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We describe a technique for self consistently characterizing both the quantum state of a single qubit system, and the positive-operator-valued measure (POVM) that describes measurements on the system. The method works with only ten measurements. We assume that a series of unitary transformations performed on the quantum state are fully known, while making minimal assumptions about both the density operator of the state and the POVM. The technique returns maximum-likely estimates of both the density operator and the POVM. To experimentally demonstrate the method, we perform reconstructions of over 300 state-measurement pairs and compare them to their expected density operators and POVMs. We find that 95% of the reconstructed POVMs have fidelities of 0.98 or greater, and 92% of the density operators have fidelities that are 0.98 or greater.
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Estimation of quantum states and measurements is crucial for the implementation of quantum information protocols. The standard method for each is quantum tomography. However, quantum tomography suffers from systematic errors caused by imperfect knowledge of the system. We present a procedure to simultaneously characterize quantum states and measurements that mitigates systematic errors by use of a single high-fidelity state preparation and a limited set of high-fidelity unitary operations. Such states and operations are typical of many state-of-the-art systems. For this situation we design a set of experiments and an optimization algorithm that alternates between maximizing the likelihood with respect to the states and measurements to produce estimates of each. In some cases, the procedure does not enable unique estimation of the states. For these cases, we show how one may identify a set of density matrices compatible with the measurements and use a semi-definite program to place bounds on the states expectation values. We demonstrate the procedure on data from a simulated experiment with two trapped ions.
Quantum process tomography is a necessary tool for verifying quantum gates and diagnosing faults in architectures and gate design. We show that the standard approach of process tomography is grossly inaccurate in the case where the states and measurement operators used to interrogate the system are generated by gates that have some systematic error, a situation all but unavoidable in any practical setting. These errors in tomography can not be fully corrected through oversampling or by performing a larger set of experiments. We present an alternative method for tomography to reconstruct an entire library of gates in a self-consistent manner. The essential ingredient is to define a likelihood function that assumes nothing about the gates used for preparation and measurement. In order to make the resulting optimization tractable we linearize about the target, a reasonable approximation when benchmarking a quantum computer as opposed to probing a black-box function.
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We examine the problem of finding the minimum number of Pauli measurements needed to uniquely determine an arbitrary $n$-qubit pure state among all quantum states. We show that only $11$ Pauli measurements are needed to determine an arbitrary two-qubit pure state compared to the full quantum state tomography with $16$ measurements, and only $31$ Pauli measurements are needed to determine an arbitrary three-qubit pure state compared to the full quantum state tomography with $64$ measurements. We demonstrate that our protocol is robust under depolarizing error with simulated random pure states. We experimentally test the protocol on two- and three-qubit systems with nuclear magnetic resonance techniques. We show that the pure state tomography protocol saves us a number of measurements without considerable loss of fidelity. We compare our protocol with same-size sets of randomly selected Pauli operators and find that our selected set of Pauli measurements significantly outperforms those random sampling sets. As a direct application, our scheme can also be used to reduce the number of settings needed for pure-state tomography in quantum optical systems.
Tomography of a quantum state is usually based on positive operator-valued measure (POVM) and on their experimental statistics. Among the available reconstructions, the maximum-likelihood (MaxLike) technique is an efficient one. We propose an extension of this technique when the measurement process cannot be simply described by an instantaneous POVM. Instead, the tomography relies on a set of quantum trajectories and their measurement records. This model includes the fact that, in practice, each measurement could be corrupted by imperfections and decoherence, and could also be associated with the record of continuous-time signals over a finite amount of time. The goal is then to retrieve the quantum state that was present at the start of this measurement process. The proposed extension relies on an explicit expression of the likelihood function via the effective matrices appearing in quantum smoothing and solutions of the adjoint quantum filter. It allows to retrieve the initial quantum state as in standard MaxLike tomography, but where the traditional POVM operators are replaced by more general ones that depend on the measurement record of each trajectory. It also provides, aside the MaxLike estimate of the quantum state, confidence intervals for any observable. Such confidence intervals are derived, as the MaxLike estimate, from an asymptotic expansion of multi-dimensional Laplace integrals appearing in Bayesian Mean estimation. A validation is performed on two sets of experimental data: photon(s) trapped in a microwave cavity subject to quantum non-demolition measurements relying on Rydberg atoms; heterodyne fluorescence measurements of a superconducting qubit.
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