Efficiently Learning Global Quantum Channels with Local Tomography

TL;DR

A local-to-global reconstruction framework for one-dimensional multi-qubit states and channels and it is proved that under this assumption, the required number of samples scales polynomially with the system size and the desired global reconstruction error.

Abstract

Scalable characterization of quantum processors is crucial for mitigating noise and imperfections. While randomized measurement protocols enable efficient access to local observables, inferring a globally consistent description of multi-qubit processes remains challenging. Here we introduce a local-to-global reconstruction framework for one-dimensional multi-qubit states and channels. The method is efficient provided that correlations, as quantified by the conditional mutual information, decay exponentially. In particular, we prove that under this assumption, the required number of samples scales polynomially with the system size and the desired global reconstruction error. Our approach is based on combining local shadow tomography with locally optimal recovery maps obtained by convex optimization. We supplement these rigorous guarantees by studying the performance of the protocol numerically for a system evolving under a local Lindbladian and a noisy, shallow circuit. By employing a tensor networ representation, we reconstruct channels acting on up to 50 qubits and accurately recover global diagnostics such as the process fidelity, the Choi state purity, and Pauli-weight-resolved process matrix elements. Our work thus extends the powerful toolbox local shadow tomography to scalable channel characterization with access to global properties.

Figures (3)

• Figure 1: Protocol for process matrix reconstruction. (a) In each experimental shot, we prepare $|0\rangle^{\otimes n}$, apply independently sampled single-qubit gates $U_1,\ldots,U_n$ before the unknown channel $\Lambda$ and $V_1,\ldots,V_n$ prior to measurement in the computational basis. (b) The measurements are used to estimate the reduced Choi states $\hat{J}_i$ on small subsystems. From this information, we compute the recovery maps $\mathcal{R}_i$, acting on sites $\{i-w+1, \ldots, i\}$, that incrementally append the new site $i+1$. The intermediate states supported on sites $\{1\, \ldots, i\}$ are represented as MPOs and are denoted by $K_i$.
• Figure 2: Reconstruction of Pauli-weight-resolved diagonal process matrix elements in a Heisenberg chain with $n=50$ qubits. (a) For Pauli weights $k=0,1,2$, the quantity $G_k = \sum_{ \boldsymbol{\alpha} \leq k} \chi_{\boldsymbol{\alpha}\boldsymbol{\alpha}}$ of the exact channel (solid lines) agrees well with the reconstructed channel obtained from our optimization procedure (markers). Panel (b) The corresponding relative reconstruction error $\Delta_R = |G_k^{\mathrm{E}} - G_k^{\mathrm{R}}|$ is shown, where $G_k^{\mathrm{E}}$ denotes the exact value and $G_k^{\mathrm{R}}$ represents the reconstructed value. Panels (a) and (b) assume perfect local tomography without noise. (c) and (d) show the relative reconstruction error $\Delta_R$ due to local noise, which is quantified by the error $\eta$ in the reduced Choi states. In (c), the error is evaluated at a fixed evolution time of $t=10^{-2}$, while (d) shows the result at $t=5\times 10^{-2}$.
• Figure 3: Global reconstruction of a noisy shallow circuit with $n=20$ qubits. (a) Structure of the channel $\Lambda_{U,\epsilon,\gamma} = \bigl(\bigotimes_j \mathcal{D}_\gamma^{(j)}\bigr) \circ U_\epsilon \circ U$, where $U$ is a single layer of nearest-neighbour two-qubit gates, $U_\epsilon$ denotes a coherent perturbation, and $\mathcal{D}_\gamma$ is a single-qubit dephasing channel. (b) Process fidelity with respect to the ideal unitary circuit $U$, showing $F(\Lambda_{U,\epsilon,\gamma},\Lambda_U)$ (solid line) for the exact noisy channel and $F(\hat{\Lambda}_{U,\epsilon,\gamma},\Lambda_U)$ (markers) for the optimization-based reconstruction, as a function of $\gamma$. (c) Purity of the Choi state $\mathcal{P}\bigl(J(\Lambda_{U,\epsilon,\gamma})\bigr)$ (solid lines) together with the purity of the reconstructed channel $\mathcal{P}\bigl(J(\hat{\Lambda}_{U,\epsilon,\gamma})\bigr)$(markers), as a function of $\gamma$.

Theorems & Definitions (23)

• Definition 1: Exponentially decaying CMI
• Theorem 1: Trace norm bound for the Choi state
• Corollary 2: End-to-end sample complexity
• Definition 2: Exponentially decaying CMI
• Definition 3: Markov entropy
• Lemma 3
• proof
• Definition 4: Sequentially generated states
• Definition 5: Locally optimal recovery
• Lemma 4: Continuity of the CMI