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arXiv:2604.00203

Query Learning Nearly Pauli Sparse Unitaries in Diamond Distance

Zahra Honjani, Mohsen Heidari

quant-phcs.ITSubmitted Mar 31, 2026• Announced Apr 2, 2026• 0 citations
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Abstract

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 $\tilde{O}(L_1^8/ε^{16})$.

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We study the problem of learning nearly (s,ε)(s,ε)(s,ε)-sparse unitaries, meaning that the Pauli spectrum is concentrated on at most sss components with at most εεε residual mass in Pauli ℓ1\ell_1ℓ1​-norm. This class generalizes well-studied families, including sparse unitaries, quantum kkk-juntas, 2k2^k2k-Pauli dimensional channels, and compositions of depth O(log⁡log⁡n)O(\log\log n)O(loglogn) circuits with near-Clifford circuits.
Given query access to an unknown nearly sparse unitary UUU, our goal is to efficiently (both in time and query complexity) construct a quantum channel that is close in diamond distance to UUU. We design a learning algorithm achieving this guarantee using O~(s6/ε4)\tilde{O}(s^6/ε^4)O~(s6/ε4) forward queries to UUU, and running time polynomial in relevant parameters.
A key contribution is an efficient quantum algorithm that, given query access to an arbitrary unknown unitary UUU, estimates all Pauli coefficients (up to a shared global phase) whose magnitude exceeds a given threshold θθθ, extending existing sparse recovery techniques to general unitaries.
We also study the broader class of unitaries with bounded Pauli ℓ1\ell_1ℓ1​-norm. For that class, we prove an exponential query lower bound Ω(2n/2)Ω(2^{n/2})Ω(2n/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 ℓ1\ell_1ℓ1​-norm uniformly bounded by L1L_1L1​ are learnable with O~(L18/ε16)\tilde{O}(L_1^8/ε^{16})O~(L18​/ε16).
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 nnn-qubit quantum unitaries UUU and Hamiltonians HHH, given query access to UUU or the unitary evolution of HHH, that may be dense in the Pauli basis but still admit concise Clifford decompositions. Specifically, we consider unitaries (or Hamiltonians) of the form U=∑iαiCiU = \sum_i α_i C_iU=∑i​αi​Ci​ over Cliffords CiC_iCi​ with bounded Clifford extent ∑i∣αi∣\sum_i |α_i|∑i​∣αi​∣. To extract this Clifford structure, we introduce an agnostic tomography protocol for Clifford unitaries that given query access to an unknown unitary UUU with optimal Clifford fidelity opt\textsf{opt}opt, outputs a Clifford unitary witnessing fidelity ≥opt−ε\geq \textsf{opt} - \varepsilon≥opt−ε for some error ε>0\varepsilon > 0ε>0, in time poly(n,(1/ε)log⁡(1/ε))\textsf{poly}(n,(1/\varepsilon)^{\log(1/\varepsilon)})poly(n,(1/ε)log(1/ε)). 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(2n)Ω(2^n)Ω(2n)) in the Pauli basis but are Clifford structured.
We propose several algorithms for learning unitary operators from quantum statistical queries with respect to their Choi-Jamiolkowski state. Quantum statistical queries capture the capabilities of a learner with limited quantum resources, which receives as input only noisy estimates of expected values of measurements. Our approach leverages quantum statistical queries to estimate the Fourier mass of a unitary on a subset of Pauli strings, generalizing previous techniques developed for uniform quantum examples. Specifically, we show that the celebrated quantum Goldreich-Levin algorithm can be implemented with quantum statistical queries, whereas the prior version of the algorithm involves oracle access to the unitary and its inverse. As an application, we prove that quantum Boolean functions with constant total influence or with constant degree are efficiently learnable in our model. Moreover, we prove that O(log⁡n) -juntas are efficiently learnable and constant-depth circuits are learnable query-efficiently with quantum statistical queries. On the other hand, all previous algorithms for these tasks demand significantly greater resources, such as oracle access to the unitary or direct access to the Choi-Jamiolkowski state. We also demonstrate that, despite these positive results, quantum statistical queries lead to an exponentially larger query complexity for certain tasks, compared to separable measurements to the Choi-Jamiolkowski state. In particular, we show an exponential lower bound for learning a class of phase-oracle unitaries and a double exponential lower bound for testing the unitarity of channels. Taken together, our results indicate that quantum statistical queries offer a unified framework for various unitary learning tasks, with potential applications in quantum machine learning, many-body physics and benchmarking of near-term devices.
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In the study of quantum process tomography, it has remained open how to learn a unitary channel in diamond distance with strictly parallel queries as efficiently as with sequential queries. Sequential queries allow one to exploit adaptivity to refine the estimate and adjust the learning strategy accordingly, whereas such adjustments are impossible for parallel queries. From the perspective of higher-order quantum operations, any parallel learning strategy can be simulated by a sequential one, whereas the converse does not hold in general, suggesting that sequential learning strategies are potentially more powerful. Contrary to this intuition, we present a new protocol for learning an unknown ddd-dimensional unitary channel to within ε\varepsilonε in diamond distance using O(d2/ε)\mathcal{O}(d^2/\varepsilon)O(d2/ε) parallel queries, matching the lower bound presented in [Haah, Kothari, O'Donnell, and Tang, FOCS '23]. Our protocol thus closes the gap between parallel and sequential strategies for unitary channel tomography. As applications, it achieves optimal query complexity for boundary-regime quantum channel tomography and improves the query efficiency of the best-known protocol for tomography of fermionic linear optics.