Nearly Optimal Circuit Size for Sparse Quantum State Preparation

TL;DR

This work analyzes the circuit-size complexity of sparse quantum state preparation for an $n$-qubit state with $d$ nonzero amplitudes. It introduces a permutation-based method to achieve $O\left(\frac{nd}{\log n} + n\right)$ gates without ancillas, and a novel $(n,r)$-unary encoding framework that yields $O\left(\frac{nd}{\log(m+n)} + n\right)$ gates with $m$ ancillas, plus a complete characterization $\Theta\left(\frac{nd}{\log nd} + n\right)$ when ancillas are unlimited. The paper establishes lower bounds via discretization arguments, showing near-matching results under reasonable assumptions and providing weaker bounds without assumptions. Together, these results map out the optimal scaling of circuit size for SQSP across ancilla regimes and introduce encoding techniques with potential broader applicability in quantum circuit design.

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 (the total count of elementary gates in the circuit) 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 an $n$-qubit $d$-sparse quantum state, we obtain the following results: \textbf{Without ancillary qubits:} Any $n$-qubit $d$-sparse quantum state can be prepared by a quantum circuit of size $O(\frac{nd}{\log n} + n)$ without using ancillary qubits, which improves the previous best results. It is asymptotically optimal when $d = \mathrm{poly}(n)$, and this optimality holds for a broader scope under some reasonable assumptions. \textbf{With limited ancillary qubits:} (i) Based on the first result, we prove for the first time a trade-off between the number of ancillary qubits and the circuit size: any $n$-qubit $d$-sparse quantum state can be prepared by a quantum circuit of size $O(\frac{nd}{\log (n + m)} + n)$ using $m$ ancillary qubits for any $m \in O(\frac{nd}{\log nd} + n)$. (ii) We establish a matching lower bound $Ω(\frac{nd}{\log {(n + m)} }+ n)$ under some reasonable assumptions, and obtain a slightly weaker lower bound $Ω(\frac{nd}{\log {(n + m)} + \log d} + n)$ without any assumptions. \textbf{With unlimited ancillary qubits:} Given arbitrary amount of ancillary qubits available, the circuit size for preparing $n$-qubit $d$-sparse quantum states is $Θ(\frac{nd}{\log nd} + n)$.

Theorems & Definitions (25)

• Theorem 1
• Theorem 2
• Remark 3
• Theorem 4
• Remark 5
• Theorem 6
• Theorem 7
• Lemma 8
• Lemma 9
• Lemma 10: hall2018theory