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arXiv:1911.00173 · Physical Review Research

Generalized overlap quantum state tomography1

Rajveer Nehra, Miller Eaton, Carlos González-Arciniegas, M. S. Kim, Thomas Gerrits, Adriana Lita, Sae Woo Nam, Olivier Pfister

Published Oct 7, 2020• 17 citations
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Abstract

We propose and experimentally demonstrate a quantum state tomography protocol that generalizes and improves upon the Wallentowitz-Vogel-Banaszek-Wódkiewicz point-by-point Wigner function reconstruction.The full density operator of an arbitrary quantum state is efficiently reconstructed in the Fock basis, using semidefinite programming, after interference with a small set of calibrated coherent states.This protocol is resource-and computationally efficient, is robust against noise, does not rely on approximate state displacements, and ensures the physicality of results.

DOI
10.1103/physrevresearch.2.042002

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We propose and experimentally demonstrate a quantum state tomography protocol that generalizes and improves upon the Wallentowitz-Vogel-Banaszek-Wódkiewicz point-by-point Wigner function reconstruction.The full density operator of an arbitrary quantum state is efficiently reconstructed in the Fock basis, using semidefinite programming, after interference with a small set of calibrated coherent states.This protocol is resource-and computationally efficient, is robust against noise, does not rely on approximate state displacements, and ensures the physicality of results.
We present an efficient algorithm that solves the quantum state tomography problem from an arbitrary number of projective measurements in any finite dimension ddd. The algorithm is flexible enough to allow us to impose any desired rank rrr to the state to be reconstructed, ranging from pure (r=1r=1r=1) to full rank (r=d)(r=d)(r=d) quantum states. The method exhibits successful and fast convergence under the presence of realistic errors in both state preparation and measurement stages, and also when considering overcomplete sets of observables. We demonstrate that the method outperforms semidefinite programming quantum state tomography for some sets of physically relevant quantum measurements in every finite dimension.
Reconstructing quantum states is an important task for various emerging quantum technologies. The process of reconstructing the density matrix of a quantum state is known as quantum state tomography. Conventionally, tomography of arbitrary quantum states is challenging as the paradigm of efficient protocols has remained in applying specific techniques for different types of quantum states. Here, we introduce a quantum state tomography platform based on the framework of reservoir computing. It forms a quantum neural network and operates as a comprehensive device for reconstructing an arbitrary quantum state (finite-dimensional or continuous variable). This is achieved with only measuring the average occupation numbers in a single physical setup, without the need of any knowledge of optimum measurement basis or correlation measurements.
Quantum state tomography (QST) aims at reconstructing the state of a quantum system. However, in conventional QST, the number of measurements scales exponentially with the number of qubits. Here, we propose a QST protocol, in which the introduction of a threshold allows one to drastically reduce the number of measurements required for the reconstruction of the state density matrix without compromising the result accuracy. In addition, one can also use the same approach to reconstruct an approximated density matrix tailoring the number of measurements on the available resources. We experimentally demonstrate this protocol by performing the tomography of states up to 7 qubits. We show that our approach can lead to results in agreement with those obtained by QST even when the number of measurements is reduced by more than two orders of magnitude.
Characterizing large-scale quantum systems is central to fundamental physics and essential for applications of quantum technologies. While a full characterization requires exponentially increasing resources, focusing on application-relevant information can often lead to significantly simplified analysis. Overlapping tomography is such a scheme, allowing one to obtain all the information contained in specific subsystems of multiparticle quantum systems in an efficient manner, but the ultimate limits of this approach remain elusive. We present protocols for overlapping tomography that are optimal with respect to the number of measurement settings. First, by providing algorithmic approaches based on graph theory we find the minimal number of Pauli settings, relating overlapping tomography to the problem of covering arrays in combinatorics. This significantly reduces the number of measurement settings, showing for instance that two-body overlapping tomography of nearest neighbors in qubit systems with planar topologies can always be performed with nine Pauli settings. Second, we prove that using general projective measurements, all k-body marginals can be reconstructed with only 3^{k} settings, independently of the system size. Finally, we demonstrate the practical applicability of our methods in a six-photon experiment. Our results will find applications in learning noise and interaction patterns in quantum computers as well as characterizing fermionic systems in quantum chemistry.
Quantum state tomography (QST) via local measurements on reduced density matrices (LQST) is a promising approach but becomes impractical for large systems. To tackle this challenge, we developed an efficient quantum state tomography method inspired by quantum overlapping tomography [Phys. Rev. Lett. 124, 100401 (2020)PRLTAO0031-900710.1103/PhysRevLett.124.100401], which utilizes parallel measurements (PQST). In contrast to LQST, PQST significantly reduces the number of measurements and offers more robustness against shot noise. Experimentally, we demonstrate the feasibility of PQST in a treelike superconducting qubit chip by designing high-efficiency circuits, preparing W states, ground states of Hamiltonians, and random states, and then reconstructing these density matrices using full quantum state tomography (FQST), LQST, and PQST. Our results show that PQST reduces measurement cost, achieving fidelities of 98.68% and 95.07% after measuring 75 and 99 observables for six-qubit and nine-qubit W states, respectively. Furthermore, the reconstruction of the largest density matrix of the 12-qubit W state is achieved with the similarity of 89.23% after just measuring 243 parallel observables, while 3^{12}=531 441 complete observables are needed for FQST. Consequently, PQST will be a useful tool for future tasks such as the reconstruction, characterization, benchmarking, and properties learning of states.