In this work, we consider the fundamental task of estimating the purity of an unknown state to within $\textit{multiplicative}$ error $\varepsilon$. When one can perform general collective measurements, $Θ\left(\frac{\sqrt{d}}{\varepsilon^2} + \frac{d}{\varepsilon}\right)$ copies are known to be necessary and sufficient for this problem [AISW20]. However, implementing collective measurements on such a large number of copies can be experimentally demanding, and we thus aim to characterize the copy complexity of this problem with incoherent measurements.
In this setting, the only non-trivial result is a non-adaptive algorithm that uses $O\left(
\frac{d}{\varepsilon^2} + \frac{d^2}{\varepsilon}
\right)$ copies [PTTW26], which is polynomially larger than the best-known lower bound. Our first result is a new algorithm performing non-adaptive incoherent measurements that succeeds using $O\left(
\frac{d}{\varepsilon^2} + \frac{d^{3/2}}{\varepsilon}
\right)$ copies, improving on the latter term in the copy complexity. Moreover, we show that for any algorithm restricted to non-adaptive measurements, the above copy complexity is optimal.
We also develop a new adaptive estimator for the purity of a state that improves on the above complexity in the high-precision regime, i.e., for $\varepsilon = o(1/d)$. We also show that the non-adaptive and adaptive estimators together yield the optimal complexity for incoherent purity estimation; in particular, we show that the copy complexity of this problem is $$
Θ\left(\min\left\{
\frac{d}{\varepsilon^2} + \frac{d^{3/2}}{\varepsilon}, \frac{d^2}{\varepsilon} + \frac{\sqrt{d}}{\varepsilon^2}\right\}
\right).
$$
In this work, we consider the fundamental task of estimating the purity of an unknown state to within multiplicative error ε. When one can perform general collective measurements, Θ(ε2d+εd) copies are known to be necessary and sufficient for this problem [AISW20]. However, implementing collective measurements on such a large number of copies can be experimentally demanding, and we thus aim to characterize the copy complexity of this problem with incoherent measurements.
In this setting, the only non-trivial result is a non-adaptive algorithm that uses $O\left(
\right)$ copies, improving on the latter term in the copy complexity. Moreover, we show that for any algorithm restricted to non-adaptive measurements, the above copy complexity is optimal.
We also develop a new adaptive estimator for the purity of a state that improves on the above complexity in the high-precision regime, i.e., for ε=o(1/d). We also show that the non-adaptive and adaptive estimators together yield the optimal complexity for incoherent purity estimation; in particular, we show that the copy complexity of this problem is $$
We study the sample complexity of shadow tomography in the high-precision regime under realistic measurement constraints. Given an unknown d-dimensional quantum state ρ and a known set of observables {Oi}i=1m, the goal is to estimate expectation values {tr(Oiρ)}i=1m to accuracy ε in Lp-norm, using possibly adaptive measurements that act on O(polylog(d)) number of copies of ρ at a time. We focus on the regime where ε is below an instance-dependent threshold.
Our main contribution is an instance-optimal characterization of the sample complexity as Θ~(Γp/ε2), where Γp is a function of {Oi}i=1m defined via an optimization formula involving the inverse Fisher information matrix. Previously, tight bounds were known only in special cases, e.g. Pauli shadow tomography with L∞-norm error. Concretely, we first analyze a simpler oblivious variant where the goal is to estimate an observable of the form ∑i=1mαiOi with ∥α∥q=1 (where q is dual to p) revealed after the measurement. For single-copy measurements, we obtain a sample complexity of Θ(Γpob/ε2). We then show Θ~(Γp/ε2) is necessary and sufficient for the original problem, with the lower bound applying to unbiased, bounded estimators. Our upper bounds rely on a two-step algorithm combining coarse tomography with local estimation. Notably, Γ∞ob=Γ∞. In both cases, allowing c-copy measurements improves the sample complexity by at most Ω(1/c).
Our results establish a quantitative correspondence between quantum learning and metrology, unifying asymptotic metrological limits with finite-sample learning guarantees.
We consider the problem of quantum state certification, where we are given the description of a mixed state σ∈Cd×d,n copies of a mixed state ρ∈Cd×d, and ε>0, and we are asked to determine whether ρ=σ or whether ∥ρ−σ∥1>ε. When σ is the maximally mixed state d1Id, this is known as mixedness testing. We focus on algorithms which use incoherent measurements, i.e. which only measure one copy of ρ at a time. Unlike those that use entangled, multi-copy measurements, these can be implemented without persistent quantum memory and thus represent a large class of protocols that can be run on current or near-term devices. For mixedness testing, there is a folklore algorithm which uses incoherent measurements and only needs O(d3/2/ε2) copies. The algorithm is non-adaptive, that is, its measurements are fixed ahead of time, and is known to be optimal for non-adaptive algorithms. However, when the algorithm can make arbitrary incoherent measurements, the best known lower bound is only Ω(d4/3/ε2) [5], and it has been an outstanding open problem to close this polynomial gap. In this work: •We settle the copy complexity of mixedness testing with incoherent measurements and show that Ω(d3/2/ε2) copies are necessary. This fully resolves open questions of [15] and [5].•We show that the instance-optimal bounds for state certification to general σ first derived in [7] for non-adaptive measurements also hold for arbitrary incoherent measurements. Qualitatively, our results say that adaptivity does not help at all for these problems. Our results are based on new techniques that allow us to reduce the problem to understanding the concentration of certain matrix martingales, which we believe may be of independent interest.
