We consider the problem of learning low-degree quantum objects up to $\varepsilon$-error in $\ell_2$-distance. We show the following results: $(i)$ unknown $n$-qubit degree-$d$ (in the Pauli basis) quantum channels and unitaries can be learned using $O(1/\varepsilon^d)$ queries (independent of $n$), $(ii)$ polynomials $p:\{-1,1\}^n\rightarrow [-1,1]$ arising from $d$-query quantum algorithms can be classically learned from $O((1/\varepsilon)^d\cdot \log n)$ many random examples $(x,p(x))$ (which implies learnability even for $d=O(\log n)$), and $(iii)$ degree-$d$ polynomials $p:\{-1,1\}^n\to [-1,1]$ can be learned through $O(1/\varepsilon^d)$ queries to a quantum unitary $U_p$ that block-encodes $p$. Our main technical contributions are new Bohnenblust-Hille inequalities for quantum channels and completely bounded~polynomials.
We consider the problem of learning low-degree quantum objects up to ε-error in ℓ2-distance. We show the following results: (i) unknown n-qubit degree-d (in the Pauli basis) quantum channels and unitaries can be learned using O(1/εd) queries (independent of n), (ii) polynomials p:{−1,1}n→[−1,1] arising from d-query quantum algorithms can be classically learned from O((1/ε)d⋅logn) many random examples (x,p(x)) (which implies learnability even for d=O(logn)), and (iii) degree-d polynomials p:{−1,1}n→[−1,1] can be learned through O(1/εd) queries to a quantum unitary Up that block-encodes p. Our main technical contributions are new Bohnenblust-Hille inequalities for quantum channels and completely bounded~polynomials.
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 ℓ1-norm. This class generalizes well-studied families, including sparse unitaries, quantum k-juntas, 2k-Pauli dimensional channels, and compositions of depth O(loglogn) 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 O~(s6/ε4) 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 ℓ1-norm. For that class, we prove an exponential query lower bound 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-norm uniformly bounded by L1 are learnable with 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 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=∑iαiCi over Cliffords Ci with bounded Clifford extent ∑i∣αi∣. 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 opt, outputs a Clifford unitary witnessing fidelity ≥opt−ε for some error ε>0, in time 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)) in the Pauli basis but are Clifford structured.
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. We prove: 1. Local Hamiltonians: We give a tolerant testing protocol to decide if H is ε1-close to k-local or ε2-far from k-local, with O(1/(ε2−ε1)4) queries, solving open questions posed in a recent work by Bluhm et al. For learning a k-local H up to error ε, we give a protocol with query complexity exp(O(k2+klog(1/ε))) independent of n, by leveraging the non-commutative Bohnenblust-Hille inequality. 2. Sparse Hamiltonians: We give a protocol to test if H is ε1-close to being s-sparse (in the Pauli basis) or ε2-far from being s-sparse, with O(s6/(ε22−ε12)6) queries. For learning up to error ε, we show that O(s4/ε8) queries suffice. 3. Learning without memory: The learning results stated above have no dependence on n, but require n-qubit quantum memory. We give subroutines that allow us to learn without memory; increasing the query complexity by a (logn)-factor in the local case and an n-factor in the sparse case. 4. Testing without memory: We give a new subroutine called Pauli hashing, which allows one to tolerantly test s-sparse Hamiltonians with O(s14/(ε22−ε12)18) queries. A key ingredient is showing that s-sparse Pauli channels can be tolerantly tested under the diamond norm with O(s2/(ε2−ε1)6) queries. Along the way, we prove new structural theorems for local and sparse Hamiltonians. We complement our learning results with polynomially weaker lower bounds. Furthermore, our algorithms use short time evolutions and do not assume prior knowledge of the terms in the support of the Pauli spectrum.
Block encoding embeds a matrix as a sub-block of a unitary matrix and serves as a fundamental input model for quantum algorithms based on quantum singular value transformation, enabling polynomial transformations of matrices encoded in unitary operators. Block encoding of unknown quantum states can be useful for quantum learning; however, the fundamental limits on converting between unknown quantum states and their block-encoding unitary channels remain poorly understood. In this paper, we investigate this convertibility in both directions. First, we prove that implementing an ε-approximate block-encoding unitary channel of an unknown quantum state requires Ω(1/ε) copies of the state, matching known upper bounds up to logarithmic factors. Second, we show that recovering a rank-r, d-dimensional quantum state ρ given query access to its block-encoding unitary channel generally requires Ω((1/λmax(ρ))d/r) queries, where λmax(ρ) is the maximum eigenvalue of ρ, revealing an unavoidable dependence on the dimension of the state. Our results identify inherent limitations of block encoding as a representation of unknown quantum states and reveal a separation between learning properties of a quantum state and generating the state itself. Using our techniques, we further establish lower bounds for specific state-generation tasks, including ground-state preparation and Gibbs-state preparation.
