Pauli Channel Learning

• Pauli channel learning is the task of inferring quantum channel parameters by representing noise as classical probability distributions over Pauli operators.
• It leverages entanglement-assisted schemes and Bell-basis measurements to transform quantum process problems into tractable classical sampling tasks.
• The approach highlights clear tradeoffs between entangled and non-entangled regimes, impacting sample complexity and resource requirements in noise estimation.

Searching arXiv for recent and foundational papers on Pauli channel learning and related process-learning settings. Pauli channel learning is the task of inferring the parameters, structure, or operationally relevant functionals of a quantum channel that is diagonal in the Pauli basis or equivalently a convex mixture of Pauli conjugations. Across the literature, the subject spans several closely related formulations: direct estimation of Pauli error rates or Pauli eigenvalues, learning under diamond-, $l_p$-, or relative-error criteria, online and adaptive prediction of channel statistics, self-consistent gate-set identification under SPAM and gauge constraints, and structured learning for local, sparse, or Pauli-Lindblad noise models. A central theme is that Pauli structure converts a quantum process-learning problem into a classical distribution-learning problem in some settings, while in other settings access restrictions, SPAM noise, or identifiability constraints produce sharp separations between easy and hard regimes (Chen et al., 2021, Trinh et al., 2024, Chen et al., 2022).

1. Formal models and parameterizations

An $n$-qubit Pauli channel is commonly written as a mixture over Pauli operators. One formulation is

$$\mathcal{P}(\rho)=\sum_{z,x\in\{0,1\}^n} p_{z,x}\,P^{z,x}\rho P^{z,x\dagger},$$

where $P^{z,x} = (+i)^{z\cdot x} Z^z X^x$, $p_{z,x}\in[0,1]$, and $\sum_{z,x} p_{z,x}=1$ (Raza et al., 2024). Equivalent notation appears throughout the literature as

$\mathcal P(\rho)=\sum_{P\in\{\mathds I,X,Y,Z\}^{\otimes n}} p(P)\,P\rho P$

or

$$\Lambda(\rho)=\sum_{a\in \mathbb{Z}_2^{2n}} p_a\, P_a \rho P_a,$$

with the underlying unknown being a classical probability distribution over $4^n$ Pauli strings (Fawzi et al., 2023, Chen et al., 2021).

A second parameterization uses the Pauli basis as an eigenbasis. In this representation,

$$\Lambda(\rho)=\frac{1}{2^n}\sum_{b\in \mathbb{Z}_2^{2n}} \lambda_b \,\operatorname{Tr}(P_b\rho)\,P_b,$$

where the $n$0 are the Pauli eigenvalues or Pauli fidelities (Chen et al., 2021, Chen et al., 2023). The two parameterizations are related by the Walsh–Hadamard transform,

$n$1

or, equivalently,

$n$2

with the symplectic inner product encoding commutation and anticommutation (Chen et al., 2021, Chen et al., 2022).

The Choi representation is particularly important for process learning. For Pauli channels,

$n$3

where $n$4 are Bell-basis projectors, so Bell-basis measurements on the Choi state expose the Pauli error distribution directly (Raza et al., 2024). The Pauli transfer matrix (PTM) offers another unifying language: $n$5 For Pauli channels, the PTM is diagonal, so PTM learning specializes to learning the Pauli eigenvalues (Caro, 2022).

Pauli twirling connects general channels to the Pauli setting. One definition is

$n$6

whose Choi matrix is the Bell-basis pinching of $n$7 (Raza et al., 2024). Several works therefore treat “Pauli channel learning” either as direct learning of a Pauli channel or as learning the Pauli projection of a more general process (Flammia et al., 2019, Flammia et al., 2021).

2. Entanglement-assisted learning and the collapse to classical sampling

A foundational fact is that maximally entangled inputs and Bell-basis measurements can transform Pauli channel learning into exact classical sampling. If one prepares $n$8 EPR pairs, applies the channel to one half, and performs Bell measurements pairwise, then the Bell outcomes identify the Pauli error. In one formulation, the tuple of Bell outcomes is exactly a sample $n$9 from the Pauli error distribution $$\mathcal{P}(\rho)=\sum_{z,x\in\{0,1\}^n} p_{z,x}\,P^{z,x}\rho P^{z,x\dagger},$$0 (Trinh et al., 2024). In another, Bell measurements on the Choi state of a Pauli channel yield samples from $$\mathcal{P}(\rho)=\sum_{z,x\in\{0,1\}^n} p_{z,x}\,P^{z,x}\rho P^{z,x\dagger},$$1 (Raza et al., 2024).

