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We discuss quantum state tomography via a stepwise reconstruction of the eigenstates of the mixed states produced in experiments. Our method is tailored to the experimentally relevant class of nearly pure states or simple mixed states, which exhibit dominant eigenstates and thus lend themselves to low-rank approximations. The developed scheme is applicable to any pure-state tomography method, promoting it to mixed-state tomography. Here, we demonstrate it with machine learning-inspired pure-state tomography based on neural-network representations of quantum states. The latter have been shown to efficiently approximate generic classes of complex (pure) states of large quantum systems. We test our method by applying it to experimental data from trapped ion experiments with four to eight qubits.
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Measurement of the energy eigenvalues (spectrum) of a multi-qubit system has recently become possible by qubit tunneling spectroscopy (QTS). In the standard QTS experiments, an incoherent probe qubit is strongly coupled to one of the qubits of the system in such a way that its incoherent tunneling rate provides information about the energy eigenvalues of the original (source) system. In this paper, we generalize QTS by coupling the probe qubit to many source qubits. We show that by properly choosing the couplings, one can perform projective measurements of the source system energy eigenstates in an arbitrary basis, thus performing quantum eigenstate tomography. As a practical example of a limited tomography, we apply our scheme to probe the eigenstates of a kink in a frustrated transverse Ising chain.
Quantum state tomography (QST) is a crucial ingredient for almost all aspects of experimental quantum information processing. As an analog of the imaging technique in the quantum settings, QST is born to be a data science problem, where machine learning techniques, noticeably neural networks, have been applied extensively. In this work, we build an integrated all-optical setup for neural network QST, based on an all-optical neural network (AONN). Our AONN is equipped with built-in nonlinear activation function, which is based on electromagnetically induced transparency. Experiment results demonstrate the validity and efficiency of the all-optical setup, indicating that AONN can mitigate the state-preparation-and-measurement error and predict the phase parameter in the quantum state accurately. Given that optical setups are highly desired for future quantum networks, our all-optical setup of integrated AONN-QST may shed light on replenishing the all-optical quantum network with the last brick.
Quantum State Tomography is the task of determining an unknown quantum state by making measurements on identical copies of the state. Current algorithms are costly both on the experimental front -- requiring vast numbers of measurements -- as well as in terms of the computational time to analyze those measurements. In this paper, we address the problem of analysis speed and flexibility, introducing textit{Neural Adaptive Quantum State Tomography} (NA-QST), a machine learning based algorithm for quantum state tomography that adapts measurements and provides orders of magnitude faster processing while retaining state-of-the-art reconstruction accuracy. Our algorithm is inspired by particle swarm optimization and Bayesian particle-filter based adaptive methods, which we extend and enhance using neural networks. The resampling step, in which a bank of candidate solutions -- particles -- is refined, is in our case learned directly from data, removing the computational bottleneck of standard methods. We successfully replace the Bayesian calculation that requires computational time of $O(mathrm{poly}(n))$ with a learned heuristic whose time complexity empirically scales as $O(log(n))$ with the number of copies measured $n$, while retaining the same reconstruction accuracy. This corresponds to a factor of a million speedup for $10^7$ copies measured. We demonstrate that our algorithm learns to work with basis, symmetric informationally complete (SIC), as well as other types of POVMs. We discuss the value of measurement adaptivity for each POVM type, demonstrating that its effect is significant only for basis POVMs. Our algorithm can be retrained within hours on a single laptop for a two-qubit situation, which suggests a feasible time-cost when extended to larger systems. It can also adapt to a subset of possible states, a choice of the type of measurement, and other experimental details.
Work extraction from the Gibbs ensemble by a cyclic operation is impossible, as represented by the second law of thermodynamics. On the other hand, the eigenstate thermalization hypothesis (ETH) states that just a single energy eigenstate can describe a thermal equilibrium state. Here we attempt to unify these two perspectives and investigate the second law at the level of individual energy eigenstates, by examining the possibility of extracting work from a single energy eigenstate. Specifically, we performed numerical exact diagonalization of a quench protocol of local Hamiltonians and evaluated the number of work-extractable energy eigenstates. We found that it becomes exactly zero in a finite system size, implying that a positive amount of work cannot be extracted from any energy eigenstate, if one or both of the pre- and the post-quench Hamiltonians are non-integrable. We argue that the mechanism behind this numerical observation is based on the ETH for a non-local observable. Our result implies that quantum chaos, characterized by non-integrability, leads to a stronger version of the second law than the conventional formulation based on the statistical ensembles.
We propose to use neural networks to estimate the rates of coherent and incoherent processes in quantum systems from continuous measurement records. In particular, we adapt an image recognition algorithm to recognize the patterns in experimental signals and link them to physical quantities. We demonstrate that the parameter estimation works unabatedly in the presence of detector imperfections which complicate or rule out Bayesian filter analyses.
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