We determine the optimal sample complexity of low-rank quantum state tomography when each measurement may act jointly on at most $t$ samples. For sufficiently small $\varepsilon$, estimating an unknown state on $\mathbb{C}^d$ of rank at most $r$ to trace norm error $\varepsilon$ with constant success probability requires, and is achievable with, $$Θ\left(\frac{dr}{\varepsilon^2}\mathop{\mathrm{max}}\left\{1,\frac{r}{\sqrt{t}}\right\}\right)$$ samples. The lower bound allows the protocol to choose each joint measurement adaptively using all previous classical outcomes; the matching upper bound is nonadaptive. Thus joint measurements on at most $t$ samples improve the complexity of algorithms making single-sample measurements by at most a factor $\sqrt{t}$. Further, measuring order $r^2$ samples jointly is necessary and sufficient to attain the unrestricted collective rate.
For the lower bound, we vary the support of a state with fixed uniform spectrum and bound the Fisher information trace of every joint measurement on $t$ samples. The adaptive Fisher chain rule and the van Trees inequality then give the trace norm lower bound. For the upper bound, we construct and analyze a nonadaptive tomography protocol based on a Gaussian joint measurement. An explicit second moment identity and a conditional Gaussian law outside the state's support give a rank-dependent error analysis, yielding the matching rate.
We determine the optimal sample complexity of low-rank quantum state tomography when each measurement may act jointly on at most t samples. For sufficiently small ε, estimating an unknown state on Cd of rank at most r to trace norm error ε with constant success probability requires, and is achievable with, Θ(ε2drmax{1,tr}) samples. The lower bound allows the protocol to choose each joint measurement adaptively using all previous classical outcomes; the matching upper bound is nonadaptive. Thus joint measurements on at most t samples improve the complexity of algorithms making single-sample measurements by at most a factor t. Further, measuring order r2 samples jointly is necessary and sufficient to attain the unrestricted collective rate.
For the lower bound, we vary the support of a state with fixed uniform spectrum and bound the Fisher information trace of every joint measurement on t samples. The adaptive Fisher chain rule and the van Trees inequality then give the trace norm lower bound. For the upper bound, we construct and analyze a nonadaptive tomography protocol based on a Gaussian joint measurement. An explicit second moment identity and a conditional Gaussian law outside the state's support give a rank-dependent error analysis, yielding the matching rate.
We study tomography for quantum channels with input dimension d1, output dimension d2, and Kraus rank at most r, to within diamond norm error ε, using adaptive experiments that retain no quantum memory between channel queries.
- For quantum channels whose non-zero Choi eigenvalues are bounded below by Ω(d1/r), we establish optimal query upper and lower bounds Θ(d1d2r2/ε2). The upper bound is achieved by a nonadaptive algorithm that uses the estimator from [Surawy-Stepney et al., Quantum (2022)], together with a new diamond-norm analysis. The lower bound applies to arbitrary adaptive incoherent protocols and follows from a new local family of channels and a uniform one-query Fisher-information bound.
- For general channels, we establish an upper bound O(d1d2r2log(2d1)/ε2), nearly matching the above lower bound Ω(d1d2r2/ε2). To achieve this, we generalize the above nonadaptive algorithm by adapting the input state over O(log(2d1)) rounds with the Matrix Multiplicative Weight Update algorithm.
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) and 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)) 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.
We study state tomography when each measurement acts on at most k fresh copies and no quantum memory is retained between blocks. We prove a lower bound matching the upper bound in [arXiv:2510.07788]. Thus the copy complexity of estimating an arbitrary d-dimensional state to trace distance ε is, up to absolute constant factors, max{d3/(kε2),d2/ε2} for every k and all sufficiently small ε. This removes the earlier restriction that k be small as a function of the accuracy. The lower bound applies to arbitrary measurements within each block and adaptive choices between blocks. The lower bound already applies in a small neighborhood of any state whose smallest eigenvalue is of order 1/d, even when the center is known.
