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Entropy

Variational Quantum Process Tomography of Non-Unitaries

Shichuan Xue, Yizhi Wang, Yong Liu, Weixu Shi, Junjie Wu

Published Jan 1, 2023• 6 citations
Publisher

Abstract

Quantum process tomography is a fundamental and critical benchmarking and certification tool that is capable of fully characterizing an unknown quantum process. Standard quantum process tomography suffers from an exponentially scaling number of measurements and complicated data post-processing due to the curse of dimensionality. On the other hand, non-unitary operators are more realistic cases. In this work, we put forward a variational quantum process tomography method based on the supervised quantum machine learning framework. It approximates the unknown non-unitary quantum process utilizing a relatively shallow depth parametric quantum circuit and fewer input states. Numerically, we verified our method by reconstructing the non-unitary quantum mappings up to eight qubits in two cases: the weighted sum of the randomly generated quantum circuits and the imaginary time evolution of the Heisenberg XXZ spin chain Hamiltonian. Results show that those quantum processes could be reconstructed with high fidelities (>99%) and shallow depth parametric quantum circuits (d≤8), while the number of input states required is at least two orders of magnitude less than the demands of the standard quantum process tomography. Our work shows the potential of the variational quantum process tomography method in characterizing non-unitary operators.

DOI
10.3390/e25010090

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Quantum process tomography is a fundamental and critical benchmarking and certification tool that is capable of fully characterizing an unknown quantum process. Standard quantum process tomography suffers from an exponentially scaling number of measurements and complicated data post-processing due to the curse of dimensionality. On the other hand, non-unitary operators are more realistic cases. In this work, we put forward a variational quantum process tomography method based on the supervised quantum machine learning framework. It approximates the unknown non-unitary quantum process utilizing a relatively shallow depth parametric quantum circuit and fewer input states. Numerically, we verified our method by reconstructing the non-unitary quantum mappings up to eight qubits in two cases: the weighted sum of the randomly generated quantum circuits and the imaginary time evolution of the Heisenberg XXZ spin chain Hamiltonian. Results show that those quantum processes could be reconstructed with high fidelities (>99%) and shallow depth parametric quantum circuits (d≤8), while the number of input states required is at least two orders of magnitude less than the demands of the standard quantum process tomography. Our work shows the potential of the variational quantum process tomography method in characterizing non-unitary operators.
Quantum process tomography is an experimental technique to fully characterize an unknown quantum process. Standard quantum process tomography suffers from exponentially scaling of the number of measurements with the increasing system size. In this work, we put forward a quantum machine learning algorithm which approximately encodes the unknown unitary quantum process into a relatively shallow depth parametric quantum circuit. We demonstrate our method by reconstructing the unitary quantum processes resulting from the quantum Hamiltonian evolution and random quantum circuits up to eight qubits. Results show that those quantum processes could be reconstructed with high fidelity, while the number of input states required are at least two orders of magnitude less than required by the standard quantum process tomography.
The impressive pace of advance of quantum technology calls for robust and scalable techniques for the characterization and validation of quantum hardware. Quantum process tomography, the reconstruction of an unknown quantum channel from measurement data, remains the quintessential primitive to completely characterize quantum devices. However, due to the exponential scaling of the required data and classical post-processing, its range of applicability is typically restricted to one- and two-qubit gates. Here, we present a technique for performing quantum process tomography that addresses these issues by combining a tensor network representation of the channel with a data-driven optimization inspired by unsupervised machine learning. We demonstrate our technique through synthetically generated data for ideal one- and two-dimensional random quantum circuits of up to 10 qubits, and a noisy 5-qubit circuit, reaching process fidelities above 0.99 using several orders of magnitude fewer (single-qubit) measurement shots than traditional tomographic techniques. Our results go far beyond state-of-the-art, providing a practical and timely tool for benchmarking quantum circuits in current and near-term quantum computers.