Optimal single-copy estimation of quantum state moments: why Tr(ρ3) and Tr(ρ4) are equally hard
Estimating nonlinear properties of an unknown quantum state with restrictive experimental accessibility is a fundamental problem in quantum learning. We study the sample complexity of estimating the state moments Tr(ρt), allowing arbitrary adaptive single-copy measurements. While purity estimation has optimal sample complexity Θ(d) at constant additive error, the optimal dimension dependence for higher moments has remained unclear. We resolve this question for every fixed integer t≥2: for any sufficiently small constant additive error, the optimal sample complexity is Θ(d1+⌈log2t⌉⌈log2t⌉) with constant success probability. The complexity thus follows a dyadic hierarchy: for each integer h≥1, all moment orders 2h−1<t≤2h share the same exponent. In particular, the third and fourth moments both require Θ(d2/3) copies. Our results also apply to multistate overlaps, weighted and partial-transpose moments, fixed-rank spectrum estimation, and distributed estimation of nonlinear quantities. We also establish an Ω(d1−1/t) lower bound for arbitrary nonadaptive single-copy protocols, matching the known upper bound and demonstrating an advantage from adaptivity for every t≥4. Our upper bound uses adaptive state filtering to reduce higher-order moment estimation to lower-order moment estimation in a smaller state space. For the lower bound, we directly compare moment-matched ensembles through a smooth interpolation, bypassing the maximally mixed state as an intermediate hypothesis and yielding matching bounds beyond d.
We prove that the spectrum of an unknown d-dimensional quantum state can be estimated to error ε in total variation distance using O(d2min{(εlogd)41,(εlogd)21}) copies. This matches the recent lower bound of Wang. When restricted to unentangled measurements, we give an algorithm with an additional factor of d in copy complexity, which we conjecture to be optimal.
We develop a framework for recovering the small eigenvalues of a quantum state by matching Chebyshev moments. We bound the variance of each Chebyshev moment estimate in terms of scalar derivatives of the corresponding polynomial, using classical and quantum Efron--Stein decompositions. Different rescalings of the Chebyshev polynomials balance approximation error and variance, yielding two regimes in our copy complexity bound.
We consider the classic question of state tomography: given copies of an unknown quantum state ρ∈Cd×d, output ρ which is close to ρ in some sense, e.g. trace distance or fidelity. When one is allowed to make coherent measurements entangled across all copies, Θ(d2/ε2) copies are necessary and sufficient to get trace distance ε [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) 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) 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-close in infidelity to ρ using only O(d3/γ) copies, which is optimal for incoherent measurements. In contrast, it is known [18] that any nonadaptive algorithm requires Ω(d3/γ2) copies. While it is folklore that in 2 dimensions, one can achieve a scaling of O(1/γ), to the best of our knowledge, our algorithm is the first to achieve the optimal rate in all dimensions.
We revisit the basic problem of quantum state certification: given copies of unknown mixed state ρ∈Cd×d and the description of a mixed state σ, decide whether σ=ρ or ∥σ−ρ∥tr≥ε. When σ is maximally mixed, this is mixedness testing, and it is known that Ω(dΘ(1)/ε2) copies are necessary, where the exact exponent depends on the type of measurements the learner can make [OW15, BCL20], and in many of these settings there is a matching upper bound [OW15, BOW19, BCL20]. Can one avoid this dΘ(1) dependence for certain kinds of mixed states σ, e.g. ones which are approximately low rank? More ambitiously, does there exist a simple functional f:Cd×d→R≥0 for which one can show that Θ(f(σ)/ε2) copies are necessary and sufficient for state certification with respect to any σ? Such instance-optimal bounds are known in the context of classical distribution testing, e.g. [VV17]. Here we give the first bounds of this nature for the quantum setting, showing (up to log factors) that the copy complexity for state certification using nonadaptive incoherent measurements is essentially given by the copy complexity for mixedness testing times the fidelity between σ and the maximally mixed state. Surprisingly, our bound differs substantially from instance optimal bounds for the classical problem, demonstrating a qualitative difference between the two settings.