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

TL;DR

This paper presents optimal quantum circuits for controlled quantum state preparation and unitary synthesis, achieving tight bounds on depth and size with any number of ancillary qubits, advancing quantum algorithm implementation efficiency.

Contribution

It introduces optimal constructions for CQSP and QSP with tight bounds, and significantly improves unitary synthesis circuit depth, resolving open complexity questions.

Findings

Optimal quantum circuit depth for CQSP and QSP established.

Quadratic reduction in unitary synthesis circuit depth achieved.

Circuit complexity bounds are proven to be tight and optimal.

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 $O\left(n2^{n/2}+\frac{n^{1/2}2^{3n/2}}{m^{1/2}}\right)$.