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We examine the question of whether quantum mechanics places limitations on the ability of small quantum devices to learn. We specifically examine the question in the context of Bayesian inference, wherein the prior and posterior distributions are encoded in the quantum state vector. We conclude based on lower bounds from Grovers search that an efficient blackbox method for updating the distribution is impossible. We then address this by providing a new adaptive form of approximate quantum Bayesian inference that is polynomially faster than its classical analogue and tractable if the quantum system is augmented with classical memory or if the low-order moments of the distribution are protected using a repetition code. This work suggests that there may be a connection between fault tolerance and the capacity of a quantum system to learn from its surroundings.
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The Kepler, K2, and Transiting Exoplanet Survey Satellite (TESS) missions have provided a wealth of confirmed exoplanets, benefiting from a huge effort from the planet-hunting and follow-up community. With careful systematics mitigation, these missions provide precise photometric time series, which enable detection of transiting exoplanet signals. However, exoplanet hunting can be confounded by several factors, including instrumental noise, search biases, and host star variability. In this Letter, we discuss strategies to overcome these challenges using newly emerging techniques and tools. We demonstrate the power of new, fast open-source community tools (e.g., lightkurve, starry, celerite, exoplanet), and discuss four high signal-to-noise ratio (S/N) exoplanets that showcase specific challenges present in planet detection: K2-43c, K2-168c, K2-198c, and K2-198d. These planets have been undetected in several large K2 planet searches, despite having transit signals with S/N > 10. Two of the planets discussed here are new discoveries. In this work we confirm all four as true planets. Alongside these planet systems, we discuss three key challenges in finding small transiting exoplanets. The aim of this Letter is to help new researchers understand where planet detection efficiency gains can be made, and to encourage the continued use of K2 archive data. The considerations presented in this Letter are equally applicable to Kepler, K2, and TESS, and the tools discussed here are available for the community to apply to improve exoplanet discovery and fitting.
We extensively test a recent protocol to demonstrate quantum fault tolerance on three systems: (1) a real-time simulation of five spin qubits coupled to an environment with two-level defects, (2) a real-time simulation of transmon quantum computers, and (3) the 16-qubit processor of the IBM Q Experience. In the simulations, the dynamics of the full system is obtained by numerically solving the time-dependent Schrodinger equation. We find that the fault-tolerant scheme provides a systematic way to improve the results when the errors are dominated by the inherent control and measurement errors present in transmon systems. However, the scheme fails to satisfy the criterion for fault tolerance when decoherence effects are dominant.
We numerically investigate quantum quenches of a nonintegrable hard-core Bose-Hubbard model to test the accuracy of the microcanonical ensemble in small isolated quantum systems. We show that, in a certain range of system size, the accuracy increases with the dimension of the Hilbert space $D$ as $1/D$. We ascribe this rapid improvement to the absence of correlations between many-body energy eigenstates as well as to the eigenstate thermalization. Outside of that range, the accuracy is found to scale as $1/sqrt{D}$ and improves algebraically with the system size.
Learning problems form an important category of computational tasks that generalizes many of the computations researchers apply to large real-life data sets. We ask: what concept classes can be learned privately, namely, by an algorithm whose output does not depend too heavily on any one input or specific training example? More precisely, we investigate learning algorithms that satisfy differential privacy, a notion that provides strong confidentiality guarantees in contexts where aggregate information is released about a database containing sensitive information about individuals. We demonstrate that, ignoring computational constraints, it is possible to privately agnostically learn any concept class using a sample size approximately logarithmic in the cardinality of the concept class. Therefore, almost anything learnable is learnable privately: specifically, if a concept class is learnable by a (non-private) algorithm with polynomial sample complexity and output size, then it can be learned privately using a polynomial number of samples. We also present a computationally efficient private PAC learner for the class of parity functions. Local (or randomized response) algorithms are a practical class of private algorithms that have received extensive investigation. We provide a precise characterization of local private learning algorithms. We show that a concept class is learnable by a local algorithm if and only if it is learnable in the statistical query (SQ) model. Finally, we present a separation between the power of interactive and noninteractive local learning algorithms.
Classical machine learning has succeeded in the prediction of both classical and quantum phases of matter. Notably, kernel methods stand out for their ability to provide interpretable results, relating the learning process with the physical order parameter explicitly. Here, we exploit quantum kernels instead. They are naturally related to the fidelity and thus it is possible to interpret the learning process with the help of quantum information tools. In particular, we use a support vector machine (with a quantum kernel) to predict and characterize quantum phase transitions. The general theory is tested in the Ising chain in transverse field. We show that for small-sized systems, the algorithm gives accurate results, even when trained away from criticality. Besides, for larger sizes we confirm the success of the technique by extracting the correct critical exponent $ u$. The characterization is completed by computing the kernel alignment between the quantum and ideal kernels. Finally, we argue that our algorithm can be implemented on a circuit based on a varational quantum eigensolver.
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