This equivalence produces sharp complexity consequences. For learning under general $$\mathcal{P}(\rho)=\sum_{z,x\in\{0,1\}^n} p_{z,x}\,P^{z,x}\rho P^{z,x\dagger},$$2 loss with entangled input, the query complexity is exactly the classical distribution-learning complexity on $$\mathcal{P}(\rho)=\sum_{z,x\in\{0,1\}^n} p_{z,x}\,P^{z,x}\rho P^{z,x\dagger},$$3 symbols. The resulting bounds are piecewise in $$\mathcal{P}(\rho)=\sum_{z,x\in\{0,1\}^n} p_{z,x}\,P^{z,x}\rho P^{z,x\dagger},$$4 and $$\mathcal{P}(\rho)=\sum_{z,x\in\{0,1\}^n} p_{z,x}\,P^{z,x}\rho P^{z,x\dagger},$$5, with the notable special cases

$$\mathcal{P}(\rho)=\sum_{z,x\in\{0,1\}^n} p_{z,x}\,P^{z,x}\rho P^{z,x\dagger},$$6

(Trinh et al., 2024). The same reduction yields optimal testing procedures for white-noise testing, entropy and support-size estimation, and diamond-distance estimation between two Pauli channels. In particular, the number of channel uses needed to estimate the noise level via entropy or support size is

$$\mathcal{P}(\rho)=\sum_{z,x\in\{0,1\}^n} p_{z,x}\,P^{z,x}\rho P^{z,x\dagger},$$7

and the same scaling holds for estimating the diamond distance between two Pauli channels (Trinh et al., 2024).

The observation that one channel use is information-theoretically equivalent to one sample from the hidden Pauli distribution also implies that adaptivity provides no additional power in this entangled-input model. The formal statement is that, in estimating any property of $$\mathcal{P}(\rho)=\sum_{z,x\in\{0,1\}^n} p_{z,x}\,P^{z,x}\rho P^{z,x\dagger},$$8 using samples drawn from $$\mathcal{P}(\rho)=\sum_{z,x\in\{0,1\}^n} p_{z,x}\,P^{z,x}\rho P^{z,x\dagger},$$9, adaptivity does not improve efficiency compared to non-adaptive strategies (Trinh et al., 2024). This does not eliminate adaptive learning in other models; rather, it identifies a regime in which the quantum problem has already reduced to classical post-processing.

This entanglement-assisted viewpoint also underlies early exponential-separation results for Pauli eigenvalue estimation. With an $P^{z,x} = (+i)^{z\cdot x} Z^z X^x$0-qubit ancilla, all Pauli eigenvalues can be estimated to additive error $P^{z,x} = (+i)^{z\cdot x} Z^z X^x$1 using $P^{z,x} = (+i)^{z\cdot x} Z^z X^x$2 samples, while ancilla-free models exhibit exponentially worse behavior (Chen et al., 2021). A later refinement with logarithmic quantum memory showed that concatenating protocols can estimate every eigenvalue to error $P^{z,x} = (+i)^{z\cdot x} Z^z X^x$3 using only $P^{z,x} = (+i)^{z\cdot x} Z^z X^x$4 ancilla qubits and $P^{z,x} = (+i)^{z\cdot x} Z^z X^x$5 measurements (Chen et al., 2023). This suggests that the principal resource is not merely entanglement per se, but the ability to preserve and process the right quantum correlations across channel uses.

3. Learning without entanglement

When entanglement between the channel input and an ancilla is forbidden, Pauli channel learning becomes markedly harder. One precise formulation proves that any entanglement-free scheme estimating every Pauli eigenvalue to error $P^{z,x} = (+i)^{z\cdot x} Z^z X^x$6 with success probability at least $P^{z,x} = (+i)^{z\cdot x} Z^z X^x$7 must use

$P^{z,x} = (+i)^{z\cdot x} Z^z X^x$8

measurement rounds (Chen et al., 2023). This bound is tight, matching the best known upper bound based on minimal stabilizer covering. The same work proves that separable-system/ancilla strategies and classical-memory-assisted schemes with mid-circuit measurements and classical feedforward are equivalent in power for sample-complexity purposes (Chen et al., 2023).

