We define a quantum learning task called agnostic tomography , where given copies of an arbitrary state ρ and a class of quantum states C , the goal is to output a succinct description of a state that approximates ρ at least as well as any state in C (up to some small error ε ). This task generalizes ordinary quantum tomography of states in C and is more challenging because the learning algorithm must be robust to perturbations of ρ .We give an efficient agnostic tomography algorithm for the class C of n -qubit stabilizer product states. Assuming ρ has fidelity at least τ with a stabilizer product state, the algorithm runs in time n O ( log ( 2 / τ ) ) / ε 2 , which is p o l y ( n / ε ) for any constant τ .
We define a quantum learning task called agnostic tomography , where given copies of an arbitrary state ρ and a class of quantum states C , the goal is to output a succinct description of a state that approximates ρ at least as well as any state in C (up to some small error ε ). This task generalizes ordinary quantum tomography of states in C and is more challenging because the learning algorithm must be robust to perturbations of ρ .We give an efficient agnostic tomography algorithm for the class C of n -qubit stabilizer product states. Assuming ρ has fidelity at least τ with a stabilizer product state, the algorithm runs in time n O ( log ( 2 / τ ) ) / ε 2 , which is p o l y ( n / ε ) for any constant τ .
We study the task of agnostic tomography: given copies of an unknown n-qubit state ρ which has fidelity τ with some state in a given class C, find a state which has fidelity ≥τ−ε with ρ. We give a new framework, stabilizer bootstrapping, for designing computationally efficient protocols for this task, and use this to get new agnostic tomography protocols for the following classes: Stabilizer states: We give a protocol that runs in time poly(n,1/ε)⋅(1/τ)O(log(1/τ)), answering an open question posed by Grewal, Iyer, Kretschmer, Liang [43] and Anshu and Arunachalam [6]. Previous protocols ran in time exp(Θ(n)) or required τ>cos2(π/8). States with stabilizer dimension n−t: We give a protocol that runs in time n3⋅(2t/τ)O(log(1/ε)), extending recent work on learning quantum states prepared by circuits with few non-Clifford gates, which only applied in the realizable setting where τ=1 [33, 40, 49, 66]. Discrete product states: If C=K⊗n for some μ-separated discrete set K of single-qubit states, we give a protocol that runs in time (n/μ)O((1+log(1/τ))/μ)/ε2. This strictly generalizes a prior guarantee which applied to stabilizer product states [42]. For stabilizer product states, we give a further improved protocol that runs in time (n2/ε2)⋅(1/τ)O(log(1/τ)). As a corollary, we give the first protocol for estimating stabilizer fidelity, a standard measure of magic for quantum states, to error ε in n3quasipoly(1/ε) time.
We give a general framework for tomography of states that have bounded-extent with respect to a structured class of states. Let C be a family of n-qubit states such that: (i)C is succinctly representable and (ii) there is a weak agnostic learner of C. We give a tomography protocol for an unknown state ∣ψ⟩ that is promised to admit a decomposition of the form ∣ψ⟩=∑ici∣φi⟩, where ∣φi⟩∈C with bounded ℓ1-norm of the coefficients (which we call extent). Our main contribution is to show that a weak agnostic learner for C can be boosted into a tomography algorithm for states with bounded extent with respect to C.
Our reduction is black-box and applies broadly across model classes. As an application, when C is the class of stabilizer states, we obtain tomography algorithms for states with stabilizer extent ξ up to trace distance ε, in time poly(n,(ξ/ε)log(ξ/ε)), which is improvable to $ \textsf{poly}(n,ξ,1/\varepsilon)$ assuming the algorithmic polynomial Freiman-Ruzsa conjecture in the high-doubling regime. When the unknown state ∣ψ⟩ is arbitrary, we give an algorithmic decomposition result in the spirit of a weak regularity lemma for quantum states with respect to C and show that the structure in ∣ψ⟩ that is explainable by C can be efficiently learned. Our main conceptual message is that agnostic learning of a structured base class automatically yields learnability of its low-complexity linear span.
