Perturbative Complexity of Interacting Theory

We present a systematic method to expand the quantum complexity of interacting theory in series of coupling constant. The complexity is evaluated by the operator approach in which the transformation matrix between the second quantization operators of reference state and the target state defines the quantum gate. We start with two coupled oscillators and perturbatively evaluate the geodesic length of the associated group manifold of gate matrix. Next, we generalize the analysis to $N$ coupled oscillators which describes the lattice $\lambda\phi^4$ theory. Especially, we introduce simple diagrams to represent the perturbative series and construct simple rules to efficiently calculate the complexity. General formulae are obtained for the higher-order complexity of excited states. We present several diagrams to illuminate the properties of complexity and show that the interaction correction to complexity may be positive or negative depending on the magnitude of reference-state frequency.


Introduction
Achieving a better understanding of physics behind a black-hole horizon is important if one wants to precisely describe the bulk geometry in terms of the information of boundary CFT [1,2,3,4,5,6,7]. In the context of the eternal AdS-Schwarzchild black hole, for instance, a related question concerns the size of a wormhole growing linearly with time; this phenomenon has been conjectured to dual to the growth of "complexity" of the dual CFT [8]. In the complexity=volume (CV) conjecture [8], the complexity is dual to the volume of an extremal codimension-one bulk surface anchored to a time slice of the boundary. In the complexity=action (CA) conjecture [9,10,11,12], one identifies the complexity with a gravitational action evaluated on the Wheeler-DeWitt (WDW) patch, anchored also on a time slice of the boundary.
Several efforts were made to provide a definition of the complexity in the field theory [13,14,15,16,17]. The complexity in there is defined as the number of operations {O I } needed to transform a reference state |ψ R to a target state |ψ T . These operators are also called as quantum gates: the more gates one needs, the more complex the target state is. One can define an affine parameter "s" associated to an unitary operator U (s) and use a set of function, Y I (s), to character the quantum circuit. The unitary operation connecting the reference state and target state is where P is a time ordering along s. The complexity C and circuit depth D[U ] (cost function) are [13] C = Min Above definitions were shown to be consistent with a gravitational computation [13]. The initial studies in field theory considered the Gaussian ground state wavefunctions in reference state and target state [13,15,16]. The theories studied so far are the free field theory or exponential type wavefunction in interacting model [17]. The operator approach had also been used in [14,16] to study the complexity of fermion theory. In our previous paper [18] we adopt the operator approach, in which the transformation matrix between the second quantization operators of reference state and target state is regarded as the quantum gate, to evaluate the complexity in free scalar field theory. Since that in the operator approach we need not to use the explicit form of the wave function we can study the complexity in the excited states 1 . We first examined the system in which the reference state is two oscillators with same frequency ω f while the target state is two oscillators with frequency ω 1 and ω 2 . We explicitly calculated the complexity in several excited states and proved that the square of geodesic length in the general state |N 1 , N 2 is The results was furthermore extended to the N couple harmonic oscillators which correspond to the lattice version of free scalar field, see sec.5 of [18].
In this paper we extend [18] by including interactions to further study the complexity using the operator approach. We present a systematic method to evaluate the complexity of the λφ 4 field theory by the perturbation of small coupling constant. An outline of the paper is as follows.
In section II, as that in [13] we describes the lattice scalar field as coupled oscillators. In section III we consider two coupled oscillators and find that, to the λ n order the square distance of excited state between target and reference state is and R (n) 2 are described in (3.24). In section IV we generalize it to the case of N coupled oscillators which correspond to the lattice version of λφ 4 theory. We use new kind of simple diagrams, figures 3, 4 and 5, to represent the perturbative series and construct simple rules, figures 1 and 2, to calculate the complexity therein. We find that the diagrams are classified into three classes : odd N , odd N 2 , and even N 2 . We explicitly calculate the complexity in the cases of N=2,3,4, 5 to any order of λ. Using these experiences we then in section V derive the general formulas of complexity in (5.5), (5.12), and (5.17). Then, we present several diagrams to illuminate the properties of complexity and find that the interaction correction to complexity may be positive or negative depending on the magnitude of reference-state frequency. We conclude in Sec. 6.

Interacting Scalar Field and Coupled Oscillators
The d-dimensional massive scalar Hamiltonian with aλφ 4 interaction is Placing the theory on a square lattice with lattice spacing δ, one has whereâ i are unit vectors pointing toward the spatial directions of the lattice. By redefining the Hamiltonian becomes

4)
When n is an one dimensional vector the Hamiltonian describes an infinite family of coupled one dimensional oscillators. We will extensively study the one dimensional oscillators in this paper while the extension to higher dim is just to replace the site index " i " to " i ", as that described in [13].

