Many-body Localization Transition: Schmidt Gap, Entanglement Length & Scaling

Many-body localization has become an important phenomenon for illuminating a potential rift between non-equilibrium quantum systems and statistical mechanics. However, the nature of the transition between ergodic and localized phases in models displaying many-body localization is not yet well understood. Assuming that this is a continuous transition, analytic results show that the length scale should diverge with a critical exponent $\nu \ge 2$ in one dimensional systems. Interestingly, this is in stark contrast with all exact numerical studies which find $\nu \sim 1$. We introduce the Schmidt gap, new in this context, which scales near the transition with a exponent $\nu > 2$ compatible with the analytical bound. We attribute this to an insensitivity to certain finite size fluctuations, which remain significant in other quantities at the sizes accessible to exact numerical methods. Additionally, we find that a physical manifestation of the diverging length scale is apparent in the entanglement length computed using the logarithmic negativity between disjoint blocks.Comment: 8 pages, 7 figure

Machine Learning Assisted Many-Body Entanglement Measurement

Entanglement not only plays a crucial role in quantum technologies, but is key to our understanding of quantum correlations in many-body systems. However, in an experiment, the only way of measuring entanglement in a generic mixed state is through reconstructive quantum tomography, requiring an exponential number of measurements in the system size. Here, we propose a machine learning assisted scheme to measure the entanglement between arbitrary subsystems of size $N_A$ and $N_B$, with $\mathcal{O}(N_A + N_B)$ measurements, and without any prior knowledge of the state. The method exploits a neural network to learn the unknown, non-linear function relating certain measurable moments and the logarithmic negativity. Our procedure will allow entanglement measurements in a wide variety of systems, including strongly interacting many body systems in both equilibrium and non-equilibrium regimes.Comment: 16 pages, 10 figures, including appendi

One-step replica symmetry breaking in the language of tensor networks

We develop an exact mapping between the one-step replica symmetry breaking cavity method and tensor networks. The two schemes come with complementary mathematical and numerical toolboxes that could be leveraged to improve the respective states of the art. As an example, we construct a tensor-network representation of Survey Propagation, one of the best deterministic k-SAT solvers. The resulting algorithm outperforms any existent tensor-network solver by several orders of magnitude. We comment on the generality of these ideas, and we show how to extend them to the context of quantum tensor networks

Many-Body Entanglement in Classical & Quantum Simulators

Entanglement is not only the key resource for many quantum technologies, but essential in understanding the structure of many-body quantum matter. At the interface of these two crucial areas are simulators, controlled systems capable of mimicking physical models that might escape analytical tractability. Traditionally, these simulations have been performed classically, where recent advancements such as tensor-networks have made explicit the limitation entanglement places on scalability. Increasingly however, analog quantum simulators are expected to yield deep insight into complex systems. This thesis advances the field in across various interconnected fronts. Firstly, we introduce schemes for verifying and distributing entanglement in a quantum dot simulator, tailored to specific experimental constraints. We then confirm that quantum dot simulators would be natural candidates for simulating many-body localization (MBL) - a recently emerged phenomenon that seems to evade traditional statistical mechanics. Following on from that, we investigate MBL from an entanglement perspective, shedding new light on the nature of the transition to it from a ergodic regime. As part of that investigation we make use of the logarithmic negativity, an entanglement measure applicable to many-body mixed states. In order to tie back into quantum simulators, we then propose an experimental scheme to measure the logarithmic negativity in realistic many-body settings. This method uses choice measurements on three or more copies of a mixed state along with machine learning techniques. We also introduce a fast method for computing many-body entanglement in classical simulations that significantly increases the size of system addressable. Finally, we introduce quimb, an open-source library for interactive but efficient quantum information and many-body calculations. It contains general purpose tensor-network support alongside other novel algorithms

Fast and converged classical simulations of evidence for the utility of quantum computing before fault tolerance

A recent quantum simulation of observables of the kicked Ising model on 127 qubits [Nature 618, 500 (2023)] implemented circuits that exceed the capabilities of exact classical simulation. We show that several approximate classical methods, based on sparse Pauli dynamics and tensor network algorithms, can simulate these observables orders of magnitude faster than the quantum experiment, and can also be systematically converged beyond the experimental accuracy. Our most accurate technique combines a mixed Schr\"odinger and Heisenberg tensor network representation with the Bethe free entropy relation of belief propagation to compute expectation values with an effective wavefunction-operator sandwich bond dimension ${>}16,000,000$, achieving an absolute accuracy, without extrapolation, in the observables of ${<}0.01$, which is converged for many practical purposes. We thereby identify inaccuracies in the experimental extrapolations and suggest how future experiments can be implemented to increase the classical hardness.Comment: This can be regarded as the full version of the preliminary note in arXiv:2306.1637

Scale Invariant Entanglement Negativity at the Many-Body Localization Transition

The exact nature of the many-body localization transition remains an open question. An aspect which has been posited in various studies is the emergence of scale invariance around this point, however the direct observation of this phenomenon is still absent. Here we achieve this by studying the logarithmic negativity and mutual information between disjoint blocks of varying size across the many-body localization transition. The two length scales, block sizes and the distance between them, provide a clear quantitative probe of scale invariance across different length scales. We find that at the transition point, the logarithmic negativity obeys a scale invariant exponential decay with respect to the ratio of block separation to size, whereas the mutual information obeys a polynomial decay. The observed scale invariance of the quantum correlations in a microscopic model opens the direction to probe the fractal structure in critical eigenstates using tensor network techniques and provide constraints on the theory of the many-body localization transition.Comment: 8 pages, 5 figure

Quantum Delocalized Interactions

Classical mechanics obeys the intuitive logic that a physical event happens at a definite spatial point. Entanglement, however, breaks this logic by enabling interactions without a specific location. In this work we study these delocalized interactions. These are quantum interactions that create less locational information than would be possible classically, as captured by the disturbance induced on some spatial superposition state. We introduce quantum games to capture the effect and demonstrate a direct operational use for quantum concurrence in that it bounds the nonclassical performance gain. We also find a connection with quantum teleportation, and demonstrate the games using an IBM quantum processor