自由费米子的电路复杂性分析

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"自由费米子的电路复杂性——一项关于自由费米子场论和高斯态的电路复杂性的研究。" 这篇论文探讨了在量子信息理论和量子计算领域中的一个重要概念——电路复杂性，特别是在自由费米子场论的背景下。电路复杂性是衡量构建一个特定量子态所需的最简单量子门序列的复杂度，这在理解量子系统演化、黑洞信息处理以及量子计算的效率等方面具有重要意义。作者首先介绍了他们的研究基础，即电路复杂性的定义，它是基于特殊正交变换群上的测地距离，该群配备了一个右不变度量。这个定义与玻色子系统的电路复杂性进行了比较和对比，揭示了两者的异同。在玻色子系统中，复杂性通常与构建特定态所需的最小操作数量相关，而在费米子系统中，由于泡利不相容原理，这个概念会有所变化。接着，论文提出了一种全面的数学框架，该框架能够计算任意两个费米子高斯态之间的电路复杂性。高斯态是量子场论中一类重要的量子态，它们具有简洁的统计特性，这使得计算其复杂性成为可能。作者应用这个框架来研究四维自由Dirac场，并计算了针对不同空间非纠缠参考状态的Dirac基态的电路复杂性。Dirac场是量子场论中描述自旋半整数粒子（如电子）的基本模型。此外，论文还证明了这个框架可以用来评估自由狄拉克场激发态的复杂性，这意味着它不仅适用于基态，也适用于包含粒子或反粒子的激发态。这扩展了电路复杂性理论的应用范围，对于理解和量化这些量子态的结构提供了新的工具。最后，作者讨论了他们的方法与基于Fubini-Study度量的替代方法之间的关系，Fubini-Study度量是量子态纯度的一个度量，通常用于比较和分析量子态。他们还探讨了电路复杂性与全息原理（如AdS/CFT对应）的关联，全息原理是一个强大的工具，它将高维引力理论与低维量子场论联系起来。同时，他们提出了可能的研究扩展方向，暗示这个领域的未来工作可能涉及更复杂的量子场论模型和更多的应用。这篇论文的发表，对于理解自由费米子系统的复杂性提供了新的见解，同时也为量子信息科学和量子计算的进一步发展奠定了理论基础。通过深入研究电路复杂性，我们可以更深入地探索量子世界，尤其是在解决量子计算的效率问题和黑洞信息悖论方面。

JHEP07(2018)139

for every Gaussian state |Gi, we can choose a canonical basis ξ

≡ (q

, p

), such that

G ≡ 1. This means the bilinear form G does not contain information that is invariant

under changing the canonical basis or put simply: “All Gaussian states look the same if

we can choose the right basis for each individual state.” This changes of course, if we have

two Gaussian states |Gi and |

Gi in the same system

and force ourselves to represent

the two-point functions G and

G with respect to the same canonical basis. Again, we

can choose a basis, such that G ≡ 1, but we will not be able to accomplish the same

for

G. The remaining freedom of choosing a canonical basis is described by the group

U(N) = Sp(2N, R) ∩ SO(2N ) consisting of canonical transformation (i.e., MΩM

= Ω)

that simultaneously orthogonal with respect to G (i.e., MGM

= G). The invariant

information about the relation between the original state |ψi and the transformed state

ψi is completely captured by the eigenvalues of the relative covariance matrix

∆

with g = G

−1

, (3.12)

i.e., G

= δ

. In particular, any quantities that depend on the two states in a

Sp(2N, R)-invariant way, e.g., their inner product,

can be computed purely from ∆.

This will apply to the complexity provided that we choose a geometry that is Sp(2N, R)-

invariant, e.g., we do not introduce penalty factors which conﬂict with the group structure.

For our Bogoliubov transformation (3.5), we have spec(∆) = (e

, e

−2r

). We say that |

ψi

arises from a one-mode squeezing of |ψi with squeezing parameter r. For bosonic Gaussian

states, understanding one-mode squeezing is the key to relate any two states. That is, for

any two bosonic Gaussian states |ψi and |

ψi with N degrees of freedom, there exists a

normal mode basis (q

, ··· , q

, p

, ··· , p

), such that |

ψi is the result of N independent

one-mode squeezing operations in the N diﬀerent normal modes [39, 51, 52]. This is related

to the Iwasawa (or KAN) decomposition of Sp(2N, R), e.g., see [56, 57].