We study the complexity of two closely related learning problems, one quantum and one classical. In the quantum setting, we consider agnostic tomography for the natural class of product mixed states. Given N copies of an n-qubit state ρ, the goal is to output a nearly optimal product mixed state approximation in trace distance. While recent work has focused on pure-state ansatz (e.g., product or stabilizer states), no polynomial-time guarantees were previously known for mixed-state ansatz. In the classical setting, we study robust learning of binary product distributions: given samples from an unknown distribution on 0,1n, the goal is to output a nearly optimal product approximation. Our main contributions are as follows. (1) We give a semi-agnostic tomography algorithm for product mixed states with polynomial sample and computational complexity achieving error O(optlog(1/opt)), where opt is the trace distance to the best product approximation. This is the first efficient algorithm with any nontrivial agnostic guarantee for mixed-state ansatz, using only single-qubit, single-copy measurements. We also prove a Quantum Statistical Query lower bound showing near-optimality, and an unconditional lower bound demonstrating that adaptivity is necessary under single-qubit measurements. (2) We give a semi-agnostic algorithm for robustly learning binary product distributions with matching guarantees and establish a Statistical Query lower bound, essentially resolving the efficient robust learnability of this class and improving on prior work since Diakonikolas et al. (2016).
Learning a classical description of a quantum state is a fundamental task in quantum computation. Among the most important classes of quantum states are stabilizer states, which play a central role in quantum error correction and fault-tolerant computation. To mitigate the effects of realistic noise, agnostic learning of stabilizer states has emerged as a natural and well-motivated problem. Recently, Chen \textit{et al.} [STOC'25, p. 429-438] resolved this problem for qubit systems by using a stabilizer bootstrapping framework. However, the agnostic learning of qudit stabilizer states remains largely unexplored, since the qudit setting introduces fundamental structural differences that prevent a direct generalization of existing qubit techniques. In this paper, we successfully generalize the stabilizer bootstrapping framework to qudit systems and present the first efficient quantum algorithm for agnostic learning of qudit stabilizer states. Specifically, given copies of an unknown n-qudit pure state ∣ψ⟩ that has fidelity τ with some stabilizer state, our algorithm outputs a stabilizer state ∣φ⟩ such that ∣⟨φ∣ψ⟩∣2≥τ−ε with high probability. The algorithm uses only single-copy and four-copy measurements, and its sample and time complexity scale as (d/τ)O(d2log(1/τ))⋅poly(n,1/ε), where the dimension d is an odd prime. As a direct corollary, our algorithm enables efficient estimation of the magic of a quantum state, as quantified by its stabilizer fidelity. Completing the picture, we also present a streamlined algorithm for the high-fidelity regime τ>cos2(π/8), establishing a qudit analogue of the threshold-based approach in prior qubit work.
Characterizing a quantum system by learning its state or evolution is a fundamental problem in quantum physics and learning theory with a myriad of applications. Recently, as a new approach to this problem, the task of agnostic state tomography was defined, in which one aims to approximate an arbitrary quantum state by a simpler one in a given class. Generalizing this notion to quantum processes, we initiate the study of agnostic process tomography: given query access to an unknown quantum channel Φ and a known concept class C of channels, output a quantum channel that approximates Φ as well as any channel in the concept class C , up to some error. In this work, we propose several natural applications for this new task in quantum machine learning, quantum metrology, classical simulation, and error mitigation. In addition, we give efficient agnostic process tomography algorithms for a wide variety of concept classes, including Pauli strings, Pauli channels, quantum junta channels, low-degree channels, and a class of channels produced by Q A C 0 circuits. The main technical tool we use is Pauli spectrum analysis of operators and superoperators. We also prove that, using ancilla qubits, any agnostic state tomography algorithm can be extended to one solving agnostic process tomography for a compatible concept class of unitaries, immediately giving us efficient agnostic learning algorithms for Clifford circuits, Clifford circuits with few T gates, and circuits consisting of a tensor product of single-qubit gates. Together, our results provide insight into the conditions and new algorithms necessary to extend the learnability of a concept class from the standard tomographic setting to the agnostic one.
We establish proper agnostic learning of matrix product states and tree tensor networks. Given copies of an arbitrary quantum state ρ, our algorithms return a state ∣ψ⟩ of chosen bond dimension such that ⟨ψ∣ρ∣ψ⟩ is within ε of the optimum over the model class, without assuming that ρ itself belongs to or is well approximated by that class. The main idea is to use improper learning to compress the mixed-state objective, reducing proper agnostic learning to optimization against a finite collection of explicitly specified pure states. We then introduce a comparator-dual compression procedure that reduces the bond dimension of these targets while uniformly preserving their overlaps with all bounded-bond comparators, with an error independent of system size. For matrix product states, this gives polynomial copy complexity in the system size, local dimension, bond dimension, and 1/ε, together with polynomial runtime in the system size for fixed local dimension, bond dimension, and accuracy. The same framework yields proper agnostic learning of tree tensor networks on bounded-degree trees, with both copy complexity and runtime polynomial in the system size when the remaining parameters are fixed.