Two Coupled Oscillators
First we consider a simple case of two coupled oscillators (M = 1): the Hamiltonian is In the second quantization, we define The state wavefunction is ψ(

Kinetic Term of Two Coupled Oscillators
The kinetic term has a diagonal form: where the constant terms are irrelevant to the following discussions. We choose the reference state with the associated kinetic term given by [13] Note that one can obtain In the operator approach, the gate matrices defined in (1.1) for operators which are simply the U(1) group elements. Using (1.2), the square distance between target and reference state for the gate matrix Y i is D 2 i = (Y i ) 2 . As the ground state is annihilated by a 1 , a 2 , i.e. a 1 a 2 |0, 0 = 0 for target state, and a → ω 2 ω f a 2 . The squared distance between target and reference state calculated from the two gate matrices is given by , 0 , the square distance between target and reference state is (3.10) This matches with the result obtained earlier in [18]. Recall that the state wavefunction is described by Ψ n (x) = 1 √ n! x|(a † ) n |0 the gate matrix of excited-state wavefunction, Ψ n (x), is thus related to the gate matrix of field operators, (a † ) n .

Interacting Term of Two Coupled Oscillators
We next study the correction to the complexity due to the interaction term: We will consider N 1 , N 2 |V |N 1 , N 2 for the excited state |N 1 , N 2 with fixed N 1 and N 2 . In this way, only the terms that have the same power of a i and a † i are relevant. Therefore we only need to consider We obtain, after dropping irrelevant terms, The associated Hamiltonian of the reference state can be chosen as In the case of zero-order of λ, The quantum gate are described by two 1 × 1 matrices, exp R . This is the case of purely kinetic term, i.e. a free theory. Now consider a perturbation to the complexity for the two coupled oscillators. At the first order of λ, we have transformations (3.20) The factors N (1,2) are within the coupling term, i.e. 3λ 2 , and we only need to consider their zero-order transform. Recall (3.18), we have to multiple them by R (0) (1,2) factors. Therefore the first-order transformations are For excited states, the first-order square distance is (3.23) Extending to higher-order interactions is straightforward. The recursion relations are with initial values R (1,2) defined in (3.19). For excited states, the n-order square distance is which is the n-order complexity of two coupled oscillators. Note that original relations (3.24) can be expanded as In this way, the perturbative series of R

Kinetic Term of N Coupled Oscillators
For N coupled oscillators, We impose a periodic boundary conditionx k+N+1 =x k . The normal coordinates are chosen to be Note that the relative sign between the Fourier series of x k and p k is important to have standard commuation relation [x k 1 , p k 2 ] = δ k 1 ,k 2 [13]. The Hamiltonian now becomes the kinetic term can be written as up to an irrelevant constant. The states in N oscillators can be defined by the creation operators a † 1 a † 2 ...a † k ...,such that ψ(x 1 , x 2 , ...) = x 1 , x 2 , ..x k ....|a † 1 a † 2 ...a † k ...|0 . As before, to find the complexity of such state we choose a reference state with the associated kinetic term given by (4.6) The square distance for the n k -th excited state is where ω k is defined in (4.3).

Interacting Term of N Coupled Oscillators : Perturbative Algorithm
We adopt the following steps to systematically study a perturbation theory of the complexity: (I) We express potential V in terms of a, a † : Then, as calculated in (3.12) and (3.13), which lead to two relations that will be extensively used in later calculations 2 The term a † j a j a † j a j in (4.10) is written as N j a † j a j in (4.12), as we did in sec. 3.2. Sec.3.2 also tells us that we will let N j → R (n−1) j N j in calculating the complexity at the n'th order of λ, (III) Adopting the series expansion (4.8), we can develop diagrammatic rules based on two basic elements, "circle" and "pair", which appear in (4.12) and (4.13). We plot them in figure  1   pairing with "j" once and assign the "pair" element A(i) 2 A(j) 2 pairing with each "i" and "j" once, then the odd N diagrams have pairings in each "j" once while the even N diagrams have pairings for each "j" twice.

