Learning Clifford-structured quantum unitaries and Hamiltonians

2026-08-10

AI Summary

This work addresses a key limitation in existing quantum learning methods, which rely on sparsity or locality assumptions in the Pauli basis and thus struggle with unitaries and Hamiltonians that are Pauli-dense yet possess underlying Clifford structure. The paper introduces a Clifford tomography protocol that requires no prior structural assumptions, efficiently approximating the optimal Clifford fidelity of an unknown unitary via query access. The approach generalizes to systems with bounded Clifford extensions and integrates Clifford decomposition, fidelity optimization, and agnostic quantum tomography. For the first time, it extends learnability from Pauli-sparse settings to densely structured Clifford scenarios, overcoming traditional sparsity constraints. The algorithm achieves high-fidelity reconstruction in polynomial time with complexity $\mathrm{poly}(n, (1/\varepsilon)^{\log(1/\varepsilon)})$, enabling efficient learning for any desired approximation error $\varepsilon$.

Abstract

Learning algorithms for structured quantum unitaries and Hamiltonians have primarily considered classes of processes that are local or sparse in the Pauli basis. We turn our attention to learning $n$-qubit quantum unitaries $U$ and Hamiltonians $H$, given query access to $U$ or the unitary evolution of $H$, that may be dense in the Pauli basis but still admit concise Clifford decompositions. Specifically, we consider unitaries (or Hamiltonians) of the form $U = \sum_i α_i C_i$ over Cliffords $C_i$ with bounded Clifford extent $\sum_i |α_i|$. To extract this Clifford structure, we introduce an agnostic tomography protocol for Clifford unitaries that given query access to an unknown unitary $U$ with optimal Clifford fidelity $\textsf{opt}$, outputs a Clifford unitary witnessing fidelity $\geq \textsf{opt} - \varepsilon$ for some error $\varepsilon > 0$, in time $\textsf{poly}(n,(1/\varepsilon)^{\log(1/\varepsilon)})$. We then apply this protocol to obtain tomography protocols for unitaries and Hamiltonians that have bounded Clifford extent. This extends learnability of Hamiltonians from those with sparse Pauli decompositions to those that are dense (i.e., has sparsity $Ω(2^n)$) in the Pauli basis but are Clifford structured.

Problem

• Clifford decomposition
• quantum unitaries
• Hamiltonians
• Pauli basis
• quantum learning

Innovation

• Clifford decomposition
• quantum unitary learning
• Hamiltonian tomography
• Clifford extent
• agnostic tomography