量子场论中算子复杂性的新洞察

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"这篇研究论文深入探讨了量子场论中算子复杂性的概念，并提供了新的理论框架。作者Run-Qiu Yang、Yu-Sen An、Chao Niu、Cheng-Yong Zhang和Keun-Young Kim在JHEP期刊上发表的文章（JHEP03(2019)161）揭示了SU（n）算子的复杂度如何通过双不变Finsler几何中的测地线长度来定义，这一特性受到量子场论对称性的约束。他们基于三个公理和一个关于连续系统复杂性的假设，进一步扩展了这一理论，将复杂度公式推广至Schatten p-范数类型。" 文章首先指出，先前的研究表明，SU（n）算子的复杂性可以通过双不变Finsler几何中的测地线长度来量化，这个长度受到量子场论对称性的影响。作者通过放松其中一个公理和一个假设，成功地将这一理论推广到更广泛的Schatten p-范数框架下，这拓宽了我们对量子系统复杂性的理解。其次，作者详细比较了双不变几何与右不变几何（如k局部几何）下的结果。他们强调了分析截面曲率的重要性，因为这有助于在不同的几何环境中统一复杂性理论。同时，他们还证明了自己的复杂性定义能够符合复杂度随时间演化的预期模式，即在一定时间内线性增长，直到达到饱和阶段。饱和时间可以通过研究SU（n）群的拓扑结构和曲率之间的关系来估算，这为理解和预测量子系统的动态复杂性提供了一个实用的工具。论文的贡献不仅在于理论上的拓展，还在于它为实际应用提供了基础，比如在量子信息处理和量子计算中，复杂性是理解和优化算法的关键参数。此外，这项工作还为未来研究开辟了新的方向，包括探索不同量子场论模型中算子复杂性的差异，以及如何利用这些理论来更好地理解物理系统的性质。这篇论文深化了我们对量子场论中算子复杂性的认识，提供了一种新的量化方法，并且展示了这一理论在实际物理问题中的潜在应用。通过结合几何和量子场论的工具，研究人员为理解复杂量子系统提供了新的视角和理论工具。

JHEP03(2019)161



Figure 2. A continuous curve c(s) connects the identity (c(0) =

I) and a particular operator

(c(1) =

O). It can be approximated by a discrete form, where every intermediate point (

) is also

an operator.

All information regarding the complexity has been encoded in the function

F so the

cost (L

[c]) of a curve c can be deﬁned as

[c] :=

i=1

C(δ

(α)

)

N→∞

−−−−→

F (H

(s))ds , (2.6)

where we assume that the curve is constructed by only δ

(r)

or only δ

(l)

. In a geometric

picture, it is the length of a given curve and

F ds is considered as a line element in a

geometry. Thus, a natural question is what kind of geometry can be allowed for complexity?

It has been shown that it is the Finsler geometry as follows.

First, it has been proven that three axioms G1,G2, and G4a implies that the function

F introduced in G4a satisﬁes

F1 (Nonnegativity)

F (H

) ≥ 0 and

F (0) = 0

F2 (Positive homogeneity) ∀λ ∈ R

F (λH

) = λ

F (H

)

F3 (Triangle inequality)

F (H

α,1

) +

F (H

α,2

) ≥

F (H

α,1

+ H

α,2

) ,

which are the deﬁning properties of so called Minkowski norm in mathematics literature.

The conditions, F1-F3, characterize properties of a norm of the vector (the generators

)) in the Lie algebra, the tangent space at the identity. The Finsler metric is nothing

but a Minkowskia norm deﬁned at all points on the base manifold. There are two natural

but diﬀerent ways to extend the Minkowski norm

F at the identity to every point on the

base manifold:

(c, ˙c) :=

F (H

) =

F ( ˙cc

−1

) , or F

(c, ˙c) :=

F (H

) =

F (c

−1

˙c) . (2.7)

Here, we introduced ‘F

(c, ˙c)’, a standard notation for the Finsler metric in mathematics

literatures. In this paper, we will simply call both

F (H

) and F

(c, ˙c) a ‘Finsler metric’

Strictly speaking, the requirements of the Minkowski norm are a little more stronger than F1-F3.

However, these diﬀerences will not aﬀect our results in this paper.

– 6 –

JHEP03(2019)161

if there is no confusion. We refer to refs. [54–57] for more detailed explanation of the

Minkowski norm and Finsler geometry.

Note that there is symmetry in the Finsler metrics. F

(c, ˙c) (F

(c, ˙c)) is right(left)-

invariant because H

) is invariant under the right(left)-translation c → cˆx(c → ˆxc)

for ∀ˆx ∈ SU(n).

At this stage, any right or left invariant Finsler metric will be allowed for ‘complexity

geometry’. By requiring that the complexity obeys some symmetries of quantum ﬁeld

theory the Finsler metric

F can be more constrained. Indeed, the unitary invariance and

the CPT symmetry of QFT respectively impose the following constraints on

[adjoint invariance]

F (H

) =

F (

†

) , (2.8)

[reversibility]

F (H

) =

F (−H

) , (2.9)

for ∀U ∈ SU(n). It has been shown that eq. (2.8) implies

[Independence of left/right generators]

F (H

) =

F (H

) , (2.10)

so the Finsler metric is bi-invariant, which means both left and right invariant.

The constraints eq. (2.8) and eq. (2.9) (together with the axioms G1-G4a) turn out

to be strong enough to determine the complexity of SU(n) operators uniquely (up to an

overall constant λ) as

F (c(s), ˙c(s)) =

F (H) = λTr

H(s)H

†

(s), H(s) := H

(s) or H

(s) , (2.11)

where c(s) is a curve in SU(n) group and λ is a positive constant. Note that the subscript

α in H

is dropped because H

or H

give the same Finsler metric (see also eq. (2.10)).

Finally, the complexity of an operator is deﬁned by the minimal length (or minimal

cost) of the curves connecting

I and

O) := min

L[c]| ∀c(s), c(0) =

I, c(1) =

. (2.12)

Therefore, we are left with the variational problem minimizing

H(s)H

†

(s)ds, with c(0) =

I, and c(1) =

O . (2.13)

This minimization problem is simpliﬁed thanks to bi-invariance of the Finsler metric ex-

plained below eq. (2.10). It has been shown in refs. [58, 59] that, if the Finsler metric

is bi-invariant, the curve c(s) is a geodesic if and only if there is a constant generator

H(s) =

H such that

˙c(s) =

Hc(s) or c(s) = exp(s

H) . (2.14)

In our case

H = ln

O, by the boundary condition

O = c(1) = exp(

H). Therefore,

O) = min

†

| ∀

H = ln

, (2.15)

where ‘min’ is not in the sense of a variational problem. It picks up the minimal value

among the multi-values of ln

– 7 –

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