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arXiv:1810.09484 · Physical Review A

Optimal choice of state tomography quorum formed by projection operators

Violeta N. Ivanova-Rohling, Niklas Rohling

Published Sep 24, 2019• 7 citations
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Abstract

A minimal set of measurement operators for quantum state tomography has in the nondegenerate case ideally eigenbases which are mutually unbiased. This is different for the degenerate case. Here we consider the situation where the measurement operators are projections on individual pure quantum states. This corresponds to maximal degeneracy. We present numerically optimized sets of projectors and find that they significantly outperform those which are taken from a set of mutually unbiased bases.

DOI
10.1103/physreva.100.032332

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A minimal set of measurement operators for quantum state tomography has in the nondegenerate case ideally eigenbases which are mutually unbiased. This is different for the degenerate case. Here we consider the situation where the measurement operators are projections on individual pure quantum states. This corresponds to maximal degeneracy. We present numerically optimized sets of projectors and find that they significantly outperform those which are taken from a set of mutually unbiased bases.
Abstract Finding optimal measurement schemes in quantum state tomography is a fundamental problem in quantum computation. It is known that for non-degenerate operators the optimal measurement scheme is based on mutually unbiassed bases. This paper is a follow up from our previous work, where we use standard numerical approaches to look for optimal measurement schemes, where the measurement operators are projections on individual pure quantum states. In this paper we demonstrate the usefulness of several machine learning techniques – reinforcement learning and parallel machine learning approaches, to discover measurement schemes, which are significantly better than the ones discovered by standard numerical methods in our previous work. The high-performing quorums of projection operators we have discovered have complex structure and symmetries, which may imply that the optimal solution will possess such symmetries.
Abstract Quantum state tomography (QST) represents an essential tool for the characterization, verification, and validation (QCVV) of quantum processors. Only for a few idealized scenarios, there are analytic results for the optimal measurement set for QST. E.g., in a setting of non-degenerate measurements, an optimal minimal set of measurement operators for QST has eigenbases which are mutually unbiased. However, in other set-ups, dependent on the rank of the projection operators and the size of the quantum system, the optimal choice of measurements for efficient QST needs to be numerically approximated. We have generalized this problem by introducing the framework of customized efficient QST. Here we extend customized QST and look for the optimal measurement set for QST in the case where some of the quantum gates applied in the measurement process are noisy. To achieve this, we use two distinct noise models: first, the depolarizing channel, and second, over- and under-rotation in single-qubit and to two-qubit gates (for further information, please see Methods). We demonstrate the benefit of using entangling gates for the efficient QST measurement schemes for two qubits at realistic noise levels, by comparing the fidelity of reconstruction of our optimized QST measurement set to the state-of-the-art scheme using only product bases.
Abstract We present a framework that formulates the quest for the most efficient quantum state tomography (QST) measurement set as an optimization problem which can be solved numerically, where the optimization goal is the maximization of the information gain. This approach can be applied to a broad spectrum of relevant setups including measurements restricted to a subsystem. To illustrate the power of this method we present results for the six-dimensional Hilbert space constituted by a qubit–qutrit system, which could be realized e.g. by the 14N nuclear spin-1 and two electronic spin states of a nitrogen-vacancy center in diamond. Measurements of the qubit subsystem are expressed by projectors of rank three, i.e. projectors on half-dimensional subspaces. For systems consisting only of qubits, it was shown analytically that a set of projectors on half-dimensional subspaces can be arranged in an informationally optimal fashion for QST, thus forming so-called mutually unbiased subspaces. Our method goes beyond qubits-only systems and we find that in dimension six such a set of mutually-unbiased subspaces can be approximated with a deviation irrelevant for practical applications.
When used in quantum state estimation, projections onto mutually unbiased bases have the ability to maximize information extraction per measurement and to minimize redundancy. We present the first experimental demonstration of quantum state tomography of two-qubit polarization states to take advantage of mutually unbiased bases. We demonstrate improved state estimation as compared to standard measurement strategies and discuss how this can be understood from the structure of the measurements we use. We experimentally compared our method to the standard state estimation method for three different states and observe that the infidelity was up to 1.84 ± 0.06 times lower by using our technique than it was by using standard state estimation methods.
We propose an efficient scheme for qutrit state tomography based on optimal unambiguous discrimination of three nonorthogonal pure qutrit states. In our scheme, we first introduce an ancillary qubit initially prepared in a known state and apply a joint unitary transformation on the whole system composed of a qutrit state and an ancillary qubit state. Afterwards, we make a local von Neumann measurement (VNM) on the ancillary qubit. If it is projected onto the state |0〉, we finally perform mutually unbiased measurements (MUMs) on the collapsed qutrit state arbitrarily chosen from a complete set of four mutually unbiased bases (MUBs). With only six properly-chosen projectors, the real and imaginary parts of non-diagonal elements can be entirely obtained. Together with the diagonal elements obtained directly from one VNM in the computational bases, the density matrix of an arbitrary unknown qutrit state can be fully determined. Compared with the previous work (R. Salazar and A. Delgado 2012), our scheme consumes much less resources. In comparison with MUBs-based tomography scheme (H. Yuan, et al., 2016), our scheme requires more measurements and less projectors, but pays the price of a loss of certain fraction identically prepared states. Our proposal can be extended to the case of other d-dimensional quantum states if their d MUBs are exactly known.