Earlier lower bounds had already identified substantial barriers in more restricted incoherent models. Without entangled ancillas and without joint measurements across copies, non-adaptive algorithms require

$P^{z,x} = (+i)^{z\cdot x} Z^z X^x$9

measurements to learn an $p_{z,x}\in[0,1]$0-qubit Pauli channel in diamond norm, while adaptive strategies require at least

$p_{z,x}\in[0,1]$1

for any $p_{z,x}\in[0,1]$2, and

$p_{z,x}\in[0,1]$3

for $p_{z,x}\in[0,1]$4 in the one-use-per-step regime (Fawzi et al., 2023). These lower bounds remain robust even when the learner can reuse the channel multiple times between measurements and interleave the uses with arbitrary unital operations (Fawzi et al., 2023).

The contrast between entanglement-assisted and entanglement-free regimes is especially sharp for Pauli eigenvalue estimation. With entanglement, the sample complexity can be $p_{z,x}\in[0,1]$5 per eigenvalue, independent of $p_{z,x}\in[0,1]$6; without entanglement, $p_{z,x}\in[0,1]$7 rounds are required (Chen et al., 2023). This separation supplies a concrete learning-theoretic manifestation of entanglement as a resource.

At the same time, unentangled learning is not uniformly intractable under every metric. A different line of work reduces Pauli error estimation to Population Recovery and gives an algorithm that learns the Pauli error rates of an $p_{z,x}\in[0,1]$8-qubit channel to precision $p_{z,x}\in[0,1]$9 in $\sum_{z,x} p_{z,x}=1$0 using just

$\sum_{z,x} p_{z,x}=1$1

applications of the channel, with only unentangled state preparation and measurements (Flammia et al., 2021). In the low-noise regime where the no-error outcome occurs with probability $\sum_{z,x} p_{z,x}=1$2, the same approach achieves multiplicative precision $\sum_{z,x} p_{z,x}=1$3 using

$\sum_{z,x} p_{z,x}=1$4

applications (Flammia et al., 2021). This suggests that hardness depends strongly on the target quantity: estimating all eigenvalues in high probability is exponentially hard without entanglement, whereas estimating Pauli error rates in $\sum_{z,x} p_{z,x}=1$5 can be near-optimal up to logarithmic factors.

4. Online learning, shadow tomography, and adaptive prediction

A distinct direction studies Pauli channels in adversarial online learning and adaptive query models. In the online setting of (Raza et al., 2024), the adversary chooses a sequence of channel test operators $\sum_{z,x} p_{z,x}=1$6, the learner outputs hypothesis channels $\sum_{z,x} p_{z,x}=1$7, and losses are evaluated using a convex, $\sum_{z,x} p_{z,x}=1$8-Lipschitz loss. For Pauli channels, the prediction

$\sum_{z,x} p_{z,x}=1$9

reduces online process learning to online learning of the classical probability vector $\mathcal P(\rho)=\sum_{P\in\{\mathds I,X,Y,Z\}^{\otimes n}} p(P)\,P\rho P$0 (Raza et al., 2024).

The main regret guarantee is

$\mathcal P(\rho)=\sum_{P\in\{\mathds I,X,Y,Z\}^{\otimes n}} p(P)\,P\rho P$1

proved באמצעות multiplicative weights update on the $\mathcal P(\rho)=\sum_{P\in\{\mathds I,X,Y,Z\}^{\otimes n}} p(P)\,P\rho P$2-dimensional error distribution (Raza et al., 2024). The dependence on $\mathcal P(\rho)=\sum_{P\in\{\mathds I,X,Y,Z\}^{\otimes n}} p(P)\,P\rho P$3 rather than $\mathcal P(\rho)=\sum_{P\in\{\mathds I,X,Y,Z\}^{\otimes n}} p(P)\,P\rho P$4 arises because the regret scales with $\mathcal P(\rho)=\sum_{P\in\{\mathds I,X,Y,Z\}^{\otimes n}} p(P)\,P\rho P$5, not with the ambient dimension itself. In the realizable mistake-bound setting, there exists an online learner making at most

$\mathcal P(\rho)=\sum_{P\in\{\mathds I,X,Y,Z\}^{\otimes n}} p(P)\,P\rho P$6

$\mathcal P(\rho)=\sum_{P\in\{\mathds I,X,Y,Z\}^{\otimes n}} p(P)\,P\rho P$7-mistakes, and a matching lower bound of $\mathcal P(\rho)=\sum_{P\in\{\mathds I,X,Y,Z\}^{\otimes n}} p(P)\,P\rho P$8 mistakes is unavoidable for constant $\mathcal P(\rho)=\sum_{P\in\{\mathds I,X,Y,Z\}^{\otimes n}} p(P)\,P\rho P$9 (Raza et al., 2024).

