Efficient synthesis of arbitrary quantum states and unitaries from a universal fault-tolerant gate-set e.g. Clifford+T is a key subroutine in quantum computation. As large quantum algorithms feature many qubits that encode coherent quantum information but remain idle for parts of the computation, these should be used if it minimizes overall gate counts, especially that of the expensive T-gates. We present a quantum algorithm for preparing any dimension-$N$ pure quantum state specified by a list of $N$ classical numbers, that realizes a trade-off between space and T-gates. Our scheme uses $\mathcal{O}(\log{(N/\epsilon)})$ clean qubits and a tunable number of $\sim(\lambda\log{(\frac{\log{N}}{\epsilon})})$ dirty qubits, to reduce the T-gate cost to $\mathcal{O}(\frac{N}{\lambda}+\lambda\log{\frac{N}{\epsilon}}\log{\frac{\log{N}}{\epsilon}})$. This trade-off is optimal up to logarithmic factors, proven through an unconditional gate counting lower bound, and is, in the best case, a quadratic improvement in T-count over prior ancillary-free approaches. We prove similar statements for unitary synthesis by reduction to state preparation. Underlying our constructions is a T-efficient circuit implementation of a quantum oracle for arbitrary classical data.

中文翻译：

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在状态准备和酉合成中用 T 门换取脏量子位


从通用容错门集（例如）有效合成任意量子态和酉态Clifford+T 是量子计算中的一个关键子程序。由于大型量子算法具有许多编码相干量子信息的量子位，但在部分计算中保持空闲状态，因此如果可以最大限度地减少总门数，尤其是昂贵的 T 门，则应使用这些算法。我们提出了一种量子算法，用于准备由 $N$ 经典数列表指定的任何维度 $N$ 纯量子态，实现空间和 T 门之间的权衡。我们的方案使用 $\mathcal{O}(\log{(N/\epsilon)})$ 干净的量子位和可调数量的 $\sim(\lambda\log{(\frac{\log{N}}{\ epsilon})})$ 脏量子位，将 T 门成本降低到 $\mathcal{O}(\frac{N}{\lambda}+\lambda\log{\frac{N}{\epsilon}}\ log{\frac{\log{N}}{\epsilon}})$。这种权衡在对数因子上是最佳的，通过无条件门计数下限证明，并且在最好的情况下，与先前的无辅助方法相比，T 计数有二次改进。我们通过还原为状态准备来证明酉综合的类似陈述。我们构建的基础是针对任意经典数据的量子预言机的 T 高效电路实现。