Terminal-Register Certification for Finite-Measurement Learning of Multiscale Quantum States

We study stabilizer state testing and learning with limited coherent quantum memory. Here an algorithm sequentially receives copies of an unknown $n$-qubit state, but may keep only $k$ qubits of coherent quantum memory between measurements. With unrestricted memory, seminal work of Gross, Nezami and Walter showed how to test $n$-qubit stabilizer states using $6$ copies, which is dimension independent, unlike the learning complexity of $Θ(n)$. We show that this testing-vs-learning separation is lost under memory constraints. More concretely we show that (1) The sample complexity of testing stabilizer states in the $k$-qubit memory framework is $Θ(n-k)$. Our upper bound goes via a novel connection to the hidden shift problem and the lower bound is proven using a novel approach to average case bounds on likelihood ratios via combinatorics of the stochastic orthogonal group. (2) The sample complexity of learning stabilizer states with $k$ qubits of memory, in the non-adaptive framework, is $Θ(n^2/k)$. As a further application of our techniques, we prove an exponential lower bound for purity testing even when the memory may be left coherent throughout the protocol. Our main results identify coherent quantum memory as the resource enabling the usual separation between stabilizer testing and learning. In particular, even with $k=0.99n$ qubits of memory, there is no constant-copy stabilizer tester; furthermore for $k=cn$ qubits of memory (for $0< c < 1$), stabilizer testing is as hard as learning, with both requiring $Θ(n)$ copies.

Whether quantum computation can reduce the amount of data sampled from an unknown probability distribution required to learn a prediction rule is a fundamental question in quantum machine learning. Quantum PAC learning studies this question using quantum data as a quantum state whose squared amplitudes encode the unknown distribution from which classical learning data are sampled. With only copies of such quantum data, the optimal worst-case sample complexity asymptotically matches that of classical PAC learning. In contrast, access to both a state-preparation unitary for this state and its inverse can improve the query-complexity dependence on the accuracy parameter in realizable learning. However, it has remained unclear whether forward-only access allows such an improvement. In this work, taking the worst case over compatible state-preparation unitaries and their finite ambient dimensions, we show that the optimal forward-only query complexities of realizable and agnostic learning are, respectively, $Θ((d+\log(1/δ))/\varepsilon)$ and $Θ((d+\log(1/δ))/\varepsilon^2)$, where $d$ is the VC dimension of the concept class, $\varepsilon$ the accuracy parameter, and $δ$ the failure probability. These bounds match the optimal sample complexities with classical data or quantum data copies. To prove them, we establish a reduction using $q$ copies of the prepared state to approximate the Haar-averaged output of any $q$-query forward-only algorithm. These results show that forward-only access cannot provide an asymptotic query-complexity advantage over learning from classical data or quantum data copies in this worst-case setting, and establish the essential role of inverse access in the known realizable-setting improvement. Our reduction also provides a new framework for analyzing limitations of forward state-preparation access via state-copy lower bounds.

We extend quantum state tomography with minimal cumulative disturbance, first investigated in [arXiv:2406.18370], to arbitrary finite-dimensional pure states. A learner sequentially receives fresh copies of an unknown pure state, chooses a rank-one projector for each copy using the previous outcomes, and performs the corresponding two-outcome projective measurement. The goal is to learn the state while keeping the chosen projectors close to the unknown state in order to minimize disturbance. The qubit solution relies on the special geometry of the Bloch sphere and does not extend directly to qudits, where pure states form a curved manifold. We show that this obstruction can be overcome by working locally on the pure-state manifold. The algorithm proceeds in epochs. In each epoch, it fixes a current estimate, measures pairs of nearby rank-one projectors obtained by moving in opposite tangent directions, and takes differences of the corresponding outcomes. This gives an exact linear observation of the tangent component of the error. The resulting local linear models are combined with a robust variance-adaptive estimator and a hot-start regularization that transfers precision across epochs. For every unknown pure state in dimension $d$, after $T$ measured copies, our protocol achieves cumulative regret $\mathcal{O}(d^3\log^2 T)$, and at each intermediate time $t\leq T$ its current estimate has online infidelity $\mathcal{O}(d^3\log(T)/t)$. Hence, pure-state tomography with essentially no cumulative disturbance is not a peculiarity of qubits but a geometric phenomenon that persists for qudits.