Circuit complexity has become a helping hand to not only the high-energy physics community but also to the people from other branches as well [1–24]. This quantum information theory technique has been significantly used recently to probe many features which were previously difficult. Though this concept is a computation tool, its contribution in the field of physics of late is massive. It provides a way to probe physics behind the horizon of black holes through the use of the “Complexity=Volume” and “Complexity=Action” conjectures [25–29]. Since then it has been extensively used in quantum field theory and in studies involving AdS/CFT correspondence [30]. These holographic approaches connect a probe on the gravity side with a concept of quantum information theory.

In the recent past, along with the out of time ordered correlation functions [31–35], it has formed the web of quantum chaos. It has been found to reveal essential information like Lyapunov exponent, scrambling time etc., required to diagnose chaos in a system. Many interesting works have been done using this tool in wide areas of physics. It was studied for cosmological islands in [36], where the authors showed that entanglement entropy from circuit complexity via a famous relation proposed in [25] resembles the page curve in some particular regime where a universal relation between circuit complexity, out-of-time-ordered-correlation (OTOC) and entanglement entropy can be written. It was used to study early Universe chaos within the framework of bouncing cosmology [37]. People have also computed circuit complexity in the context of supersymmetric quantum field theory [38]. The connection between entanglement and emergence of space-time has been an active area of study, where the entanglement entropy is the minimum cross-sectional area of an Einstein-Rosen Bridge (ERB). However, classically the ERB continues to grow for a very long time, whereas the dual thermodynamic system comes to a thermal equilibrium quickly. This led Susskind to introduce a new variable namely ‘complexity’ which could be responsible for the ERB growth [25]. Complexity of a quantum system is a real quantity and its growth rate has been conjectured to be proportional to the entropy of the black hole based on these observations. It was very recently shown in [39] that there exists some relationship between entangling power and circuit complexity. Most importantly, if the entanglement entropy grows linearly with time, the geometric circuit complexity also grows linearly.

In this paper, we will study the evolution of complexity with respect to the entanglement entropy of the black hole gas model [40]. An important feature of this model is that the total entropy of the black hole gas is directly proportional to the volume of the system instead of the area and the system behaves like a thermodynamic gas. We will compute the most common measures for determining the entanglement between the squeezed states of black hole gas, namely the von Neumann and Rényi entropy, which quantifies the amount of uncertainty linked to the density matrix. We will measure the entanglement entropy by constructing an effective thermal representation for the reduced single-mode state from the two-mode squeezed state which varies linearly with respect to squeezing parameter r. Then by using the scale factor predicted by the black hole gas model in the flat space-time metric as a dynamical variable we will study the evolution of complexity in three spatial dimensions and compare it with the entanglement entropy in terms of squeezed state parameters. One of the reasons we are interested in the black hole gas is that its equation of state describes a universe right after the big bang and before the start of inflation, if one wants to avoid such an equation of state governing radiations of very high densities then one needs to have inflation right from the Planck scale. The key highlights of this paper are as follows: (a)

The behavior of circuit complexity calculated from two different approaches viz the covariance matrix method and Nielsen’s wave function method has been studied with respect to scale factor for the black hole gas model. We observe interesting behaviors for different spatial dimensions.

In this section, we review a model proposed in Ref. [40], where the author has studied the state of a system moving towards maximal entropy S. Note that this system is unlike the inflationary model where we have a low entropy state after the inflation because the positive energy of the matter content is compensated by the gravitational potential. Here, we provide a quick derivation of the equation of state that describes the preinflationary Universe. We will consider a configuration where we find entropy S(E,V) of a system in a toroidal box of volume V in the limit E→∞. At low enough energies the matter phase corresponds to radiation whose entropy as a function of E and V in dimension d can be given by S≈Vρdd+1. By putting more energy into the box one can look for a configuration where the system turns into a black hole of radius R whose entropy is given by Shole=AG.(1)One might suspect that for a given box of radius R, Eq. (1) describes the state with maximum entropy for energy E=Ebh, since throwing more energy into the black hole will only result in increasing the Hubble expansion. However, if we let go the constraint that the energy inside the box shouldn’t be greater than the mass of the black hole inside the box then one could arrive at a configuration where the entropy of the system is greater than (1). This could be achieved by putting N number of black holes, each of radius R in a lattice instead of a single box of volume V. The number of black holes in such a configuration is given as Nhole=(VRd).(2)Therefore the total entropy of the system is S=NholeShole=(VRd)(Rd-1G)=VRG.(3)We notice that (3) is in contrast with (1) where the entropy is directly proportional to the volume rather than the area of the horizon. The energy leading to such a state is undoubtedly greater than Ebh and could be expressed as follows: E=NholeEhole=(VRd)(Rd-2G)=VR2G.(4)Substituting the value of R from (4) to (3) we get S=(VEG)-12(5)and for ρ=E/V we get, S=KρGV.(6)This contrast in the definition of entropy is subject to the constraint that microstates cannot expand freely to a larger size unlike in asymptotically flat space where the entropy is given by the area law. Also the resulting lattice configuration with E>Ebh, having N number of black holes, would not collapse to form one large black hole as the entropy corresponding to the lattice configuration is larger than a single black hole state in a box of volume V.

