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Nearly Optimal Circuit Size for Sparse Quantum State Preparation

Authors Lvzhou Li , Jingquan Luo

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LIPIcs.ICALP.2025.113.pdf
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ACM Subject Classification
  • Theory of computation → Circuit complexity
  • Theory of computation → Quantum computation theory
Keywords
  • Quantum computing
  • quantum state preparation
  • circuit complexity

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Abstract

Quantum state preparation is a fundamental and significant subroutine in quantum computing. In this paper, we conduct a systematic investigation of 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:  
- Without ancillary qubits: Any n-qubit d-sparse quantum state can be prepared by a quantum circuit of size O(nd/(log n) + n) without using ancillary qubits, which improves the previous best results. It is asymptotically optimal when d = poly(n), and this optimality holds for a broader scope under some reasonable assumptions. 
- 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((nd)/(log(n + m)) + n) using m ancillary qubits for any m ∈ O((nd)/(log nd) + n). (ii) We establish a matching lower bound Ω((nd)/(log(n+m))+n) under some reasonable assumptions, and obtain a slightly weaker lower bound Ω((nd)/(log(n+m)+log d) + n) without any assumptions. 
- With unlimited ancillary qubits: Given an arbitrary amount of ancillary qubits available, the circuit size for preparing n-qubit d-sparse quantum states is Θ((nd)/(log nd) + n).

Cite As Get BibTex

Lvzhou Li and Jingquan Luo. Nearly Optimal Circuit Size for Sparse Quantum State Preparation. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 113:1-113:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.113

Author Details

Lvzhou Li
  • Institute of Quantum Computing and Software, School of Computer Science and Engineering, Sun Yat-sen University, Guangzhou, China
Jingquan Luo
  • Institute of Quantum Computing and Software, School of Computer Science and Engineering, Sun Yat-sen University, Guangzhou, China

Funding

Lvzhou Li and Jingquan Luo were supported by the National Key Research and Development Program of China (Grant No.2024YFB4504004), the National Natural Science Foundation of China (Grants No.92465202 and No.62272492), the Guangdong Provincial Quantum Science Strategic Initiative (Grant No.GDZX2303007), and the Guangzhou Science and Technology Program (Grant No.2024A04J4892).

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