Low-depth quantum state preparation

A crucial subroutine in quantum computing is to load the classical data of $N$ complex numbers into amplitude a superposed $n=\ensuremath{\lceil}{log}_{2}N\ensuremath{\rceil}$-qubit state. It has been proven that any algorithm universally implementing this would need at least $O(N)$ constant weight operations. However, proof assumes only $n$ qubits are used, whereas circuit depth could be reduced by extending space and allowing ancillary qubits. Here we investigate space-time tradeoff state preparation with data. We propose algorithms $O({n}^{2})$ encode using single- two-qubit gates, local measurements Different variances proposed different runtime. In particular, present scheme $O({N}^{2})$ qubits, depth, average runtime, which exponentially improves conventional bound. While requires more it consists blocks simultaneously act on number most $O(n)$ entangled. also prove fundamental lower bound $\mathrm{\ensuremath{\Omega}}(n)$ for minimum runtime an arbitrary aligning our $O({n}^{2})$. The expected have wide applications both near-term universal computing.