Optimal (controlled) quantum state preparation and improved unitary synthesis by quantum circuits with any number of ancillary qubits

Pei Yuan and Shengyu Zhang

Tencent Quantum Laboratory, Tencent, Shenzhen, Guangdong 518057, China

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Abstract

As a cornerstone for many quantum linear algebraic and quantum machine learning algorithms, controlled quantum state preparation (CQSP) aims to provide the transformation of $|i\rangle |0^n\rangle \to |i\rangle |\psi_i\rangle $ for all $i\in \{0,1\}^k$ for the given $n$-qubit states $|\psi_i\rangle$. In this paper, we construct a quantum circuit for implementing CQSP, with depth $O\left(n+k+\frac{2^{n+k}}{n+k+m}\right)$ and size $O(2^{n+k})$ for any given number $m$ of ancillary qubits. These bounds, which can also be viewed as a time-space tradeoff for the transformation, are optimal for any integer parameters $m,k\ge 0$ and $n\ge 1$. When $k=0$, the problem becomes the canonical quantum state preparation (QSP) problem with ancillary qubits, which asks for efficient implementations of the transformation $|0^n\rangle|0^m\rangle \to |\psi\rangle |0^m\rangle$. This problem has many applications with many investigations, yet its circuit complexity remains open. Our construction completely solves this problem, pinning down its depth complexity to $\Theta(n+2^{n}/(n+m))$ and its size complexity to $\Theta(2^{n})$ for any $m$. Another fundamental problem, unitary synthesis, asks to implement a general $n$-qubit unitary by a quantum circuit. Previous work shows a lower bound of $\Omega(n+4^n/(n+m))$ and an upper bound of $O(n2^n)$ for $m=\Omega(2^n/n)$ ancillary qubits. In this paper, we quadratically shrink this gap by presenting a quantum circuit of the depth of $O\left(n2^{n/2}+\frac{n^{1/2}2^{3n/2}}{m^{1/2}}~~\right)$.

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[9] Xiao-Ming Zhang, Tongyang Li, and Xiao Yuan, "Quantum State Preparation with Optimal Circuit Depth: Implementations and Applications", Physical Review Letters 129 23, 230504 (2022).

[10] Anton S. Albino, Lucas Q. Galvão, Ethan Hansen, Mauro Q. Nooblath Neto, and Clebson Cruz, "Quantum algorithm for finding minimum values in a Quantum Random Access Memory", arXiv:2301.05122, (2023).

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[12] Bojia Duan and Chang-Yu Hsieh, "Hamiltonian-based data loading with shallow quantum circuits", Physical Review A 106 5, 052422 (2022).

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