Full quantum tomography of high-dimensional quantum systems is experimentally infeasible due to the exponential scaling of the number of required measurements on the number of qubits in the system. However, several ideas were proposed recently for predicting the limited number of features for these states, or estimating the expectation values of operators, without the need for full state reconstruction. These ideas go under the general name of shadow tomography. Here we provide an experimental demonstration of property estimation based on classical shadows proposed in [H.-Y. Huang et al., Nat. Phys.

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We study the recovery of Hermitian low rank matrices X ∈ C n×n from undersampled measurements via nuclear norm minimization. We consider the particular scenario where the measurements are Frobenius inner products with random rank-one matrices of the form a j a * j for some measurement vectors a 1 , . . . , am, i.e., the measurements are given by y j = tr(Xa j a * j ). The case where the matrix X = xx * to be recovered is of rank one reduces to the problem of phaseless estimation (from measurements, y j = | x, a j | 2 via the PhaseLift approach, which has been introduced recently. We derive bounds for the number m of measurements that guarantee successful uniform recovery of Hermitian rank r matrices, either for the vectors a j , j = 1, . . . , m, being chosen independently at random according to a standard Gaussian distribution, or a j being sampled independently from an (approximate) complex projective t-design with t = 4. In the Gaussian case, we require m ≥ Crn measurements, while in the case of 4-designs we need m ≥ Crn log(n). Our results are uniform in the sense that one random choice of the measurement vectors a j guarantees recovery of all rank r-matrices simultaneously with high probability. Moreover, we prove robustness of recovery under perturbation of the measurements by noise. The result for approximate 4-designs generalizes and improves a recent bound on phase retrieval due to Gross, Kueng and Krahmer. In addition, it has applications in quantum state tomography. Our proofs employ the so-called bowling scheme which is based on recent ideas by Mendelson and Koltchinskii.Date: October 25, 2014.

Bell's Theorem shows that quantum mechanical correlations can violate the constraints that the causal structure of certain experiments impose on any classical explanation. It is thus natural to ask to which degree the causal assumptions -e.g. "locality" or "measurement independence" -have to be relaxed in order to allow for a classical description of such experiments. Here, we develop a conceptual and computational framework for treating this problem. We employ the language of Bayesian networks to systematically construct alternative causal structures and bound the degree of relaxation using quantitative measures that originate from the mathematical theory of causality. The main technical insight is that the resulting problems can often be expressed as computationally tractable linear programs. We demonstrate the versatility of the framework by applying it to a variety of scenarios, ranging from relaxations of the measurement independence, locality and bilocality assumptions, to a novel causal interpretation of CHSH inequality violations.The paradigmatic Bell experiment [1] involves two distant observers, each with the capability to perform one of two possible experiments on their shares of a joint system. Bell observed that even absent of any detailed information about the physical processes involved, the causal structure of the setup alone implies strong constraints on the correlations that can arise from any classical description [2]. The physically wellmotivated causal assumptions are: (i) measurement independence: experimenters can choose which property of a system to measure, independently of how the system has been prepared; (ii) locality: the results obtained by one observer cannot be influenced by any action of the other (ideally space-like separated) experimenter. The resulting constraints are Bell's inequalities [1]. Quantum mechanical processes subject to the same causal structure can violate these constraints -a prediction that has been abundantly verified experimentally [3][4][5][6][7]. This effect is commonly referred to as quantum nonlocality.It is now natural to ask how stable the effect of quantum non-locality is with respect to relaxations of the causal assumptions. Which "degree of measurement dependence", e.g., is required to reconcile empirically observed correlations with a classical and local model? Such questions are not only, we feel, of great relevance to foundational questions -they are also of interest to practical applications of non-locality, e.g. in cryptographic protocols. Indeed, eavesdroppers can (and do [8]) exploit the failure of a given cryptographic device to be constrained by the presumed causal structure to compromise its security. At the same time, it will often be difficult to ascertain that causal assumptions hold exactly -which makes it important to develop a systematic quantitative theory.Several variants of this question have recently attracted considerable attention [9][10][11][12][13][14][15][16][17][18][19][20]. For example, measurement dependence has been found ...

ABSTRACT. The problem of retrieving phase information from amplitude measurements alone has appeared in many scientific disciplines over the last century. PhaseLift is a recently introduced algorithm for phase recovery that is computationally tractable, numerically stable, and comes with rigorous performance guarantees. PhaseLift is optimal in the sense that the number of amplitude measurements required for phase reconstruction scales linearly with the dimension of the signal. However, it specifically demands Gaussian random measurement vectors -a limitation that restricts practical utility and obscures the specific properties of measurement ensembles that enable phase retrieval. Here we present a partial derandomization of PhaseLift that only requires sampling from certain polynomial size vector configurations, called t-designs. Such configurations have been studied in algebraic combinatorics, coding theory, and quantum information. We prove reconstruction guarantees for a number of measurements that depends on the degree t of the design. If the degree is allowed to grow logarithmically with the dimension, the bounds become tight up to polylog-factors. Beyond the specific case of PhaseLift, this work highlights the utility of spherical designs for the derandomization of data recovery schemes.

Achieving error rates that meet or exceed the fault-tolerance threshold is a central goal for quantum computing experiments, and measuring these error rates using randomized benchmarking is now routine. However, direct comparison between measured error rates and thresholds is complicated by the fact that benchmarking estimates average error rates while thresholds reflect worst-case behavior when a gate is used as part of a large computation. These two measures of error can differ by orders of magnitude in the regime of interest. Here we facilitate comparison between the experimentally accessible average error rates and the worst-case quantities that arise in current threshold theorems by deriving relations between the two for a variety of physical noise sources. Our results indicate that it is coherent errors that lead to an enormous mismatch between average and worst case, and we quantify how well these errors must be controlled to ensure fair comparison between average error probabilities and fault-tolerance thresholds.

We propose an ordered set of experimentally accessible conditions for detecting entanglement in mixed states. The k-th condition involves comparing moments of the partially transposed density operator up to order k. Remarkably, the union of all moment inequalities reproduces the Peres-Horodecki criterion for detecting entanglement. Our empirical studies highlight that the first four conditions already detect mixed state entanglement reliably in a variety of quantum architectures. Exploiting symmetries can help to further improve their detection capabilities. We also show how to estimate moment inequalities based on local random measurements of single state copies (classical shadows) and derive statistically sound confidence intervals as a function of the number of performed measurements. Our analysis includes the experimentally relevant situation of drifting sources, i.e. non-identical, but independent, state copies.

We study the performance of classical and quantum machine learning (ML) models in predicting outcomes of physical experiments. The experiments depend on an input parameter x and involve execution of a (possibly unknown) quantum process E. Our figure of merit is the number of runs of E required to achieve a desired prediction performance. We consider classical ML models that perform a measurement and record the classical outcome after each run of E, and quantum ML models that can access E coherently to acquire quantum data; the classical or quantum data are then used to predict the outcomes of future experiments. We prove that for any input distribution DðxÞ, a classical ML model can provide accurate predictions on average by accessing E a number of times comparable to the optimal quantum ML model. In contrast, for achieving an accurate prediction on all inputs, we prove that the exponential quantum advantage is possible. For example, to predict the expectations of all Pauli observables in an n-qubit system ρ, classical ML models require 2 ΩðnÞ copies of ρ, but we present a quantum ML model using only OðnÞ copies. Our results clarify where the quantum advantage is possible and highlight the potential for classical ML models to address challenging quantum problems in physics and chemistry.