The main ingredient is a uniform Fisher information bound for one measurement block that depends only on the smallest eigenvalue of the state. The proof avoids the perturbative expansion responsible for the restriction in [arXiv:2402.16353]. Fano's inequality for metric balls and a log-Sobolev comparison between mutual and Fisher information then reduce the adaptive protocol to this block bound [arXiv:1607.00550, arXiv:1902.08582].
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.
In this work, we consider the fundamental task of quantum state certification: given copies of an unknown quantum state ρ, test whether it matches some target state σ or is ε-far from it. For certifying d-dimensional states, Θ(d/ε2) copies of ρ are known to be necessary and sufficient. However, the algorithm achieving this complexity makes fully entangled measurements over all O(d/ε2) copies of ρ. Often, one is interested in certifying states to a high precision; this makes such joint measurements intractable even for low-dimensional states. Thus, we study whether one can obtain optimal rates for quantum state certification and related testing problems while only performing measurements on t copies at once, for some 1<t≪d/ε2. While it is well-understood how to use intermediate entanglement to achieve optimal quantum state learning, the only protocol known to achieve optimal testing is the one using fully entangled measurements.
Our main result is a smooth copy complexity upper bound for state certification as a function of t, which achieves a near-optimal rate at t=d2. In the high-precision regime, i.e., for ε<d1, this is a strict improvement over the entanglement used by the aforementioned optimal protocol. We also extend our techniques to develop new algorithms for the related tasks of mixedness testing and purity estimation, and show tradeoffs achieving the optimal rates for these problems at t=d2 as well. Our algorithms are based on novel reductions from testing to learning and leverage recent advances in quantum state tomography in a non-black-box fashion. We complement our testing upper bounds with a (nearly) matching lower bound that holds in the high-precision regime, implying that one must take t≥Ω(d2) to achieve the optimal rate for certification.
How many samples of a quantum state are required to learn a complete description of it? As we will see in this thesis, the fine-grained answer depends on the measurements available to the learner, but in general it is at least Ω(d^2/ϵ^2) where d is the dimension of the state and ϵ the trace distance accuracy. Optimal algorithms for this task -- known as quantum state tomography -- make use of powerful, yet highly impractical entangled measurements, where some joint measurement is performed on all copies of the state. What can be accomplished without such measurements, where one must perform measurements on individual copies of the states? \n \nIn Chapter 2 we show a relationship between the recently proposed quantum online learning framework and quantum state tomography. Specifically, we show that tomography can be accomplished using online learning algorithms in a black-box manner and O(d^4/ϵ^4) two-outcome measurements on separate copies of the state. The interpretation of this approach is that the experimentalist uses informative measurements to teach the learner by helping it make "mistakes" on measurements as early as possible. \n \nWe move on to proving lower bounds on tomography in Chapter 3. First, we review a known lower bound for entangled measurements as well as a Ω(d^3/ϵ^2) lower bound in the setting where non-entangled measurements are made non-adaptively, both due to Ref.[18]. We then derive a novel bound of Ω(d^4/ϵ^2) samples when the learner is further restricted to observing a constant number of outcomes (e.g., two-outcome measurements). This implies that the folklore "Pauli tomography" algorithm is optimal in this setting. \n \nUnderstanding the power of adaptive measurements, where measurement choices can depend on previous outcomes, is currently an open problem. In Chapter 4 we present two scenarios in which adapting on previous outcomes makes no difference to the number of samples required. In the first, the learner is limited to adapting on at most o(d^2/ϵ^2) of the previous outcomes. In the second, measurements are drawn from some set of at most exp(O(d)) measurements. In particular, this second lower bound implies that adaptivity makes no difference in the regime of efficiently implementable measurements, in the context of quantum computing. \n \nFinally, we apply the above technique to the problems of classical shadows and shadow tomography to obtain similar lower bounds. Here, one is interested only in determining the expectations of some fixed set of observables. We once again find that, for the worst-case input of observables, adaptivity makes no difference to the sample complexity when considering efficient, non-entangled measurements. As a corollary, we find a straightforward algorithm for shadow tomography is optimal in this setting.