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Characterising quantum processes is a key task in the development of quantum technologies, especially at the noisy intermediate scale of today’s devices. One method for characterising processes is randomised benchmarking, which is robust against state preparation and measurement (SPAM) errors, and can be used to benchmark Clifford gates. A complementing approach asks for full tomographic knowledge. Compressed sensing techniques achieve full tomography of quantum channels essentially at optimal resource efficiency. So far, guarantees for compressed sensing protocols rely on unstructured random measurements and can not be applied to the data acquired from randomised benchmarking experiments. It has been an open question whether or not the favourable features of both worlds can be combined. In this work, we give a positive answer to this question. For the important case of characterising multi-qubit unitary gates, we provide a rigorously guaranteed and practical reconstruction method that works with an essentially optimal number of average gate fidelities measured respect to random Clifford unitaries. Moreover, for general unital quantum channels we provide an explicit expansion into a unitary 2-design, allowing for a practical and guaranteed reconstruction also in that case. As a side result, we obtain a new statistical interpretation of the unitarity – a figure of merit that characterises the coherence of a process. In our proofs we exploit recent representation theoretic insights on the Clifford group, develop a version of Collins’ calculus with Weingarten functions for integration over the Clifford group, and combine this with proof techniques from compressed sensing.

Quantum Monte Carlo (QMC) methods are the gold standard for studying equilibrium properties of quantum many-body systems. However, in many interesting situations, QMC methods are faced with a sign problem, causing the severe limitation of an exponential increase in the runtime of the QMC algorithm. In this work, we develop a systematic, generally applicable, and practically feasible methodology for easing the sign problem by efficiently computable basis changes and use it to rigorously assess the sign problem. Our framework introduces measures of non-stoquasticity that—as we demonstrate analytically and numerically—at the same time provide a practically relevant and efficiently computable figure of merit for the severity of the sign problem. Complementing this pragmatic mindset, we prove that easing the sign problem in terms of those measures is generally an NP-complete task for nearest-neighbor Hamiltonians and simple basis choices by a reduction to the MAXCUT-problem.

One of the outstanding problems in non-equilibrium physics is to precisely understand when and how physically relevant observables in many-body systems equilibrate under unitary time evolution. General equilibration results show that equilibration is generic provided that the initial state has overlap with sufficiently many energy levels. But results not referring to typicality which show that natural initial states actually fulfill this condition are lacking. In this work, we present stringent results for equilibration for systems in which Rényi entanglement entropies in energy eigenstates with finite energy density are extensive for at least some, not necessarily connected, sub-system. Our results reverse the logic of common arguments, in that we derive equilibration from a weak condition akin to the eigenstate thermalization hypothesis, which is usually attributed to thermalization in systems that are assumed to equilibrate in the first place. We put the findings into the context of studies of many-body localization and many-body scars.Over recent years the study of the relaxation to equilibrium of complex many-body systems has attracted great attention. This interest can be motivated from at least two points of view. From a foundational viewpoint, it is desirable to understand how statistical equilibrium ensembles emerge within the framework of unitary quantum mechanics -without introducing any external probability measures. It is then necessary to first explain how systems undergoing unitary evolution attain equilibrium at all. A key ingredient to explain this behavior has been found to be the dynamical build-up of entanglement from low-entangled initial states and results showing equilibration under quite general conditions have been derived [1][2][3][4][5][6][7][8][9][10][11][12]. The increase of entanglement over time is a generic feature of complex quantum systems and leads to an increase of the entropy of subsystems over time reminiscent to the second law of thermodynamics.From a more concrete perspective, the recent interest in the study of non-equilibrium dynamics is motivated by the fact that such dynamics can now be realized in well-controlled experiments, for example in ion traps or optical lattices [13][14][15][16][17][18]. Moreover, the discovery of many-body localized systems [19], which equilibrate [20] but fail to thermalize [18], shows that there remains much to be understood about the equilibration behavior of complex quantum systems. Despite the great progress in understanding the equilibration behavior of manybody systems, rigorous results showing that systems with natural initial states equilibrate to high precision based on concrete physical properties have been lacking.In this article we aim to fill this gap, by taking a new perspective to the problem. To do this, we carefully reconsider the entanglement content of energy eigenstates in complex, interacting many-body systems and devise a working-definition of "entanglement-ergodic" systems whose energy eigenstates at finite energy densi...