A two-stage solution to quantum process tomography: error analysis and optimal design

TL;DR

The paper introduces a two-stage analytical solution for quantum process tomography (QPT) that applies to both trace-preserving and non-trace-preserving dynamics by exploiting a tensor structure to achieve $O(MLd^2)$ time and $O(ML)$ storage. It provides an explicit analytical error upper bound and proposes optimal input states and measurement operators (e.g., SIC and MUB configurations) to minimize the bound and maximize robustness, with a closed-form estimation that bypasses heavy convex optimization. The approach is validated through numerical simulations and IBM quantum-device experiments, showing favorable scalability and performance compared to convex methods, especially for unitary-like processes. The results offer practical pathways to efficient and robust QPT, with a clear route to extending the framework to ancillary-based tomographic schemes and broader process classes.

Abstract

Quantum process tomography is a critical task for characterizing the dynamics of quantum systems and achieving precise quantum control. In this paper, we propose a two-stage solution for both trace-preserving and non-trace-preserving quantum process tomography. Utilizing a tensor structure, our algorithm exhibits a computational complexity of $O(MLd^2)$ where $d$ is the dimension of the quantum system and $ M $, $ L $ represent the numbers of different input states and measurement operators, respectively. We establish an analytical error upper bound and then design the optimal input states and the optimal measurement operators, which are both based on minimizing the error upper bound and maximizing the robustness characterized by the condition number. Numerical examples and testing on IBM quantum devices are presented to demonstrate the performance and efficiency of our algorithm.

Figures (7)

• Figure 1: Procedure for our TSS algorithm for QPT which has four steps. In Step 1, using \ref{['lsl']}, we reconstruct the parameterization matrix of all of the output states $\hat{A}$ directly from measurement data which is different from the QST in qci8022944. Here $\hat{A}$ is not needed be a physical estimate because it is only an intermediate product. In Step 2, we utilize \ref{['simls']}. Step 3 and Step 4 address the positive semidefinite constraint and partial trace constraint, respectively, by solving Problems \ref{['subproblem1']} and \ref{['subproblem2']}. These two steps together constitute our two-stage solution.
• Figure 2: Log-log plot of MSE versus the total number of copies $N_t$ using SIC states, MUB states, random states and natural basis states. (a) TP process; (b) non-TP process.
• Figure 3: An example of the IBM Composer. The input state is $\frac{I-\sigma_{z}}{2}\otimes \frac{I + \sigma_{x}}{2}$ and the measurement operators are $\frac{I \pm \sigma_{y}}{2} \otimes \frac{I \pm \sigma_{x}}{2}$.
• Figure 4: Log-log plot of MSE and infidelity versus the total number of copies $N_t$ for the CNOT process using simulation (Sim) by MATLAB and $ibmq\_qasm\_simulator$ (QASM).
• Figure 5: MSE and infidelity versus the total number of copies $N_t$ for the CNOT process using $ibmq\_quito$ 5-qubit system.