Quantum overlapping tomography (QOT), which focuses on reconstructing subsystems of quantum systems, has emerged as a promising approach for quantum state learning, especially when full-state tomography is infeasible. QOT has attracted considerable interest and seen substantial development, but has not yet reached its ultimate limit. The authors introduce a unified framework for optimal QOT by mapping the problem to the ``clique cover'' model from graph theory. This framework provides superlative efficiency and experimental feasibility in measurement strategies. This validation of QOT's utility paves the way for advanced quantum system characterization and state-property learning.
Quantum overlapping tomography (QOT), which focuses on reconstructing subsystems of quantum systems, has emerged as a promising approach for quantum state learning, especially when full-state tomography is infeasible. QOT has attracted considerable interest and seen substantial development, but has not yet reached its ultimate limit. The authors introduce a unified framework for optimal QOT by mapping the problem to the ``clique cover'' model from graph theory. This framework provides superlative efficiency and experimental feasibility in measurement strategies. This validation of QOT's utility paves the way for advanced quantum system characterization and state-property learning.
Partial tomography, which focuses on reconstructing reduced density matrices (RDMs), has emerged as a promising approach for characterizing complex quantum systems, particularly when full state tomography is impractical. Recently, overlapping tomography has been proposed as an efficient method for determining all k-qubit RDMs using logarithmic polynomial measurements, though it has not yet reached the ultimate limit. Here, we introduce a unified framework for optimal quantum overlapping tomography by mapping the problem to the clique cover model. This framework provides the most efficient and experimentally feasible measurement schemes to date, significantly reducing the measurement costs. Our approach is also applicable to determining RDMs with different topological structures. Moreover, we experimentally validate the feasibility of our schemes on practical nuclear spin processor using average measurements and further apply our method to noisy data from a superconducting quantum processor employing projection measurements. The results highlight the strong power of overlapping tomography, paving the way for advanced quantum system characterization and state property learning in the future.
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 k-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 n-qutrit systems, we prove that pairwise tomography requires at most 8+56⌈log8n⌉ 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.
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.
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 4×4×2×2 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.
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.
Reconstructing the full quantum state of a many-body system requires the estimation of a number of parameters that grows exponentially with system size. Nevertheless, there are situations in which one is only interested in a subset of these parameters, and a full reconstruction is not needed. A paradigmatic example is a scenario in which one aims at determining all the reduced states only up to a given size. Overlapping tomography provides constructions to address this problem with a number of product measurements much smaller than what is obtained when performing independent tomography of each reduced state. There are, however, many relevant physical systems with a natural notion of locality where one is mostly interested in the reduced states of neighboring particles. In this work, we study this form of local overlapping tomography. First of all, we show that, contrary to its full version, the number of product-measurement settings needed for local overlapping tomography does not grow with system size. Then, we present strategies for qubit and fermionic systems in selected lattice geometries. The developed methods find a natural application in the estimation of many-body systems prepared in current quantum simulators or quantum computing devices, where interactions are often local.