We provide more sample-efficient versions of some basic routines in quantum data analysis, along with simpler proofs. Particularly, we give a quantum ”Threshold Search” algorithm that requires only O((log2 m)/є2) samples of a d-dimensional state ρ. That is, given observables 0 ≤ A1, A2, …, Am ≤ 1 such that (ρ Ai) ≥ 1/2 for at least one i, the algorithm finds j with (ρ Aj) ≥ 1/2−є. As a consequence, we obtain a Shadow Tomography algorithm requiring only O((log2 m)(logd)/є4) samples, which simultaneously achieves the best known dependence on each parameter m, d, є. This yields the same sample complexity for quantum Hypothesis Selection among m states; we also give an alternative Hypothesis Selection method using O((log3 m)/є2) samples.
We provide more sample-efficient versions of some basic routines in quantum data analysis, along with simpler proofs. Particularly, we give a quantum ”Threshold Search” algorithm that requires only O((log2 m)/є2) samples of a d-dimensional state ρ. That is, given observables 0 ≤ A1, A2, …, Am ≤ 1 such that (ρ Ai) ≥ 1/2 for at least one i, the algorithm finds j with (ρ Aj) ≥ 1/2−є. As a consequence, we obtain a Shadow Tomography algorithm requiring only O((log2 m)(logd)/є4) samples, which simultaneously achieves the best known dependence on each parameter m, d, є. This yields the same sample complexity for quantum Hypothesis Selection among m states; we also give an alternative Hypothesis Selection method using O((log3 m)/є2) samples.
Shadow Tomography is a fundamental problem in quantum information theory. Given multiple copies of an unknown d-dimensional quantum state ρ and a known collection of observables E1,…,EM, the goal is to estimate all expectation values {Tr(ρEi)}i=1M to additive accuracy ε with probability at least 1−δ.
An elusive open question from the seminal shadow tomography work of Aaronson is whether this task admits a dimension-independent sample complexity with only polylogarithmic dependence on M, as suggested by the best-known lower bounds. In this work, we propose two different quantum protocols for shadow tomography with the best sample complexity O(ε2log(M)log(M/δ)), which is polylogarithmic in the number of observables and independent of the dimension of the unknown state, thereby answering Aaronson's original question while also providing an exponential improvement in the prior best dimension independent sample complexity of shadow tomography from Sinha (STOC 2025) and, more recently, Chen, O'Donnell, Pelecanos, and Wright.
Our approach first reduces the general shadow tomography problem to a finite-ensemble estimation problem via a minimax argument. We then develop an observable-independent protocol that repeatedly applies the pretty-good measurement while updating the prior distribution over the finite ensemble according to the measurement outcomes. A tail analysis of the resulting estimation error yields simultaneous accuracy guarantees for all observables and a cubic-logarithmic upper bound. We also introduce a refined recovery-label measurement for the same finite ensemble, which yields the bound in our main theorem.
We provide more sample-efficient versions of some basic routines in quantum data analysis, along with simpler proofs. Particularly, we give a quantum "Threshold Search" algorithm that requires only O((log2m)/ϵ2) samples of a d-dimensional state ρ. That is, given observables 0≤A1,A2,...,Am≤1 such that tr(ρAi)≥1/2 for at least one i, the algorithm finds j with tr(ρAj)≥1/2−ϵ. As a consequence, we obtain a Shadow Tomography algorithm requiring only O~((log2m)(logd)/ϵ4) samples, which simultaneously achieves the best known dependence on each parameter m, d, ϵ. This yields the same sample complexity for quantum Hypothesis Selection among m states; we also give an alternative Hypothesis Selection method using O~((log3m)/ϵ2) samples.
