Circuit complexity for free fermions

Abstract

We study circuit complexity for free fermionic field theories and Gaussian states. Our definition of circuit complexity is based on the notion of geodesic distance on the Lie group of special orthogonal transformations equipped with a right-invariant metric. After analyzing the differences and similarities to bosonic circuit complexity, we develop a comprehensive mathematical framework to compute circuit complexity between arbitrary fermionic Gaussian states. We apply this framework to the free Dirac field in four dimensions where we compute the circuit complexity of the Dirac ground state with respect to several classes of spatially unentangled reference states. Moreover, we show that our methods can also be applied to compute the complexity of excited states. Finally, we discuss the relation of our results to alternative approaches based on the Fubini-Study metric, the relevance to holography and possible extensions.Let us also note that there is no need to consider a tolerance with this continuous description, since the Y (s) can always be adjusted to produce exactly the desired transformation (2.1). 4 Note that our notation here differs slightly from that in [32] where the overall factor of −i was absorbed in the O I , which were then anti-Hermitian operators. 5 We define the boundary conditions more precisely below in the discussion around eq. (2.11).8 To correctly account for the dimensions of q and p, these expressions should include a specific scale, e.g., a = 1 √ 2 (ω 1 q + i p/ω 1 ) yields a properly dimensionless annihilation operator. One effect of the Bogoliubov transformations (3.5) is then to scale this scale, e.g., ω 1 → e r ω 1 with ϕ = ϑ = 0 in eq. (3.7). See [38] for further discussion.9 Note that we can change a toã = e iϕ a without changing the vacuum, which corresponds to a U(1) subgroup of Bogoliubov transformations that do not change the vacuum. For N bosonic degrees of freedom, there is the freedom of unitarily mixing all N annihilation operators (and creation operators respectively) among themselves, leading to a U(N ) subgroup of different choices of a i that all define the same vacuum. 20 Of course, the symmetric combinations are trivial, since {ξ a , ξ b } = G ab ≡ δ ab .

Aspects of the first law of complexity

“…However, it is natural that this restricted basis should form a closed algebra, and typically, the O I provide a representation of a Lie algebra g, i.e., [O I , O J ] = if IJ K O K . For example, a GL(N , R) group appears in evaluating the complexity of the ground state of a free scalar field [57], and the latter was extended to a Sp(2N , R) group in examining the corresponding thermofield double state [64] -see also [60]. 5 In the following, we will find that the affine symplectic group, i.e., R 2N Sp(2N , R) plays a central role in evaluating the complexity of the coherent states of interest.…”

“…(2.6). Another interesting suggestion in [60] was to construct a family of new cost functions using the Schatten norm (e.g., see [128][129][130])…”

“…This approach was first applied to a concrete quantum field theory calculation in [57], where the authors adapted Nielsen's approach to evaluate the complexity of the vacuum state of a free scalar field theory. These calculations have been extended in a number of interesting ways in the past few years, e.g., [58][59][60][61][62][63][64][65][66][67][68][69][70][71][72][73][74][75][76], but we will be particularly interested in [65] where the same techniques were applied to explore the complexity of coherent states in the same QFT.…”

Holographic complexity in Vaidya spacetimes. Part I

“…One possible way to improve the comparison of the holographic and QFT quenches would be to consider CFT-to-CFT quenches for a free scalar (in which the initial and final masses both vanish) using the protocol described in section 3.2 of [88]. Another simple extension of this work would be to study the complexity for a mass quench of a free fermion, using the techniques of [46].…”

“…Some initial studies of this question appear in [98,99], which examine the evolution of the complexity through a mass quench in a free scalar field theory (analogous to those studied in [86][87][88]). A remarkable feature of these quenches is that the scalar field remains in a Gaussian state throughout the entire process, and so methods developed in [44][45][46] can still be applied to evaluate the complexity. The comparison of our holographic results with those in [98] is not straightforward since, e.g., the initial and final masses are nonvanishing (i.e., neither the initial nor final scalar theories are CFTs).…”