Learning Clifford-structured quantum unitaries and Hamiltonians
2026-08-10 • Computational Complexity
Computational Complexity
AI summaryⓘ
The authors study how to learn certain complex quantum operations called unitaries and Hamiltonians that can be very complicated when described using the usual building blocks but have simpler underlying structures related to Clifford operations. They develop a new method to reconstruct these operations by finding the closest simpler Clifford process, even when the original operation looks very dense or complicated. Their approach works efficiently and extends previous learning methods that were limited to simpler, sparse cases. This means they can now learn a broader class of quantum processes that were previously hard to analyze.
quantum unitaryHamiltonianClifford groupClifford decompositionquantum tomographyPauli basisquantum learning algorithmsClifford fidelitysparsityquantum complexity
Authors
Arkopal Dutt, Dale Jacobs, John Jeang, Saeed Mehraban, Vladimir Podolskii
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.