\section{Introduction}

Quantum simulation is one of the most important applications of quantum computing because the time evolution of many physical systems can be realized with a number of quantum gates scaling logarithmically with the system size, opening up the possibility of making calculations in regimes inaccessible on classical computers \cite{feynman1982simulating,SLoyd,RevModPhys.86.153}. Many quantum algorithms have been developed to simulate efficiently the dynamics of quantum systems, such as single particle non-relativistic systems \cite{OPPROP:PR877,Zalka08011998,Strini2008}, many-body systems \cite{boghosian1998simulating}, fermionic systems \cite{PhysRevLett.79.2586,PhysRevA.64.022319} and many others \cite{yung2014}.

Although many efficient algorithms have been discovered for the simulation of the time evolution of quantum systems, the initialization of the quantum register to a desired initial state remains a challenge. Ideally, this state would be physically relevant while being efficiently implemented on the quantum computer. For general initial states, this cannot be achieved because it involves the execution of diagonal unitary gates requiring $O(2^{n+1})$ operations, where $n$ is the number of qubits \cite{PhysRevLett.91.027902,PhysRevA.71.052330,bullock2004asymptotically}. This can be improved by using Walsh basis functions techniques and by approximating the unitary operation \cite{1367-2630-16-3-033040}. Therefore, these ``brute-force'' techniques entail an exponential number of gates and thus, deteriorate the performance of any quantum simulation algorithm.

A quantum algorithm for the initialization of the quantum register has also been formulated for the simulation of real space non-relativistic quantum mechanics \cite{Zalka08011998,kaye2004quantum,grover2002creating}. The initial state is constructed incrementally by adding qubit contributions while conserving the probability distribution. This method also requires diagonal unitaries for general states, but may be efficient for a certain class of function.

Another popular approach for the initialization is the phase-approximation method pioneered by Abrams and Lloyd \cite{PhysRevLett.83.5162}. The Abrams and Lloyd technique (ALT) yields eigenvalues and eigenstates of time-independent Hamiltonian operators by evolving a trial state in time and by separating its spectral component using a quantum Fourier transform on an ancilla qubit register. If the trial state is polynomially close to the eigenstate, i.e. if the overlap between the trial state and the eigenstate is bounded by a polynomial function of the problem size, success can be achieved in polynomial time because the success probability of the algorithm is proportional to this overlap. Conversely, this probability can be exponentially small if the eigenstate component in the trial state is unimportant compared to other modes, although this can be improved by means of the adiabatic-state-preparation algorithm \cite{Aspuru-Guzik1704}. The ALT has been employed in the context of chemical physics to assess the possibility of determining eigenstates of molecular systems on quantum computers \cite{Aspuru-Guzik1704,Kassal02122008}. Despite its very interesting properties and success, this scheme usually requires a relatively large number of ancilla qubits to properly resolve the eigenenergies and eigenstates.

In this article, we present an alternative approach to the initialization problem which bears some resemblance to the ALT but differs on two main aspects:  (1) an apodization technique is introduced to improve the accuracy of the generated states, (2) a non-unitary operation is introduced to decrease the number of required ancilla qubits.

The goal of our algorithm is to set the value of the amplitude of a quantum register with the resulting state $|\Psi_{\rho}\rangle$ having spectral components within a specified energy range.
 Our algorithm takes advantage of the efficient time evolution to initialize the register by a spectral filtering technique: a trial function is evolved in time and filtered to keep only the desired spectral components. It can be seen as a quantum generalization of the  Feit-Fleck spectral method, originally developed to evaluate eigenenergies and eigenstates of the Schr\"{o}dinger equation in a static potential by solving the time-dependent dynamics \cite{Feit1982412}. The latter has been used successfully in a wide range of applications on classical computers \cite{feit1983,Mocken2004558,FillionGourdeau2014559}.

The quantum algorithm presented in the following has some interesting properties. First, it requires only two ancilla qubits. Second, assuming that the overlap between the trial state and the desired state is not negligible, it yields the desired state with a relatively high probability (see Eq. \eqref{eq:prob_success}) and it can be implemented efficiently. Third, it can accommodate states with many spectral components, such as wave packets. However, the weight of each component is connected to their value in the trial function, prior to filtering. Finally, the implementation naturally allows for an apodization function which improves the filtering and the accuracy of the desired states. Hereinafter, we shall assess each of these properties.

This article is separated as follows. Section \ref{sec:quantum_sim} reviews some basic facts on quantum simulations. In Sec. \ref{sec:qu_spec_filt}, the quantum implementation of the spectral filtering method is presented. Sec. \ref{sec:complexity_an} is devoted to the complexity analysis and the resource requirements for the algorithm. A simple example where the ground state of the harmonic oscillator is generated from the filtering method is considered in Sec. \ref{sec:harm_osci}. Finally, the conclusion is found in Sec. \ref{sec:conclusion}.

