Quantum Process Tomography with Digital Twins of Error Matrices

TL;DR

Quantum Process Tomography (QPT) is challenged by SPAM errors that induce self-consistency issues, biasing gate characterizations when using standard protocols. The authors introduce EM-QPT, which mitigates SPAM by reconstructing effective probes through the identity-process error matrix $\tilde{\chi}^I$, and further enhance it with ML-QPT by learning a digital twin of SPAM errors via a variational autoencoder (VAE) to generate statistically faithful error matrices. Numerical simulations and superconducting-qubit experiments show substantial fidelity improvements over standard QPT, with ML-QPT achieving higher precision and robustness, including resistance to anomalies. The work offers a scalable, practical toolkit for high-precision gate diagnostics and hardware benchmarking, with potential extensions to larger multi-qubit systems and ancilla-assisted QPT, and provides open-source code for community use.

Abstract

Accurate and robust quantum process tomography (QPT) is crucial for verifying quantum gates and diagnosing implementation faults in experiments aimed at building universal quantum computers. However, the reliability of QPT protocols is often compromised by faulty probes, particularly state preparation and measurement (SPAM) errors, which introduce fundamental inconsistencies in traditional QPT algorithms. We propose and investigate enhanced QPT for multi-qubit systems by integrating the error matrix in a digital twin of the identity process matrix, enabling statistical refinement of SPAM error learning and improving QPT precision. Through numerical simulations, we demonstrate that our approach enables highly accurate and faithful process characterization. We further validate our method experimentally using superconducting quantum gates, achieving at least an order-of-magnitude fidelity improvement over standard QPT. Our results provide a practical and precise method for assessing quantum gate fidelity and enhancing QPT on a given hardware.

Figures (13)

• Figure 1: Digital twin-enhanced quantum process tomography.(a) The variational autoencoder (VAE) consists of an encoder and a decoder built with deep neural networks. The input training data $\textbf{x}$ is mapped by the encoder into a parametric probability distribution $\mathcal{N}(\textbf{z}; \mu, \sigma^2)$. The latent variable $\textbf{z}$ is sampled from this distribution and used to reconstruct the output $\textbf{x}'$ through the decoder and a pre-designed quantum processing layer (QProcess). (b) The digital twin is applied to reconstruct the error matrix $\textbf{x}^* \to \chi^{I}_{\text{DT}}$ using a trained VAE, enhancing EM-QPT for a quantum process $\mathcal{E}$.
• Figure 2: Numerical results for EM-QPT of one- and two-qubit gates. (a) Average process infidelity of single-qubit gates using std-QPT (left panel) and EM-QPT (right panel), as a function of coherent and incoherent errors. (b, c) Fidelity for the evenly mixed error regime for single- and two-qubit gates, respectively. Solid curves and squares are the average results over $10^2$ gate samples, with shadow and error bar showing one standard deviation. The horizontal line is the statistical error $1/N_{\text{shot}}$ with the shot number $N_{\text{shot}}=10^4$.
• Figure 3: Experimental results for one- and two-qubit gates. (a) Performance of std-QPT (left) and EM-QPT (right) as a function of coherent and incoherent errors. (b) The points from (a) with $\lambda_1=\lambda_2$. Each data point in (a)/(b) is averaged over $15$/$10^2$ QPT experiments for all 24 single-qubit Clifford gates. (c) Infidelity distribution over $10^2$ QPT experiments of a CZ gate estimated with std-QPT, EM-QPT, and ML-QPT. The inner box plot indicates the median (white horizontal line) and the interquartile range (black box). Here, SPAM errors $\sim 6%$ and the RB fidelity is $99.23%$.
• Figure 4: Precision and sensitivity. (a) Top: infidelity distribution of EM-QPT and ML-QPT, based on the training dataset and digital twin, compared with std-QPT (dashed blue), reference (dot-dashed black), and RB (solid green). Bottom: CDFs used to calculate the $W_1$ distance from the reference distribution. (b) Top: standard deviations $\sigma(\mathbb{P}_{(i)})$ of the fidelity distributions based on EM-QPT and ML-QPT for a testing dataset comprising $10^2$ QPT experiments on X gates. Bottom: normalized $W_1$ distance of EM-/ML-QPT. The blue and green shaded regions indicate cases where the $W_1$ distance is smaller than that of std-QPT and RB, respectively. Parameters: $\xi_{\rm max} = 5, \xi_{\rm min} = 1, \xi_0 = 4$.
• Figure 5: Schematic diagram of QPT and implementation of noise channels. (a) Quantum process tomography with active reset (dashed rectangle) for the initialization of the qubit in the ground state $|0\rangle$. $U_1$ and $U_2$ are the sets of unitary rotations responsible for initial state preparation and measurement projectors, respectively. $\mathcal{E}$ is the process under study. (b) Incoherent noise channel, where the amplitude of the readout pulse (left graph) are scaled by a factor of $(1-\lambda_1)$, resulting a biased readout threshold (vertical green line in the right graph). Readout signal histograms visualize the separation between the peaks from the ground $|0\rangle$ (blue) and excited $|1\rangle$ (red) states for different $\lambda_1$ values. (c) Coherent noise channel, the amplitude $A_U$ of the unitary waveforms (left graph) is modified according to $A_U = A_0(1+r)$ with a uniformly sampled factor $r\sim \mathcal{U}(-\lambda_2,\lambda_2)$ (right graph).