit instances is

$E_0 \approx 2E(\lambda)-E(2\lambda),$6

while for the tensor-grid construction it is

$E_0 \approx 2E(\lambda)-E(2\lambda),$7

If each $E_0 \approx 2E(\lambda)-E(2\lambda),$8 has variance $E_0 \approx 2E(\lambda)-E(2\lambda),$9, then the LRE estimator variance is approximately $m=1$0. Under a fixed shot budget $m=1$1, optimal allocation is $m=1$2, giving sampling overhead

$m=1$3

whereas equal allocation gives

$m=1$4

The paper’s practical guidance is correspondingly conservative: keep $m=1$5 small, reduce $m=1$6 by chunking layers together, increase $m=1$7 if variance dominates, and allocate shots proportional to $m=1$8 (Russo et al., 2024).

The original numerical study reports scenarios in which LRE outperforms standard Richardson extrapolation. For GHZ-like circuits followed by their inverse, with observable $m=1$9, mean absolute errors at depth 2 were Unmitigated $S=\{1,2\}$0, RE $S=\{1,2\}$1, and LRE $S=\{1,2\}$2; at depth 5 they were Unmitigated $S=\{1,2\}$3, RE $S=\{1,2\}$4, and LRE $S=\{1,2\}$5; at depth 8 they were Unmitigated $S=\{1,2\}$6, RE $S=\{1,2\}$7, and LRE $S=\{1,2\}$8 (Russo et al., 2024). The same study states that LRE again outperformed RE and unmitigated baselines on randomized circuits $S=\{1,2\}$9 with high CNOT density. At the same time, it emphasizes the bias–variance trade-off: increasing degree $$E_0 \approx \frac{3}{2}E(\lambda)-\frac{1}{2}E(3\lambda),$$0 decreases bias but increases variance, and with small $$E_0 \approx \frac{3}{2}E(\lambda)-\frac{1}{2}E(3\lambda),$$1 the variance of LRE can dominate (Russo et al., 2024).

4. Hybrid quantum neural networks and system-level evaluation

A system-level evaluation of LRE in hybrid quantum neural networks implements the method through Mitiq on PennyLane circuits simulated with Qiskit Aer noise models (Njiki et al., 19 Apr 2026). The studied quantum subnetwork uses three qubits and four logical stages: data encoding via AngleEmbedding with $$E_0 \approx \frac{3}{2}E(\lambda)-\frac{1}{2}E(3\lambda),$$2 rotations, followed by three StronglyEntanglingLayers consisting of parameterized single-qubit rotations and $$E_0 \approx \frac{3}{2}E(\lambda)-\frac{1}{2}E(3\lambda),$$3 entanglers. The classical part is a linear layer followed by softmax, mapping measured Z-basis expectation values to class probabilities. In this setting, “layers” refer to the trainable StronglyEntanglingLayers; the encoding layer is not trainable and is typically not folded (Njiki et al., 19 Apr 2026).

The training pipeline uses Adam with initial learning rate $$E_0 \approx \frac{3}{2}E(\lambda)-\frac{1}{2}E(3\lambda),$$4 halved every 5 epochs, 20 total epochs, and batch size 5. Data preprocessing uses StandardScaler and label encoding with a 75/25 train-test split. Qiskit Aer noise models are applied before and after each transpiled gate for noise probabilities $$E_0 \approx \frac{3}{2}E(\lambda)-\frac{1}{2}E(3\lambda),$$5. Each circuit evaluation uses 8192 shots, and each configuration is averaged over 3 repetitions (Njiki et al., 19 Apr 2026).

The five evaluated channels are depolarizing, amplitude damping, phase damping, bit flip, and phase flip, with the following parameterizations:

$$E_0 \approx \frac{3}{2}E(\lambda)-\frac{1}{2}E(3\lambda),$$6

for depolarizing noise,

$$E_0 \approx \frac{3}{2}E(\lambda)-\frac{1}{2}E(3\lambda),$$7

with

$$E_0 \approx \frac{3}{2}E(\lambda)-\frac{1}{2}E(3\lambda),$$8

for amplitude damping, and

$$E_0 \approx \frac{3}{2}E(\lambda)-\frac{1}{2}E(3\lambda),$$9

for bit-flip and phase-flip noise, respectively (Njiki et al., 19 Apr 2026).

The layerwise estimator used conceptually is

$$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$00

where $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$01 denotes the expectation when only layer $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$02 is folded to scale factor $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$03 and all other layers remain at baseline scale 1. The implementation uses layerwise unitary folding, and the paper notes that exact options such as the specific scale set $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$04 or a fold_layerwise=True flag are not enumerated, although Mitiq v0.45.1 is employed and first-order settings such as $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$05 or $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$06 are described as common choices (Njiki et al., 19 Apr 2026).

