In this work, we propose a testing procedure to distinguish between the different approaches for computing complexity. Our test does not require a direct comparison between the approaches and thus avoids the issue of choice of gates, basis, etc. The proposed testing procedure employs the information-theoretic measures Loschmidt echo and Fidelity; the idea is to investigate the sensitivity of the complexity (derived from the different approaches) to the evolution of states. We discover that only circuit complexity obtained directly from the wave function is sensitive to time evolution, leaving us to claim that it surpasses the other approaches. We also demonstrate that circuit complexity displays a universal behaviour — the complexity is proportional to the number of distinct Hamiltonian evolutions that act on a reference state. Due to this fact, for a given number of Hamiltonians, we can always find the combination of states that provides the maximum complexity; consequently, other combinations involving a smaller number of evolutions will have less than maximum complexity and, hence, will have resources. Finally, we explore the evolution of complexity in non-local theories; we demonstrate the growth of complexity is sustained over a longer period of time as compared to a local theory.

Article funded by SCOAP3

0$, but we allow $\hat q $ to have either sign. This Hamiltonian is more general than a free scalar field theory discretized on a lattice; depending on the choice of parameters, a variety of interesting behaviors arise.\footnote{E.g. writing $q^2= (a^2+b^2)$ and $\hat q= 2 a\, b$ ($a,b \in {\bf R}$), one has the bosonic analog of the Su-Schreiffer-Heeger model~\cite{topo1, topomechanics}.} For us, this provides a convenient/natural medium to explore our testing procedure. Eq.~(\ref{hamilton}) is readily analyzed by expanding the position and momentum operators in Fourier modes (normal modes) as \begin{equation} \label{normal} x_l = \frac{1}{\sqrt{N}} \sum_k e^{-i\, \frac{2\pi k \, l}{N}} \tilde x_k \,, \ \ p_l = \frac{1}{\sqrt{N}} \sum_k e^{-i\, \frac{2\pi k \, l}{N}} \tilde p_k \,, \end{equation} where $0\leq k \leq (N-1)$ with $N$ being the total number of (lattice) sites; one obtains\footnote{We have used the orthogonality condition $$\frac{1}{N}\sum_{l=0}^{N-1}\exp[- i\,2\pi\,(k-k')l/N]=\delta_{k,k'}.$$} \begin{equation} H(q,\hat q) = \frac{1}{2} \sum_k \left[ \tilde p_{k}\tilde p_{-k} + \omega_k^2~ \tilde x_{k}\tilde x_{-k}\right] , \label{HBfourier} \end{equation} where $\omega_k^2 = q^2 + \hat q \cos (\frac{2\,\pi\, k}{N})$ and $\omega_k=\omega_{-k}$. Eq.~(\ref{HBfourier}) is then diagonalized by introducing creation and annihilation operators\footnote{$[a^{\phantom \dagger}_k, a_k^{\dagger}]=1$ $\forall$ $k$.} \begin{equation} H(q,\hat q) = \sum_k \omega_k \left( a^{\dagger}_k a^{\phantom \dagger}_k + 1/2 \right) \,, \label{Hdiagonal} \end{equation} where \begin{equation} \tilde x_k = \frac{1}{\sqrt{2\omega_k} } \left( a^{\phantom \dagger}_k + a^{\dagger}_{-k} \right) \,, \ \ \tilde p_k = \frac{1}{i} \sqrt{\frac{\omega_k}{2}} \left( a^{\phantom \dagger}_k - a^{\dagger}_{-k} \right). \label{creationannihilation} \end{equation} \pagebreak We are interested in studying quenches in the above model --- the quench protocol we employ is \begin{subequations} \begin{alignat}{2} H & = H(q,\hat q)& \ \ \ \ \ {\rm for} \ t & \leq 0 \label{H} \\ H & = H_1(q_1,\hat q_1)& \ \ \ {\rm for} \ t & > 0 \label{H1} \,, \end{alignat} \end{subequations} where $(q,\hat q)$ and $ (q_1,\hat q_1)$ are different. For $t \leq 0$, we prepare the system in the ground state of $H(q,\hat q)$; then we evolve the state by $U_1(t)=\exp[-i\, H_1(q_1,\hat q_1)\, t]$. In what follows, we consider the evolution of the complexity following the quench --- we consider the complexity between the initial state and the time evolved state. In the following section, we compute the complexity for this model by the different approaches. ]]>

0$) in terms of the eigenoperators of $H$ (the Hamiltonian for $t \leq 0$). As per eq.~(\ref{creationannihilation}), we have\footnote{Note that $\omega_{1,k}$ and $\omega_k$ are functions of $(q_1,\hat q_1) $ and $(q,\hat q)$, respectively.} \begin{align} \begin{aligned} \textrm{for}\,\, H(q,\hat q) :&&\,\,\,\, \left( \begin{array}{c} \tilde x_k \\ \tilde p_k \\ \end{array} \right) & =\frac{1}{{\sqrt{2 \omega _k}} }\left( \begin{array}{cc} 1 & 1 \\ -i\, \omega _k & i\, \omega _k \\ \end{array} \right) \left( \begin{array}{c} a_k \\ a^{\dagger}_{-k} \\ \end{array} \right),\\ \textrm{for}\,\, H_1(q_1,\hat q_1) :&&\,\,\,\, \left( \begin{array}{c} \tilde x_k \\ \tilde p_k \\ \end{array} \right) & =\frac{1}{{\sqrt{2 \omega _{1,k}}} }\left( \begin{array}{cc} 1 & 1 \\ -i\, \omega _{1,k} & i\, \omega _{1,k} \\ \end{array} \right) \left( \begin{array}{c} a_{1,k} \\ a^{\dagger}_{1,-k} \\ \end{array} \right) \,; \end{aligned} \end{align} from this, one obtains the Bogoliubov transformation relating $(a_{1,k}, a^{\dagger}_{1,-k})$ to $(a_{k}, a^{\dagger}_{-k})$: \begin{align} \begin{split} \left( \begin{array}{c} a_{1,k} \\ a^{\dagger}_{1,-k} \\ \end{array} \right)=\left( \begin{array}{cc} \mathcal{U}_k & \mathcal{V}_k \\ \mathcal{V}_k & \mathcal{U}_k \\ \end{array} \right) \left( \begin{array}{c} a_{k} \\ a^{\dagger}_{-k} \\ \end{array} \right), \end{split} \end{align} where \begin{equation*} \mathcal{U}_{k}=\frac{\omega_{1,k}+\omega_k}{2\sqrt{\omega_{1,k}\omega_{k}}} \,, \ \ \mathcal{V}_{k}=\frac{\omega_{1,k}-\omega_k}{2\sqrt{\omega_{1,k}\omega_{k}}} \,, \end{equation*} with $|\mathcal{U}_k|^2-|\mathcal{V}_k|^2=1$. Hence, we obtain \begin{align} \begin{split} \label{Ham} H_1(q_1,\hat q_1)=\sum_{k=0}^{N-1}\omega_{1,k}[(\mathcal {U}_{k}^2+\mathcal{V}_{k}^2)\tau_{k}^{z} + \mathcal{U}_{k}\mathcal{V}_{k} \tau_{k}^{+} + \mathcal{U}_{k}\mathcal{V}_{k}\tau_{k}^{-}], \end{split} \end{align} where \begin{equation} \left\{ \tau_{k}^{+} = a_{k}^{\dagger}a_{-k}^{\dagger} \,, \ \ \tau_{k}^{-} = a^{\phantom \dagger}_{-k} a^{\phantom \dagger}_{k} \,, \ \ \tau_{k}^{z} = \left(a_k^{\dagger}a^{\phantom \dagger}_k + a^{\phantom \dagger}_{-k}a_{-k}^{\dagger} \right)/2 \right\} \end{equation} satisfy an $\SU(1,1)$ algebra \begin{equation} [\tau_{k}^{z}, \tau_{k}^{\pm}]=\pm \tau_{k}^{\pm } \,, \ \ [\tau_{k}^{+},\tau_{k}^{-}]=-2\tau_{k}^{z} \,. \end{equation} As discussed above, we take the ground state of $ H(q,\hat q,q') $ as our reference state; this is given by \begin{equation} \label{ground} |\psi_0\rangle=\prod_{k=0}^{N-1}|k,-k\rangle \,, \end{equation} where $|k,-k\rangle$ denotes the Fock vacuum for modes $k$ and $(-k)$. We are interested in the complexity of