Low-depth Quantum State Preparation

• URL: http://arxiv.org/abs/2102.07533v3
• Date: Thu, 12 Aug 2021 12:55:25 GMT
• Title: Low-depth Quantum State Preparation
• Authors: Xiao-Ming Zhang, Man-Hong Yung and Xiao Yuan
• Abstract summary: A crucial subroutine in quantum computing is to load the classical data of $N$ complex numbers into the amplitude of a superposed $n=lceil logNrceil$-qubit state. Here we investigate this space-time tradeoff in quantum state preparation with classical data. We propose quantum algorithms with $mathcal O(n2)$ circuit depth to encode any $N$ complex numbers.
• Score: 3.540171881768714
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: A crucial subroutine in quantum computing is to load the classical data of $N$ complex numbers into the amplitude of a superposed $n=\lceil \log_2N\rceil$-qubit state. It has been proven that any algorithm universally implementing this subroutine would need at least $\mathcal O(N)$ constant weight operations. However, the proof assumes that only $n$ qubits are used, whereas the circuit depth could be reduced by extending the space and allowing ancillary qubits. Here we investigate this space-time tradeoff in quantum state preparation with classical data. We propose quantum algorithms with $\mathcal O(n^2)$ circuit depth to encode any $N$ complex numbers using only single-, two-qubit gates and local measurements with ancillary qubits. Different variances of the algorithm are proposed with different space and runtime. In particular, we present a scheme with $\mathcal O(N^2)$ ancillary qubits, $\mathcal O(n^2)$ circuit depth, and $\mathcal O(n^2)$ average runtime, which exponentially improves the conventional bound. While the algorithm requires more ancillary qubits, it consists of quantum circuit blocks that only simultaneously act on a constant number of qubits and at most $\mathcal O(n)$ qubits are entangled. We also prove a fundamental lower bound $\mit\Omega(n)$ for the minimum circuit depth and runtime with arbitrary number of ancillary qubits, aligning with our scheme with $\mathcal O(n^2)$. The algorithms are expected to have wide applications in both near-term and universal quantum computing.

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