Classical verification of quantum learning allows classical clients to reliably leverage quantum computing advantages by interacting with untrusted quantum servers. Yet, current quantum devices available in practice suffers from a variety of noises and whether existed classical verification protocols carry over to noisy scenarios remains unclear. Here, we propose an efficient classical error rectification algorithm to reconstruct the noise-free results given by the quantum Fourier sampling circuit with practical constant-level noises. In particular, we prove that the error rectification algorithm can restore the heavy Fourier coefficients by using a small number of noisy samples that scales logarithmically with the problem size. We apply this algorithm to the agnostic parity learning task with uniform input marginal and prove that this task can be accomplished in an efficient way on noisy quantum devices with our algorithm. In addition, we prove that a classical client with access to the random example oracle can verify the agnostic parity learning results from the noisy quantum prover in an efficient way, under the condition that the Fourier coefficients are sparse. Our results demonstrate the feasibility of classical verification of quantum learning advantages with noises, which provide a valuable guide for both theoretical studies and practical applications with current noisy intermediate scale quantum devices.
Classical verification of quantum learning allows classical clients to reliably leverage quantum computing advantages by interacting with untrusted quantum servers. Yet, current quantum devices available in practice suffers from a variety of noises and whether existed classical verification protocols carry over to noisy scenarios remains unclear. Here, we propose an efficient classical error rectification algorithm to reconstruct the noise-free results given by the quantum Fourier sampling circuit with practical constant-level noises. In particular, we prove that the error rectification algorithm can restore the heavy Fourier coefficients by using a small number of noisy samples that scales logarithmically with the problem size. We apply this algorithm to the agnostic parity learning task with uniform input marginal and prove that this task can be accomplished in an efficient way on noisy quantum devices with our algorithm. In addition, we prove that a classical client with access to the random example oracle can verify the agnostic parity learning results from the noisy quantum prover in an efficient way, under the condition that the Fourier coefficients are sparse. Our results demonstrate the feasibility of classical verification of quantum learning advantages with noises, which provide a valuable guide for both theoretical studies and practical applications with current noisy intermediate scale quantum devices.
We develop a framework for learning from noisy quantum experiments in which fault-tolerant devices access uncharacterized systems through noisy couplings. Introducing the complexity class NBQP ("noisy BQP''), we model noisy fault-tolerant quantum computers that cannot generally error-correct the oracle systems they query. Using this class, we prove that while noise can eliminate the exponential quantum learning advantages of unphysical, noiseless learners, a superpolynomial gap remains between NISQ and fault-tolerant devices. Turning to canonical learning tasks in noisy settings, we find that the exponential two-copy advantage for purity testing collapses under local depolarizing noise. Nevertheless, we identify a setting motivated by AdS/CFT in which noise-resilient physical structure restores this quantum learning advantage. We then analyze noisy Pauli shadow tomography, deriving lower bounds characterizing how instance size, quantum memory and noise jointly control sample complexity, and design algorithms with parametrically matching scalings. We study similar tradeoffs in quantum metrology, and show that the Heisenberg-limited sensitivity of existing error-correction-based protocols persists only up to a timescale inverse-polynomial in the error rate per probe qubit. Together, our results demonstrate that the primitives underlying quantum-enhanced experiments are fundamentally fragile to noise, and that realizing meaningful quantum advantages in future experiments will require interfacing noise-robust physical properties with available algorithmic techniques.
It is not known whether a classical polynomial-time verifier can verify arbitrary polynomial-time quantum computations with an efficient quantum honest prover without computational assumptions. We show that, relative to a classical oracle, such verification is possible for a language beyond the polynomial hierarchy that is decided exactly in quantum polynomial time. The protocol is public-coin, with perfect completeness and soundness error at most 1/4. To our knowledge, no language with such a proof system was previously proved to lie outside AM relative to an oracle.
For every unbounded polynomial-time computable function, the oracle can be chosen so that the message count and the oracle query counts of the quantum algorithm, verifier, and honest prover are bounded by this function of the input length. The message count is optimal up to a constant factor, since for a suitable oracle every interactive proof system for the language uses at least a constant fraction of the messages of our protocol on infinitely many lengths, with a constant depending on its verifier.
The language is a variant of recursive Fourier sampling in which code tables encode secrets that address data tables. The tables are evaluated recursively, and the oracle stores only the leaves. The quantum algorithm recovers every secret exactly, and the verifier checks the prover's secrets recursively against random code entries. The recursion restricts the input in a correlated way, as the same secret fixes the code bits and chooses the free coordinate, so Håstad's switching lemma does not apply. We prove a switching theorem for these restrictions, which may be of independent interest. For instances of sufficient height, no constant-depth circuit of size 2poly(n) changes its output with probability above 2−Ω(n) when the answer of a random instance is flipped by changing a single leaf.
Quantum computing holds unparalleled potentials to enhance machine learning. However, a demonstration of quantum learning advantage has not been achieved so far. We make a step forward by rigorously establishing a noise-robust, unconditional quantum learning advantage in expressivity, inference speed, and training efficiency, compared to commonly-used classical models. Our proof is information-theoretic and pinpoints the origin of this advantage: entanglement can be used to reduce the communication required by non-local tasks. In particular, we design a task that can be solved with certainty by quantum models with a constant number of parameters using entanglement, whereas commonly-used classical models must scale linearly to achieve a larger-than-exponentially-small accuracy. We show that the quantum model is trainable with constant resources and robust against constant noise. Through numerical and trapped-ion experiments on IonQ Aria, we demonstrate the desired advantage. Our results provide valuable guidance for demonstrating quantum learning advantages with current noisy intermediate-scale devices.