the time-evolved state \begin{equation} \label{evolv} |\psi_1(t)\rangle= U_1(t)|\psi_0 (t=0)\rangle. \end{equation} To evaluate this, we employ the decomposition\footnote{Now $\tau_k^{-}$ simply annihilates the state $|k, -k\rangle$ for all values of $k$. So the action of $\exp(\gamma_k^{-}\tau_k^{-})$ on $|k, k\rangle$ is trivial. Also, $\tau_k^{-} |k, -k\rangle= \frac{1}{2} |k,-k\rangle$. Then, $\exp((\ln \gamma_k^0)\tau_k^{z}) |k,-k\rangle=(\gamma_k^0)^{1/2}|k, -k\rangle$. This produces just an overall phase and can be absorbed inside the normalization ($\mathcal{N}_k(t)$) of the state. Non trivial effects comes from the $\exp(\gamma_k^{+}\tau_k^{+})$.}~\cite{perelomov} \begin{align} \begin{split} \label{BCH} \exp(\beta_k \, \tau_k^z+\alpha_k^{+}\tau_k^{+}+\alpha_k^{-}\tau_k^{-})=\exp(\gamma_k^{+}\tau_k^{+})\exp ((\ln \gamma_k^0)\tau_k^{z})\exp(\gamma_k^{-}\tau_k^{-}), \end{split} \end{align} where \begin{equation} \label{def1} \gamma_k^0=\left(\cosh\left(\mu_k\right)-\frac{\beta_k}{2\mu_k}\sinh\left(\mu_k\right)\right)^{-2} \,, \ \ \gamma_k^{\pm}=\left(\frac{\alpha_k^{\pm}}{\mu_k}\right)\left( \frac{\sinh(\mu_k)}{\cosh(\mu_k)-\frac{\beta_k}{2\,\mu_k}\sinh(\mu_k)} \right) \end{equation} with $\mu_k^2=(\beta_k^2/4)-\alpha_k^{+}\alpha_k^{-}$; one obtains \begin{equation} \label{evolv1} |\psi_1(t)\rangle = \prod_{k=0}^{N-1}\mathcal{N}_{k}(t)\exp(\gamma_{1, k}^{+}(t)\, a^{\dagger}_{k}a^{\dagger}_{-k})|k,-k\rangle \,, \end{equation} where \begin{equation} \label{def} \gamma_{1, k}^{+}= \left(\frac{\alpha_{1,k}^{+}}{ \mu_{1,k}}\right) \left( \frac{\sinh\left(\mu_{1,k}\right)}{\cosh\left(\mu_{1,k}\right)-\frac{\beta_{1,k}}{2\,\mu_{1,k}}\sinh\left(\mu_{1,k}\right)} \right) \,, \ \ \mu_{1,k}^2=\frac{\beta_{1,k}^2}{4}-\alpha_{1,k}^{+}\alpha_{1,k}^{-} \,, \\ \end{equation} with \begin{equation*} \beta_{1,k}=-i\, t\, \omega_{1,k}\, \left(\mathcal{U}_k^2+\mathcal{V}_k^2\right) \,, \ \ \alpha_{1,k}^{+}=\alpha_{1,k}^{-}=-i\,t\,\omega_{1,k} \, \mathcal{U}_k\mathcal{V}_k \,. \end{equation*} The state eq.~(\ref{evolv1}) can be thought of as an $\SU(1,1)$ coherent state; the state manifold can be given a Riemannian structure~\cite{bookt} --- considering the class of states \begin{equation} \label{class} |\psi\{\gamma_{k,\tau}(t)\}\rangle =\prod_{k=0}^{N-1}\mathcal{N}_k(t)\exp(\gamma_{k,\tau}(t) a^{\dagger}_k\, a^{\dagger}_{-k})|k,-k\rangle \end{equation} labeled by the parameter $\tau$ and evaluating the Fubini-Study line element \begin{equation} \left( \frac{ds}{d\tau} \right)^2 = \langle \frac{d\psi}{d\tau} \mid \frac{d\psi}{d\tau} \rangle - \langle \frac{d\psi}{d\tau} \mid \psi \rangle \langle \psi \mid \frac{d\psi}{d\tau} \rangle \,, \label{FS} \end{equation} one obtains \begin{equation} \label{met} ds^2=\sum_{k=0}^{N-1} ds_k^2=\sum_{k=0}^{N-1}\frac{|d\gamma_{k,\tau}|^2}{(1-|\gamma_{k,\tau}|^2)^2}. \end{equation} For each value of $k$, one has $H^2$ in the $CP^1$ representation. For a given $k$, the distance is naturally defined by\footnote{For simplicity, we defined range of $\tau$ from $0$ to $1$; one can reparametrize it to redefine its initial and final value.