Quantum state tomography (QST) is a crucial tool for characterizing quantum states. However, performing full QST becomes impractical for reconstructing multiqubit density matrices since datasets and computational costs grow exponentially with qubit number. To bypass full QST, we introduce selective and efficient QST (SEEQST)—a method that enables efficient estimation of multiple selected elements of an arbitrary multiqubit density matrix. We show that any N -qubit density matrix can be partitioned into 2 N disjoint subsets, each containing 2 N elements. With SEEQST, any such subset can be accurately estimated from just two experiments with only single-qubit measurements. The complexity for estimating any subset remains constant regardless of Hilbert-space dimension, so, if desired, SEEQST can reconstruct the full density matrix, using 2 N + 1 − 1 experiments, where standard methods would use 3 N experiments. We provide a circuit decomposition for the SEEQST experiments, demonstrating that their maximum circuit depth scales logarithmically with N assuming all-to-all connectivity.
Quantum state tomography (QST) is a crucial tool for characterizing quantum states. However, performing full QST becomes impractical for reconstructing multiqubit density matrices since datasets and computational costs grow exponentially with qubit number. To bypass full QST, we introduce selective and efficient QST (SEEQST)—a method that enables efficient estimation of multiple selected elements of an arbitrary multiqubit density matrix. We show that any N -qubit density matrix can be partitioned into 2 N disjoint subsets, each containing 2 N elements. With SEEQST, any such subset can be accurately estimated from just two experiments with only single-qubit measurements. The complexity for estimating any subset remains constant regardless of Hilbert-space dimension, so, if desired, SEEQST can reconstruct the full density matrix, using 2 N + 1 − 1 experiments, where standard methods would use 3 N experiments. We provide a circuit decomposition for the SEEQST experiments, demonstrating that their maximum circuit depth scales logarithmically with N assuming all-to-all connectivity.
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.
In quantum information transformation and quantum computation, the most critical issues are security and accuracy. These features, therefore, stimulate research on quantum state characterization. A characterization tool, quantum state tomography, reconstructs the density matrix of an unknown quantum state. Theoretically, reconstructing an unknown state using this method can be arbitrarily accurate. However, this is not so practical owing to the huge burden of measurements and data processing for large numbers of qubits. Even comprising an efficient estimator and a precise algorithm, an optimal tomographic framework can also be overburdened owing to the exponential growth of the measurements. Moreover, the consequential postprocessing of huge amounts of data challenges the capacity of computers. Thus, it is crucial to build an efficient framework that requires fewer measurements but yields an expected accuracy. To this end, we built a tomography schema by which only a few Pauli measurements enable an accurate tomographic reconstruction. Subsequently, this schema was verified as efficient and accurate through numerical simulations on the tomography of multiqubit quantum states. Furthermore, it was proven to be robust through numerical simulations on a noisy superconducting qubit system. Therefore, the tomography schema paves an alternative effective way to reconstruct the density matrix of a quantum state owing to its efficiency and accuracy, which are essential for quantum state tomography.
Characterizing quantum states is essential for validating quantum devices, yet conventional quantum state tomography becomes prohibitively expensive as system size grows. Direct tomography offers a distinct route by enabling selective access to individual complex density-matrix elements, with a particular advantage for sparse target states and some verification tasks. Here we introduce a direct quantum state tomography scheme combining strong-measurement estimation with a fan-out coupling architecture. It enables mutually commuting interactions between system qubits and a single meter qubit, thereby achieving constant circuit depth, independent of system size. Notably, the involutory fan-out coupling reduces to the identity under repetition, enabling straightforward noise scaling for quantum error mitigation. We experimentally validate the scheme on a superconducting quantum processor via the IBM Quantum Platform, demonstrating four-qubit state reconstruction and single-circuit GHZ-state fidelity estimation up to 20 qubits with error mitigation. Consistent results with standard tomography and improved efficiency establish our scheme as a promising approach to reconstructing full quantum states and scalable verification tasks.
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.
Abstract Quantum state tomography (QST) is a widely employed technique for characterizing the state of a quantum system. However, it is plagued by two fundamental challenges: computational and experimental complexity grows exponentially with the number of qubits, rendering experimental implementation and data post-processing arduous even for moderately sized systems. Here, we introduce gradient-descent (GD) algorithms for the post-processing step of QST in discrete- and continuous-variable systems. To ensure physically valid state reconstruction at each iteration step of the algorithm, we use various density-matrix parameterizations: Cholesky decomposition, Stiefel manifold, and projective normalization. These parameterizations have the added benefit of enabling a rank-controlled ansatz, which simplifies reconstruction when there is prior information about the system. We benchmark the performance of our GD-QST techniques against state-of-the-art methods, including constrained convex optimization, conditional generative adversarial networks, and iterative maximum likelihood estimation. Our comparison focuses on time complexity, iteration counts, data requirements, state rank, and robustness against noise. We find that rank-controlled ansatzes in our stochastic mini-batch GD-QST algorithms effectively handle noisy and incomplete data sets, yielding significantly higher reconstruction fidelity than other methods. Simulations achieving full-rank seven-qubit QST in under three minutes on a standard laptop, with 18 GB of RAM and no dedicated GPU, highlight that GD-QST is computationally more efficient and outperforms other techniques in most scenarios, offering a promising avenue for characterizing noisy intermediate-scale quantum devices. Our Python code for GD-QST algorithms is publicly available at github.com/mstorresh/GD-QST .
Quantum state tomography (QST) is a central task in quantum information, and its efficiency is commonly characterized by sample complexity. Although collective measurements on multiple copies achieve optimal performance, they are difficult to implement on near-term devices, motivating the study of single-copy approaches. Here, we introduce an ancilla-free single-copy QST protocol based on logarithmic-depth local circuits on an n-qubit system. For rank-r states in dimension d=2n, our protocol achieves trace-norm error ε using O(dr2logd/ε2) copies, matching the single-copy lower bound up to a logarithmic factor. For full-rank mixed states, it removes this logarithmic overhead and achieves the optimal scaling O(d3/ε2), with nearly optimal classical runtime for explicit matrix output. These results show that sample-optimal QST can be realized using experimentally accessible shallow-depth measurements.