Query Learning Nearly Pauli Sparse Unitaries in Diamond Distance

• URL: http://arxiv.org/abs/2604.00203v1
• Date: Tue, 31 Mar 2026 20:13:10 GMT
• Title: Query Learning Nearly Pauli Sparse Unitaries in Diamond Distance
• Abstract summary: This class generalizes well-studied families, including sparse unitaries, quantum $k$-juntas, $2k$-Pauli dimensional channels, and compositions of depth $O(loglog n)$ circuits.
• Score: 6.852394426719304
• License: http://creativecommons.org/licenses/by/4.0/
• 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 $\tilde{O}(L_1^8/ε^{16})$.

Related papers

• Query-Optimal Estimation of Unitary Channels via Pauli Dimensionality [0.8594140167290097]
  We study process tomography of unitary channels whose Pauli spectrum is supported on a small subgroup.<n>We present an algorithm that achieves this using $O(2k/epsilon)$ queries.<n>We also give a computationally efficient algorithm for learning compositions of depth-$O(log n)$ circuits with near-Clifford circuits.
  arXiv  Detail & Related papers  (2025-09-30T18:40:42Z)
• Testing and learning structured quantum Hamiltonians [4.137418441736385]
  We consider the problems of testing and learning an unknown $n$qubit Hamiltonian $H$ from queries to its evolution operator $e-iHt$ under the normalized Frobenius norm.
  arXiv  Detail & Related papers  (2024-10-31T17:54:13Z)
• Learning low-degree quantum objects [5.2373060530454625]
  We show how to learn low-degree quantum objects up to $varepsilon$-error in $ell$-distance. Our main technical contributions are new Bohnenblust-Hille inequalities for quantum channels and completely boundedpolynomials.
  arXiv  Detail & Related papers  (2024-05-17T17:36:44Z)
• Basic quantum subroutines: finding multiple marked elements and summing numbers [1.1265248232450553]
  We show how to find all $k$ marked elements in a list of size $N$ using the optimal number $O(sqrtN k)$ of quantum queries.
  arXiv  Detail & Related papers  (2023-02-20T19:11:44Z)
• Optimal algorithms for learning quantum phase states [8.736370689844682]
  We show that the sample complexity of learning an unknown degree-$d$ phase state is $Theta(nd)$ if we allow separable measurements. We also consider learning phase states when $f$ has sparsity-$s$, degree-$d$ in its $mathbbF$ representation.
  arXiv  Detail & Related papers  (2022-08-16T17:15:06Z)
• Exponential Separation between Quantum and Classical Ordered Binary Decision Diagrams, Reordering Method and Hierarchies [68.93512627479197]
  We study quantum Ordered Binary Decision Diagrams($OBDD$) model. We prove lower bounds and upper bounds for OBDD with arbitrary order of input variables. We extend hierarchy for read$k$-times Ordered Binary Decision Diagrams ($k$-OBDD$) of width.
  arXiv  Detail & Related papers  (2022-04-22T12:37:56Z)
• Low-degree learning and the metric entropy of polynomials [44.99833362998488]
  We prove that any (deterministic or randomized) algorithm which learns $mathscrF_nd$ with $L$-accuracy $varepsilon$ requires at least $Omega(sqrtvarepsilon)2dlog n leq log mathsfM(mathscrF_n,d,|cdot|_L,varepsilon) satisfies the two-sided estimate $$c (1-varepsilon)2dlog
  arXiv  Detail & Related papers  (2022-03-17T23:52:08Z)
• TURF: A Two-factor, Universal, Robust, Fast Distribution Learning Algorithm [64.13217062232874]
  One of its most powerful and successful modalities approximates every distribution to an $ell$ distance essentially at most a constant times larger than its closest $t$-piece degree-$d_$. We provide a method that estimates this number near-optimally, hence helps approach the best possible approximation.
  arXiv  Detail & Related papers  (2022-02-15T03:49:28Z)
• The Curse of Passive Data Collection in Batch Reinforcement Learning [82.6026077420886]
  In high stake applications, active experimentation may be considered too risky and thus data are often collected passively. While in simple cases, such as in bandits, passive and active data collection are similarly effective, the price of passive sampling can be much higher when collecting data from a system with controlled states.
  arXiv  Detail & Related papers  (2021-06-18T07:54:23Z)
• Learning a Latent Simplex in Input-Sparsity Time [58.30321592603066]
  We consider the problem of learning a latent $k$-vertex simplex $KsubsetmathbbRdtimes n$, given access to $AinmathbbRdtimes n$. We show that the dependence on $k$ in the running time is unnecessary given a natural assumption about the mass of the top $k$ singular values of $A$.
  arXiv  Detail & Related papers  (2021-05-17T16:40:48Z)
• Small Covers for Near-Zero Sets of Polynomials and Learning Latent Variable Models [56.98280399449707]
  We show that there exists an $epsilon$-cover for $S$ of cardinality $M = (k/epsilon)O_d(k1/d)$. Building on our structural result, we obtain significantly improved learning algorithms for several fundamental high-dimensional probabilistic models hidden variables.
  arXiv  Detail & Related papers  (2020-12-14T18:14:08Z)
• An Optimal Separation of Randomized and Quantum Query Complexity [67.19751155411075]
  We prove that for every decision tree, the absolute values of the Fourier coefficients of a given order $ellsqrtbinomdell (1+log n)ell-1,$ sum to at most $cellsqrtbinomdell (1+log n)ell-1,$ where $n$ is the number of variables, $d$ is the tree depth, and $c>0$ is an absolute constant.
  arXiv  Detail & Related papers  (2020-08-24T06:50:57Z)
• Model-Free Reinforcement Learning: from Clipped Pseudo-Regret to Sample Complexity [59.34067736545355]
  Given an MDP with $S$ states, $A$ actions, the discount factor $gamma in (0,1)$, and an approximation threshold $epsilon > 0$, we provide a model-free algorithm to learn an $epsilon$-optimal policy. For small enough $epsilon$, we show an improved algorithm with sample complexity.
  arXiv  Detail & Related papers  (2020-06-06T13:34:41Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.