} \begin{equation} s_k=\int_{0}^{1} d\tau\, \frac{1}{1-|\gamma_{k,\tau}|^2}\Big |\frac{d\gamma_{k,\tau}}{d\tau}\Big|. \end{equation} The full state manifold has the form $H^2\times H^2\times \cdots$ --- the distance can be defined as \begin{equation} s=\sqrt{\sum_{k=0}^{N-1} s_k^2} \,. \label{distancefunction} \end{equation} In this (Fubini-Study) approach, the complexity geodesic distance between the reference and target states~(\ref{com1}) --- it follows from eq.~(\ref{distancefunction}) that the complexity is \begin{equation} \label{com1} \mathcal{C}_{FS}= \sqrt{\sum_{k=0}^{N-1}\mathcal{C}_{k}^2} \,, \end{equation} where $\mathcal{C}_k$ is the geodesic distance for a particular $k$. To proceed, we write \begin{equation} \gamma_{k,\tau}=|\gamma_k| \exp(i\,\phi_k) \,, \ \ |\gamma_k|=\tanh\Big(\frac{\theta_k}{2}\Big) \,; \end{equation} one obtains \begin{equation} ds^2=\frac{1}{4}\,\sum_{k=0}^{N-1} (d\theta_k^2+\sinh(\theta_k)^2 d\phi_k^2). \end{equation} Considering two points $(\theta_{1,k},\phi_{1,k})$ and $(\theta_{2,k},\phi_{2,k})$, the complexity~(\ref{com1}) takes the form \begin{equation} \label{genFS} \mathcal{C}_{FS} = \frac{1}{2}\sqrt{ \sum_{k=0}^{N-1}\Big( \arcosh\Big[\cosh(\theta_{1,k})\cosh(\theta_{2,k}) - \sinh(\theta_{1,k})\sinh(\theta_{2,k})\cos(\phi_{1,k}-\phi_{2,k})\Big] \Big)^2} \,. \end{equation} For reference and target states given by eqs.~(\ref{ground}) and~(\ref{evolv1}), respectively, $\theta_{1,k}=0$ and $\theta_{2,k}=2\arctanh |\gamma_{1,k}|$ with $\gamma_{1,k}$ defined in~(\ref{def}),\footnote{For a more detailed discussion about the choice of reference state, see appendix~\ref{TypSec_B}.} the complexity takes the form \begin{equation} \label{circfs} \mathcal{C}_{FS}=\sqrt{\sum_{k=0}^{N-1} (\arctanh |\gamma_{1,k}|)^2}. \end{equation} ]]>

=\alpha_3=\cos \theta_k(\tau=0). $ \hidewidth \end{aligned} \end{align} These three generators satisfy the following commutation relation, \begin{align} \begin{split} [M_{11}, M_{22}]=2\,M_{22},\quad [M_{11},M_{33}]=-2\,M_{33},\quad [M_{22},M_{33}]= M_{11}. \end{split} \end{align} From these representations (induced on the covariance matrix) we can now identify the operators as~\cite{Camargo-ml-2018eof}, \begin{align} \begin{split} M_{11}\rightarrow \frac{i}{2}(\tilde x_k\,\tilde p_{k}+\tilde p_{k}\,\tilde x_k),\quad M_{22}\rightarrow \frac{i}{2}\tilde x_k^2,\quad M_{33}\rightarrow \frac{i}{2}\tilde p_{k}^2. \end{split} \end{align} Finally, in terms of these operators we get, \begin{align} \begin{split} \tilde U^{k}\left(\tau\right)=\exp\left[\frac{i\,\rho_{k}\left(\tau=1\right)\,\tau }{2}\left\{\sin \theta_k\left(\tau=0\right) (\tilde x_k \, \tilde p_k+\tilde p_k\, \tilde x_k)+\cos \theta_k(\tau=0) (\tilde x_k^2+\tilde p_k^2)\right\}\right]. \end{split} \end{align} Note that even though we start with the full $\GL(2, R)$ generators, the optimal circuit is composed only of the generators belonging to the $\SL(2, R)$ sub-group. \paragraph{A brief comparison.} At this point, we pause and make a brief comparison between the three methods and comment on the structure of the optimal circuit. In all three methods, for each value of $k$ we have restricted ourselves to the space of GL(2, R) unitaries. Then we have performed an optimization to find the best possible unitary which leads to the minimal depth. Both the Fubini-Study (section~\ref{TypSec_3.1}) and covariance matrix methods (section~\ref{TypSec_3.2}) give a set of operators which satisfy SU(1,1) and SL(2, R) algebras, respectively; the optimal circuit for both these two cases