Efficient Learning of Non-Interacting Fermion Distributions

We give an efficient classical algorithm that recovers the distribution of a non-interacting fermion state over the computational basis. For a system of $n$ non-interacting fermions and $m$ modes, we show that $O(m^2 n^4 \log(m/\delta)/ \varepsilon^4)$ samples and $O(m^4 n^4 \log(m/\delta)/ \varepsilon^4)$ time are sufficient to learn the original distribution to total variation distance $\varepsilon$ with probability $1 - \delta$. Our algorithm empirically estimates the one- and two-mode correlations and uses them to reconstruct a succinct description of the entire distribution efficiently.Comment: 7 page

Efficient Learning of Quantum States Prepared With Few Non-Clifford Gates II: Single-Copy Measurements

Recent work has shown that $n$-qubit quantum states output by circuits with at most $t$ single-qubit non-Clifford gates can be learned to trace distance $\epsilon$ using $\mathsf{poly}(n,2^t,1/\epsilon)$ time and samples. All prior algorithms achieving this runtime use entangled measurements across two copies of the input state. In this work, we give a similarly efficient algorithm that learns the same class of states using only single-copy measurements.Comment: 22 pages. arXiv admin note: text overlap with arXiv:2305.1340

Improved Stabilizer Estimation via Bell Difference Sampling

We study the complexity of learning quantum states in various models with respect to the stabilizer formalism and obtain the following results: - We prove that $\Omega(n)$ $T$-gates are necessary for any Clifford+$T$ circuit to prepare computationally pseudorandom quantum states, an exponential improvement over the previously known bound. This bound is asymptotically tight if linear-time quantum-secure pseudorandom functions exist. - Given an $n$-qubit pure quantum state $|\psi\rangle$ that has fidelity at least $\tau$ with some stabilizer state, we give an algorithm that outputs a succinct description of a stabilizer state that witnesses fidelity at least $\tau - \varepsilon$. The algorithm uses $O(n/(\varepsilon^2\tau^4))$ samples and $\exp\left(O(n/\tau^4)\right) / \varepsilon^2$ time. In the regime of $\tau$ constant, this algorithm estimates stabilizer fidelity substantially faster than the na\"ive $\exp(O(n^2))$-time brute-force algorithm over all stabilizer states. - In the special case of $\tau > \cos^2(\pi/8)$, we show that a modification of the above algorithm runs in polynomial time. - We improve the soundness analysis of the stabilizer state property testing algorithm due to Gross, Nezami, and Walter [Comms. Math. Phys. 385 (2021)]. As an application, we exhibit a tolerant property testing algorithm for stabilizer states. The underlying algorithmic primitive in all of our results is Bell difference sampling. To prove our results, we establish and/or strengthen connections between Bell difference sampling, symplectic Fourier analysis, and graph theory.Comment: 40 pages, 2 figure

Efficient Learning of Quantum States Prepared With Few Non-Clifford Gates

We give an algorithm that efficiently learns a quantum state prepared by Clifford gates and $O(\log(n))$ non-Clifford gates. Specifically, for an $n$-qubit state $\lvert \psi \rangle$ prepared with at most $t$ non-Clifford gates, we show that $\mathsf{poly}(n,2^t,1/\epsilon)$ time and copies of $\lvert \psi \rangle$ suffice to learn $\lvert \psi \rangle$ to trace distance at most $\epsilon$. This result follows as a special case of an algorithm for learning states with large stabilizer dimension, where a quantum state has stabilizer dimension $k$ if it is stabilized by an abelian group of $2^k$ Pauli operators. We also develop an efficient property testing algorithm for stabilizer dimension, which may be of independent interest.Comment: 23 page