3.2 Two fermions

We now turn to the case of fermionic Gaussian states. In this case, the space of Gaussian

states for N fermionic degrees of freedom is given by the quotient M

f,N

= O(2N)/U(N),

which has dimension N(N − 1), e.g., [52]. Of course, this space is a small submanifold

within the full 2

-dimensional Hilbert space H of the fermionic system. Further, it is not

preserved by general unitary transformations U(2

) acting on H, but only the subgroup

O(2N) corresponding to Bogoliubov transformations. That is, the most straightforward

way to think of characterizing the fermionic Gaussian states is in terms of the annihilation

and creation operators. With N fermionic pairs (a

, a

†

) satisfying {a

, a

†

} = δ

, the corre-

sponding Gaussian state is again deﬁned by a

|ψi = 0 and the Bogoliubov transformations

mixing these fermionic operators map Gaussian states to Gaussian states.

Of course, this is the situation where we are examining circuit complexity of states since we have both

the target state and the reference state.

Note that one could have just as easily deﬁned

∆ = G ˜g with ˜g =

−1

. However, one then has

∆ = ∆

−1

and due to the fact that ∆ is symplectic, the two have the same spectrum. This is discussed in more detail

in section 5.

For bosonic states, we ﬁnd the simple formula |hG|

Gi|

= det

√

2∆

1/4

√

1+∆

derived in [52].

– 11 –

JHEP07(2018)139

In analogy to eq. (3.4) for the bosons, we begin by deﬁning a set of Hermitian fermionic

operators given by

√



†

+ a



and p

√



†

− a



, (3.13)

which are commonly referred to as Majorana modes. In contrast to the analogous bosonic

operators, they do not consist of conjugate pairs (q

, p

), but rather they are governed by

the anti-commutation relations: {q

, q

} = δ

= {p

, p

} and {q

, p

} = 0. Turning to the

covariance matrix (3.1), if we choose the Majorana basis ξ

≡ (q

, ··· , q

, p

, ··· , p

), the

symmetric component becomes simply

= hψ|{ξ

, ξ

}|ψi = δ

. (3.14)

This result holds for any Gaussian state since G simply encodes the canonical anti-

commutation relations G

= {ξ

, ξ

} ≡ δ

(which are preserved by the Bogoliubov

transformations). Further, as we will see below, this matrix G

provides a useful pos-

itive deﬁnite metric. Hence, in the fermionic case, the nontrivial component of eq. (3.1) is

the antisymmetric two-point correlator

Ω

= −i hψ|[ξ

, ξ

] |ψi, (3.15)

which characterizes the corresponding Gaussian state |ψi. Given eq. (3.13) above, we may

evaluate this matrix for the state |ψi annihilated by a

Ω ≡

0 1

−1 0

, (3.16)

where 1 and 0 are N ×N identity and zero matrices, respectively. We note that this Ω

coincides with the form of the symplectic form (3.2) for bosons.

Now in analogy with our discussion of bosons, a pair (a

, a

†

) and (˜a

, ˜a

†

) deﬁnes two

distinct Gaussian states satisfying a

|ψi = 0 and ˜a

ψi = 0. Hence understanding the

group of transformations mapping fermionic Gaussian states amongst themselves is again

understanding the Bogoliubov transformations acting on the fermionic annihilation and

creation operators. It is simplest to work with the Majorana basis, i.e.,

≡ (˜q

, ˜p

) and

≡ (q

, p

), where the Bogoliubov transformations act as a linear transformation. Again,

we deﬁne the inverse transformation M, such that ξ

= M

. The condition of preserving

the anti-commutation relations translates into

(M G M

)

= M

)

= G

. (3.17)

Recalling that G

≡ δ

in the Majorana basis, eq. (3.17) makes evident the O(2N) group

structure, which we referred to above. Of course, the transformation of the states is now

encoded in the transformation of the antisymmetric two-point correlator

Ω

= (MΩM

)

= M

Ω

)

. (3.18)

– 12 –

JHEP07(2018)139

Hence in a discussion of the circuit complexity of fermionic Gaussian states, we can

represent the unitary circuits and gates with the appropriate orthogonal transforma-

tions and their generators, and describe their action on the states in terms of the above

transformation.

To make this discussion more concrete, let us consider a simple example. However (as

we now show), the simplest case of a single pair, i.e., N = 1, turns out to be trivial. In

this case, the most general Bogoliubov transformation is

˜a = α a + β a

†

, (3.19)

˜a

†

= α

∗

†

+ β

∗

a .

Demanding that the anti-commutation relation is preserved, i.e., {˜a, ˜a

†

} = 1, yields

|α|

+ |β|

= 1 . (3.20)

However, fermionic creation and annihilation operators also need to satisfy ˜a

= (˜a

†

)

= 0.

Computing this explicitly for above transformation leads to a second requirement

˜a

= αβ{a, a

†

} = 2 α β = 0 . (3.21)

This means up to an overall phase, the only possible transformations are α = 1, β = 0 or

α = 0, β = 1. That is, ˜a = a or we swap the role of creation and annihilation operators

with ˜a = a

†

. With N = 1, the space of Gaussian states is M = O(2)/U(1), where the U(1)

corresponds to the overall complex phase, but this space simply consists of two points.

