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Efficient Self-Consistent Quantum Comb Tomography on the Product Stiefel Manifold

Xinlin He, Zetong Li, Congcong Zheng, Sixuan Li, Xutao Yu, Zaichen Zhang

TL;DR

The paper tackles self-inconsistency and computational burden in quantum comb tomography for non-Markovian dynamics. It introduces the Comb-Instrument-State (CIS) set, parameterized on a product Stiefel manifold, enabling unconstrained Riemannian optimization that respects CP, CPTP, and causal constraints. Using a Stiefel-adapted ADAM optimizer, the method achieves accurate, scalable reconstructions with fewer resources, outperforming isometry-based QCT in simulations. These results suggest a practical path toward efficient learning of quantum combs on systems with bounded memory.

Abstract

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.

Efficient Self-Consistent Quantum Comb Tomography on the Product Stiefel Manifold

TL;DR

The paper tackles self-inconsistency and computational burden in quantum comb tomography for non-Markovian dynamics. It introduces the Comb-Instrument-State (CIS) set, parameterized on a product Stiefel manifold, enabling unconstrained Riemannian optimization that respects CP, CPTP, and causal constraints. Using a Stiefel-adapted ADAM optimizer, the method achieves accurate, scalable reconstructions with fewer resources, outperforming isometry-based QCT in simulations. These results suggest a practical path toward efficient learning of quantum combs on systems with bounded memory.

Abstract

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.

Paper Structure

This paper contains 14 sections, 1 theorem, 25 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Manifold Geometry of Comb-Instrument-State Set
  3. Tomography Framework based on Riemannian Optimization
  4. Framework of self-consistent QCT
  5. Complexity
  6. Numerical Experiment Results
  7. Conclusions
  8. Proof of Theorem 1
  9. Parameterization of the comb.
  10. Parameterization of instruments.
  11. Parameterization of initial states.
  12. Riemannian gradient on Stiefel manifold
  13. Stiefel ADAM algorithm
  14. Complexity analysis

Key Result

Theorem 1

The CIS set $\Gamma =\{\mathcal{C}^{ (N)},\mathfrak{J},\mathfrak{S}\}$ has the parameterization of where $d_{a_{t+1}}\ge \mathrm{rank} (\Upsilon^{ (t+1)})$, $d_{e_{t,v,x}}\ge \mathrm{rank} (A^{ (t)}_{v,x})$, $d_{r_{u}}\ge\mathrm{rank} (\rho^{ (0)}_u)$, with orthogonal constraints $V^{ (t)\dagger}V^{ (t)}=\mathds{1}_{d_{\mathtt{i}_{t}}d_{{a_{t}}}}$, $J^{ (t)\dagger}_{v}J^{ (t)}_{v}=\mathds{1}_{d_{

Figures (3)

  • Figure 1: The general framework of self-consistent quantum comb tomography (QCT). There is an operational open quantum process based on a quantum comb (the green part). The initial state $\rho^{ (0)}$ may contain system–environment entanglement. The maps $\mathcal{A}^{ (t)}$ denote the experimentally accessible operations, and the comb governs the joint system–environment evolution. They jointly form a comb-instrument-state (CIS) set, with certain geometric properties: every component of the CIS set is on a (product) Stiefel manifold. The tomography framework contains 2 steps. After collecting the data from the quantum device, the second step is mainly based on the principle of manifold optimization. $\mathcal{F} (\Gamma)$ is the loss function, which is related to the intermediate state $\eta^{ (t)}_{\mathtt{i}}, \eta^{ (t)}_{\mathtt{o}}$ and the output probabilities $p^{ (t)}$ in the process. Finally, the core optimization steps are the calculation of the Riemannian gradient and retraction of the product manifold.
  • Figure 2: Numerical simulation results. (a) Comparison of loss values with the iQCT method. The bars show the average optimal loss under a certain quantum comb ancillary dimension. (b) Difference of Pauli transfer matrix (PTM) between practice and tomographic results. $\Delta_{PTM}$ is the average norm of the PTM difference of the whole instrument set.
  • Figure 3: Pauli transfer matrix (PTM) tomography results in numerical simulations. There are the knowledge and tomographic results of the entire state set and the selected instrument set. For the left part, every $\rho_u$ is related to an initial state. For the right part, the first line is the knowledge, and the second line is the results at $t=0$, corresponding to the instrument's practice. The third and fourth lines are the same.

Theorems & Definitions (2)

  • Definition 1: CIS set
  • Theorem 1