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Proceedings/Proceedings of the ... International Conference on Software Engineering and Knowledge Engineering

Quantum Software Models: Quantum Modules Tomography and Recovery Theorem

Iaakov Exman, Ariel Zvulunov

Published Jul 1, 2023• 2 citations
Publisher

Abstract

Quantum Tomography partially measures and then recovers the remaining density matrix quantum state, in order to verify that a certain device -processor or detector -indeed outputs the intended quantum state.However, single matrix element measurements rapidly increase in number with the density matrix dimension and are error-prone.This work proposes a novel quantum software viewpoint on quantum tomography.Instead of individual matrix elements, measurements and density matrix recovery are performed on higher-abstraction modules, i.e. semantically meaningful groups of matrix elements.Quantum Modules Tomography potentially reduce the necessary number of explicit measurements, while still allowing recovery of the whole density matrix.A Recovery Theorem is formulated and proved: density matrix diagonal measurements suffice to recover the whole system density matrix, in terms of modules.The recovery procedure is illustrated by a few case studies.

DOI
10.18293/seke2023-214

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Quantum Tomography partially measures and then recovers the remaining density matrix quantum state, in order to verify that a certain device -processor or detector -indeed outputs the intended quantum state.However, single matrix element measurements rapidly increase in number with the density matrix dimension and are error-prone.This work proposes a novel quantum software viewpoint on quantum tomography.Instead of individual matrix elements, measurements and density matrix recovery are performed on higher-abstraction modules, i.e. semantically meaningful groups of matrix elements.Quantum Modules Tomography potentially reduce the necessary number of explicit measurements, while still allowing recovery of the whole density matrix.A Recovery Theorem is formulated and proved: density matrix diagonal measurements suffice to recover the whole system density matrix, in terms of modules.The recovery procedure is illustrated by a few case studies.
The reconstruction of quantum states from experimental measurements, often achieved using quantum state tomography (QST), is crucial for the verification and benchmarking of quantum devices. However, performing QST for a generic unstructured quantum state requires an enormous number of state copies that growsexponentiallywith the number of individual quanta in the system, even for the most optimal measurement settings. Fortunately, many physical quantum states, such as states generated by noisy, intermediate-scale quantum computers, are usually structured. In one dimension, such states are expected to be well approximated by matrix product operators (MPOs) with a matrix/bond dimension independent of the number of qubits, therefore enabling efficient state representation. Nevertheless, it is still unclear whether efficient QST can be performed for these states in general. In other words, there exist no rigorous bounds on the number of state copies required for reconstructing MPO states that scales polynomially with the number of qubits. In this paper, we attempt to bridge this gap and establish theoretical guarantees for the stable recovery of MPOs using tools from compressive sensing and the theory of empirical processes. We begin by studying two types of random measurement settings: Gaussian measurements and Haar random projective measurements. We show that the information contained in an MPO with a constant bond dimension can be preserved using a number of random measurements that depends onlylinearlyon the number of qubits, assuming no statistical error of the measurements. We then study MPO-based QST with Haar random projective measurements that can in principle be implemented on quantum computers. We prove that only apolynomialnumber of state copies in the number of qubits is required to guarantee bounded recovery error of an MPO state. Remarkably, such recovery can be achieved by measuring the state in each random basis only once, despite the large statistical error associated with the outcome of each measurement. Our work may be generalized to accommodate random local or t-design measurements that are more practical to implement on current quantum computers. It may also facilitate the discovery of efficient QST methods for other structured quantum states.
Reconstruction of density matrices is important in NMR quantum computing. An analysis is made for a 2-qubit system using the error matrix method. It is found that the state tomography method determines well the parameters that are necessary for reconstructing the density matrix in NMR quantum computations. Analysis is also made for a simplified state tomography procedure that uses fewer read-outs. The result of this analysis with the error matrix method demonstrates that a satisfactory accuracy in the density matrix reconstruction can be achieved even in a measurement with the number of read-outs being largely reduced.
In standard optical tomographic methods, the off-diagonal elements of a density matrix ρ are measured indirectly. Thus, the reconstruction of ρ, even if it is based on linear inversion, typically magnifies small errors in the experimental data. Recently, an optimal tomography solution measuring all the elements of ρ one-by-one without error magnification has been theoretically proposed. We implemented this method for two-qubit polarization states. For comparison, we also experimentally implemented other well-known tomographic protocols, either based solely on local measurements (of, e.g., the Pauli operators and James-Kwiat-Munro-White projectors) or with mutually unbiased bases requiring both local and global measurements. We reconstructed seventeen separable, partially and maximally entangled two-qubit polarization states. Our experiments show that our method has the highest stability against errors in comparison to other quantum tomographies. In particular, we demonstrate that each optimally-reconstructed state is embedded in an uncertainty circle of the smallest radius, both in terms of trace distance and disturbance. We explain how to experimentally estimate uncertainty radii for all the implemented tomographies and show that, for each reconstructed state, the relevant uncertainty circles intersect indicating the approximate location of the corresponding physical density matrix.
We present an algebraic algorithm for quantum state tomography that leverages measurements of certain observables to estimate structured entries of the underlying density matrix. Under low-rank assumptions, the remaining entries can be obtained solely using standard numerical linear algebra operations. The proposed algebraic matrix completion framework applies to a broad class of generic, low-rank mixed quantum states and, compared with state-of-the-art methods, is computationally efficient while providing deterministic recovery guarantees.
Quantum state tomography aims at reconstructing the state of a quantum system. The standard approach is based on a tomographically complete set of observables and a maximum likelihood estimation of the density matrix. However, the number of measurements scales exponentially with the number of qubits or qudits of the system. By leveraging some properties of any physical density matrix, we propose a method in which the number of necessary measurements depends on the state under investigation but without the need for any a priori information. In some cases, this can lead to a significant reduction of the resources necessary for state characterization.