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arXiv:2404.16541 · New Journal of Physics

Optimal depth and a novel approach to variational unitary quantum process tomography

Vladlen Galetsky, Pol Juliá Farré, Soham Ghosh, Christian Deppe, Roberto Ferrara

Published Jul 1, 2024• 5 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1367-2630/ad5df1

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In this work, we present two new methods for Variational Quantum Circuit (VQC) Process Tomography onto n qubits systems: PT_VQC and U-VQSVD.Compared to the state of the art, PT_VQC halves in each run the required amount of qubits for process tomography and decreases the required seed states from 4nto 2n, ensuring high-fidelity reconstruction of the targeted unitary U. It is worth noting that, for a fixed reconstruction accuracy, PT_VQC achieves faster convergence per iteration compared to Quantum Deep Neural Network (QDNN) and tensor network schemes.U-VQSVD utilizes variational singular value decomposition to extract eigenvectors (up to a global phase) and their associated eigenvalues from an unknown unitary representing a general channel. We assess the performance of U-VQSVD by executing an attack on a non-unitary channel Quantum Physical Unclonable Function (QPUF), outperforming an uninformed impersonation attack 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 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.
Singular value decomposition is central to many problems in engineering and scientific fields. Several quantum algorithms have been proposed to determine the singular values and their associated singular vectors of a given matrix. Although these algorithms are promising, the required quantum subroutines and resources are too costly on near-term quantum devices. In this work, we propose a variational quantum algorithm for singular value decomposition (VQSVD). By exploiting the variational principles for singular values and the Ky Fan Theorem, we design a novel loss function such that two quantum neural networks (or parameterized quantum circuits) could be trained to learn the singular vectors and output the corresponding singular values. Furthermore, we conduct numerical simulations of VQSVD for random matrices as well as its applications in image compression of handwritten digits. Finally, we discuss the applications of our algorithm in recommendation systems and polar decomposition. Our work explores new avenues for quantum information processing beyond the conventional protocols that only works for Hermitian data, and reveals the capability of matrix decomposition on near-term quantum devices.
Universal compilation is a training process that compiles a trainable unitary into a target unitary and it serves vast potential applications from quantum dynamic simulations to optimal circuits with deep-compressing, device benchmarking, quantum error mitigation, and so on. Here, we propose a universal compilation-based variational algorithm for the preparation and tomography of quantum states in low-depth quantum circuits. We apply the Fubini-Study distance to be a trainable cost function under various gradient-based optimizers, including the quantum natural gradient approach. We evaluate the performance of various unitary topologies and the trainability of different optimizers for getting high efficiency. In practice, we address different circuit ansatzes in quantum state preparation, including the linear and graph-based ansatzes for preparing different entanglement target states such as representative GHZ and W states. We also discuss the effect of the circuit depth, barren plateau, readout noise in the model, and the error mitigation solution. We next evaluate the reconstructing efficiency in quantum state tomography via various popular circuit ansatzes and reveal the crucial role of the circuit depth in the robust fidelity. The results are comparable with the shadow tomography method, a similar fashion in the field. Our work expresses the adequate capacity of the universal compilation-based variational algorithm to maximize the efficiency in the quantum state preparation and tomography. Further, it promises applications in quantum metrology and sensing and is applicable in the near-term quantum computers for verification of the circuits fidelity and various quantum computing tasks.
Abstract We present a quantum-analogous experimental demonstration of variational quantum process tomography using an optical processor. This approach leverages classical one-hot encoding and unitary decomposition to perform the variational quantum algorithm on a photonic platform. We create the first benchmark for variational quantum process tomography evaluating the performance of the quantum-analogous experiment on the optical processor against several publicly accessible quantum computing platforms, including IBM’s 127-qubit Sherbrooke processor, QuTech’s five-qubit Tuna-5 processor, and Quandela’s 12-mode Ascella quantum optical processor. We evaluate each method using process fidelity, cost function convergence, and processing time per iteration for variational quantum circuit depths of d = 3 and d = 6. Our results indicate that the optical processors outperform their superconducting counterparts in terms of fidelity and convergence behavior reaching fidelities of 0.8 after nine iterations, particularly at higher depths, where the noise of decoherence and dephasing affect the superconducting processors significantly. We further investigate the influence of any additional quantum optical effects in our platform relative to the classical one-hot encoding. From the process fidelity results it shows that the (classical) thermal noise in the phase-shifters dominates over other optical imperfections, such as mode mismatch and dark counts from single-photon sources. The benchmarking framework and experimental results demonstrate that photonic processors are strong contenders for near-term quantum algorithm deployment, particularly in hybrid variational contexts. This analysis is valuable not only for state and process tomography but also for a wide range of applications involving variational quantum circuit based algorithms.