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

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 |ii |0ni → |ii |ψii for all i ∈ {0, 1}k for the given n-qubit states |ψii. In this paper, we construct a quantum circuit for implementing CQSP, with depth On + k + n+2nk++kmand size O(2n+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 ≥ 0 and n ≥ 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 |0ni |0mi → |ψi |0mi. 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 Θ(n + 2n/(n + m)) and its size complexity to Θ(2n) 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 Ω(n + 4n/(n + m)) and an upper bound of O(n2n) for m = Ω(2n/n) ancillary qubits. In this paper, we quadratically shrink this gap by presenting a quantum circuit of the depth of O ( n2n/2 + n1/m212/32n/2 ) .