Following the first paper on quantum algorithms for SDP-solving by Brandão and Svore in 2016, rapid developments has been made on quantum optimization algorithms. Recently Brandão et al. improved the quantum SDP-solver in the so-called quantum state input model, where the input matrices of the SDP are given as purified mixed states. They also gave the first non-trivial application of quantum SDP-solving by obtaining a more efficient algorithm for the problem of shadow tomography (proposed by Aaronson in 2017). In this paper we improve on all previous quantum SDP-solvers. Mainly we construct better Gibbs-samplers for both input models, which directly gives better bounds for SDP-solving. For an SDP with m constraints involving n×n matrices, our improvements yield an O((m+nγ)sγ4) upper bound on SDP-solving in the sparse matrix input model and an O((m+B2.5γ3.5)Bγ4) upper bound in the quantum state input model. We then apply these results to the problem of shadow tomography to simultaneously improve the best known upper bounds on sample complexity due to Aaronson and complexity due Brandao et al. Furthermore, we apply our quantum SDP-solvers to the problems of quantum state discrimination and E-optimal design. In both cases we beat the classical lower bound in terms of some parameters, at the expense of heavy dependence on some other parameters. Finally we prove two lowers bounds for solving SDPs using quantum algorithms: (1) $\tildeΩ(\sqrt{m}B/\eps)$ in the quantum state input model, and (2) $\tildeΩ(\sqrt{m}α/\eps)$ in the quantum operator input model. These lower bounds show that the m factor and the polynomial dependence on the parameters B,α, and $1/\eps$ are necessary.
In the quantum state tomography problem, one wishes to estimate an unknown d-dimensional mixed quantum state ρ, given few copies. We show that O(d/ε) copies suffice to obtain an estimate ρ that satisfies ||ρ − ρ||F2 ≤ ε (with high probability). An immediate consequence is that O((ρ) · d/ε2) ≤ O(d2/ε2) copies suffice to obtain an ε-accurate estimate in the standard trace distance. This improves on the best known prior result of O(d3/ε2) copies for full tomography, and even on the best known prior result of O(d2log(d/ε)/ε2) copies for spectrum estimation. Our result is the first to show that nontrivial tomography can be obtained using a number of copies that is just linear in the dimension.
We study the power of quantum memory for learning properties of quantum systems and dynamics, which is of great importance in physics and chemistry. Many state-of-the-art learning algorithms require access to an additional external quantum memory. While such a quantum memory is not required a priori, in many cases, algorithms that do not utilize quantum memory require much more data than those which do. We show that this trade-off is inherent in a wide range of learning problems. Our results include the following: (1) We show that to perform shadow tomography on an n-qubit state rho with M observables, any algorithm without quantum memory requires Ω(min(M,2n)) samples of rho in the worst case. Up to logarithmic factors, this matches the upper bound of [HKP20] and completely resolves an open question in [Aar18, AR19]. (2) We establish exponential separations between algorithms with and without quantum memory for purity testing, distinguishing scrambling and depolarizing evolutions, as well as uncovering symmetry in physical dynamics. Our separations improve and generalize prior work of [ACQ21] by allowing for a broader class of algorithms without quantum memory. (3) We give the first tradeoff between quantum memory and sample complexity. We prove that to estimate absolute values of all n-qubit Pauli observables, algorithms with k<n qubits of quantum memory require at least Ω(2(n−k)/3) samples, but there is an algorithm using n-qubit quantum memory which only requires O(n) samples. The separations we show are sufficiently large and could already be evident, for instance, with tens of qubits. This provides a concrete path towards demonstrating real-world advantage for learning algorithms with quantum memory.
In Online Shadow Tomography, we are given copies of an unknown d-dimensional quantum state ρ, an adversary (adaptively) proposes a sequence of bounded observables A(1),…,A(m), and after each A(t) is given we must estimate Tr(A(t)ρ) to within ±ε. This is the direct quantum generalization of the classical problem of Adaptive Data Analysis. Prior results for online Shadow Tomography were suboptimal in all three parameters m,d,ε, lagging behind the best known and classical rates, for which there is some evidence of optimality. In this work, we finally close this gap, giving a pair of algorithms matching the classical rates. Our first algorithm is the first to achieve o(log2m)-dependence together with poly(log(d)/ε); moreover, it improves all three exponents even in the Offline Shadow Tomography setting. Our second algorithm is known to be optimal among bounds independent of d, and improves the best prior result by a mlogm factor.
The key to our proof is a new framework for quantifying post-measurement damage, based on the quantum Efron-Stein decomposition.