The same work also gives a shadow tomography procedure specialized to Pauli channels. Given an unknown $$\Lambda(\rho)=\sum_{a\in \mathbb{Z}_2^{2n}} p_a\, P_a \rho P_a,$$0-qubit Pauli channel and adaptively chosen test operators $$\Lambda(\rho)=\sum_{a\in \mathbb{Z}_2^{2n}} p_a\, P_a \rho P_a,$$1, the algorithm outputs estimates $$\Lambda(\rho)=\sum_{a\in \mathbb{Z}_2^{2n}} p_a\, P_a \rho P_a,$$2 satisfying

$$\Lambda(\rho)=\sum_{a\in \mathbb{Z}_2^{2n}} p_a\, P_a \rho P_a,$$3

with probability at least $$\Lambda(\rho)=\sum_{a\in \mathbb{Z}_2^{2n}} p_a\, P_a \rho P_a,$$4, using

$$\Lambda(\rho)=\sum_{a\in \mathbb{Z}_2^{2n}} p_a\, P_a \rho P_a,$$5

copies of the channel (Raza et al., 2024). The key idea is Bell sampling of the Choi state, which yields exact samples from the Pauli error distribution, followed by classical adaptive data analysis (Raza et al., 2024).

This adaptive-shadow result should be distinguished from the statement that adaptivity does not help once maximally entangled inputs and Bell measurements are available (Trinh et al., 2024). The latter is a query-complexity equivalence for distribution estimation and testing, whereas the former concerns adaptive query answering with explicit regret and copy-complexity guarantees. The two are consistent: both exploit the same classicalization of the Pauli channel, but they analyze different learning objectives.

5. Structured noise, locality, and sparse models

A major branch of the literature restricts Pauli channels to physically structured families. One early result showed that an arbitrary $$\Lambda(\rho)=\sum_{a\in \mathbb{Z}_2^{2n}} p_a\, P_a \rho P_a,$$6-qubit Pauli channel can be learned to relative precision using

$$\Lambda(\rho)=\sum_{a\in \mathbb{Z}_2^{2n}} p_a\, P_a \rho P_a,$$7

measurements, while any set of $$\Lambda(\rho)=\sum_{a\in \mathbb{Z}_2^{2n}} p_a\, P_a \rho P_a,$$8 Pauli errors can be estimated to relative precision with

$$\Lambda(\rho)=\sum_{a\in \mathbb{Z}_2^{2n}} p_a\, P_a \rho P_a,$$9

measurements (Flammia et al., 2019). For channels whose Pauli error distribution is a Markov random field with at most $4^n$0-local correlations, the entire channel can be learned with

$4^n$1

measurements (Flammia et al., 2019).

The notion of local Pauli noise was later expanded to the case where the conditional-independence structure is itself unknown. In that setting, the noise distribution $4^n$2 is assumed to be a Gibbs distribution over a bounded-degree hypergraph, and the algorithm learns both the hypergraph structure and the parameters (Rouzé et al., 2023). The headline claim is that structure learning uses only $4^n$3 samples, up to locality-dependent constants, after which the learned structure can be used to obtain a full channel description close in diamond distance from $4^n$4 samples (Rouzé et al., 2023). The same framework is SPAM-robust and uses only single-qubit Cliffords (Rouzé et al., 2023).

Sparse Pauli-Lindblad models form another important structured class. One representation is

$4^n$5

with sparse local generators $4^n$6 (Belkin et al., 2 Jun 2026). Extended sparse Pauli-Lindblad learning has motivated new methods such as Pauli-rotation twirling, graph coloring, and uniform covering arrays to reduce the number of unique fidelities and learning bases that must be estimated (Berg et al., 2023). In that framework, the model is parameterized through fidelities

$4^n$7

with parameter recovery posed as a nonnegative least-squares problem

$4^n$8

(Berg et al., 2023).

