Learning Quantum Processes and Hamiltonians via the Pauli Transfer Matrix
Matthias C. Caro
TL;DR
The paper demonstrates that learning an unknown $n$-qubit quantum process via Pauli transfer matrices is exponentially easier with a quantum memory than without one. By recasting PTM learning as shadow tomography on the Choi state and employing Pauli shadow tomography, it achieves $O\left((n+\log(1/\delta))/\varepsilon^4\right)$ copies to learn all PTM entries, while memoryless strategies require $\Omega\left(4^n/\varepsilon^2\right)$ queries. It extends these results to predicting expectation values under sparsity constraints, and shows how PTM learning serves as a subroutine for learning arbitrary Hamiltonians via short-time dynamics and polynomial interpolation. The work situates these results within shadow tomography, classical shadows, and channel-learning literature, and provides explicit lower bounds for memoryless models under channel-promises such as doubly-stochasticity and entanglement-breaking behavior. Overall, the results highlight a robust quantum memory–enabled advantage for learning highly complex quantum dynamics, with practical implications for efficiently characterizing quantum processes and Hamiltonians on near-term devices.
Abstract
Learning about physical systems from quantum-enhanced experiments, relying on a quantum memory and quantum processing, can outperform learning from experiments in which only classical memory and processing are available. Whereas quantum advantages have been established for a variety of state learning tasks, quantum process learning allows for comparable advantages only with a careful problem formulation and is less understood. We establish an exponential quantum advantage for learning an unknown $n$-qubit quantum process $\mathcal{N}$. We show that a quantum memory allows to efficiently solve the following tasks: (a) learning the Pauli transfer matrix of an arbitrary $\mathcal{N}$, (b) predicting expectation values of bounded Pauli-sparse observables measured on the output of an arbitrary $\mathcal{N}$ upon input of a Pauli-sparse state, and (c) predicting expectation values of arbitrary bounded observables measured on the output of an unknown $\mathcal{N}$ with sparse Pauli transfer matrix upon input of an arbitrary state. With quantum memory, these tasks can be solved using linearly-in-$n$ many copies of the Choi state of $\mathcal{N}$, and even time-efficiently in the case of (b). In contrast, any learner without quantum memory requires exponentially-in-$n$ many queries, even when querying $\mathcal{N}$ on subsystems of adaptively chosen states and performing adaptively chosen measurements. In proving this separation, we extend existing shadow tomography upper and lower bounds from states to channels via the Choi-Jamiolkowski isomorphism. Moreover, we combine Pauli transfer matrix learning with polynomial interpolation techniques to develop a procedure for learning arbitrary Hamiltonians, which may have non-local all-to-all interactions, from short-time dynamics. Our results highlight the power of quantum-enhanced experiments for learning highly complex quantum dynamics.