are made up of scaling operators $(i\, (\tilde x_k\tilde p_k+\tilde p_k\tilde x_k))$ and $(i\,\tilde x_k^2, i\, \tilde p_k^2)$ operators. These operators are local operators in normal mode basis. On the other hand, the geodesic analysis done in the context of circuit complexity (section~\ref{TypSec_3.2}) forced us to a different set of operators ($\hat O_{k} $) as shown in~(\ref{difop}), except the scaling operators ($i\, (\tilde x_k\tilde p_k+\tilde p_k\tilde x_k)$). The $\hat O_k$ operators (as mentioned in~(\ref{difop})) are slightly more non-local compared to the $i\,\tilde x_k^2$ and $i\, p_k^2$ operators in the normal mode basis. We should note that when the wave function is a real Gaussian, then we do not need any of these extra operators. Only the scaling operator $O_k$ (mentioned in~(\ref{difop})) is sufficient. The expressions for complexity coming from the covariance matrix method~(\ref{genCov}) and circuit complexity method~(\ref{genCirc}) are basically identical, given the same reference state. However, when the target wave function is a complex Gaussian, they are different as it is evident from~(\ref{genCirc}) and~(\ref{genCov}). It seems, the advantage of using Fubini-Study and covariance matrix methods is that we get the optimal circuit made from local operators, whereas the circuit complexity method tends to prefer slightly more non-local operators. However, in the next section, we will establish that the circuit complexity (from wave function) has an advantage over the other two methods as it can capture the evolution of states. To end this section, we stress that complexity depends on both the choice of reference state and gates (and also on the measure used). For a fixed reference state and fixed measure, the value of the complexity will depend on the underlying unitary gates. In the next section we compare the complexity obtained from the different methods --- this will not be a comparison between their magnitudes, but rather a comparison of their sensitivity to a particular test that we propose. ]]>

& \mathcal{C} (\psi_1, \tilde \psi_1) \end{eqnarray} Therefore, although the closeness of states between ($\psi_0$ and $\psi_2$) is same as the closeness between ($\tilde \psi_1$ and $\psi_1$), the complexity of $\psi_2$ with respect to $\psi_0$ is larger than the complexity of $\psi_1$ with respect to $\tilde \psi_1$. We further plot $|\mathcal{C}_{\text{LE}}(\tilde U)-\mathcal{C}_{\text{F}}(\tilde U)|$ with respect to time. From figure~\ref{Diff2} we observe that this difference becomes constant quite fast and just fluctuates around this constant value even at large time. It will be interesting to do further numerical analysis to explore the late time behaviour and investigate their physical implications in a future work. \begin{figure} \centering \resizebox*{1\textwidth}{!}{\scalebox{0.60}{\includegraphics[trim = 0 5 0 10, clip]{LEvsF.pdf}} \scalebox{0.60}{\includegraphics[trim = 0 5 0 10, clip]{LEvsF1.pdf}}}\relax ]]>

