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arXiv:2001.05806 · Physical Review Applied

Quantum Pure State Tomography via Variational Hybrid Quantum-Classical Method

Tao Xin, Xinfang Nie, Xiangyu Kong, Jingwei Wen, Dawei Lu, Jun Li

Published Feb 7, 2020• 31 citations
arXivHTMLPDF

Abstract

To obtain a complete description of a quantum system, one usually employs standard quantum state tomography, which, however, requires an exponential number of measurements to be performed and hence is impractical when the system's size grows large. In this paper, we introduce a self-learning tomographic scheme based on the variational hybrid quantum-classical method. The key part of the scheme is a learning procedure, in which we learn a control sequence capable of driving the unknown target state coherently to a simple fiducial state, so that the target state can be directly reconstructed by applying the control sequence in reverse. In this manner, the state tomography problem is converted to a state-to-state transfer problem. To solve the latter problem, we use a closed-loop-learning control approach. Our scheme is experimentally tested using techniques of four-qubit nuclear magnetic resonance. The experimental results indicate that the proposed tomographic scheme can handle a broad class of states, including entangled states in the field of quantum information, as well as dynamical states of quantum many-body systems common to condensed-matter physics.

DOI
10.1103/physrevapplied.13.024013

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To obtain a complete description of a quantum system, one usually employs standard quantum state tomography, which, however, requires an exponential number of measurements to be performed and hence is impractical when the system's size grows large. In this paper, we introduce a self-learning tomographic scheme based on the variational hybrid quantum-classical method. The key part of the scheme is a learning procedure, in which we learn a control sequence capable of driving the unknown target state coherently to a simple fiducial state, so that the target state can be directly reconstructed by applying the control sequence in reverse. In this manner, the state tomography problem is converted to a state-to-state transfer problem. To solve the latter problem, we use a closed-loop-learning control approach. Our scheme is experimentally tested using techniques of four-qubit nuclear magnetic resonance. The experimental results indicate that the proposed tomographic scheme can handle a broad class of states, including entangled states in the field of quantum information, as well as dynamical states of quantum many-body systems common to condensed-matter physics.
Quantum state tomography is a key process in most quantum experiments. In this work, we employ quantum machine learning for state tomography. Given an unknown quantum state, it can be learned by maximizing the fidelity between the output of a variational quantum circuit and this state. The number of parameters of the variational quantum circuit grows linearly with the number of qubits and the circuit depth, so that only polynomial measurements are required, even for highly entangled states. After that, a subsequent classical circuit simulator is used to transform the information of the target quantum state from the variational quantum circuit into a familiar format. We demonstrate our method by performing numerical simulations for the tomography of the ground state of a one-dimensional quantum spin chain, using a variational quantum circuit simulator. Our method is suitable for near-term quantum computing platforms, and could be used for relatively large-scale quantum state tomography for experimentally relevant quantum states.
We propose a quantum tomography scheme for pure states which adopts machine learning methods, along with a built-in fidelity estimation approach to assess the reliability of the tomographic state. We prove the validity of the scheme theoretically and perform numerically simulated experiments on several typical target quantum states such as W, cluster and dimer states. We find that the required number of measurements to meet the convergence criterion does not grow exponentially in the number of qubits, thus the scheme achieves high efficiency that is crucial for large-scale quantum states tomography in a laboratory.
We introduce a self-learning tomographic technique in which the experiment guides itself to an estimate of its own state. Self-guided quantum tomography uses measurements to directly test hypotheses in an iterative algorithm which converges to the true state. We demonstrate through simulation on many qubits that Self-guided quantum tomography is a more efficient and robust alternative to the usual paradigm of taking a large amount of informationally complete data and solving the inverse problem of postprocessed state estimation.
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.
Abstract Modern day quantum simulators can prepare a wide variety of quantum states but the accurate estimation of observables from tomographic measurement data often poses a challenge. We tackle this problem by developing a quantum state tomography scheme which relies on approximating the probability distribution over the outcomes of an informationally complete measurement in a variational manifold represented by a convolutional neural network. We show an excellent representability of prototypical ground- and steady states with this ansatz using a number of variational parameters that scales polynomially in system size. This compressed representation allows us to reconstruct states with high classical fidelities outperforming standard methods such as maximum likelihood estimation. Furthermore, it achieves a reduction of the estimation error of observables by up to an order of magnitude compared to their direct estimation from experimental data.
Quantum state tomography is the experimental procedure of determining an unknown state. It is not only essential for the verification of resources and processors of quantum information but is also important in its own right with regard to the foundation of quantum mechanics. Standard methods have been elusive for large systems because of the enormous number of observables to be measured and the exponential complexity of data post-processing. Here, we propose a new scheme of quantum state tomography that requires the measurement of only three observables (acting jointly on the system and pointer) regardless of the size of the system. The system is coupled to a "pointer" of single qubit, and the wavefunction of the system is "reaped" onto the pointer upon the measurement of the system. Subsequently, standard two-state tomography on the pointer and classical post-processing are used to reconstruct the quantum state of the system. We also developed an efficient and scalable iterative maximum likelihood algorithm to estimate states from statistically incomplete data.