We consider the problem of establishing limitations on the power of quantum query algorithms via the completely bounded polynomial method. In particular, we prove several optimal functional inequalities involving different notions of completely bounded polynomials. These inequalities lead to limiting theorems for the power of quantum query algorithms that improve on prior works.
1. An optimal root-influence bound for block-multilinear polynomials. Prior work showed that block-multilinear polynomials p of degree t satisfy a root-influence bound, ∥p∥cb≥∑iInfi[p]/t2, which is stronger than the bound appearing in the Aaronson-Ambainis conjecture. We find the optimal constant in that inequality: ∥p∥cb≥∑iInfi[p]/t. Since the amplitudes of quantum algorithms that query disjoint blocks of inputs-such as t-fold forrelation- are block-multilinear polynomials with ∥p∥cb≤1, our inequality shows that they satisfy t≥∑iInfi[p]. We prove that this inequality yields both a more efficient classical simulation than prior results based on the Aaronson-Ambainis argument, and a qualitative improvement: all classical queries are nonadaptive.
2. Optimal Fourier growth of the highest level of quantum query algorithms. We show that for every polynomial p defined on {−1,1}n of degree 2t, the Fourier Growth at the level 2t, namely ∥p2t∥ℓ1, satisfies ∥p2t∥ℓ1≤(en/(2t−1))22t−1∥p∥cb. This is optimal up to the factor e, as witnessed by 2t-fold forrelation. As quantum query algorithms that make t queries (to the whole input) satisfy ∥p∥cb≤1, this yields a Fourier growth bound for these algorithms, partially resolving a question by Girish (STOC, 2026).
We consider the problems of testing and learning an n -qubit Hamiltonian H=∑xλxσx expressed in its Pauli basis, from queries to its evolution operator U=e−iHt . To this end, we prove the following results. Testing : We give a tolerant testing protocol to decide if a Hamiltonian is ε1 -close to k -local or ε2 -far from k -local in the ℓ2 norm of the coefficients, with O(1/(ε2−ε1)4) queries, thereby solving two open questions posed in a recent work by Bluhm, Caro and Oufkir (Bluhm, A., Caro, M.C., Oufkir, A.). We give a protocol for testing whether a Hamiltonian is ε1 -close to being s -sparse or ε2 -far from being s -sparse in the ℓ2 norm of the coefficients, with O(s6/(ε22−ε12)6) queries. Learning : We give a protocol to ε -learn unstructured Hamiltonian in the ℓ∞ norm of the coefficients with O(1/ε4) queries. Combining this with the non-commutative Bohnenblust-Hille inequality, we obtain an algorithm for learning k -local Hamiltonians in ℓ2 norm of the coefficients that only uses O(exp(k2+klog(1/ε))) queries. For Hamiltonians that are s -sparse in the Pauli basis, we can learn them in the ℓ2 norm with O(s2/ε4) queries. Learning without quantum memory : The learning results stated above have no dependence on the system size n , but require n -qubit quantum memory. We give subroutines that allow us to reproduce all the above learning results without quantum memory; squaring the query complexity and paying a (logn) -factor in the local case and an n -factor in the sparse case. Testing without quantum memory : We give a new subroutine called Pauli hashing , which allows one to tolerantly test s -sparse Hamiltonians in the ℓ2 norm using O~(s14/(ε22−ε12)18) query complexity. A key ingredient is showing that s -sparse Pauli channels can be tested in a tolerant fashion as being ε1 -close to being s -sparse or ε2 -far under the diamond norm, using O~(s2/(ε2−ε1)6) queries via Pauli hashing. In order to prove these results, we prove new structural theorems for local Hamiltonians, sparse Pauli channels and sparse Hamiltonians. We complement our learning algorithms with lower bounds that are polynomially weaker. Furthermore, our algorithms use short time evolutions and do not assume prior knowledge of the terms on which the Pauli spectrum is supported on, i.e., we do not require prior knowledge of the support of the Hamiltonian.