Learning only task-relevant channel information can be substantially cheaper than full tomography. For bounded-degree $4^n$9-local observables $$\Lambda(\rho)=\frac{1}{2^n}\sum_{b\in \mathbb{Z}_2^{2n}} \lambda_b \,\operatorname{Tr}(P_b\rho)\,P_b,$$0, one method estimates only the low-weight Pauli eigenvalues $$\Lambda(\rho)=\frac{1}{2^n}\sum_{b\in \mathbb{Z}_2^{2n}} \lambda_b \,\operatorname{Tr}(P_b\rho)\,P_b,$$1 needed to construct a corrected observable $$\Lambda(\rho)=\frac{1}{2^n}\sum_{b\in \mathbb{Z}_2^{2n}} \lambda_b \,\operatorname{Tr}(P_b\rho)\,P_b,$$2 satisfying $$\Lambda(\rho)=\frac{1}{2^n}\sum_{b\in \mathbb{Z}_2^{2n}} \lambda_b \,\operatorname{Tr}(P_b\rho)\,P_b,$$3 (Chen et al., 2023). The sample complexity for this partial learning task is

$$\Lambda(\rho)=\frac{1}{2^n}\sum_{b\in \mathbb{Z}_2^{2n}} \lambda_b \,\operatorname{Tr}(P_b\rho)\,P_b,$$4

with a matching lower bound

$$\Lambda(\rho)=\frac{1}{2^n}\sum_{b\in \mathbb{Z}_2^{2n}} \lambda_b \,\operatorname{Tr}(P_b\rho)\,P_b,$$5

for learning all $$\Lambda(\rho)=\frac{1}{2^n}\sum_{b\in \mathbb{Z}_2^{2n}} \lambda_b \,\operatorname{Tr}(P_b\rho)\,P_b,$$6 with $$\Lambda(\rho)=\frac{1}{2^n}\sum_{b\in \mathbb{Z}_2^{2n}} \lambda_b \,\operatorname{Tr}(P_b\rho)\,P_b,$$7 (Chen et al., 2023). A plausible implication is that, for error mitigation and observable prediction, “Pauli channel learning” is often best understood as a task-specific parameter estimation problem rather than as complete channel identification.

6. Self-consistent learning, gauge freedom, and gate-set noise

In gate-set settings, the object of interest is not a single isolated channel but a family of Pauli noise channels attached to state preparation, measurement, and Clifford gates. Here full identification is obstructed by gauge freedom. One formulation models noisy operations as

$$\Lambda(\rho)=\frac{1}{2^n}\sum_{b\in \mathbb{Z}_2^{2n}} \lambda_b \,\operatorname{Tr}(P_b\rho)\,P_b,$$8

with all $$\Lambda(\rho)=\frac{1}{2^n}\sum_{b\in \mathbb{Z}_2^{2n}} \lambda_b \,\operatorname{Tr}(P_b\rho)\,P_b,$$9 Pauli channels (Chen et al., 2024). Another writes $n$00 and studies which Pauli fidelities $n$01 are learnable in the presence of unknown SPAM noise (Chen et al., 2022).

A basic criterion for individual gate-noise parameters is that

$n$02

where $n$03 records the support pattern of a Pauli operator (Chen et al., 2022). More generally, learnable linear functions of logarithmic fidelities correspond exactly to the cycle space of the pattern transfer graph, while unlearnable information corresponds to the cut space (Chen et al., 2022). This produces the decomposition

$n$04

with cycle benchmarking learning precisely the cycle-space invariants (Chen et al., 2022).

The gate-set generalization in (Chen et al., 2024) develops the same theme using algebraic graph theory. The pattern transfer graph has one root vertex for SPAM and one vertex for each nonzero $n$05-bit pattern. Under the natural identification between parameters and graph edges, learnable functions correspond exactly to the cycle space $n$06, while gauge directions correspond to the cut space $n$07, yielding

$n$08

for the complete model (Chen et al., 2024). For reduced ansatzes $n$09, the learnable and gauge spaces are

$n$10

(Chen et al., 2024).

