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Optimal depth and a novel approach to variational quantum process tomography

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

Published Dec 8, 2024• 0 citations
Publisher

Abstract

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.

DOI
10.1109/globecom52923.2024.10901365

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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.
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 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.
Full characterization of quantum states and processes is a fundamental requirement for verification and benchmarking of quantum devices. It has been realized in systems with few components, but for larger systems it becomes unfeasible because of the exponential growing with the system size of the number of measurements and the amount of computational power required to process them. A new approach for quantum state tomography and quantum process tomography, requiring a limited number of measurements, has been recently introduced by Maciel et al. [Quantum Inf. Comput. 12, 0442 (2012); Int. J. Mod. Phys. C 22, 1361 (2011).], namely variational quantum-process tomography (VQT). Here we adopt the VQT approach for the experimental characterization of two-qubit quantum processes and compare the reconstructed maps with those obtained by standard tomographic methods. The results demonstrate the high performance of this approach and propose VQT as a powerful alternative to the standard quantum process tomography.
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.