Efficient Learning Algorithms for Noisy Quantum State and Process Tomography

TL;DR

A provably efficient and structure-agnostic learning framework for noisy $n$-qubit quantum circuits under generic noise with arbitrary noise strength and extends to quantum process tomography, obtaining a unified protocol applicable to both unital and non-unital channels.

Abstract

Efficiently characterizing large quantum states and processes is a central yet notoriously challenging task in quantum information science, as conventional tomography methods typically require resources that grow exponentially with system size. Here, we introduce a provably efficient and structure-agnostic learning framework for noisy $n$-qubit quantum circuits under generic noise with arbitrary noise strength. We first develop a sample-efficient learning algorithm for unital noisy quantum states. Building on this result, we extend the framework to quantum process tomography, obtaining a unified protocol applicable to both unital and non-unital channels. The resulting approach is input-agnostic and does not rely on assumptions about specific input distributions. Our theoretical analysis shows that both state and process learning require only polynomially many samples and polynomial classical post-processing in the number of qubits, while achieving near-unit success probability over ensembles generated by local random circuits. Numerical simulations of two-dimensional Hamiltonian dynamics further demonstrate the accuracy and robustness of the approach, including for structured circuits beyond the random-circuit setting assumed in the theoretical analysis. These results provide a scalable and practically relevant route toward characterizing large-scale noisy quantum devices, addressing a key bottleneck in the development of quantum technologies.

Figures (6)

• Figure 1: (a) Illustration of the noisy quantum state learning, wherein a trained model $\hat{\rho}$ is generated by leveraging the adaptive measurement result from the target noisy quantum state $\rho$. (b) Depiction of the noisy quantum process learning. Here, the noisy quantum process $\mathcal{C}(\gamma)$ represents a $d$-depth quantum circuit with noise strength $\gamma$, and $O$ represents an unknown measurement operator. The task is to learn a function $f$ such that $\left| f(\cdot)-{\rm Tr}[O\mathcal{C}(\gamma,\cdot)] \right|\leq\epsilon$ for all input quantum states $\rho_{\rm{in}}$, with efficient sample complexity. (c) Outline of the fundamental principle underlying our learning algorithm.(d) The proposed learning algorithm can be applied to the quantum error mitigation task, more details are provided in Appendix \ref{['sec:apply']}.
• Figure 2: (a) QST results for various numbers of qubits and $l'$. Each circuit is 20 layers accompanied by depolarizing noise of strength 0.02 and fixed $\theta_h=\frac{\pi}{4}$. The grid illustrates the $3\times5$$2D$ transverse field Ising model. (b). Learning of the $\rho$ generated by sweeping $\theta_h$ from $0$ to $\frac{\pi}{2}$; the circuit size $2\times 5$, $45$ layers, depolarizing noise strength 0.02. The heat-map shows a $25\times25$ sub-matrix of the matrix $\langle i|\left(\rho-\hat{\rho}\right)|j\rangle$ at $\theta_h=\frac{\pi}{2}$, where basis $i,j\in\{0,1\}^n$ (the full matrix in Appendix \ref{['sec:expr']}). (c). QPT results for different qubit numbers and $l'$, where the circuit depth is 5 layers and the depolarizing noise strength is 0.01. (d). QPT for the $2\times5$ system under 2 kinds of noise and other settings identical to c.
• Figure 3: (a) The procedure of the ZNE. (b) The numerical result of the ZNE-QEM using the proposed learning algorithm.
• Figure 4: The demonstration of a layer of the state preparation circuit for $n$ qubits. The structure consists of parameterized single-qubit rotations followed by a cyclic CNOT entangling layer.
• Figure 5: The experiment result of QPT where the input state is generated from the form as Fig \ref{['fig:spc']}. Set $l'=2$ and the process Eq. \ref{['equ:sim']} with $5$ layers is at the depolarizing noise strength $0.01$.

Theorems & Definitions (26)

• Theorem 1: Noisy Quantum State Learning
• Theorem 2: Noisy Quantum Process Learning
• Definition 1: Architecture, restatement of Ref. haferkampLinearGrowthQuantum2022
• Definition 2: Random Quantum Circuit, restatement of Ref.haferkampLinearGrowthQuantum2022
• Definition 3: Random noisy quantum circuit
• Definition 4: Schatten $\tau-$Norm
• Definition 5: The Squared Normalized Frobenius Norm
• Definition 6: Hamming Weight of Pauli Operators
• Lemma 1: Unified Representation of Noisy Quantum State
• Lemma 2: The unital noisy state truncation