一种多项式时空启发式T计数算法
A polynomial time and space heuristic algorithm for T-count
摘要：
这项工作的重点是减少物理成本的实现量子算法时，使用最先进的容错量子纠错码，特别是那些实现T门消耗更多的资源比其他门的门设置。更具体地说，我们考虑了一组酉群，它可以由一个由Clifford+T门集（一个通用门集）组成的量子线路精确实现。我们的主要兴趣是计算一个给定的$n$ -量子比特酉$U$的电路，使用最小可能的T门数（称为酉$U$的T计数）。我们考虑的问题COUNT-T，其优化版本的目的是找到T计数的$U$。在其决策版本中，目标是决定T计数是否至多为某个正整数$m$。给定COUNT-T的一个预言，我们可以在$U$的T-计数和维数的时间多项式中计算T-计数最优电路。我们给出了一个可证明的经典算法，该算法求解时间上的COUNT-T（决策）$O\left(N^{2(c-1)\lceil\frac{m}{c}\rceil}\text{poly}(m,N)\right)$和空间上的$O\left(N^{2\lceil\frac{m}{c}\rceil}\text{poly}(m,N)\right)$，其中$N=2^n$和$c\geq 2$。这给出了用中间相遇技术的变体来解决这个问题的空间-时间权衡。我们还介绍了一个渐近更快的乘法方法，剃掉了$N^{0.7457}$的整体复杂性的一个因素。最后，除了对严格算法的改进之外，我们还给出了一种启发式算法，该算法在某些假设下输出T计数最优电路，并且具有空间和时间复杂度$\text{poly}(m,N)$。虽然我们的启发式方法仍然随着量子比特的数量呈指数级缩放（尽管具有较低的指数，但是通过使用$m$从指数级缩放到多项式缩放，有很大的改进。
英文摘要：
This work focuses on reducing the physical cost of implementing quantum algorithms when using the state-of-the-art fault-tolerant quantum error correcting codes, in particular, those for which implementing the T gate consumes vastly more resources than the other gates in the gate set. More specifically, we consider the group of unitaries that can be exactly implemented by a quantum circuit consisting of the Clifford+T gate set, a universal gate set. Our primary interest is to compute a circuit for a given $n$-qubit unitary $U$, using the minimum possible number of T gates (called the T-count of unitary $U$). We consider the problem COUNT-T, the optimization version of which aims to find the T-count of $U$. In its decision version the goal is to decide if the T-count is at most some positive integer $m$. Given an oracle for COUNT-T, we can compute a T-count-optimal circuit in time polynomial in the T-count and dimension of $U$. We give a provable classical algorithm that solves COUNT-T (decision) in time $O\left(N^{2(c-1)\lceil\frac{m}{c}\rceil}\text{poly}(m,N)\right)$ and space $O\left(N^{2\lceil\frac{m}{c}\rceil}\text{poly}(m,N)\right)$, where $N=2^n$ and $c\geq 2$. This gives a space-time trade-off for solving this problem with variants of meet-in-the-middle techniques. We also introduce an asymptotically faster multiplication method that shaves a factor of $N^{0.7457}$ off of the overall complexity. Lastly, beyond our improvements to the rigorous algorithm, we give a heuristic algorithm that outputs a T-count-optimal circuit and has space and time complexity $\text{poly}(m,N)$, under some assumptions. While our heuristic method still scales exponentially with the number of qubits (though with a lower exponent, there is a large improvement by going from exponential to polynomial scaling with $m$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
