Learning low-degree quantum objects

TL;DR

This work tackles the problem of learning low-degree quantum objects up to $\varepsilon$-error in $\ell_2$ distance, demonstrating that structured quantum objects can be learned with complexity independent of the ambient system size $n$ under degree constraints. The authors develop new Bohnenblust-Hille inequalities, including a completely bounded form and a non-commutative channel variant, to tightly control coefficient magnitudes of quantum channels, unitaries, and quantum-query polynomials. They then leverage these inequalities to design efficient sampling-based learning algorithms: for channels and unitaries, for quantum polynomials from $d$-query algorithms, and for classical polynomials accessed quantumly via block-encodings, achieving substantial speedups over prior bounds and, in some cases, $n$-independent sample complexities. The results reveal deep connections between functional-analytic inequalities and quantum learning theory, with implications for device characterization, quantum process tomography, and classical-quantum separations in learning tasks.

Abstract

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.

Figures (3)

• Figure 1: Quantum query algorithms considered in \ref{['theo:learnqpolynomials']}.
• Figure 2: Quantum circuit for estimating the Fourier coefficient $\widehat{U}(\alpha)$ for any $\alpha \in \{0,1,2,3\}^n$. Gate $G$ on the ancilla qubit is set to $I$ for estimating $\mathrm{Re}(\widehat{U}(\alpha))$ and is set to the $S$ gate for estimating $\mathrm{Im}(\widehat{U}(\alpha))$. The circuit $U_{\mathrm{EPR}}$ corresponds to the state preparation unitary of $n$-qubit EPR pairs over $2n$ qubits which contains $n$ sequential $\textsf{CNOT}$ gates with control on the $j$th qubit in the first $n$-qubit register and target on the $j$th qubit in the second $n$-qubit register, ordered with increasing $j \in [n]$.
• Figure 3: Triangles of norm comparisons. In the left triangle we display the norm comparisons implied by the Littlewood littlewood1930bounded and Grothendieck inequalities grothendieck1953resume and our \ref{['theo:BHGf']} for real bilinear maps. In the right triangle we depict the best upper bound for the $\textsf{BH}$ constant bayart2014bohr, the no extension of the Grothendieck inequality perez2008unbounded, and our \ref{['theo:BHGf']} for $d$-linear tensors.

Theorems & Definitions (27)

• Theorem 1
• Proposition 1
• Theorem 2
• Theorem 4
• Proposition 4
• Definition 1: Degree of a matrix
• Definition 2: Degree of a superoperator
• Theorem 6: defant2019fourier
• Lemma 7
• Theorem 8: Bohnenblust-Hille inequality for $S_1\to S_\infty$ maps