Efficiently characterizing large quantum states and processes is a central yet notoriously challenging task in quantum information science, as conventional tomography methods typically require resources that grow exponentially with system size. Here, we introduce a structure-agnostic learning framework for noisy n-qubit quantum circuits under~i.i.d.~single-qubit noise. We first prove that quantum states with unital noise channels admit an efficient learnable representation in the logarithmic-depth regime. We then extend this framework to quantum process tomography under constant noise, deriving a unified protocol that applies to both unital and non-unital noisy channels and retains efficient guarantees for logarithmic-depth circuits. This process-learning formulation is input-agnostic and imposes no distributional assumptions on the input quantum states. We further study a more general regime with arbitrary noise strength. In this setting, low-weight Pauli propagation induces a terminal truncation whose threshold depends logarithmically on the inverse accuracy, leading to quasi-polynomial complexity and near-unit success probability in the average case. In contrast to the preceding two results, this arbitrary-noise guarantee does not impose any restriction on the circuit depth, and therefore covers arbitrary-depth circuits, including both the noiseless limit (γ=0) and the strong-decoherence regime (γ=Θ(1)). Numerical simulations of two-dimensional Hamiltonian dynamics further demonstrate the accuracy and robustness of the approach, including for structured circuits beyond the random-circuit setting assumed in the theoretical analysis. These results provide a scalable and practically relevant route toward characterizing large-scale noisy quantum devices, addressing a key bottleneck in the development of quantum technologies.
In a work by Raz (J. ACM and FOCS 16), it was proved that any algorithm for parity learning on n bits requires either Ω(n2) bits of classical memory or an exponential number (in n) of random samples. A line of recent works continued that research direction and showed that for a large collection of classical learning tasks, either super-linear classical memory size or super-polynomially many samples are needed. All these works consider learning algorithms as classical branching programs, which perform classical computation within bounded memory. However, these results do not capture all physical computational models, remarkably, quantum computers and the use of quantum memory. It leaves the possibility that a small piece of quantum memory could significantly reduce the need for classical memory or samples and thus completely change the nature of the classical learning task. Despite the recent research on the necessity of quantum memory for intrinsic quantum learning problems like shadow tomography and purity testing, the role of quantum memory in classical learning tasks remains obscure. In this work, we study classical learning tasks in the presence of quantum memory. We prove that any quantum algorithm with both, classical memory and quantum memory, for parity learning on n bits, requires either Ω(n2) bits of classical memory or Ω(n) bits of quantum memory or an exponential number of samples. In other words, the memory-sample lower bound for parity learning remains qualitatively the same, even if the learning algorithm can use, in addition to the classical memory, a quantum memory of size c n (for some constant c>0). Our result is more general and applies to many other classical learning tasks. Following previous works, we represent by the matrix M: A × X → {−1,1} the following learning task. An unknown x is sampled uniformly at random from a concept class X, and a learning algorithm tries to uncover x by seeing streaming of random samples (ai, bi = M(ai, x)) where for every i, ai∈ A is chosen uniformly at random. Assume that k,ℓ,r are integers such that any submatrix of M of at least 2−k·|A| rows and at least 2−ℓ·|X| columns, has a bias of at most 2−r. We prove that any algorithm with classical and quantum hybrid memory for the learning problem corresponding to M needs either (1) Ω(k · ℓ) bits of classical memory, or (2) Ω(r) qubits of quantum memory, or (3) 2Ω(r) random samples, to achieve a success probability at least 2−O(r). Our results refute the possibility that a small amount of quantum memory significantly reduces the size of classical memory needed for efficient learning on these problems. Our results also imply improved security of several existing cryptographical protocols in the bounded-storage model (protocols that are based on parity learning on n bits), proving that security holds even in the presence of a quantum adversary with at most c n2 bits of classical memory and c n bits of quantum memory (for some constant c>0).
Learning problems involving quantum data are natural candidates for demonstrating an advantage in quantum machine learning. Recent results indicate that, for certain tasks and under noiseless conditions, coherent processing of quantum data outperforms fixed-measurement schemes followed by classical processing. It remained uncertain whether this performance gap persists at a finite scale, and in the presence of noise that is unavoidable with current quantum devices. In this work, we present simulations and analysis of the performance of existing hardware on a learning problem known to exhibit asymptotic advantage, now subjected to noisy quantum data. Comparing coherent quantum processing directly against fixed-measurement schemes, our results demonstrate a clear performance separation at a scale of just 30 to 40 noisy qubits. Already at this scale, the fundamental bottleneck is no longer classical computation but data acquisition; matching the noisy coherent protocol with measure-first strategies would still require months or even years of measurements. By systematically evaluating hardware constraints such as state preparation, gate errors, readout errors, connectivity, and coherence times, we provide evidence that a demonstration of such a strong learning advantage is accessible on near-term devices.