Now, from the first law of thermodynamics we can show T=(∂S∂E)-1=2KEGV,(7)p=T(∂S∂V)=EV=ρ.(8)Now from (8) we see that equation of state takes the form ρ=wp with w=1. The solution for the black hole gas model can be obtained by starting with the FLRW flat metric in (1+d) dimensions which is given by the following line element ds2=-dt2+a2(t)dx→2.(9)Solving the Friedmann equation for black hole gas in the (d+1)-dimensional flat metric with scale factor a(t) we get a(t)=a0t1/d.(10)The above FLRW flat metric can be written in terms of the conformal time scale by using the following conversion relation dτ=dta(t).(11)Integrating both sides of the above equation we get the following relationship between the physical time t and the conformal time τ in the arbitrary (d+1)-dimensional black hole gas, t={exp(a0τ)d=1(a0(1-d)d)dd-1τdd-1d>1.(12)

This relationship is extremely useful for the present computation which helps us the directly translate the information in terms of desired conformal time from physical time.

In this conformal time coordinates, the flat FLRW line element gets transformed as ds2=a2(τ)(-dτ2+dx→2).(13)

Hence, the above solution of the black hole gas scale factor can be written in terms of conformal time as follows: a(τ)={a0exp(a0τ)d=1a0(a0(1-d)d)1d-1τ1d-1d>1.(14)The corresponding Hubble parameter with respect to the conformal time scale can also be written as H(τ)={a0d=11d-11τd>1.(15)Now we will describe an equivalent scenario inside a black hole gas where one can obtain the above mentioned scale factor and Hubble parameter in any arbitrary (d+1) dimensions. In this scenario we embed a scalar field within the framework of Einstein gravity having the previously mentioned spatially flat FLRW space-time, where the solution of the scale factor is exactly same as mentioned earlier. In this framework the representative action of the scenario is described as S=12∫dd+1x-g[R-(∂ϕ)2-V(ϕ)],(16)where we have fixed the Planck mass Mp=1 for the present computation. The first term in the action represents the usual Einstein Hilbert term, the second term represents the kinetic term of the embedded scalar field. and the last term represents the potential function in a black hole gas V(ϕ) in any arbitrary (d+1) dimensions. We have found the following two possibilities of the potential functions are allowed in the present context which can finally give rise to same same scale factor which we mentioned earlier, V(ϕ)={3d2exp(-2dϕ)Choice I3d2exp(2dϕ)Choice II.(17)Here it is important to note that, since we have embedded the scalar field in the homogeneous and isotropic spatially flat FLRW background it turns out to be the field is only a function of the time coordinate.

Further solving the Klein Gordon field equation in d+1 dimension spatially flat FLRW background the dynamical solution of the homogeneous and isotropic background scalar field ϕ in terms of the conformal time coordinate can be expressed as: ϕ(τ) = ∓2d-1ln(a0(d-1)dτ).(18)This solution actually representing the dynamics of the field inside the black hole gas in (d+1)-dimensional spatially flat FLRW background.

Now using this solution one can find out the following conformal time dependence of the potential function in the context of black hole gas V(τ)={3exp(-2a0τ)d=13d2(a0(1-d)dτ)-2dd-1d>1.(19)In Figs. 1 and 2, we plotted the two choices of fields with respect to the conformal time. In choice I, we see that for d=1 the field keeps on increasing monotonically and for higher dimensions i.e., d=2, 3 the field value decreases slowly initially with respect to conformal time and then rapidly as we increase the value of conformal time. The exact opposite behavior can be seen for choice II. For d=1 the field value keeps on decreasing monotonically, where as for d=2, 3 the field value grows steadily and then at a faster rate with conformal time.

In Figs. 3 and 4, we plotted potential for the two choices against the field. As can be seen from Eq. (17), the potential function takes the form of exponentially decreasing and increasing function for first and second choices respectively.

In Fig. 5 we have drawn the potential function with respect to conformal time. We have taken the logarithm for the potential. It can be seen that for d=1, the potential decreases continuously in a straight line whereas for d=2, 3 the potential rises quickly after going through a slow rise period in the beginning.