\mathcal{C}_{\text{F}_1}^{(\psi_3 , \psi_1)}(\tilde U) > \mathcal{C}_{\text{F}_2}^{(\psi_2 , \tilde \psi_2)}(\tilde U) \end{equation} The superscripts denote the pair of wave functions for which we compute the complexity. Now there are several comments are in order. \begin{itemize} \item We made an important assumption here that, each evolution (forward and backward) are done by a different Hamiltonian. Let us try to elaborate this point. For the case of LE ($\langle \psi_0 | \psi_4\rangle $) in the above example, $|\psi_4\rangle$ is being generated from $|\psi_0\rangle$ by 4 evolutions by 4 different Hamiltonians.\footnote{We thank the referee for raising this point.} One can obviously generalize this argument for any number of evolutions. Now once we fix the set of the Hamiltonians entering in LE, we can easily compute various types of Fidelity by distributing this same set of Hamiltonians in various different ways such that quantum mechanically all of these overlaps will be the same. For the case stated earlier, there will be two distinct types of fidelities namely, $\langle \psi_1 | \psi_3\rangle$ and $\langle \tilde \psi_2 | \psi_2\rangle $. \item Another point is that, for the purpose of these computations our starting point is always the ground state of the Hamiltonian (for example $|\psi_0\rangle$ in the equation~(\ref{4evolve})) which is of the form~(\ref{hamilton}). All the other states are constructed by evolving this ground state. \item{If we do not impose the above constraints (especially the first one) our arguments will fail. To illustrate this let us consider the two evolution case as discussed in the section~\ref{TypSec_4} as an example, however, this can be generalized for any number of evolutions. Now if we consider the following scenario: \begin{eqnarray} \label{r} |\psi_0\rangle = e^{iH_1 t} |\phi_0\rangle \equiv |\phi_1\rangle, \cr e^{iH'_1 t} e^{-iH_1 t} |\psi_0\rangle = |\psi_2\rangle =e^{i H'_1 t} |\phi_0\rangle \equiv |\tilde \phi_1\rangle,\cr |\tilde \psi_1\rangle=e^{-i H'_1 t} |\psi_0\rangle=|\tilde \psi_1\rangle\equiv e^{-i H'_1t}e^{i H_1 t}|\phi_0\rangle\equiv |\phi_2\rangle, \end{eqnarray} then (under time reversal) the LE, $\langle \psi_2|\psi_0\rangle$ will become fidelity $\langle \tilde \phi_1'|\phi_1\rangle$ in terms of $\phi$'s and vice versa. Consequently our conclusion in~(\ref{conclude}) will be reversed: \begin{equation} \mathcal{C}(\psi_2, \psi_0) > \mathcal{C}(\tilde \psi_1,\psi_1) \leftrightarrow \mathcal{C}(\phi_2, \phi_0) < \mathcal{C}(\tilde \phi_1,\phi_1) \end{equation} Now notice that, the second line of the equation~(\ref{r}) gives $|\psi_2\rangle \equiv |\tilde \phi_1\rangle$, when we substitute the first equation as follows: \begin{align} \label {r1} e^{iH'_1 t} e^{-iH_1 t} |\psi_0\rangle & =e^{iH'_1 t} e^{-iH_1 t} e^{iH_1 t} |\phi_0\rangle \cr |\psi_2\rangle & = e^{i H'_1 t} |\phi_0\rangle \end{align} \noindent Therefore, \begin{equation} \label{r2} |\psi_2\rangle \equiv |\tilde \phi_1\rangle \end{equation} Note that~(\ref{r1}) is leading to~(\ref{r2}), since we are using the hamiltonian $H_1$ to backward evolve $|\phi_0\rangle$ to get $|\psi_0\rangle$. This hamiltonian is also used to forward evolve $|\psi_0\rangle$ on the second line of~(\ref{r1}). Now as we have restricted ourselves (mentioned in the first bullet point) to the case where each of these forward and backward evolutions has been done with different Hamiltonians this situation can easily be avoided. Hence our conclusion will be valid. } \item{ Last but the not the least, the result in~(\ref{sr}) does not depend on whether we are performing a forward or a backward evolution and also does not depend on the degree to which these Hamiltonians differ from each other.