Methods for self-consistent quantum tomography

Takanori Sugiyama, Shinpei Imori, Fuyuhiko Tanaka

Cyril Stark

Takanori Sugiyama, Shinpei Imori, Fuyuhiko Tanaka

Quantum tomography is a class of characterization methods frequently used in current experiments, but its standard protocols suffer from unreliability originated from preknowledge assumptions. Self-consistent quantum tomography is an approach to avoid the problem, which treats every quantum operation in a characterization experiment as unknown objects to be characterized. As compensation for the beneficence, it leads to a problem that its characterization results cannot be determined uniquely only from experimental data due to the existence of experimentally undetectable gauge degrees of freedom, and we need to introduce a criterion to fix the gauge. Here, we propose to use a regularization technique to fix the gauge. First, we derive a sufficient condition on a characterization experiment to obtain all information of objects to be characterized except for the gauge. Second, we propose a self-consistent data-processing method with regularization and physicality constraints. A careless use of regularization can lead to non-negligible bias on the characterization result. As a solution for the concern, we propose a concrete way to tune the strength of the regularization, and mathematically prove that the method provides characterization results that converge to the gauge-equivalence class of the quantum operations of interest at the limit of data going to infinity. The asymptotic convergence guarantees the reliability of the method. We also derive the asymptotic convergence rate, which would be optimal. These theoretical results hold for any finite-dimensional quantum systems. Finally, as its first numerical implementation, we show numerical results on one-qubit system, which confirm the theoretical results and prove that the method proposed is practical.

Xinlin He, Zetong Li, Congcong Zheng +3 more

Characterizing non-Markovian quantum dynamics is currently hindered by the self-inconsistency and high computational complexity of existing quantum comb tomography (QCT) methods. In this work, we propose a self-consistent framework that unifies the quantum comb, instrument set, and initial states into a single geometric entity, termed as the Comb-Instrument-State (CIS) set. We demonstrate that the CIS set naturally resides on a product Stiefel manifold, allowing the tomography problem to be solved via efficient unconstrained Riemannian optimization while automatically preserving physical constraints. Numerical simulations confirm that our approach is computationally scalable and robust against gate definition errors, significantly outperforming conventional isometry-based QCT methods. Our work indicates the potential to efficiently learn quantum comb with fewer computational resources.

Giacomo Mauro D’Ariano, Matteo G. A. Paris, Massimiliano Federico Sacchi

Marco Cattaneo, Matteo A. C. Rossi, Keijo Korhonen +4 more

We propose an estimation method for quantum measurement tomography (QMT) based on semidefinite programming (SDP) and discuss how it may be employed to detect experimental imperfections, such as shot noise and/or faulty preparation of the input states on near-term quantum computers. Moreover, if the positive operator-valued measure (POVM) we aim to characterize is informationally complete, we put forward a method for self-consistent tomography, i.e., for recovering a set of input states and POVM effects that is consistent with the experimental outcomes and does not assume any a priori knowledge about the input states of the tomography. Contrary to many methods that have been discussed in the literature, our approach does not rely on additional assumptions such as low noise or the existence of a reliable subset of input states.