In Fig. 6 we plotted the scale factor of the black hole gas model with respect to the conformal time scale. The plots have been done by fixing the value of the constant a0 to 1. We observe a significant difference in the behavior of the scale factor for the spatial dimension d=1 and the higher spatial dimensions. The scale factor corresponding to the spatial dimension d=1 shows increasing behavior in the late time scales. The scale factor for higher spatial dimensions shows a decreasing behavior. The decrease is linear for the spatial dimension d=2 whereas it is nonlinear for spatial dimension d=3.

In Fig. 7 we plotted the behavior of the Hubble parameter (H(τ)=a′(τ)a(τ)) with respect to the conformal time scale. It can be seen that the Hubble parameter is just a positive constant for the spatial dimension d=1, whereas it exhibits a decreasing behavior approaching negative infinity at late conformal time scales (near τ=0) for the other spatial dimensions. Moreover, the value taken by the Hubble parameter for the higher spatial dimensions is always negative.

One could also arrive at (6) by using the T and S duality symmetries of string theory; this seems interesting because it does not require the state of black hole gas to be a state near the big bang since microstates of stringy theory could also describe the microstates of black holes.

One of the challenges in quantum information processing is to find out the efficient circuit for implementing a unitary operation U which can be used to solve a computational problem like Search Algorithm or Shor’s factoring [41–43]. In computer science, a similar term called complexity [44,45] exists, which can be defined as the minimum number of computational gates required to implement a certain algorithm. If we extend this definition to quantum version as minimum number of quantum gates out of basic unitary gates [46] in order to implement a unitary operation U, we get quantum complexity [47,48]. Therefore, understanding the difficulty of implementing such unitary operation U, as a sequence of logical gates, is very helpful and at the same time challenging.

In Refs. [49–51], the authors introduced a geometric approach to compute quantum circuit complexity based on the idea that finding the optimal quantum circuit is equivalent to problems of computing geodesics in Riemannian geometry. Here, we define a Riemannian metric on the space of n-qubit operations, and the distance d(I,U) between the identity and target unitary operation U is equivalent to the number of quantum gates, which is then identified as the circuit complexity. It was shown that minimizing this distance d(I,U) i.e., finding geodesic length, gives a good measure of complexity. Thus, one can employ well developed tools of Riemannian geometry such as the Levi-Civita connection, geodesics, curvature, etc. to analyze the quantum circuit complexity. It is important to note that, even though in Refs. [49–51] geometric techniques were introduced to study complexity, in Ref. [52], the authors have used previous techniques from the theory of symmetric spaces to study time-optimal control of quantum evolution.

Let U be a transformation which transforms reference state |ψR⟩ to the target state |ψT⟩ via |ψT⟩=U|ψR⟩.(49)The unitary transform U, in the language of quantum computation has an order of unitary gates Qi such that U=Q1Q2…Qd where d is the depth of the circuit. We can also introduce the tolerance ε which tells us whether the transformation is successful |||ψT⟩-U|ψR⟩||2≤ε.(50)This makes sense because in any practical implications, it is difficult to represent the unitary transformation U exactly as a combination of discrete unitary Qi operations.

Obviously, there exists infinite number of ways to achieve this target state |ψT⟩ from the reference state. The circuit complexity is then the depth of the optimal circuit out of this infinite possibilities.

Motivated from the theory of Hamiltonian control problem, authors in Refs. [49–51] introduced a geometric approach to compute this circuit complexity which was later used to compute complexity in various quantum mechanical and quantum field theoretic models. Instead of directly counting discrete set of gates required for constructing U, Nielsen’s approach is geometric. In this method, with a time-dependent Hamiltonian H(t) one constructs unitary U as U=P←exp[-i∫01dτH(τ)]where  H(τ)=∑IYI(τ)OI.(51)Here, the Hermitian operators OI form the basis for time dependent Hamiltonian H(τ). The path-ordering operator P is another version of the time-ordering operator which indicates that the circuit, made out of noncommuting operators, is built from right to left. The right to left application of operators is a choice of convention. The control functions YI(τ) can be thought as particular gates added at a time s represented by Eq. (51).

One can then construct paths in the space of unitaries as U=P←exp[-i∫0τdτ′H(τ′)].(52)The most interesting case is when the trajectory satisfies the boundary conditions U(τ=0)=1 and U(τ=1)=U. These OI and YI(τ) satisfy YI(τ)OI=∂τU(τ)U-1(τ).(53)Equation (53) is actually just a time-dependent Schrödinger equation in the form when one solves it via time-ordered exponentials.