Interacting Term of N Coupled Oscillators : Some Calculations
We now take several values of N as examples to plot the diagrams and use (4.12) and (4.13) to calculate the associated complexity. General formulae will be presented in the next section.
• N=2: 0 Figure 6: N=2 diagram As shown in figure 6 the series expansion (4.8) is (4.14) We have used (4.12) and (4.13). The above result matches with (3.14). The associated complexity can be evaluated to any order in λ: with initial values R (1,2) defined in (3.19). For excited states, the n-order squared distance is , which is the n-order complexity of 2 coupled oscillators. While above results exactly match (3.24) we have expressed them in the new form that helps us to identify rules for computing a general N result.
We have recurrent relations For excited states, D (n)2 , which is the n-order complexity of 3 coupled oscillators.
• N=4: We have recurrent relations , which is the n-order complexity of 4 coupled oscillators.
• N=5: 0 Figure 9: N=5 diagram As shown in figure 9 the series expansion (4.8) is We have recurrent relations For excited states, D (n)2 , which is the n-order complexity of 5 coupled oscillators.
• N=6: 0 Figure 10: N=6 diagram As shown in figure 10 the series expansion (4.8) is • N=7: 0 Figure 11: N=7 diagram As shown in figure 11 the series expansion (4.8) is • N=8: 0 Figure 12: N=8 diagram As shown in figure 12 the series expansion (4.8) is With these experiences we will in the next section derive general formulae of the complexity for any N to any order in λ.

Complexity of N Coupled Oscillators : General Formulae
From the above analysis and relations (4.12) and (4.13), we find where "cirle" and "pair" can be read from diagrams; see figures 3, 4, and 5.
• Odd N : We recall, from Sec 4.2, the odd N case is simplest as it only has one "circle" located at N , and each "pair" is independent to each other (figure 3). Eq(5.1) becomes the n-order complexity is • Odd N 2 : These cases have two "circle" located at N 2 and N , pairing with each other ( figure  4). The potential is The remaining contributions are those from pure "pairing" sites. Recalling the figure 2 and the relation (4.13) we can evaluate the corresponding potential. The result is By adding the kinematic term (4.5) and defining the recursion relations the n-order complexity is • Even N 2 : These cases have two "circle" locate at N 2 and N , pairing with each other, and two "circle" locate at N 4 and 3N 4 , pairing with each other as well ( figure 5). The potential is Again, the remaining contributions are those from pure "pairing" sites. We find The above result is the same as the odd N 2 , i.e. (5.7), but drop the "circle" at N 4 and 3N 4 since the potential of the two "circle" has been considered in V "circle" even N 2 . By adding the kinematic term (4.5) and defining the recursion relations These general formulae allow one to obtain higher-order complexity for excited states at any N coupled oscillators, which is a lattice version of λφ 4 theory.

Complexity of N Coupled Oscillators : Numerical Results
We now use above formulas to perform numerical calculations and plot several diagrams to illuminate the properties of complexity.
(1) We plot in figure 13 the complexity for various lattice site number N .  Figure 16 : Complexity v.s. coupling constant λ. Left-hand diagram is that with ω = 10, ω f = 0.01. Right-hand diagram is that with ω = ω f = 1.
It shows that the complexity may increase or decrease while increasing coupling constant λ.
The property of how the complexity depends on λ can be see, for example, from eq.(3.26) and eq.(3.27). The interaction correction to complexity in the two relations is proportional the coefficient of 3λ 2ω f , which is negative for large ω f and become positive for small ω f . Figure 16 is consistent with this argument.

Concluding Remarks
We adopt operator approach to compute the complexity of the lattice λφ 4 scalar theory. A perturbation algorithm has been developed for computing the complexity to obtain the general formulae (5.5), (5.12), and (5.17) which can be used to obtain higher-order complexity of excited states for any N lattice sites. The interaction correction to complexity may be positive or negative depending on the magnitude of reference-state frequency.
We conclude the paper by the remark : Our algorithm is based on a simple relation λa † j a j a † j a j → λN j a † j a j → λN j R (n−1) j a † j a j (6.1) in which the first arrow is due to the perturbation property while the second one is use to calculate the complexity. The relation is explained in sec.3.2. The similar relation could be found in many other systems. For examples : • It is easily to see that our method could be used in interacting Fermion theory.
• For the theory which has two different field operators a j and b j and associated interaction is λ φ 2 ξ 2 the relation will become in which the fields φ and ξ could be Boson or Fermion field.
• For the λφ 6 theory the relation will become λa † j a j a † j a j a † j a j → λ(N j ) 2 a † j a j → λ(N j R (n−1) j ) 2 a † j a j (6.3) Of course, the associated diagrams and basic rules in each case shall be slightly modified.
In this way, our algorithm can be applied to many quantum field theories and several manybody models in condense matter. We will study the problem in the next series of paper.