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arXiv:2508.13959 · Annual Symposium on Foundations of Computer Science

Adversarially Robust Quantum State Learning and Testing

Maryam Aliakbarpour, Vladimir Braverman, Nai-Hui Chia, Yuhan Liu

Published Dec 14, 2025• 0 citations
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Abstract

Quantum state learning is a fundamental problem in physics and computer science. As near-term quantum devices are error-prone, it is important to design error-resistant algorithms. Apart from device errors, other unexpected factors could also affect the algorithm, such as careless human read-out error, or even a malicious hacker deliberately altering the measurement results. Thus, we want our algorithm to work even in the worst case when things go against our favor.We consider the practical setting of single-copy measurements and propose the $\gamma$-adversarial corruption model where an imaginary adversary can arbitrarily change $\gamma$-fraction of the measurement outcomes. This is stronger than the $\gamma$-bounded SPAM noise model, where the post-measurement state changes by at most $\gamma$ in trace distance. Under our stronger model of corruption, we design an algorithm using non-adaptive measurements that can learn an unknown rank- r state up to $\tilde{O}(\gamma \sqrt{r})$ in trace distance, provided that the number of copies is sufficiently large. We further prove an information-theoretic lower bound of $\Omega(\gamma \sqrt{r})$ for non-adaptive measurements, demonstrating the optimality of our algorithm. Our upper and lower bounds also hold for quantum state testing, where the goal is to test whether an unknown state is equal to a given state or far from it. Our results are intriguingly optimistic and pessimistic at the same time. For general states, the error is dimension-dependent and $\gamma \sqrt{d}$ in the worst case, meaning that only corrupting a very small fraction $(1 / \sqrt{d})$ of the outcomes could totally destroy any non-adaptive learning algorithm. However, for constant-rank states that are useful in many quantum algorithms, it is possible to achieve dimension-independent error, even in the worst-case adversarial setting.