\section{Quantum simulations}
\label{sec:quantum_sim}

In this section, a brief review of quantum simulations on digital quantum computers is given. The main goal here is to present the notation and to describe the objective of our initialization algorithm.

To simulate a quantum system on a quantum computer, one has to find a set of quantum gates that approximates the exact evolution operator
\begin{align}
\label{eq:evol_op}
\hat{U}(0,T) = \exp\left[-iT \hat{H}  \right],
\end{align}
where $T$ is the final time and $\hat{H}$ is the Hamiltonian describing the physical system under investigation. For the more general case of a time-dependent Hamiltonian, this would become a time-ordered exponential \footnote{For time-dependent Hamiltonian, the evolution operator is $\hat{U}(0,T) = \mathcal{T}\exp\left[-i \int_{0}^{T}dt \hat{H}(t)  \right]$, where $\mathcal{T}$ is the time-ordering operator.}.

The evolution operator usually cannot be evaluated exactly and one has to resort to some approximation scheme. One popular method is based on the Trotterization of the exponential where the total time is split into $N_{t}$ small intervals of duration $\Delta t = T/N_{t}$, as
$\hat{U}(0,T) = U_{1} \cdots U_{N_{t}}$.
Then, each evolution operator $U_{i}:=\hat{U}((i-1)\Delta t, i \Delta t)$ is approximated by the Trotter-Suzuki exponential product $\hat{U}_{\mathrm{approx}}$ with an accuracy $O(\Delta t^{q+1})$, where $q \in \mathbb{N}^{+}$ gives the order of the approximation. The order can be improved to arbitrarily large value \cite{suzuki1993general}. The resulting scheme is a product of unitary transformations which can be simulated on a quantum computer using quantum gates \cite{nielsen2010quantum}. These gates are applied on a quantum register made of $n$ qubits where the state of the register is given by a $2^{n}$-dimensional vector in the Hilbert space $\mathcal{H}_{n} = \bigotimes_{i=1}^{n} \mathcal{H}_{1}$, where $\mathcal{H}_{1}$ is the Hilbert space of one qubit. The state of the register is then expressed as
\begin{align}
|\psi_{n} \rangle
&=   \sum_{k = 1}^{2^{n}}
\alpha_{k}
 | k \rangle,
\end{align}
where $(\alpha_{k})_{k=1,\cdots,2^{n}}$ are complex amplitudes and $| k \rangle$ represents the tensor product of $n$ qubit states, the sum is carried over all binary strings. Similarly, the state of the physical system to be simulated is described by a vector in a Hilbert space $\mathcal{H}$. Such vectors can be approximately written as
\begin{align}
|\Psi(t) \rangle \approx \sum_{k=1}^{2^{n}} b_{k}(t) |\phi_{k} \rangle,
\end{align}
where $b_{k}(t)$ are time-dependent coefficients and $|\phi_{k} \rangle$ are orthonormal basis vectors that span a Hilbert space $\mathcal{H}_{\rm approx}$ with dimension $2^{n}$, matching the dimension of the quantum register. It is assumed that vectors in the Hilbert space $\mathcal{H}_{\rm approx}   \subseteq \mathcal{H}$ are accurate approximations of vectors in the full Hilbert space of the physical system $\mathcal{H}$. This occurs when $n$ is large enough, assuming the convergence of the discrete Hilbert space to some region in the Hilbert space of the physical system. Then, the coefficients $b_{k}$ can be directly mapped to the coefficients $\alpha_{k}$ as $b_{k} \mapsto \alpha_{k}$, allowing the quantum computer to store the discretized state of the physical system under consideration. At the same time, the approximate evolution operator expressed in the physical basis as $(U_{\rm approx})_{jk} = \langle \phi_{j}|\hat{U}_{\rm approx}|\phi_{k} \rangle$ is mapped to quantum gates. Applying these operations on the quantum register, the coefficients $\alpha_{k}$ store the approximate time evolution of the state. When the number of gates obtained from the mapping scales like $\mathrm{poly}(n)$ for a given constant precision $\epsilon \ll 1$, we shall say that the resulting algorithm is efficient. This whole strategy is the essence of quantum simulations on digital quantum computers.

The main goal of the algorithm presented in this article is the initialization of the quantum register $|\psi_{n}\rangle$ to an initial state relevant for quantum simulations. Of course, this step must occur before the actual time evolution is implemented. The filtering technique presented in the next section sets the value of the coefficients $\alpha_{k}$, ensuring that the register represents a discretized eigenstate of a given time-independent Hamiltonian. These initial eigenstates are typical in numerical simulations of quantum systems on classical computers in atomic, molecular and optics physics, for example \cite{doi:10.1021/j100319a003,tannor2007introduction}.

