Simultaneous reconstruction of quantum process and noise via corrupted sensing

TL;DR

This work addresses scalable quantum process tomography under corrupted measurements by introducing a corrupted-sensing framework that jointly reconstructs the quantum process and sparse noise. It develops two complementary representations: a Choi-state approach with a generalized restricted isometry property (GRIP) for the extended sensing matrix, and a process-matrix (Chi) approach that recovers sparse process matrices under a measurement model $\mathbf{y}=\Phi\bm{\chi}+\mathbf{v}+\mathbf{z}$. The authors prove GRIP conditions and propose convex estimators that enforce physical constraints (positive semidefinite Chi, trace-preserving conditions) to enable simultaneous recovery, then validate the method with extensive numerical simulations on 2-, 3-, and 4-qubit gates. The findings demonstrate significant reductions in experimental configurations while maintaining high-fidelity reconstructions, underscoring the approach's practical relevance for scalable quantum characterization. The work also highlights future directions, including explicit error bounds, alternative analytical tools for GRIP, and extensions to measurement tomography.

Abstract

Quantum processes, including quantum gates and channels, are integral to various quantum information tasks, making the efficient characterization of these processes and their underlying noise critically important. Here, we propose a framework for quantum process tomography in the presence of corrupted noise that is able to simultaneously reconstruct the process and corrupted noise. Firstly, within the Choi-state representation, we derive the corresponding generalized restricted isometry property and demonstrate the simultaneous reconstruction of various quantum gates under sparse noise. Moreover, in comparison with the Choi-state scheme, the process-matrix representation is employed to simultaneously reconstruct sparse noise and a broader range of target quantum gates. Our results demonstrate that significant reduction in experimental configurations is achievable even under corrupted noise.

Figures (9)

• Figure 1: Schematic procedure of the corrupted sensing QPT based on the Choi-state representation. The maximally entangled state $\hbox{$| \Psi \rangle$}$ is prepared and sent into the quantum process $\mathcal{E}$. The output Choi state $\rho_{\mathcal{E}}$ is then measured to obtain the noisy data ${\bf y} = \{y_1, \ldots, y_m\}$. We randomly select $m$ Pauli observables $\{\mathcal{W}_1, \ldots, \mathcal{W}_m\}$, with $m \leq d^4$. By using the recovery algorithm, the Choi state $\rho_{\mathcal{E}}$ and the measurement noise ${\bf v}$ can be reconstructed simultaneously, and ${\bf z}$ denotes any other potential noise.
• Figure 2: Fidelity $F(\hat{\rho}_{\mathcal{E}_{\rm cn}}, \rho_{\mathcal{E}_{\rm cn}})$ and MSE $T_{\text{MSE}}$ as functions of the number of sampled Pauli observables $\mathcal{W}$ (denoted by $m$). The error bars are obtained from $100$ runs of tomography, with each run selecting $m$ Pauli observables randomly. The purple and blue points represent the fidelity between the reconstructed Choi state and true Choi state of the CNOT gate, for sparsity ratios ${\eta = 0.05}$ and $0.1$, respectively. The red and green points show the MSE between the reconstructed sparse Gaussian noise and true sparse Gaussian noise, for sparsity ratios ${\eta = 0.05}$ and $0.1$, respectively. The Gaussian noise has a mean of zero, a standard deviation of $1$. The regularization parameters are chosen as ${\tau_1 = 0.01m, \tau_2 = 10^{-2}}$.
• Figure 3: Fidelity $F(\hat{\rho}_{\mathcal{E}_{\rm Tof}}, \rho_{\mathcal{E}_{\rm Tof}})$ and MSE $T_{\text{MSE}}$ as functions of the number of sampled Pauli observables $\mathcal{W}$ (denoted by $m$). The error bars are obtained from $50$ runs of tomography, with each run selecting $m$ Pauli observables randomly. The purple and blue points represent the fidelity between the reconstructed Choi state and true Choi state of Toffoli gate, for sparsity ratios ${\eta = 0.05}$ and $0.1$, respectively. The red and green points show the MSE between the reconstructed sparse Gaussian noise and true sparse Gaussian noise, for sparsity ratios ${\eta = 0.05}$ and $0.1$, respectively. The Gaussian noise has a mean of zero, a standard deviation of $1$. The regularization parameters are chosen as ${\tau_1 = 0.01m, \tau_2 = 10^{-2}}$.
• Figure 4: Schematic procedure of the corrupted sensing QPT based on the process-matrix representation. Consider preparing different initial states $\rho_o$, applying the process $\mathcal{E}$, and then measuring the outputs using a set of observables $M_o$ to get the noisy data ${\bf y}\!:\{y_1,\cdots,y_m\}$. Here, we select $m$ configurations $(\rho_o,M_o), o= 1,\cdots, m\leq kl$ randomly. By using the recovery algorithm, the process matrix $\chi$ and the measurement noise ${\bf v}$ can be reconstructed simultaneously, and ${\bf z}$ denotes any other potential noise.
• Figure 5: Fidelity $F(\hat{\chi}_{\text{cn}},\chi_{\text{cn}})$ and MSE $T_{\text{MSE}}$ as functions of the number of configurations $m$. The error bars are obtained from $100$ runs of tomography, with each run selecting $m$ combinations randomly. The blue points represent the fidelity between the reconstructed process matrix and true process matrix of the CNOT gate. The green points show the MSE between the reconstructed sparse Gaussian noise and true sparse Gaussian noise. The Gaussian noise has a mean of zero, a standard deviation of $1$, and a sparsity level of ${s = \lfloor 0.1m \rfloor}$. The regularization parameters are chosen as ${\mu_1 = 10^{-5}, \mu_2 = 10^{-3}}$.

Theorems & Definitions (6)

• Definition 1
• Proposition 1
• proof
• Proposition 2
• proof
• Remark 1