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Learning in Quantum Mechanics https://doi.org/10.17863/CAM.118679 Salmon, Wilfred Strelchuk, Sergii Arvidsson-Shukur, David This thesis explores the interactions of learning and quantum mechanics. At its heart, learning consists of extracting information from data. We will consider two types of data; random and deterministic. When data is random, one usually tries to learn an approximation to a desired object, when it is deterministic, one usually tries to learn an exact description of an object. Chapters 2 and 3 of this thesis focus on approximate learning of classical data embedded in quantum systems. The quantum mechanical nature of the physical systems leads to inherent randomness, even if the classical data is embedded in a deterministic way. In the final chapter we turn to exact learning of quantum objects, where we explore learning succinct characterisation of quantum objects from verbose descriptions. Chapter 1 gives a brief motivation for quantum information, and gives a review of notation used throughout the thesis. Chapter 2 introduces quantum metrology, the quantum generalisation of parameter estimation. Pa- rameter estimation is a foundational area of modern statistics. In parameter estimation, one considers data generated from an instance of a parameterised model, with the aim of inferring the values of the parameters that generated the dataset. Parameter estimation is ubiquitous across modern science and thus has a rich and well-developed theory. Quantum metrology is a comparatively modern theory of parameter estimation, where classical parameters are encoded in quantum systems. Existing literature in quantum metrology focuses on the so-called asymptotic regime, corresponding to understanding the limit of attainable precision as experimental resources (and time) tend to infin- ity. We consider the non-asymptotic regime, of particular relevance when experimental resources are limited; often the case for quantum systems. We generalise the classical framework of non-asymptotic parameter estimation to quantum metrology. This generalisation prescribes a non-asymptotic, oper- ational method to determine if one measurement is better than another. We say that a measurement is admissible if there is no measurement that is strictly better than it; a measurement is optimal if it is as least as good as any other candidate measurement. Note that optimality is a much stronger condition than admissibility - there may be many incomparable admissible measurements, none of which are optimal. We prove several results within our non-asymptotic framework. First, we give a complete characterisation of when an optimal measurement exists - if and only if a parameterised state lies in a very restrictive class, called classical parameterised states. Second, we give three sufficient conditions for when an approximately optimal measurement exists, giving explicit bounds on the level of optimality. Finally, we give several necessary conditions, and one sufficient condition, for when a measurement is admissible. Given the fundamental nature of admissibility within classical parameter estimation, our results serve as a foundation for non-asymptotic quantum metrology. iii Second, we explore an emerging technique in quantum metrology, known as quantum filtering (or postselection). Quantum filtering considers how one can distil information (on classical parameters) from many copies of a parameterised quantum state into few copies of a quantum state. Recently, a filter was discovered that can losslessly compress information into arbitrarily few quantum states, as long as one has a suitable initial guess of the parameter. First, we completely characterise which filters allow for lossless compression, showing that the known lossless filter lies within a more general family. Our discovery of the family of lossless filters allows for flexibility in the design of experiments. Second, we show that filters that are optimal for noiseless systems may be sub-optimal in the presence of noise. This counter-intuitive result shows the subtlety of filter design. Third, we show that lossless compression is not achievable in classical experiments (except in trivial cases), showing that lossless compression is a genuine quantum effect. The quantum nature of lossless compression has already been explored, by appealing to established indicators of classicality, rather than operational quantities. Our proof, however, is purely operational in nature, completely characterising the limited cases where classical lossless compression is possible. Finally, we give the first practical, iterative quantum filter- ing algorithm. All existing literature on filtering assumes the asymptotic regime, and does not give a concrete way to realise a practical advantage. Our algorithm is the first result in non-asymptotic quantum filtering. In our explicit scheme, we show that the quantum effect of filtering can have a subtle interplay with the classical theory of admissibility of estimators. Chapter 3 considers probably approximately correct (PAC) learning, which underpins of all modern machine learning. We consider quantum machine learning of classical data, where classical objects are encoded in quantum states. Existing literature has demonstrated a wide range of quantum speedups in PAC learning, ranging from polynomial to exponential. However, all known improvements apply to special cases - not to the general theory. Indeed, it was recently shown that established access models do not admit a generic asymptotic quantum speedup. We consider a natural extension to the most widespread quantum access model, and argue it is applicable to most quantum scenarios of interest. We show that in this new access model, there is a generic quadratic reduction in the quan- tity of resources required for PAC learning. Furthermore, we prove that this quadratic improvement is optimal. Given the success of modern machine learning algorithms, our results pave the way for generic improvements in quantum machine learning. Chapter 4 considers classical, exact learning of elements of the stabiliser formalism. The stabiliser formalism is a cornerstone of modern quantum computation (see Section 4.1.1 for a more thorough review). We explore common algorithmic primitives that are used for learning efficient classical de- scriptions of elements of the stabiliser formalism from inefficient ones. Such primitives have seen widespread use in classical simulation of quantum computers, and in increasing the efficiency of quan- tum circuit design. We give several new algorithms for these primitives, that have asymptotic and realisable speedups over existing implementations. Given the ubiquity of our primitives, these speedups will be widely useful in a range of problems. University of Cambridge 2024-08-15 Thesis Doctoral Doctor of Philosophy (PhD) EPSRC. Hitachi https://www.repository.cam.ac.uk/handle/1810/384800 https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/21328f85-2acf-45cf-b398-1e0e154f045c/download 1d148dbcb720f5e72e9978faaccd9c4e https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/e63a1b73-bb93-455d-a0ae-23663969a5a4/download 87eda9de84448d1f82354d60eee3eb5f https://creativecommons.org/licenses/by/4.0/ Quantum Computing Quantum Metrology