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Pairwise tomography networks for many-body quantum systems

Guillermo García-Pérez, Matteo A. C. Rossi, Boris Sokolov, Elsi-Mari Borrelli, Sabrina Maniscalco

Published Oct 27, 2022• 20 citations
OpenAlex

Abstract

We introduce the concept of pairwise tomography networks to characterize quantum properties in many-body systems and demonstrate an efficient protocol to measure them experimentally. Pairwise tomography networks are generators of multiplex networks where each layer represents the graph of a relevant quantifier, such as, e.g., concurrence, quantum discord, purity, quantum mutual information, or classical correlations. We propose a measurement scheme to perform two-qubit tomography of all pairs showing exponential improvement in the number of qubits N with respect to previously existing methods. We illustrate the usefulness of our approach by means of several examples revealing its potential impact to quantum computation, communication, and simulation. We perform a proof-of-principle experiment demonstrating pairwise tomography networks of W states on IBM Q devices. DOI: 10.1103/PhysRevResearch.2.023393

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We introduce the concept of pairwise tomography networks to characterize quantum properties in many-body systems and demonstrate an efficient protocol to measure them experimentally. Pairwise tomography networks are generators of multiplex networks where each layer represents the graph of a relevant quantifier, such as, e.g., concurrence, quantum discord, purity, quantum mutual information, or classical correlations. We propose a measurement scheme to perform two-qubit tomography of all pairs showing exponential improvement in the number of qubits N with respect to previously existing methods. We illustrate the usefulness of our approach by means of several examples revealing its potential impact to quantum computation, communication, and simulation. We perform a proof-of-principle experiment demonstrating pairwise tomography networks of W states on IBM Q devices. DOI: 10.1103/PhysRevResearch.2.023393
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 is a technique in quantum information science used to reconstruct the density matrix of an unknown quantum state, providing complete information about the quantum state. It is of significant importance in fields such as quantum computation, quantum communication, and quantum simulation. However, as the size of the quantum system increases, the number of measurement settings and sampling requirements for quantum state tomography grow exponentially with the number of qubits. This not only makes experimental design and implementation more complex, but also exacerbates the consumption of experimental resources. These limitations severely hinder the application of state tomography in large-scale quantum systems. To reduce measurement settings and improve sampling efficiency, this study proposes a state tomography method based on auxiliary systems. This method can be implemented through either entanglement between the quantum system to be measured and a quantum auxiliary system or through correlation between the quantum system and a probabilistic classical auxiliary system. Measurements on the entire joint system enable more efficient extraction of information about the quantum state to be measured. This method relies on standard quantum gate operations and requires only two measurement settings, with a total sampling complexity of O(d2)O(d^2)O(d2), significantly simplifying experimental operations and measurement processes. Additionally, this study provides two schemes for measuring purity based on the proposed circuit, one of which achieves measurement precision at the Heisenberg limit. This study validates the effectiveness of the proposed method through a detailed theoretical analysis, a series of numerical simulations, and experiments.
Quantum computation depends on quantum entanglement, a correlation between subsystems that cannot occur classically. A variety of theoretical measures exist for quantifying the degree entanglement in such schemes, all of which are functions of the system density matrix. How can the entanglement be measured experimentally? Using quantum tomography techniques developed for two photon entangled states, the density matrix can be reconstructed from the appropriate experimental data. In this case the state tomography gives the complete characterization of the physical system (for the relevant degree of freedom, such as spin or polarization). It gives information on both the degree of nonclassical correlation, that is entanglement, as well as the amount of decoherence in the system. In this proceedings we discuss the general state tomography procedure required to characterize a few qubit quantum computer, for any architecture.
Quantum state tomography is essential for characterizing quantum systems, but it becomes infeasible for large systems due to exponential resource scaling. Overlapping tomography addresses this challenge by reconstructing all kkk-body marginals using few measurement settings, enabling the efficient extraction of key information for many quantum tasks. While optimal schemes are known for qubits, the extension to higher-dimensional qudit systems remains largely unexplored. Here, we investigate optimal qudit overlapping tomography, constructing local measurement settings from generalized Gell-Mann matrices. By establishing a correspondence with combinatorial covering arrays, we present two explicit constructions of optimal measurement schemes. For nnn-qutrit systems, we prove that pairwise tomography requires at most 8+56⌈log⁡8n⌉8 + 56\left\lceil \log_{8} n \right\rceil8+56⌈log8​n⌉ measurement settings, and provide an explicit scheme achieving this bound. Furthermore, we develop an efficient algorithm to determine the optimal order of these measurement settings, minimizing the experimental overhead associated with switching configurations. Compared to the worst-case ordering, our optimized schedule reduces switching costs by approximately 50\%. These results provide a practical pathway for efficient characterization of qudit systems, facilitating their application in quantum communication and computation.
Large-scale quantum systems have advanced rapidly via the exploration of more particles and higher dimensions, offering great potential for developing quantum technologies. However, their characterization becomes prohibitive with increasing local dimensionality and particle number. Here we propose high-dimensional quantum overlapping tomography based on a graph-theoretic formulation, which allows one to efficiently reconstruct few-body marginals of multipartite high-dimensional quantum systems. We experimentally realize it on a photonic four-party entangled state in a 4×4×2×24 \times 4 \times 2 \times 24×4×2×2 system. Using measurements in mutually unbiased bases, we reconstruct all six two-body marginals with only 25 projective measurement settings, compared with 94 and 225 settings for independent tomography of all two-body reduced states and full state tomography, respectively. The reconstructed marginals reveal a layered entanglement structure vital for high-dimensional quantum networks. We further show that these marginals enable more noise-resilient certification of multipartite high-dimensional entanglement than the fidelity-based criterion. Our work thus offers a scalable route for learning multidimensional quantum systems.