This graph-theoretic characterization has practical consequences. The number of experiments needed is

$n$11

and efficient ansatzes such as fully local or quasi-local noise yield polynomially many experiments (Chen et al., 2024). In fully local models, only the single-qubit depolarizing gauges survive, so

$n$12

(Chen et al., 2024). This identifies exactly which gate-set Pauli noise parameters are self-consistently learnable and which remain inherently ambiguous.

A common misconception is that enough clever experiments should recover every Pauli parameter in a gate set. The gauge-space results show that this is false: some directions are fundamentally unidentifiable unless one supplements the model with additional assumptions. An explicit attempt to do so by assuming perfect state preparation led to unphysical estimates on IBM hardware, indicating that the assumption was inaccurate (Chen et al., 2022).

7. Statistical estimators and computational regimes

The statistical estimator used for Pauli channel learning depends strongly on the assumed structure and access model. In some settings, direct inversion or empirical Pauli fidelities are natural; in others, they are suboptimal.

For 1D-local sparse Pauli-Lindblad channels, maximum likelihood estimation can be made computationally tractable because the likelihood reduces to an efficiently evaluable Bayesian network (Belkin et al., 2 Jun 2026). The exact MLE objective is

$n$13

where the likelihood is expressed as a sum over subsets of sparse generators (Belkin et al., 2 Jun 2026). By mapping Pauli errors to classical bit flips and introducing latent Bernoulli variables, the model becomes a classical Bayesian network amenable to belief propagation in 1D (Belkin et al., 2 Jun 2026). The paper reports that MLE gives about a threefold reduction in sample complexity relative to empirical Pauli fidelities in simulations, and in a 10-qubit benchmark the per-parameter MSE improved from $n$14 for EPF to $n$15 for MLE (Belkin et al., 2 Jun 2026).

Other learning frameworks emphasize geometric rather than likelihood-based estimation. In ORUC learning, a Pauli channel dressed by local unitaries is learned via simplex-constrained optimization over the Pauli probability vector and contracted quantum learning for the unitary degrees of freedom (Smart et al., 28 Jan 2025). The Pauli component is fit through

$n$16

combined with simplex projection or Riemannian gradient updates (Smart et al., 28 Jan 2025). This is not a general Pauli-channel result, but it illustrates how Pauli learning can serve as a subroutine within broader channel classes.

Memory constraints yield another computational-statistical tradeoff. A protocol using logarithmic quantum memory can estimate every eigenvalue of a Pauli channel with $n$17 measurements (Chen et al., 2023), while a later work replaces the logarithmic-memory counting subroutine with a “weakly-driven quantum walk” that lowers the quantum memory overhead to constant order while preserving the exponential advantage in measurement complexity (Wang et al., 9 Sep 2025). In that construction, the total channel-query cost of the double-stage subroutine is

$n$18

reflecting an explicit tradeoff between memory usage and query complexity (Wang et al., 9 Sep 2025). This suggests that “efficient” Pauli channel learning is multidimensional: the relevant complexity measure may be shots, measurements, total channel uses, quantum memory, or classical post-processing.

From a broader process-learning perspective, Pauli transfer matrix learning gives a different computational regime. With quantum memory, the entire PTM of an arbitrary unknown $n$19-qubit channel can be learned from

$n$20

copies of the Choi state (Caro, 2022). For Pauli channels, this specializes to diagonal PTM learning. Without quantum memory, however, PTM learning for general channels requires exponentially many queries (Caro, 2022). This places Pauli channel learning at an intersection between special-purpose tomography and general quantum process learning.

Pauli channel learning therefore denotes not a single algorithmic problem but a family of closely related inference tasks whose difficulty depends on representation, metric, experimental access, target functionality, structural assumptions, and identifiability constraints. In the simplest entanglement-assisted settings it reduces exactly to classical distribution learning (Trinh et al., 2024). In online and adaptive regimes it admits dimension-efficient regret guarantees (Raza et al., 2024). Under locality, Gibbs, or Pauli-Lindblad assumptions it becomes polynomial in the number of qubits (Flammia et al., 2019, Rouzé et al., 2023, Berg et al., 2023). In gate-set settings it is constrained by gauge freedom and cycle-space identifiability (Chen et al., 2022, Chen et al., 2024). And in experimentally realistic models with limited entanglement or memory, it exhibits sharp resource separations that make it a canonical testbed for quantum advantages in learning (Chen et al., 2021, Chen et al., 2023, Chen et al., 2023).