DOI
10.1109/focs63196.2025.00013

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Quantum state learning is a fundamental problem in physics and computer science. As near-term quantum devices are error-prone, it is important to design error-resistant algorithms. Apart from device errors, other unexpected factors could also affect the algorithm, such as careless human read-out error, or even a malicious hacker deliberately altering the measurement results. Thus, we want our algorithm to work even in the worst case when things go against our favor.We consider the practical setting of single-copy measurements and propose the γ\gammaγ-adversarial corruption model where an imaginary adversary can arbitrarily change γ\gammaγ-fraction of the measurement outcomes. This is stronger than the γ\gammaγ-bounded SPAM noise model, where the post-measurement state changes by at most γ\gammaγ in trace distance. Under our stronger model of corruption, we design an algorithm using non-adaptive measurements that can learn an unknown rank- r state up to O~(γr)\tilde{O}(\gamma \sqrt{r})O~(γr​) in trace distance, provided that the number of copies is sufficiently large. We further prove an information-theoretic lower bound of Ω(γr)\Omega(\gamma \sqrt{r})Ω(γr​) for non-adaptive measurements, demonstrating the optimality of our algorithm. Our upper and lower bounds also hold for quantum state testing, where the goal is to test whether an unknown state is equal to a given state or far from it. Our results are intriguingly optimistic and pessimistic at the same time. For general states, the error is dimension-dependent and γd\gamma \sqrt{d}γd​ in the worst case, meaning that only corrupting a very small fraction (1/d)(1 / \sqrt{d})(1/d​) of the outcomes could totally destroy any non-adaptive learning algorithm. However, for constant-rank states that are useful in many quantum algorithms, it is possible to achieve dimension-independent error, even in the worst-case adversarial setting.
We study single-copy shadow tomography in the adversarial robust setting, where the goal is to learn the expectation values of MMM observables O1,…,OMO_1, \ldots, O_MO1​,…,OM​ with ε\varepsilonε accuracy, but γγγ-fraction of the outcomes can be arbitrarily corrupted by an adversary. We show that all non-adaptive shadow tomography algorithms must incur an error of ε=Ω~(γmin⁡{M,d})\varepsilon=\tildeΩ(γ\min\{\sqrt{M}, \sqrt{d}\})ε=Ω~(γmin{M​,d​}) for some choice of observables, even with unlimited copies. Unfortunately, the classical shadows algorithm by [HKP20] and naive algorithms that directly measure each observable suffer even more. We design an algorithm that achieves an error of ε=O~(γmax⁡i∈[M]∥Oi∥HS)\varepsilon=\tilde{O}(γ\max_{i\in[M]}\|O_i\|_{HS})ε=O~(γmaxi∈[M]​∥Oi​∥HS​), which nearly matches our worst-case error lower bound for M≥dM\ge dM≥d and guarantees better accuracy when the observables have stronger structure. Remarkably, the algorithm only needs n=1γ2log⁡(M/δ)n=\frac{1}{γ^2}\log(M/δ)n=γ21​log(M/δ) copies to achieve that error with probability at least 1−δ1-δ1−δ, matching the sample complexity of the classical shadows algorithm that achieves the same error without corrupted measurement outcomes. Our algorithm is conceptually simple and easy to implement. Classical simulation for fidelity estimation shows that our algorithm enjoys much stronger robustness than [HKP20] under adversarial noise. Finally, based on a reduction from full-state tomography to shadow tomography, we prove that for rank rrr states, both the near-optimal asymptotic error of ε=O~(γr)\varepsilon=\tilde{O}(γ\sqrt{r})ε=O~(γr​) and copy complexity O~(dr2/ε2)=O~(dr/γ2)\tilde{O}(dr^2/\varepsilon^2)=\tilde{O}(dr/γ^2)O~(dr2/ε2)=O~(dr/γ2) can be achieved for adversarially robust state tomography, closing the large gap in [ABCL25] where optimal error can only be achieved using pseudo-polynomial number of copies in ddd.
We study the problems of quantum tomography and shadow tomography using measurements performed on individual, identical copies of an unknown d -dimensional state. We first revisit known lower bounds [ 23 ] on quantum tomography with accuracy ε in trace distance, when the measurement choices are independent of previously observed outcomes, i.e., they are nonadaptive. We give a succinct proof of these results through the χ 2 -divergence between suitable distributions. Unlike prior work, we do not require that the measurements be given by rank-one operators. This leads to stronger lower bounds when the learner uses measurements with a constant number of outcomes (e.g., two-outcome measurements). In particular, this rigorously establishes the optimality of the folklore “Pauli tomography” algorithm in terms of its sample complexity. We also derive novel bounds of Ω(r2d/ϵ2)\Omega (r^2 d/\epsilon ^2)Ω(r2d/ϵ2) and Ω(r2d2/ϵ2)\Omega (r^2 d^2/\epsilon ^2)Ω(r2d2/ϵ2) for learning rank r states using arbitrary and constant-outcome measurements, respectively, in the nonadaptive case. In addition to the sample complexity, a resource of practical significance for learning quantum states is the number of unique measurement settings required (i.e., the number of different measurements used by an algorithm, each possibly with an arbitrary number of outcomes). Motivated by this consideration, we employ concentration of measure of χ 2 -divergence of suitable distributions to extend our lower bounds to the case where the learner performs possibly adaptive measurements from a fixed set of exp⁡(O(d))\exp (O(d))exp(O(d)) possible measurements. This implies in particular that adaptivity does not give us any advantage using single-copy measurements that are efficiently implementable. We also obtain a similar bound in the case where the goal is to predict the expectation values of a given sequence of observables, a task known as shadow tomography. Finally, in the case of adaptive, single-copy measurements implementable with polynomial-size circuits, we prove that a straightforward strategy based on computing sample means of the given observables is optimal.