} \end{itemize} Now given these facts we next try to generalize our results. We can perform the same operations for arbitrary number of different Hamiltonians, leading to arbitrary number of evolutions for the state. We have tested this for 8 different evolutions with different Hamiltonians and interestingly, we find that the complexity corresponding to Loschmidt echo is always larger than any possible fidelity. Moreover, for different fidelities, the number of evolutions performed on reference state dictates the magnitude of their complexities. This is shown in the figure~\ref{8ham}. \begin{figure} \centering \scalebox{0.50}{\includegraphics{8evolutions.pdf} } ]]>

\mathcal{C}_{\text{F}_1}^{(\psi_{n-1} , \psi_1)}(\tilde U) > \mathcal{C}_{\text{F}_2}^{(\psi_{n-2} , \psi_2)}(\tilde U) > .. \ldots > \mathcal{C}_{\text{F}_n}^{(\psi_{n/2} , \tilde \psi_{n/2})}(\tilde U). \end{equation} This result implies, although the closeness between two states (overlap) does not change with unitary evolutions, they are very different from the perspective of a quantum circuit and the difficulty (in terms of complexity) of getting an evolved target state from the other. And our analysis also gives us a guideline about pairs for which it is easier to move between states in the sense of complexity. It tells us which pair of states will have the smallest complexity for a given set of Hamiltonians. One interesting feature of these differences in complexity for the overlaps is that they do not die away with time. The differences are small at very early time, but it become fixed as soon as the complexities saturate. Moreover, our analysis indicates an upper bound on the complexity for a given overlap evolved with a fixed number of Hamiltonians. Since as overlaps they are all the same, this comparison by complexity can be seen as comparing the same quantity from different ways, therefore, by abusing the language we can say that, any pair other than the pair with maximum complexity will have uncomplexity or resources~\cite {Hol8}. In our language, \begin{quote} \emph{The complexity corresponding to the Loschmidt Echo will always have the highest complexity. Therefore, the complexity corresponding to the Fidelity will have~resources.} \end{quote} Given the recent experimental advances one can possibly simulate these overlaps in an experimental setting~\cite{Echo, Swingleex, Swingleex1}. Our complexity analysis is providing us with the most efficient (optimal) quantum circuit needed to simulate the time evolution, hence providing a natural selection mechanism. Note that quantum mechanically all of these overlaps are the same. Therefore, this test will reduce the difficulties of experimental implementation in the sense that it can be obtained by constructing a quantum circuit with a minimal number of gates. Before ending this section, we want to further clarify what we meant by having \emph{resources}. Again, let's assume that we want to make overlap measurements (say, for two steps of evolutions) in the lab. We can measure either Loschmidt Echo or Fidelity since quantum mechanically the result is the same. Now suppose we are being \emph{supplied} with either $\psi_1$ or $\tilde \psi_1$, apart form $\psi_0$ then our result