Variational techniques have long been at the heart of atomic, solid-state, and many-body physics. They have recently extended to quantum and classical machine learning, providing a basis for representing quantum states via neural networks. These methods generally aim to minimize the energy of a given ansatz, though open questions remain about the expressivity of quantum and classical variational ansätze. The connection between variational techniques and quantum computing, through variational quantum algorithms, offers opportunities to explore the quantum complexity of classical methods. We demonstrate how the concept of non-stabilizerness, or magic, can create a bridge between quantum information and variational techniques and we show that energy accuracy is a necessary but not always sufficient condition for accuracy in non-stabilizerness. Through systematic benchmarking of neural network quantum states, matrix product states, and variational quantum methods, we show that while classical techniques are more accurate in non-stabilizerness, not accounting for the symmetries of the system can have a severe impact on this accuracy. Our findings form a basis for a universal expressivity characterization of both quantum and classical variational methods.@en

The recent progress in quantum technologies shines a bright light on the future of quantum computation. However, resource estimation for quantum computations remains a key challenge. The resource I study in this thesis is known as Magic or Non-stabilizerness and it represents the key requirement for quantum computational advantage in computation. Recent studies in quantum information suggested wide classes of quantifiers for non-stabilizerness.

In this thesis, I develop novel techniques for quantifying non-stabilizerness with the tools fromquantuminformation theory. I am specifically interested inmaking quantum resource estimation computationally as efficient as possible so that quantum resource estimation can become a routine step in both numerical and experimental exploration of quantumcomputing.

I show that by measuring the spreading of the local information in the quantumsystem, we can quantify non-stabilizerness. The measures of such information-spreading can be classified into two categories, entropic-based measures, such as mutual information, and correlator-based measures such as Out-of-Time Ordered Correlators. I investigated both classes of measures and I related them both numerically and analytically to the estimation of non-stabilizerness.

Finally, I relate non-stabilizerness quantification to classical variational methods. Classical methods designed to approximate quantum states are by construction not restricted by non-stabilizerness. The question therefore remained how well these techniques can capture quantumresources. I provide a systematic benchmark for both classical and quantum approximate methods of expressing quantum states and their tradeoff between non-stabilizerness expressivity and ground-state energy accuracy. We observed that having better energy accuracy is necessary but insufficient to have better accuracy in non-stabilizerness.

This Thesis forms a bridge between quantum information resource theory and condensed matter physics and offers a stepping stone towards further exchange between these two fields.
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The advent of quantum technologies brought forward much attention to the theoretical characterization of the computational resources they provide. A method to quantify quantum resources is to use a class of functions called magic monotones and stabilizer entropies, which are, however, notoriously hard and impractical to evaluate for large system sizes. In recent studies, a fundamental connection between information scrambling, the magic monotone mana and 2-Renyi stabilizer entropy was established. This connection simplified magic monotone calculation, but this class of methods still suffers from exponential scaling with respect to the number of qubits. In this work, we establish a way to sample an out-of-time-order correlator that approximates magic monotones and 2-Renyi stabilizer entropy. We numerically show the relation of these sampled correlators to different non-stabilizerness measures for both qubit and qutrit systems and provide an analytical relation to 2-Renyi stabilizer entropy. Furthermore, we put forward and simulate a protocol to measure the monotonic behaviour of magic for the time evolution of local Hamiltonians.

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