This means that — in contrast to a single bosonic degree of freedom — the squeezing

of a single fermionic degree of freedom is trivial. The ﬁrst non-trivial system consists of

two fermionic degrees of freedom, often interpreted as two qubits. With N = 2, the state

manifold will be

f,2

= O(4)/U(2) = S

∪ S

, (3.22)

which is two-dimensional. In this case, we consider two pairs fermionic creation and an-

nihilation operators, (a

, a

†

) and (a

, a

†

). For this example, let us consider the fermionic

Bogoliubov transformation

˜a

= α a

− β a

†

, (3.23)

˜a

†

= β

∗

+ α

∗

†

This is not the most general transformation, but the natural choice if we want to mix a

with a

†

. In fact, one can show that one can bring any Bogoliubov transformation into

this form by mixing a

with a

, and ˜a

with ˜a

via U(2), which does not change the

corresponding Gaussian states, |ψi and |

ψi.

It will be a general feature (for any N ) that the full set of fermionic Gaussian states always consists of

two disconnected components corresponding to the Z

grading of states with even and odd fermion number.

Note that neither of the two components is preferred and which corresponds to an even and odd fermion

number depends on one’s choice of the vacuum, or alternatively on one’s notion of particle.

– 13 –

JHEP07(2018)139

Further, for eq. (3.23), we may choose α to be real so that the following parametrization

works well:

α = cos ϑ , β = e

iϕ

sin ϑ . (3.24)

The induced transformation M that maps

into ξ

can then be written as

M ≡







1 0 0 0

0 cos(ϕ) 0 −sin(ϕ)

0 0 1 0

0 sin(ϕ) 0 cos(ϕ)













cos(ϑ) sin(ϑ) 0 0

−sin(ϑ) cos(ϑ) 0 0

0 0 cos(ϑ) −sin(ϑ)

0 0 sin(ϑ) cos(ϑ)













1 0 0 0

0 cos(ϕ) 0 sin(ϕ)

0 0 1 0

0 −sin(ϕ) 0 cos(ϕ)













cos(ϑ) sin(ϑ) cos(ϕ) 0 sin(ϑ)sin(ϕ)

−sin(ϑ)cos(ϕ) cos(ϑ) −sin(ϑ)sin(ϕ) 0

0 sin(ϑ)sin(ϕ) cos(ϑ) −sin(ϑ)cos(ϕ)

−sin(ϑ)sin(ϕ) 0 sin(ϑ)cos(ϕ) cos(ϑ)







(3.25)

Here, we have decomposed M as a series of rotations and so it is clear that M ∈ O(4) or

rather M ∈ SO(4), because we can continuously reach 1, and satisﬁes MGM

= G. The

antisymmetric covariance matrix

Ω = MΩM

of the transformed state |

ψi can then be

evaluated to be

Ω ≡







0 −sin(2ϑ) sin(ϕ) cos(2ϑ) sin(2ϑ) cos(ϕ)

sin(2ϑ) sin(ϕ) 0 −sin(2ϑ) cos(ϕ) cos(2ϑ)

−cos(2ϑ) sin(2ϑ) cos(ϕ) 0 sin(2ϑ) sin(ϕ)

−sin(2ϑ) cos(ϕ) −cos(2ϑ) −sin(2ϑ) sin(ϕ) 0







. (3.26)

Note that we get the same state for ϑ = 0 and ϑ = π, which is perhaps half the expected

range. This is due to the fact that the transformation with ϑ = π leads to ˜a

= −a

and ˜a

= −a

, which leaves the vacuum invariant. Therefore the state which is most

distant

from the original Gaussian state |ψi corresponds ϑ = π/2, which we see trades

the annihilation and creation operators, i.e., eq. (3.23) reduces to (˜a

, ˜a

) = (−˜a

†

, ˜a

†

) with

ϑ = π/2 (and ϕ = 0).

As mentioned in eq. (3.22), the space of states is given by the quotient M

f,2

O(4)/U(2) = S

∪ S

because we need to divide by the subgroup U(2) associated to

mixing creation and annihilations operators among themselves, respectively. In particular,

we see that the manifold of fermionic Gaussian states again consists of two disconnected

components — see footnote 15. We can only continuously deform one state to the other,

if they lie in the same component — unless we are willing to leave the space of Gaussian

states. Our choice of Bogoliubov transformations parametrized by ϑ and ϕ corresponds to

the S

connected to the identity.

Of course, we mean ‘most distant’ on the S

component connected to the identity. We cannot reach the

states on the other component along a continuous trajectory without leaving the space of Gaussian states.

– 14 –

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