Quantum process tomography is a pivotal technique in fully characterizing quantum dynamics. However, exponential scaling of the Hilbert space with the increasing system size extremely restrains its experimental implementations. Here, we put forward a more efficient, flexible, and error-mitigated method: variational entanglement-assisted quantum process tomography with arbitrary ancillary qubits. Numerically, we simulate up to eight-qubit quantum processes and show that this tomography with m ancillary qubits (0≤m≤n) alleviates the exponential costs on state preparation (from 4^{n} to 2^{n-m}), measurement settings (at least a 1 order of magnitude reduction), and data postprocessing (efficient and robust parameter optimization). Experimentally, we first demonstrate our method on a silicon photonic chip by rebuilding randomly generated one-qubit and two-qubit unitary quantum processes. Further using the error mitigation method, two-qubit quantum processes can be rebuilt with average gate fidelity enhanced from 92.38% to 95.56%. Our Letter provides an efficient and practical approach to process tomography on the noisy quantum computing platforms.
Abstract In this work, we present two new methods for variational quantum circuit (VQC) process tomography (PT) ontonqubits systems: unitary PT based on VQCs (PT_VQC) and unitary evolution-based variational quantum singular value decomposition (U-VQSVD). Compared to the state of the art, PT_VQC halves in each run the required amount of qubits for unitary PT and decreases the required state initializations from 4nto just 2n, all while ensuring high-fidelity reconstruction of the targeted unitary channelU. It is worth noting that, for a fixed reconstruction accuracy, PT_VQC achieves faster convergence per iteration step compared to quantum deep neural network and tensor network schemes. The novel U-VQSVD algorithm utilizes variational singular value decomposition to extract eigenvectors (up to a global phase) and their associated eigenvalues from an unknown unitary representing a universal channel. We assess the performance of U-VQSVD by executing an attack on a non-unitary channel quantum physical unclonable function. By using U-VQSVD we outperform an uninformed impersonation attack (using randomly generated input states) by a factor of 2 to 5, depending on the qubit dimension. For the two presented methods, we propose a new approach to calculate the complexity of the displayed VQC, based on what we denote as optimal depth.
Quantum process characterization is a fundamental task in quantum information processing, yet conventional methods, such as quantum process tomography, require prohibitive resources and lack scalability. Here, we introduce an efficient quantum process learning method specifically designed for short-time Hamiltonian dynamics. Our approach reconstructs an equivalent quantum circuit representation from measurement data of unknown Hamiltonian evolution without requiring additional assumptions and achieves polynomial sample and computational efficiency. Our results have broad applications in various directions. We demonstrate applications in quantum machine learning, where our protocol enables efficient training of variational quantum neural networks by directly learning unitary transformations. Additionally, it facilitates the prediction of quantum expectation values with provable efficiency and provides a robust framework for verifying quantum computations and benchmarking realistic noisy quantum hardware. This work establishes a new theoretical foundation for practical quantum dynamics learning, paving the way for scalable quantum process characterization in both near-term and fault-tolerant quantum computing.
Characterization of quantum processes is a preliminary step necessary in the development of quantum technology. The conventional method uses standard quantum process tomography, which requires d2{d}^{2}d2 input states and d4{d}^{4}d4 quantum measurements for a ddd-dimensional Hilbert space. These experimental requirements are compounded by the complexity of processing the collected data, which can take several orders of magnitude longer than the experiment itself. In this paper we propose an alternative self-guided algorithm for quantum process tomography, tuned for the task of finding an unknown unitary process. Our algorithm is a fully automated and adaptive process characterization technique. The advantages of our algorithm are the following: it has an inherent robustness to both statistical and technical noise; it requires less space and time since there is no postprocessing of the data; it requires only a single input state and measurement; and it provides on-the-fly diagnostic information while the experiment is running. Numerical results show our algorithm achieves the same 1/n1/n1/n scaling as standard quantum process tomography when nnn uses of the unknown process are used. We also present experimental results wherein the algorithm and its advantages are realized for the task of finding an element of SU(2).