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The European Physical Journal Special Topics

Benchmarking regularisation methods for quantum process tomography on NISQ devices

I. J. David, Ilya Sinayskiy, Francesco Petruccione

Published Dec 1, 2023• 2 citations
Publisher

Abstract

Abstract Quantum process tomography (QPT) is a crucial tool for characterizing and validating quantum devices and quantum algorithms. However, the problem of finite sampling leads to an estimated process matrix which is non-positive semi-definite (non-PSD), which can yield a reconstructed quantum channel that is non-physical. To address this problem, various methods have been proposed to correct the issue of finite sampling in the estimation of the process matrix. In this work, we perform a comparison of regularisation methods that will be used to tackle the problem of finite sampling in QPT. For this comparison we simulate some common single qubit quantum channels. We use two metrics, the minimum eigenvalue of the Choi matrix and the fidelity, to compare the effectiveness of these methods. Our results show that the spectral transformations perform the best overall in dealing with finite sampling present in reconstructing the quantum channel in the NISQ era.

DOI
10.1140/epjs/s11734-023-01067-1

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Abstract Quantum process tomography (QPT) is a crucial tool for characterizing and validating quantum devices and quantum algorithms. However, the problem of finite sampling leads to an estimated process matrix which is non-positive semi-definite (non-PSD), which can yield a reconstructed quantum channel that is non-physical. To address this problem, various methods have been proposed to correct the issue of finite sampling in the estimation of the process matrix. In this work, we perform a comparison of regularisation methods that will be used to tackle the problem of finite sampling in QPT. For this comparison we simulate some common single qubit quantum channels. We use two metrics, the minimum eigenvalue of the Choi matrix and the fidelity, to compare the effectiveness of these methods. Our results show that the spectral transformations perform the best overall in dealing with finite sampling present in reconstructing the quantum channel in the NISQ era.
Quantum Process Tomography (QPT) is a fundamental technique to characterize quantum operations. However, its precision gets hampered while running in real quantum hardware due to inherent noise. In our study, we propose a post-processing method using principal component analysis (PCA) to improve the fidelity of noisy gates. We first implemented the standard QPT on various gates ranging from single-qubit to three-qubit in simulation under depolarizing error scenarios and on Fake IBM_Brisbane and ran single and two-qubit gates (H, CX, and SQSCZ) on real quantum hardware (IBM_Brisbane); the three-qubit CCX is evaluated in simulation only. After that, the reconstructed Choi matrix is decomposed into real and imaginary components. Then, by applying a PCA-based approach, we show steady improvements in a variety of quantitative performance indicators in addition to process fidelity while also maintaining the physical constraints of the Choi matrix. Additionally, PCA performed better or similarly when compared to CVX-based maximum-likelihood reconstruction and linear inversion. Although PCA improves QPT reconstructions, it is unable to address the fundamental scalability issue of QPT since it cannot overcome the exponential scaling with qubit number.
Quantum process tomography (QPT) protocols are often designed by lifting quantum state tomography (QST) protocols through the Choi-Jamiolkowski isomorphism, at the cost of an extra step enforcing the trace-preserving (TP) constraint. Existing approaches fall into two classes. The folklore partial normalization, which sandwiches the density estimate between inverse square roots of its input marginal, has a closed form but is used as a heuristic, its only analyses being asymptotic and losing a factor polynomial in the dimension. Projection-based methods are principled but have no closed forms and scale poorly with dimension. We resolve this dichotomy by showing that the partial normalization is the exact projection onto the Choi states of quantum channels in fidelity (equivalently in Bures or purified distance). Consequently, any QST algorithm with guarantees in fidelity lifts to a QPT algorithm that inherits its sample complexity, through a single closed-form congruence that preserves the Choi rank and is computed in Kraus form without any eigendecomposition of Choi dimension. We lift two single-copy protocols and prove that the non-adaptive one (Fidelity Projected Least Squares) recovers the Choi rank and reaches infidelity ε\varepsilonε from a number of channel uses proportional to 1/ε1/\varepsilon1/ε. Numerically, for channels of low Choi rank, it improves on the state-of-the-art projected least squares in accuracy, and by orders of magnitude in the time and memory of the TP-regularization step and of the returned estimate.
