Fast Quantum Process Tomography via Riemannian Gradient Descent

TL;DR

This work addresses the challenge of scalable quantum process tomography under physical CPTP constraints by recasting QPT as a Riemannian optimization problem on the Stiefel manifold of Kraus stacks. It develops a gradient-descent framework with a Cayley-based retraction and adapts Adam-style momentum to operate on manifolds, enabling fast, data-driven tomography even with incomplete measurements. The authors provide an open-source GPU-accelerated implementation and validate the approach through simulations (random channels and a harmonic oscillator with a 64-dimensional space) and hardware experiments on IBM devices, showing accurate reconstruction with favorable runtimes. Overall, the method offers a practical path to high-fidelity QPT in larger quantum systems, with potential extensions to tensor networks and uncertainty quantification.

Abstract

Constrained optimization plays a crucial role in the fields of quantum physics and quantum information science and becomes especially challenging for high-dimensional complex structure problems. One specific issue is that of quantum process tomography, in which the goal is to retrieve the underlying quantum process based on a given set of measurement data. In this paper, we introduce a modified version of stochastic gradient descent on a Riemannian manifold that integrates recent advancements in numerical methods for Riemannian optimization. This approach inherently supports the physically driven constraints of a quantum process, takes advantage of state-of-the-art large-scale stochastic objective optimization, and has superior performance to traditional approaches such as maximum likelihood estimation and projected least squares. The data-driven approach enables accurate, order-of-magnitude faster results, and works with incomplete data. We demonstrate our approach on simulations of quantum processes and in hardware by characterizing an engineered process on quantum computers.

Figures (7)

• Figure 1: Overview of proposed quantum process tomography.
• Figure 2: Pauli-Liouville representation of simulated "noise." A channel with Pauli noise $K_\mathrm{noise} = [\sqrt{p}I\otimes I, \sqrt{(1-p)}\sigma_z\otimes\sigma_z]$ is applied after a unitary operation $U$. Process tomography is performed to fit the noise+$U$ channel from measurement data. The "noise" part defined as $\hat{P}\circ U^{-1} - \mathrm{I}$. (a) $p=0.1$, (b) $p=0.25$, and (c) $p=0.25$ with measurement data perturbed by a normal distribution with $\epsilon=0.01$.
• Figure 3: Iterative-based Cayley transform has better performance (time and accuracy) with respect to traditional Cayley transform. The accuracy (dashed lines) are computed by solving the exact exponential map and taking the norm difference with respect to approximated retraction.
• Figure 4: (a) Mean loss of the optimization as noise $\epsilon$ is increased for different channel ranks. Reconstruction error saturates as a power curve with respect to noise $\epsilon$ (note the log-log scale.) (b) Channel fidelity at $\epsilon=0.01$ for a reconstructed quantum channel with varying Kraus rank as well as access to only a fraction of data. Kraus rank is the dominant factor for fidelity, while the fraction of data is responsible for fidelity margin.
• Figure 5: Wigner functions of a probe state undergoing ideal simulation and tomographic reconstruction with a Fock space $N=64$. The data are measured values of displaced parity operator $\Pi_d(\beta)$, but with fewer sample points $\beta$ to mimic an experimental setup.