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Weighting gates in circuit complexity and holography
Progress of Theoretical and Experimental Physics Pub Date : 2021-07-21 , DOI: 10.1093/ptep/ptab098
I Akal 1
Affiliation  

Motivated by recent studies of quantum computational complexity in quantum field theory and holography, we discuss how weighting certain classes of gates building up a quantum circuit more heavily than others affects the complexity. Utilizing Nielsen’s geometric approach to circuit complexity, we investigate the effects for a regulated field theory for which the optimal circuit is a representation of $GL(N,\mathbb{R})$. More precisely, we work out how a uniformly chosen weighting factor acting on the entangling gates affects the complexity and, particularly, its divergent behavior. We show that assigning a higher cost to the entangling gates increases the complexity. Employing penalized and unpenalized complexities for the $\mathcal{F}_{\kappa=2}$ cost, we further find an interesting relation between the latter and that based on the unpenalized $\mathcal{F}_{\kappa=1}$ cost. In addition, we exhibit how imposing such penalties modifies the leading-order UV divergence in the complexity. We show that appropriately tuning the gate weighting eliminates the additional logarithmic factor, thus resulting in a simple power-law scaling. We also compare the circuit complexity with holographic predictions, specifically based on the complexity=action conjecture, and relate the weighting factor to certain bulk quantities. Finally, we comment on certain expectations concerning the role of gate penalties in defining complexity in field theory and also speculate on possible implications for holography.

中文翻译:

电路复杂性和全息术中的加权门

受最近在量子场论和全息术中对量子计算复杂性的研究的启发,我们讨论了加权某些类别的门如何比其他门更重地影响复杂性。利用尼尔森对电路复杂性的几何方法,我们研究了调节场论的影响,其中最优电路是 $GL(N,\mathbb{R})$ 的表示。更准确地说,我们计算出作用在纠缠门上的统一选择的加权因子如何影响复杂性,特别是其发散行为。我们表明,为纠缠门分配更高的成本会增加复杂性。对 $\mathcal{F}_{\kappa=2}$ 成本使用惩罚和未惩罚的复杂性,我们进一步发现后者与基于未惩罚的 $\mathcal{F}_{\kappa=1}$ 成本之间的有趣关系。此外,我们展示了施加此类惩罚如何修改复杂性中的领先顺序 UV 分歧。我们表明,适当调整门权重可以消除额外的对数因子,从而产生简单的幂律缩放。我们还将电路复杂性与全息预测进行比较,特别是基于复杂性=作用猜想,并将加权因子与某些体积相关联。最后,我们评论了关于门惩罚在定义场论复杂性中的作用的某些期望,并推测了对全息术的可能影响。我们展示了施加这样的惩罚是如何改变复杂性中领先的 UV 分歧的。我们表明,适当调整门权重可以消除额外的对数因子,从而产生简单的幂律缩放。我们还将电路复杂性与全息预测进行比较,特别是基于复杂性=作用猜想,并将加权因子与某些体积相关联。最后,我们评论了关于门惩罚在定义场论复杂性中的作用的某些期望,并推测了对全息术的可能影响。我们展示了施加这样的惩罚是如何改变复杂性中领先的 UV 分歧的。我们表明,适当调整门权重可以消除额外的对数因子,从而产生简单的幂律缩放。我们还将电路复杂性与全息预测进行比较,特别是基于复杂性=作用猜想,并将加权因子与某些体积相关联。最后,我们评论了关于门惩罚在定义场论复杂性中的作用的某些期望,并推测了对全息术的可能影响。具体基于复杂性=作用猜想,并将权重因子与某些批量相关联。最后,我们评论了关于门惩罚在定义场论复杂性中的作用的某些期望,并推测了对全息术的可能影响。具体基于复杂性=作用猜想,并将权重因子与某些批量相关联。最后,我们评论了关于门惩罚在定义场论复杂性中的作用的某些期望,并推测了对全息术的可能影响。
更新日期:2021-07-21
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