Quantum state tomography refers to the process of estimating the density matrix of an unknown system after obtaining data through a series of quantum measurements. Quantum state tomography can be divided into two processes: quantum measurement and reconstruction algorithm. From these two processes, different schemes can be designed, and they also have their own advantages and disadvantages. At the same time, quantum tomography is mainly concerned with two factors: estimation accuracy and complexity. High-precision quantum tomography is a necessary condition for quantum computing and other quantum technologies, and the scarcity of quantum resources forces us to find more effective methods to reduce estimation errors. The complexity problem of quantum tomography grows with the dimensionality of the quantum system. I simulate the behavior of a linear optics quantum computing circuit that performs the quantum Fourier transform. We investigated the possibility of using k-means clustering algorithm to predict the output of the quantum computer and compare its performance with the conditional generative adversarial network (CGAN), shadow tomography, maximum likelihood estimation. Using this simulated data, I apply the various algorithms to the order-finding problem to determine the trade-off between the required number of measurements and the accuracy. The result shows that k-means clustering can take a polynomial number of samples to access the result of the order-finding problem with more than 90 percent accuracy. The comparison of all four methods is analyzed.
Quantum state tomography refers to the process of estimating the density matrix of an unknown system after obtaining data through a series of quantum measurements. Quantum state tomography can be divided into two processes: quantum measurement and reconstruction algorithm. From these two processes, different schemes can be designed, and they also have their own advantages and disadvantages. At the same time, quantum tomography is mainly concerned with two factors: estimation accuracy and complexity. High-precision quantum tomography is a necessary condition for quantum computing and other quantum technologies, and the scarcity of quantum resources forces us to find more effective methods to reduce estimation errors. The complexity problem of quantum tomography grows with the dimensionality of the quantum system. I simulate the behavior of a linear optics quantum computing circuit that performs the quantum Fourier transform. We investigated the possibility of using k-means clustering algorithm to predict the output of the quantum computer and compare its performance with the conditional generative adversarial network (CGAN), shadow tomography, maximum likelihood estimation. Using this simulated data, I apply the various algorithms to the order-finding problem to determine the trade-off between the required number of measurements and the accuracy. The result shows that k-means clustering can take a polynomial number of samples to access the result of the order-finding problem with more than 90 percent accuracy. The comparison of all four methods is analyzed.
Multi-qubit state tomography is a key problem in quantum information technology, which has been studied extensively. In this work, we focus on multi-qubit state tomography based on neural network estimators and typical conventional estimation approaches. For multi-qubit pure states, fully connected neural networks and restricted Boltzmann machine networks are designed, respectively, to carry out state tomography of 2-qubit (low-dimensional) systems and 5-qubit (high-dimensional) systems. Comparisons are made with maximum likelihood estimation and least squares estimation, where performance indicators are selected as reconstruction accuracy, time cost and the number of parameters. Simulation results indicate that intelligent approaches have significant advantages over conventional approaches for state tomography of low-dimensional systems; for high-dimensional systems, however, the conventional approach is still dominant when the measurement is complete, while the restricted Boltzmann machine network can achieve higher reconstruction accuracy when the measurement is incomplete.
Quantum state tomography (QST), the task of estimating an unknown quantum state given measurement outcomes, is essential to building reliable quantum computing devices. Whereas computing the maximum-likelihood (ML) estimate corresponds to solving a finite-sum convex optimization problem, the objective function is not smooth nor Lipschitz, so most existing convex optimization methods lack sample complexity guarantees; moreover, both the sample size and dimension grow exponentially with the number of qubits in a QST experiment, so a desired algorithm should be highly scalable with respect to the dimension and sample size, just like stochastic gradient descent. In this paper, we propose a stochastic first-order algorithm that computes an ε-approximate ML estimate in O((DlogD)/ε2) iterations with O(D3) per-iteration time complexity, where D denotes the dimension of the unknown quantum state and ε denotes the optimization error. Our algorithm is an extension of Soft-Bayes to the quantum setup.
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.
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 .
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.
We explore the impact of quantum algorithms on classical-data problems if quantum algorithms can be implemented on quantum computers. It is underlined in this paper that both state estimator design problems for classical linear systems and quantum state tomography can be considered as classical-data problems. Our recent research indicated that the time complexity of state estimator design problems can be reduced to O(qn) in comparison to the time complexity O(n6) in traditional classical algorithms when quantum states can be efficiently prepared, the system matrix is sparse, and both the condition number κ and the reciprocal of precision ε are small in size O(poly log(n)), where n is the dimension of the state x(t) and q is the dimension of the input u(t). This is in contrast to the further observation that the time complexity of state estimator design problems is O(n2) when quantum states cannot be efficiently prepared. The time complexity of quantum state tomography with dimension d is O(d4) when classical algorithms are adopted. We have demonstrated that if the system matrix is sparse, and both the condition number κ and the reciprocal of precision ε are small in size O(poly log(d)), quantum algorithms can reduce the time complexity of quantum state tomography to O(dpoly log d) or O(d3) when quantum states can or cannot be efficiently prepared. In other words, preparing quantum states efficiently has become a bottleneck constraint for quantum acceleration. Our recent researches can be regarded as a novel attempt of enriching application area of quantum computation.