Quantum Process Tomography (QPT) is one of the most important task of checking the functionality of quantum information processing devices. However, there is a significant difficulty that the required resources grow scaling exponentially with the number of qubits during the QPT. Recently, a compressed sensing QPT (CS QPT) is proposed that can reduce the required resource significantly. But the work did not consider the outlier noises caused by the disturbance of the system or channel. In this paper, we propose a joint sparse coding QPT aiming for the reconstruction of the two or more similar quantum channel corrupted by sparse outlier noises simultaneously. The simulations verify that the method is robust to outlier noises and the obtained reconstruction matrix is superior to the QPT using compressed sensing method.
Abstract Quantum algorithms for simulating quantum systems provide a clear and provable advantage over classical algorithms in fault-tolerant settings. There is also interest in quantum algorithms and their implementation in Noisy Intermediate Scale Quantum (NISQ) settings. In these settings, various noise sources and errors must be accounted for when executing any experiments. Recently, NISQ devices have been verified as versatile testbeds for simulating open quantum systems and have been used to simulate simple quantum channels. Our goal is to solve the more complicated problem of simulating convex mixtures of single qubit Pauli channels on NISQ devices. We consider two specific cases: mixtures of Markovian channels that result in a non-Markovian channel (M + M = nM) and mixtures of non-Markovian channels that result in a Markovian channel (nM + nM = M). For the first case, we consider mixtures of Markovian single qubit Pauli channels; for the second case, we consider mixtures of Non-Markovian single qubit depolarising channels, which is a special case of the single qubit Pauli channel. We show that efficient circuits, which account for the topology of currently available devices and current levels of decoherence, can be constructed by heuristic approaches that reduce the number of CNOT gates used in our circuit. We also present a strategy for regularising the process matrix so that the process tomography yields a completely positive and trace-preserving (CPTP) channel. Key points This work simulates the convex mixtures of single qubit Markovian and non-Markovian quantum channels on NISQ devices provided by the IMBQE. The circuits used to implement the channels take into account the topolgy of the quantum device used as well as the number of CNOT gates used. We present a strategy for regularising the process matrix to ensure the quantum process tomography yields a CPTP channel. Something that is not correctly implemented in Qiskit. A method is outlined for finding mixtures of non-Markovian depolarising channels that yield a Markovian depolarising channel. It is also shown that, one cannot convexly mix two Markovian depolarising channels that leads to a non-Markovian depolarising channel.
Quantum Process Tomography (QPT) methods aim at identifying, i.e. estimating, a quantum process. QPT is a major quantum information processing tool, since it especially allows one to experimentally characterize the actual behavior of quantum gates, that may be used as the building blocks of quantum computers. We here consider unitary (possibly dense, i.e. without sparsity constraints) processes, which corresponds to isolated systems. Moreover, we develop a QPT method applicable to a significant number of qubits and hence to a high state space dimension, which allows one to tackle more complex problems. Using the unitarity of the process allows us to propose a method that first achieves part of QPT by performing an eigenanalysis of the estimated density matrix of a process output. We then complete the QPT by using another state differently. The relevance of our method is validated by means of simulations.
ABSTRACT Quantum process tomography (QPT) plays a central role in characterizing quantum gates and circuits, diagnosing quantum devices, calibrating hardware, and supporting quantum error correction. However, conventional QPT methods face challenges related to scalability and sensitivity to noise. In this work, we propose a QPT framework based on quantum compilation, which represents quantum processes using optimized Kraus operators and Choi matrices. By formulating QPT as a compilation and optimization problem, our approach significantly reducing measurement and computational overhead while maintaining reconstruction accuracy. We benchmark the method using numerical simulations of Haar‐random unitary gates and demonstrate a reliable process reconstruction. We further apply the framework to dephasing channels with both time‐homogeneous and time‐inhomogeneous noise, as well as to depolarizing and amplitude‐damping channels, where stable performance is observed across different noise regimes. These results indicate that quantum compilation–based QPT can serve as a practical alternative to standard QPT methods for quantum process characterization and device validation.