Quantum state preparation is an important subroutine for quantum computing. We show that any $n$-qubit quantum state can be prepared with a $\mathrm{\Theta }(n)$-depth circuit using only single- and two-qubit gates, although with a cost of an exponential amount of ancillary qubits. On the other hand, for sparse quantum states with $d⩾2$ nonzero entries, we can reduce the circuit depth to $\mathrm{\Theta }(\log (nd))$ with $O(nd\log d)$ ancillary qubits. The algorithm for sparse states is exponentially faster than best-known results and the number of ancillary qubits is nearly optimal and only increases polynomially with the system size. We discuss applications of the results in different quantum computing tasks, such as Hamiltonian simulation, solving linear systems of equations, and realizing quantum random access memories, and find cases with exponential reductions of the circuit depth for all these three tasks. In particular, using our algorithm, we find a family of linear system solving problems enjoying exponential speedups, even compared to the best-known quantum and classical dequantization algorithms.

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具有最佳电路深度的量子态准备：实现和应用

量子态准备是量子计算的一个重要子程序。我们证明任何$n$-qubit 量子态可以用$\mathrm{\Theta }(n)$仅使用一个和两个量子位门的深度电路，尽管辅助量子位的成本呈指数级增长。另一方面，对于具有$d⩾\mathrm{2个}$非零条目，我们可以将电路深度减少到$\mathrm{\Theta }(\mathrm{日志}(nd))$和$欧(nd\mathrm{日志}d)$辅助量子比特。稀疏状态的算法比最知名的结果呈指数级增长，辅助量子位的数量几乎是最优的，并且只会随着系统规模的增加而呈多项式增长。我们讨论了结果在不同量子计算任务中的应用，例如哈密顿模拟、求解线性方程组和实现量子随机存取存储器，并找到了所有这三个任务的电路深度呈指数减少的情况。特别是，使用我们的算法，我们发现一系列线性系统解决问题享受指数加速，甚至与最著名的量子和经典反量化算法相比也是如此。