implies that it is easier to measure Fidelity as the complexity between the states entering in Fidelity is less compared to the complexity between the states entering in Loschmidt Echo. Note that, if we do not have either of $\psi_1$ or $\tilde \psi_1$, but only $\psi_0$ then of course we will loose this advantage since then we have to also take into account the complexity of simulating $\psi_1$ or $\tilde \psi_1$ from $\psi_0$. We will have this advantage only when we are being supplied with either $\tilde \psi_1$ or $\psi_1$. This is precisely what we meant by having \emph{resources} i.e having possession of states with some extrinsic complexity (w.r.t.\ $\psi_0$). ]]>

1$ i.e.\ we consider a non-local theory. Figure~\ref{CC1} shows the early time ($t < 10$) behavior of the circuit complexity for both $\mathcal{C}_{\text{LE}}(\tilde U)$ and $\mathcal{C}_{\text{F}}(\tilde U)$ for several values of $\alpha$.\footnote{We have set $N=500$ and used the values of the parameters in eq.~(\ref{paravalue}).} Notice that the complexity grows for a substantial amount of time in these non-local theories i.e.\ we get the desired time-dependence of the complexity; the more non-local the theory is, the slower the rate of growth. [Also notice that the difference between $\mathcal{C}_{\text{LE}}(\tilde U)$ and $\mathcal{C}_{\text{F}}(\tilde U)$ becomes more pronounced as the theory becomes more non-local.] In non-local theories, the entanglement entropy exhibits volume-law scaling (compared to the area-law scaling exhibited by local theories)~\cite{TAL}; we speculate the volume-law (or area-law) scaling of the entanglement entropy is related to the growth of the complexity. We investigate this in detail in a forthcoming paper~\cite{WP}. ]]>

1$), the complexity grows more slowly. There are many interesting directions to pursue. It is important to understand the issue of the time scales, namely the relation between the equilibration time and the time for the complexity to saturate. Until now, we have worked primarily in a discretized set-up; we would like take a continuum limit to make better contact with (continuous) QFTs. Moreover, to make contact with holography, it is important to generalize our results to interacting theories; toward that goal, one starting point could be to use the construction of~\cite{Ame}. It is known that the Loschmidt echo can be used as a diagnostic for chaos, and one can extract information about the Lyapunov exponent from it~\cite{Echo, Echothesis}; it would be interesting to evaluate $\mathcal{C}_{\text{LE}}(t)$ for chaotic theories and study its relation to chaos. Furthermore, it would be valuable to extend this construction to mixed states, which would create a platform to test some of the holographic results related to sub-region complexity~\cite{Agon}. Last but not the least, tensor networks provide useful ways to represent the time evolution of wave functions~\cite{Milsted}; it would be interesting to understand if this kind of computation could shed light on the optimal network required to represent time evolution, thereby improving such constructions. One could also study the causal structure of spacetime~\cite{causnet}, and understand the connection between our construction and various path integral approaches~\cite{Taka, Vidal,Vidala,Vidalb}. ]]>