Circuit Complexity of Sparse Quantum State Preparation

• URL: http://arxiv.org/abs/2406.16142v1
• Date: Sun, 23 Jun 2024 15:28:20 GMT
• Title: Circuit Complexity of Sparse Quantum State Preparation
• Authors: Jingquan Luo, Lvzhou Li,
• Abstract summary: We show that any $n$-qubit $d$-sparse quantum state can be prepared by a quantum circuit of size $O(fracdnlog d)$ and depth $Theta(log dn)$ using at most $O(fracndlog d )$ ancillary qubits. We also establish the lower bound $Omega(fracdnlog(n + m) + log d + n)$ on the circuit size with $m$ ancillary qubits available.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Quantum state preparation is a fundamental and significant subroutine in quantum computing. In this paper, we conduct a systematic investigation on the circuit size for sparse quantum state preparation. A quantum state is said to be $d$-sparse if it has only $d$ non-zero amplitudes. For the task of preparing $n$-qubit $d$-sparse quantum states, we obtain the following results: (a) We propose the first approach that uses $o(dn)$ elementary gates without using ancillary qubits. Specifically, it is proven that any $n$-qubit $d$-sparse quantum state can be prepared by a quantum circuit of size $O(\frac{dn}{\log n} + n)$ without using ancillary qubits. This is asymptotically optimal when $d = poly(n)$, and this optimality extends to a broader scope under some reasonable assumptions. (b) We show that any $n$-qubit $d$-sparse quantum state can be prepared by a quantum circuit of size $O(\frac{dn}{\log d})$ and depth $\Theta(\log dn)$ using at most $O(\frac{n{d}}{\log d} )$ ancillary qubits, which not only reduces the circuit size compared to the one without ancillary qubits when $d = \omega(poly(n))$, but also achieves the same asymptotically optimal depth while utilizing fewer ancillary qubits and applying fewer quantum gates compared to the result given in [PRL, 129, 230504(2022)]. (ii) We establish the lower bound $\Omega(\frac{dn}{\log(n + m) + \log d} + n)$ on the circuit size with $m$ ancillary qubits available. we also obtain a slightly stronger lower bound under reasonable assumptions. (c) We prove that with arbitrary amount of ancillary qubits available, the circuit size for preparing $n$-qubit $d$-sparse quantum states is $\Theta({\frac{dn}{\log dn} + n})$.

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