Continuous-variable systems enable key quantum technologies in computation, communication, and sensing. Bosonic Gaussian states emerge naturally in various such applications, including gravitational-wave and dark-matter detection. A fundamental question is how to characterize an unknown bosonic Gaussian state from as few samples as possible. Despite decades-long exploration, the ultimate efficiency limit remains unclear. In this work, we study the necessary and sufficient number of copies to learn an nnn-mode Gaussian state, with energy less than EEE, to ε\varepsilonε trace distance with high probability. We prove a lower bound of Ω(n3/ε2)Ω(n^3/\varepsilon^2)Ω(n3/ε2) for Gaussian measurements, matching the best known upper bound up to doubly-log energy dependence, and Ω(n2/ε2)Ω(n^2/\varepsilon^2)Ω(n2/ε2) for arbitrary measurements. We further show an upper bound of O~(n2/ε2)\widetilde{O}(n^2/\varepsilon^2)O(n2/ε2) given that the Gaussian state is promised to be either pure or passive. Interestingly, while Gaussian measurements suffice for nearly optimal learning of pure Gaussian states, non-Gaussian measurements are provably required for optimal learning of passive Gaussian states. Finally, focusing on learning single-mode Gaussian states via non-entangling Gaussian measurements, we provide a nearly tight bound of Θ~(E/ε2)\widetildeΘ(E/\varepsilon^2)Θ(E/ε2) for any non-adaptive schemes, showing adaptivity is indispensable for nearly energy-independent scaling. As a byproduct, we establish sharp bounds on the trace distance between Gaussian states in terms of the total variation distance between their Wigner distributions, and obtain a nearly tight sample complexity bound for learning the Wigner distribution of any Gaussian state to ε\varepsilonε total variation distance. Our results significantly advance quantum learning theory in the bosonic regimes and have practical impact in quantum sensing and benchmarking applications.
Quantum state tomography is fundamental to quantum information processing but becomes infeasible at scale due to the exponential growth of the state space. Shadow tomography alleviates this challenge by focusing on predicting measurement outcomes rather than reconstructing the full state. Its online variant models adaptive and potentially adversarial measurement scenarios, where a learner sequentially predicts outcomes while competing with the best fixed quantum state in hindsight. We show that exploiting additional structure in the measurements leads to significantly stronger regret guarantees. In particular, under the assumption that the adversarial measurements have bounded Frobenius norm, we analyze online mirror descent and derive optimal regret bounds that depend on intrinsic structural properties, such as rank or sparsity in the standard basis, rather than the dimension of the measurement operators. As a complementary result, we also show that, even in the setting where adversarial measurements are known to be sparse in the Pauli basis commonly used in variational quantum eigensolvers and near-term quantum error mitigation, the underlying regret bound for learning quantum states is the same as that obtained in the generic setting, where the adversarial measurements are not known to possess any particular structure.
We consider the classic question of state tomography: given copies of an unknown quantum state ρ∈Cd×d\rho \in \mathbb{C}^{d \times d}ρ∈Cd×d, output ρ^\widehat{\rho}ρ​ which is close to ρ\rhoρ in some sense, e.g. trace distance or fidelity. When one is allowed to make coherent measurements entangled across all copies, Θ(d2/ε2)\Theta\left(d^{2} / \varepsilon^{2}\right)Θ(d2/ε2) copies are necessary and sufficient to get trace distance ε\varepsilonε [18], [29]. Unfortunately, the protocols achieving this rate incur large quantum memory overheads that preclude implementation on near-term devices. On the other hand, the best known protocol using incoherent (single-copy) measurements uses O(d3/ε2)O\left(d^{3} / \varepsilon^{2}\right)O(d3/ε2) copies [24], and multiple papers have posed it as an open question to understand whether or not this rate is tight [6], [18]. In this work, we fully resolve this question, by showing that any protocol using incoherent measurements, even if they are chosen adaptively, requires Ω(d3/ε2)\Omega\left(d^{3} / \varepsilon^{2}\right)Ω(d3/ε2) copies, matching the upper bound of [24]. We do so by a new proof technique which directly bounds the “tilt” of the posterior distribution after measurements, which yields a surprisingly short proof of our lower bound, and which we believe may be of independent interest. While this implies that adaptivity does not help for tomography with respect to trace distance, we show that it actually does help for tomography with respect to infidelity. We give an adaptive algorithm that outputs a state which is γ\gammaγ-close in infidelity to ρ\rhoρ using only O~(d3/γ)\widetilde{O}\left(d^{3} / \gamma\right)O(d3/γ) copies, which is optimal for incoherent measurements. In contrast, it is known [18] that any nonadaptive algorithm requires Ω(d3/γ2)\Omega\left(d^{3} / \gamma^{2}\right)Ω(d3/γ2) copies. While it is folklore that in 2 dimensions, one can achieve a scaling of O(1/γ)O(1 / \gamma)O(1/γ), to the best of our knowledge, our algorithm is the first to achieve the optimal rate in all dimensions.
We study the problems of quantum state certification, equivalence testing and independence testing. In certification, given samples of an unknown quantum state ρρρ and the description of a state σσσ, the goal is to test whether ρ=σρ=σρ=σ, or whether ρρρ and σσσ are far in a given distance measure. In equivalence testing, σσσ is also unknown and only accessible via samples. Independence testing decides whether ρAC=ρA⊗ρCρ_{AC}=ρ_A\otimesρ_CρAC​=ρA​⊗ρC​, or is far from being a product. The sample complexities of these problems are now well-understood for a decision gap ε\varepsilonε in trace distance: in the single-copy measurement setting with ddd-dimensional states, all three tasks can be solved using the same non-adaptive approach, which uses Θ(d3/2/ε2)Θ(d^{3/2}/\varepsilon^2)Θ(d3/2/ε2) samples and is optimal in general, even without adaptivity. In this work, we consider decision gaps expressed in fidelity and study possible separations between these problems and how adaptivity can help. We prove that certification with respect to fidelity for a state σσσ of rank rrr does not benefit from adaptivity and requires Θ~(r3/2/ε)\widetildeΘ(r^{3/2}/\varepsilon)Θ(r3/2/ε) samples. For equivalence testing and independence testing, we provide adaptive algorithms using O~(min⁡{d3/2/ε2,d9/4/ε})\widetilde{O}(\min\{d^{3/2}/\varepsilon^2,d^{9/4}/\varepsilon\})O(min{d3/2/ε2,d9/4/ε}) and O~(min⁡{(dAdC)3/2/ε2,dA9/4dC3/4/ε})\widetilde{O}(\min\{(d_Ad_C)^{3/2}/\varepsilon^2,d_A^{9/4}d_C^{3/4}/\varepsilon\})O(min{(dA​dC​)3/2/ε2,dA9/4​dC3/4​/ε}) samples, for dA≥dCd_A\geq d_CdA​≥dC​, respectively. Our main technique is a framework that uses partial learning and a reduction to testing in ℓ2\ell_2ℓ2​-distance, adapted from the distribution testing literature. We show that adaptivity matters for equivalence testing in fidelity by proving that Ω~(1/ε2)\widetildeΩ(1/\varepsilon^2)Ω(1/ε2) samples are necessary in the non-adaptive case even for qubits, showing a separation from certification.