Towards Optimal Circuit Size for Sparse Quantum State Preparation

• URL: http://arxiv.org/abs/2404.05147v2
• Date: Tue, 9 Apr 2024 12:06:53 GMT
• Title: Towards Optimal Circuit Size for Sparse Quantum State Preparation
• Authors: Rui Mao, Guojing Tian, Xiaoming Sun,
• Abstract summary: We consider the preparation for $n$-qubit sparse quantum states with $s$ non-zero amplitudes and propose two algorithms. The first algorithm uses $O(ns/log n + n)$ gates, improving upon previous methods by $O(log n)$. The second algorithm is tailored for binary strings that exhibit a short Hamiltonian path.
• Score: 10.386753939552872
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Compared to general quantum states, the sparse states arise more frequently in the field of quantum computation. In this work, we consider the preparation for $n$-qubit sparse quantum states with $s$ non-zero amplitudes and propose two algorithms. The first algorithm uses $O(ns/\log n + n)$ gates, improving upon previous methods by $O(\log n)$. We further establish a matching lower bound for any algorithm which is not amplitude-aware and employs at most $\operatorname{poly}(n)$ ancillary qubits. The second algorithm is tailored for binary strings that exhibit a short Hamiltonian path. An application is the preparation of $U(1)$-invariant state with $k$ down-spins in a chain of length $n$, including Bethe states, for which our algorithm constructs a circuit of size $O\left(\binom{n}{k}\log n\right)$. This surpasses previous results by $O(n/\log n)$ and is close to the lower bound $O\left(\binom{n}{k}\right)$. Both the two algorithms shrink the existing gap theoretically and provide increasing advantages numerically.

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