早期容错迭代量子相位估计在成本误差权衡下的最优资源缩放

中文摘要 AI 辅助

迭代量子相位估计算法（IPEA）相较于需要逆量子傅里叶变换的原始算法，更适合NISQ和早期容错硬件。然而，当前的误差缓解和纠错实现会导致非理想酉算子，从而在IPEA中引起错误的相位反馈。提高相位比特可靠性的一种方法是将第k次迭代中对酉算子的2^k次幂的重复次数设为N_k^*次，然后进行多数表决。然而，由于电路复杂度的差异，不同迭代会产生不同的资源成本（w_k）和不同的单次测量错误概率（q_k）。在资源约束严格的条件下，我们提出每个比特的最优测量次数{N_k^*: 1≤k≤L}是什么：arg max_{N_k} P(所有L个比特正确) 受限于 ∑_{k=1}^L w_k N_k ≤ W。我们在以下条件下获得了N^*_k的闭式表达式：W ≫ ∑_k w_k（足以每次迭代至少测量一次），w_k随k增加而增加，q_k足够小且L较大。我们通过分析由紧的集中和反集中界导出的可处理的上界和下界替代问题，并证明当q_k≪1时它们的解收敛，从而得到这些结果。我们观察到，当W≫∑_k w_k ln(∑_k w_k)时，最优N_k^*与ln(∑_k w_k)/c_k成正比，且不依赖于单个w_k，其中c_k=-ln(2√((1-q_k/2)q_k/2)) ∝ ln(1/q_k)。然而，当∑_k w_k ≪ W ≪ ∑_k w_k ln(∑_k w_k)时，分配还（加性地）受到1/c_k ln(1/w_k)的影响，对于较大的k值（即酉算子的更高次幂）尤为明显。这些结果导致了简单的资源分配经验法则，可能有助于实际系统的设计。

英文摘要

The iterative quantum phase estimation algorithm (IPEA) is better suited for NISQ and early fault-tolerant hardware compared to the original algorithm, which requires the inverse quantum Fourier transform. However, current error mitigation and correction implementations lead to imperfect unitaries, causing erroneous phase feedback in IPEA. To improve the reliability of phase bits is to repeat the k^th iteration for the 2^k-th power of the unitary $N_k^*$ times, followed by a majority decision. However, due to variations in their circuit complexities, different iterations incur varying resource costs ($w_k$) and different per-shot error probabilities ($q_k$). Given tight resource constraints, we ask what the optimal number $\{N_k^*: 1\le k \le L\}$ of shots per bit is: $$\arg \max_{\{N_k\}} \mathbb{P}(\text{all } L \text{ bits correct}) \text{ s.t.} \sum_{k=1}^L w_k N_k \le W$$ We obtain closed-form expressions for $N^*_k$ under the conditions: $W \gg \sum_k w_k $(enough for at least one shot per iteration), $w_k$ increases with $k$, $q_k$ are small enough and $L$ is large. We do so by analyzing tractable upper and lower surrogate problems derived via tight concentration and anti-concentration bounds and showing that their solutions converge when $q_k\ll 1$. We observe that when $W\gg \sum_k w_k \ln\! \left(\sum_k w_k\right)$, the optimal $N_k^*$ is proportional to $\frac{\ln\! \left(\sum_k w_k\right)}{c_k}$ and does not depend on individual $w_k$, where $c_k=-\ln\left(2\sqrt{\left(1-\frac{q_k}{2} \right) \frac{q_k}{2}}\right) \propto \ln\frac{1}{q_k}$. However, when $\sum_k w_k \ll W\ll\sum_k w_k \ln\! \left(\sum_k w_k\right)$, the allocation is also (additively) influenced by $\frac{1}{c_k}\ln\frac{1}{w_k}$ for larger values of $k$, i.e., higher powers of unitary. These results lead to simple rules of thumb for resource allocation that may benefit the design of practical systems.

发表机构

• Indian Institute of Technology Madras（马德拉斯印度理工学院）

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