Optimal algorithms for learning quantum phase states

TL;DR

This paper investigates the complexity of learning quantum phase states, providing algorithms with optimal sample complexities for various classes, and demonstrating their practical feasibility for near-term quantum applications.

Contribution

The paper introduces optimal algorithms for learning quantum phase states with different structures, achieving near-term implementability and extending to generalized and noisy states.

Findings

Sample complexity for degree-$d$ phase states is $ heta(n^d)$ with separable measurements.

Sample complexity reduces to $ heta(n^{d-1})$ with entangled measurements.

Algorithms enable learning of diagonal unitaries in the Clifford hierarchy and IQP circuits.

Abstract

We analyze the complexity of learning $n$-qubit quantum phase states. A degree-$d$ phase state is defined as a superposition of all $2^n$ basis vectors $x$ with amplitudes proportional to $(-1)^{f(x)}$, where $f$ is a degree-$d$ Boolean polynomial over $n$ variables. We show that the sample complexity of learning an unknown degree-$d$ phase state is $\Theta(n^d)$ if we allow separable measurements and $\Theta(n^{d-1})$ if we allow entangled measurements. Our learning algorithm based on separable measurements has runtime $\textsf{poly}(n)$ (for constant $d$) and is well-suited for near-term demonstrations as it requires only single-qubit measurements in the Pauli $X$ and $Z$ bases. We show similar bounds on the sample complexity for learning generalized phase states with complex-valued amplitudes. We further consider learning phase states when $f$ has sparsity-$s$, degree-$d$ in its $\mathbb{F}_2$ representation (with sample complexity $O(2^d sn)$), $f$ has Fourier-degree-$t$ (with sample complexity $O(2^{2t})$), and learning quadratic phase states with $\varepsilon$-global depolarizing noise (with sample complexity $O(n^{1+\varepsilon})$). These learning algorithms give us a procedure to learn the diagonal unitaries of the Clifford hierarchy and IQP~circuits.