The reported validation accuracies aggregated by Low, Medium, and High noise intervals are as follows:

Noise channel	Baseline (Low / Medium / High)	LRE (Low / Medium / High)
Depolarizing	1.0000 / 0.9737 / 0.6053	0.8420 / 0.8680 / 0.4740
Bit flip	0.9737 / 0.9474 / 0.9211	0.9210 / 0.8680 / 0.7630
Phase flip	1.0000 / 0.9737 / 0.9737	0.9210 / 0.9210 / 0.9470
Amplitude damping	0.9737 / 0.9737 / 0.6579	0.8420 / 0.9210 / 0.5260
Phase damping	0.9737 / 0.9737 / 0.9474	0.9470 / 0.8680 / 0.8160

In this HQNN setting, LRE largely tracks the degradation of the baseline as noise increases and generally underperforms the unmitigated baseline across intervals. The same study reports that ZNE and DDD show similarly limited benefits, while PEC shows limited gains only in the low-noise depolarizing regime. It also attributes possible instability to the fact that folding increases circuit depth, which can heighten sampling variance for fixed shot budgets and amplify gate noise (Njiki et al., 19 Apr 2026).

5. Time-layer Richardson extrapolation for finite-element Schrödinger solvers

In the finite-element literature, LRE is used for the generalized 1D time-dependent Schrödinger equation on the whole axis,

$$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$07

with Crank–Nicolson time stepping, arbitrary-order finite elements in space, and discrete transparent boundary conditions after truncation to $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$08 (Zlotnik et al., 2014). The baseline FE–CN method has error $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$09, and the extrapolation relies on the even-power expansion

$$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$10

The extrapolants are formed at aligned time layers. For fourth-order accuracy,

$$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$11

for sixth-order accuracy,

$$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$12

and for eighth-order accuracy,

$$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$13

These combinations yield

$$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$14

Alignment requires $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$15 divisible by 2, 6, or 12 for $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$16, respectively (Zlotnik et al., 2014).

The paper states that the Richardson-extrapolated solutions inherit the $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$17-type and energy-type stability properties of the underlying CN+DTBC scheme because they are linear combinations of stable solutions at aligned time layers (Zlotnik et al., 2014). It also gives explicit asymptotic cost estimates. If computing $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$18 costs $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$19, then computing $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$20 costs

$$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$21

$$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$22

$$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$23

and the additional costs are reported as $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$24 of the baseline (Zlotnik et al., 2014).

The numerical results are correspondingly strong. In Example 1, for $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$25, $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$26, and $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$27, the reported baseline-versus-extrapolated error-reduction factors are approximately $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$28, $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$29, and $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$30 in $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$31 for $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$32, and $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$33, $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$34, and $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$35 in $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$36. At $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$37, the factors become approximately $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$38, $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$39, and $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$40 in $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$41, and $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$42, $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$43, and $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$44 in $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$45 (Zlotnik et al., 2014). In Example 3, FE+DTBC+LRE achieves relative uniform-in-time and $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$46-in-space error $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$47 with $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$48, $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$49, $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$50, $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$51, compared with best $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$52 reported for a finite-difference approach using $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$53, $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$54 (Zlotnik et al., 2014).

6. Assumptions, limitations, and recurrent misconceptions

LRE is not a single universally valid procedure; its correctness depends on the error model assumed in each domain. In the quantum setting, the central assumption is that the expectation value behaves as a sufficiently smooth function of independently scalable layerwise noise variables. The literature explicitly notes that crosstalk, coherent errors, non-Markovianity, and device drift can invalidate the idealized scaling or degrade accuracy. Folding also does not scale readout errors, so measurement biases that do not depend on $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$55 persist unless combined with measurement error mitigation (Russo et al., 2024).

In the HQNN study, these limitations appear operationally rather than only formally. The paper states that amplitude damping and depolarizing noise do not always scale analytically with gate folding as assumed by Richardson extrapolation, that digital noise scaling increases circuit depth and idle time, and that replacing each expectation by an LRE aggregate multiplies evaluations and can increase gradient noise, slowing or destabilizing learning at fixed shots. It therefore recommends keeping $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$56 small, using modest scale sets such as $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$57 or $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$58, checking empirical polynomial behavior in $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$59, allocating more shots to larger $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$60, and avoiding very large $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$61 under damping-type noise (Njiki et al., 19 Apr 2026).

In the finite-element setting, the decisive assumption is the even-power time-error expansion. The paper reports a failure mode when time regularity is limited by discontinuous potentials: for $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$62 in its most demanding example, a degradation near the $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$63 level is observed for very small $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$64, and the authors attribute this to limited time regularity invalidating the expansion for very small $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$65. Suggested remedies are to restrict the extrapolation order, avoid excessively small $$E_{\text{extr}}\approx \sum_{i=0}^m c_i E(s_i\lambda),$$66, or smooth interfaces (Zlotnik et al., 2014).

A recurrent misconception is that LRE is synonymous with uniformly improved performance. The published record in the provided sources is mixed but coherent. The multivariate quantum paper reports cases in which LRE achieves superior performance compared to traditional Richardson extrapolation (Russo et al., 2024), whereas the HQNN evaluation finds that LRE generally follows the same degradation trends as the unmitigated baseline and can underperform it under several noise models (Njiki et al., 19 Apr 2026). This suggests that LRE is best understood as a structured bias-cancellation framework whose effectiveness depends on how accurately the extrapolation assumptions match the underlying error processes, the chosen partition into layers or chunks, and the available computational or sampling budget.