As we discussed before, there are an infinite number of ways of achieving a unitary transformation U. However, not all processes are optimal. In order to find out the optimal transformation, a cost function F(U,Y→(τ)) is defined. This cost function F(U,Y→(τ)) is a local functional along the trajectory of the U(τ) and tangent vectors Y→(τ)). Now, for each path the cost is defined as D(U(t))=∫01dtF(U(t),U˙(t)).(54)Nielsen showed that, the variational geometric approach of minimizing this functional is basically finding the optimized quantum circuit. Standing on the physical grounds, the cost function F should satisfy certain properties. Those are appended below. (a)

Continuity: F∈C0 i.e., F should be continuous. It is reasonable to assume continuity on physical grounds.

If one extends the continuity condition F∈C0 with F∈C∞ i.e., F is smooth; then the manifold is known as Finsler Manifold. Nielsen’s geometric approach of determining complexity is computing the geodesic in Finsler geometry, and the length of this geodesic gives the complexity. This definition of complexity is also called the geometric circuit complexity.

In literature, there are different choices of these cost functions F(U,v). These choices depends on how one defines the complexity, and the elementary gate sets for their setup. Some of the simple examples are F1(U,Y)=∑I|YI|,Fp(U,Y)=∑IpI|YI|,F2(U,Y)=∑I|YI|2,Fq(U,Y)=∑IqI|YI|2.(57)We can now give comments on various choices of these cost functions. F1; the linear cost functional, measure is the nearest concept that is close to counting individual gates in the circuit. F2; the quadratic cost functional, can be considered as the proper distance in the manifold. F1p can be thought of as a cost function where penalty factors pI are used to favor certain directions over others. This becomes reasonable when one consider elementary gates coupled only to the neighboring qubits and discard those qubits which are nonlocal. Depending on the system, one can choose different cost functions to study the circuit complexity.

One can also introduce a general class of inhomogeneous and homogeneous family of functionals represented by Fk(U,Y)=∑I|YI|k,F1k(U,Y)=∑I|YI|1k,(58)where k≥1 represents the degree of homogeneity. Fk was introduced in the context of holography to match the results obtained from “Complexity=Action” and “Complexity=Volume” conjectures.

A simple but a very rich example of entangled multi-mode field states is the two-mode squeezed vacuum state. (More details about two-mode squeezed states can be found in [53].) As already defined in the previous section, the two-mode squeezing operator is given by S^k→(ξ)=exp(ξ*c^k→c^-k→-ξc^-k→†c^k→†),(59)where ξ=rkeiϕk, rk and ϕk are known as squeezing parameters and 0≤rk<∞ and 0≤ϕk≤2π. The two-mode squeezed vacuum state, which will act as our target state is given by the action of two-mode squeezing operator S^k→(ξ) on the two-mode vacuum state (initial state), |0⟩k→|0⟩-k→=|0,0⟩. |ψsq⟩k→,-k→=S^k→(ξ)|0,0⟩=exp(ξ*c^k→c^-k→-ξc^-k→†c^k→†)|0,0⟩.(60)In terms of the number of states, one can show that the state of two-mode squeezed states is given by |ψsq⟩k→,-k→=1coshrk∑n=0∞(-1)neinθ(tanhrk)n|nk,n-k⟩).(61)We are now in the position to compute circuit complexity of the reference and target state of the two-mode squeezed states and compare it to the entanglement entropy. For this, we need to write our reference and target quantum states as Gaussian wave functions. The auxiliary position and momentum variables are q^k→=12Ωk(c^k→†+c^k→),(62)p^k→=iΩk2(c^k→†-c^k→),(63)with, [q^k→,p^k′→]=iδ3(k→-k′→). The reference state i.e., the two-mode vacuum state, in the position space can be expressed as a Gaussian wave function as follows: ψR(qk→,q-k→)=⟨qk→,q-k→|0⟩k→,-k→=(Ωkπ)14exp(-Ωk2(qk→2+q-k→2)).(64)The target state, the two-mode squeezed state, in the position space has the wave function ψsq(qk→,q-k→)=⟨qk→,q-k→|ψsq⟩k→=eA(qk→2+q-k→2)-Bqk→q-k→coshrkπ1-e-4iϕktanh2rk,(65)where, A and B are the coefficients and are functions of squeezing parameters rk and ϕk A=Ωk2e-4iϕktanh2rk+1e-4iϕktanh2rk-1,B=2Ωke-2iϕktanhrke-4iϕktanh2rk-1.(66)It is helpful to define three other terms for simplifying the complexity calculation Σk→=-2A+B,Σ-k→=-2A-B,ωk→=ω-k→=Ωk2.(67)Three methods of computing complexity were discussed in [54]. For our case, two methods i.e., computing complexity via covariance matrix method and Nielsen’s method are relevant. As we discussed before in Eqs. (57) and (58), complexity depends on the choice of cost functions. Let C1 be the circuit complexity corresponding to linear cost functional F1, C2 to quadratic cost functional F2, and Ck to k family of functionals Fk.