Abstract Quantum tomography is a crucial tool for characterizing quantum states and devices and estimating nonlinear properties of the systems. Performing full quantum state tomography on an N qubit system requires an exponentially increasing overhead with O(3^N) distinct Pauli measurement settings to resolve all complex phases and reconstruct the density matrix. However, many potential quantum computing applications, such as linear system solves, require only real-valued amplitudes. We introduce a readout method for real-valued quantum states that reduces measurement settings required for state vector reconstruction to O(N); the post-processing cost remains exponential Ω(2^N). This approach offers a substantial speedup over conventional tomography. We experimentally validate our method up to 10 qubits on the latest available IBM quantum processor and demonstrate that it accurately extracts key properties such as entanglement and magic. Our method also outperforms the standard SWAP test for state overlap estimation. This calculation resembles a numerical integration in certain cases and can be applied to extract nonlinear properties, which are important in application fields. We further implement the method to readout the solution from a quantum linear solver.
Abstract Quantum tomography is a crucial tool for characterizing quantum states and devices and estimating nonlinear properties of the systems. Performing full quantum state tomography on an N qubit system requires an exponentially increasing overhead with O(3^N) distinct Pauli measurement settings to resolve all complex phases and reconstruct the density matrix. However, many potential quantum computing applications, such as linear system solves, require only real-valued amplitudes. We introduce a readout method for real-valued quantum states that reduces measurement settings required for state vector reconstruction to O(N); the post-processing cost remains exponential Ω(2^N). This approach offers a substantial speedup over conventional tomography. We experimentally validate our method up to 10 qubits on the latest available IBM quantum processor and demonstrate that it accurately extracts key properties such as entanglement and magic. Our method also outperforms the standard SWAP test for state overlap estimation. This calculation resembles a numerical integration in certain cases and can be applied to extract nonlinear properties, which are important in application fields. We further implement the method to readout the solution from a quantum linear solver.
Quantum computing shows promise for addressing computationally intensive problems but is constrained by the exponential resource requirements of general quantum state tomography (QST), which fully characterizes quantum states through parameter estimation. We introduce the QST with Chebyshev polynomials, an approximate tomography method for pure quantum states encoding complex-valued functions. This method reformulates tomography as the estimation of Chebyshev expansion coefficients, expressed as inner products between the target quantum state and Chebyshev basis functions, measured using the Hadamard test circuit. By treating the truncation order of the Chebyshev polynomials as a controllable parameter, the method provides a practical balance between efficiency and accuracy. For quantum states encoding functions dominated by large-scale features, such as those representing fluid flow fields, appropriate truncation enables faithful reconstruction of the dominant components via quantum circuits with linear depth, while keeping both measurement repetitions and post-processing independent of qubit count, in contrast to the exponential scaling of full measurement-based QST methods. Validation on analytic functions and numerically generated flow-field data demonstrates accurate reconstruction and effective extraction of large-scale features, indicating the method's suitability for systems governed by macroscopic dynamics.
This paper introduces a novel and efficient technique for quantum state estimation, coined as low-rank matrix-completion quantum state tomography for characterizing pure quantum states, as it requires only non-entangling bases and 2n+1 local Pauli operators. This significantly reduces the complexity of the process and increases the accuracy of the state estimation, as it eliminates the need for the entangling bases, which are experimentally difficult to implement on quantum devices. The required minimal post-processing, improved accuracy and efficacy of this matrix-completion-based method make it an ideal benchmarking tool for investigating the properties of quantum systems, enabling researchers to verify the accuracy of quantum devices, characterize their performance, and explore the underlying physics of quantum phenomena. Our numerical results demonstrate that this method outperforms contemporary techniques in its ability to accurately reconstruct multi-qubit quantum states on real quantum devices, making it an invaluable contribution to the field of quantum state characterization and an essential step toward the reliable deployment of intermediate- and large-scale quantum devices.
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 less 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 multi-qubit quantum states. Furthermore, this schema was proven to be robust through numerical simulations on a noisy superconducting qubit system. Therefore, the tomography schema paves an alternatively effective way to reconstruct the density matrix of a quantum state owing to its efficiency and accuracy, which are essential for quantum state tomography.
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), 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.
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.
Quantum device characterization via state tomography plays an important role in both validating quantum hardware and processing quantum information, unfortunately with the exponential number of the measurements. As one of the main stream quantum platforms, superconducting quantum computing (SQC) system at least requires 3n measurement settings consisted of single-qubit readout operators in reconstructing a n-qubit state. In this work, I add the 2-qubit evolutions as the readout operators, and propose an optimal tomographic scheme with the cost-reduced measurements using the integer programming. In detail, I present the minimum number of required measurements to fully reconstruct a state for the Nearest-Neighbor, 2-Dimensional, and All-to-All connectivities on SQC qubits. It is shown that this method can reduce the number of measurements by over 60% compared with the previous state tomography on SQC systems. It is expected that the experimentalist from SQC field can directly utilize the ready-made results for reconstructing quantum states involved in their research. Besides, this method can be applied to reduce the complexity of traditional state tomography in some quantum platforms including but not limited to SQC systems.