Query Learning Nearly Pauli Sparse Unitaries in Diamond Distance
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917375874433024 |
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| author | Honjani, Zahra Heidari, Mohsen |
| author_facet | Honjani, Zahra Heidari, Mohsen |
| contents | We study the problem of learning nearly $(s,ε)$-sparse unitaries, meaning that the Pauli spectrum is concentrated on at most $s$ components with at most $ε$ residual mass in Pauli $\ell_1$-norm. This class generalizes well-studied families, including sparse unitaries, quantum $k$-juntas, $2^k$-Pauli dimensional channels, and compositions of depth $O(\log\log n)$ circuits with near-Clifford circuits.
Given query access to an unknown nearly sparse unitary $U$, our goal is to efficiently (both in time and query complexity) construct a quantum channel that is close in diamond distance to $U$. We design a learning algorithm achieving this guarantee using $\tilde{O}(s^6/ε^4)$ forward queries to $U$, and running time polynomial in relevant parameters.
A key contribution is an efficient quantum algorithm that, given query access to an arbitrary unknown unitary $U$, estimates all Pauli coefficients (up to a shared global phase) whose magnitude exceeds a given threshold $θ$, extending existing sparse recovery techniques to general unitaries.
We also study the broader class of unitaries with bounded Pauli $\ell_1$-norm. For that class, we prove an exponential query lower bound $Ω(2^{n/2})$. We introduce a more relaxed accuracy metric which is the diamond distance restricted to a set of input states. Then, we show that, under this metric, unitaries with Pauli $\ell_1$-norm uniformly bounded by $L_1$ are learnable with $\tilde{O}(L_1^8/ε^{16})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_00203 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Query Learning Nearly Pauli Sparse Unitaries in Diamond Distance Honjani, Zahra Heidari, Mohsen Quantum Physics Information Theory We study the problem of learning nearly $(s,ε)$-sparse unitaries, meaning that the Pauli spectrum is concentrated on at most $s$ components with at most $ε$ residual mass in Pauli $\ell_1$-norm. This class generalizes well-studied families, including sparse unitaries, quantum $k$-juntas, $2^k$-Pauli dimensional channels, and compositions of depth $O(\log\log n)$ circuits with near-Clifford circuits. Given query access to an unknown nearly sparse unitary $U$, our goal is to efficiently (both in time and query complexity) construct a quantum channel that is close in diamond distance to $U$. We design a learning algorithm achieving this guarantee using $\tilde{O}(s^6/ε^4)$ forward queries to $U$, and running time polynomial in relevant parameters. A key contribution is an efficient quantum algorithm that, given query access to an arbitrary unknown unitary $U$, estimates all Pauli coefficients (up to a shared global phase) whose magnitude exceeds a given threshold $θ$, extending existing sparse recovery techniques to general unitaries. We also study the broader class of unitaries with bounded Pauli $\ell_1$-norm. For that class, we prove an exponential query lower bound $Ω(2^{n/2})$. We introduce a more relaxed accuracy metric which is the diamond distance restricted to a set of input states. Then, we show that, under this metric, unitaries with Pauli $\ell_1$-norm uniformly bounded by $L_1$ are learnable with $\tilde{O}(L_1^8/ε^{16})$. |
| title | Query Learning Nearly Pauli Sparse Unitaries in Diamond Distance |
| topic | Quantum Physics Information Theory |
| url | https://arxiv.org/abs/2604.00203 |