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

• URL: http://arxiv.org/abs/2202.11302v3
• Date: Tue, 16 May 2023 11:49:27 GMT
• Title: Optimal (controlled) quantum state preparation and improved unitary synthesis by quantum circuits with any number of ancillary qubits
• Authors: Pei Yuan, Shengyu Zhang
• Abstract summary: Controlled quantum state preparation (CQSP) aims to provide the transformation of $|irangle |0nrangle to |irangle |psi_irangle $ for all $iin 0,1k$ for the given $n$-qubit states. We construct a quantum circuit for implementing CQSP, with depth $Oleft(n+k+frac2n+kn+k+mright)$ and size $Oleft(2n+kright)$
• Score: 20.270300647783003
• License: http://creativecommons.org/licenses/by/4.0/
• 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 $O\left(2^{n+k}\right)$ 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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