Optimal algorithms for learning quantum phase states

• URL: http://arxiv.org/abs/2208.07851v2
• Date: Wed, 3 May 2023 05:22:04 GMT
• Title: Optimal algorithms for learning quantum phase states
• Authors: Srinivasan Arunachalam, Sergey Bravyi, Arkopal Dutt, Theodore J. Yoder
• Abstract summary: 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.
• Score: 8.736370689844682
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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