\section{Quantum spectral filtering method}
\label{sec:qu_spec_filt}

The basic ingredient of the spectral filtering method is the observation that an exact state $|\varphi_{\rho}\rangle$ with a spectral content $\rho$ can be approximated from
\begin{align}
\label{eq:eigen_state}
|\varphi_{\rho}\rangle \approx |\Psi_{\rho}\rangle = \frac{1}{T} \int_{0}^{T} dt w_{\rho}(t) e^{iE_{\rho}t} |\Psi_{\rm trial}(t)\rangle,
\end{align}
where $T$ is is final time of the calculation, $w_{\rho}(t)$ is the window (apodization) function, $E_{\rho}$ is the central frequency and $|\Psi_{\rm trial}(t)\rangle$ is a time-dependent state initialized to an arbitrary trial value. This last formula is a simple consequence of the trial state eigendecomposition, as demonstrated in Appendix \ref{app:filtering}.

Here, $w_{\rho}(t)$ is the window function that selects the wanted spectral component and that accounts for the fact that a finite time evolution is performed. It is normalized such that $w_{\rho}(t)\in [0,1]$ and $\max_{t\in [0,T]} w_{\rho}(t) =1$. For an infinite time evolution, the spectrum would become a weighted Dirac comb. On the other hand, if the window function is rectangular ($w_{\rho}(t) = 1$ for $t \in [0,T]$), the spectral peaks will be given by a sequence of $\mathrm{sinc}(E_{\rho})$ functions which have large spectral leakage, i.e. they have large spectral components outside the central frequency. This phenomenon occurs in the standard ALT. A window function which vanishes smoothly close to $t=0$ and $t=T$ reduces spectral leakage by improving the suppression of unwanted modes. As shown in Eq. \ref{eq:spec_coeff} of Appendix \ref{app:filtering}, the magnitude of unwanted modes is suppressed by the line shape centered on energy $E_{\rho}$, given by
\begin{align}
\label{eq:line_shape}
L_{\rho}(E_{\rho}-E_{m}) = \frac{1}{T} \int_{0}^{T}dt e^{i(E_{\rho}-E_{m})t} w_{\rho}(t),
\end{align}
where $E_{m}$ is the (discrete) energy of the state (see Appendix \ref{app:filtering}).
The strength of the suppression is dictated by the choice of the window function. Having a large suppression factor is important when the overlap of the trial state and the desired state is small as $ A:=|\langle \varphi_{\rho}|\Psi_{\rm trial}(0)\rangle| \ll 1 $. In this case, the spectral coefficient of the desired state $|a_{\rho}|^{2}$ in the trial state also has a small contribution. For this mode to dominate after the filtering procedure, the suppression factor has to be at least as high as the ratio between $|a_{\rho}|^{2}$ and the maximum spectral coefficient of other modes. Then, a window function with a high suppression factor could, in principle, filter the unwanted mode successfully and in turn, this would increase the accuracy of the eigenstate estimation. However, as demonstrated in the following, $A$ is also proportional to the total probability of success of the quantum algorithm.

The algorithm described below requires the energy $E_{\rho}$ of the desired state to be known beforehand. For some cases, this can be obtained from analytical or classical computational methods. There also exist efficient quantum computation approaches whereby the spectrum is calculated semi-classically from the autocorrelation function obtained by performing a measurement  of $\langle \sigma_{x,y}\rangle$, where $\sigma_{x,y}$ are Pauli matrices, on one added ancilla qubit \cite{PhysRevLett.81.5672,PhysRevA.65.042323}. An explicit implementation of this approach relevant to the initialization is given in Appendix \ref{app:en_spec}.

Then, the determination of the eigenstate proceeds in two stages: (1) choosing a trial state and (2) evolving this trial state in time while evaluating Eq. \eqref{eq:eigen_state}. These steps can be implemented on a quantum computer by supplementing the quantum register with an additional qubit $|c \rangle$. The Hilbert space of this qubit serves to label whether the quantum register $| \psi_{n} \rangle $ stores the trial function $|\Psi_{\rm trial}(t)\rangle$  or the constructed initial state $|\Psi_{\rho}\rangle$. The quantum register should be initialized in the state $|\psi_{n+1}\rangle:=|c\rangle \otimes| \psi_{n} \rangle =|0\rangle \otimes |00 \cdots 0\rangle$.

In the first step, the trial function is  constructed from controlled gates, as displayed in Fig. \ref{fig:init_ff}. At this point, the quantum register is in the state $| \psi_{n+1} \rangle =|0\rangle \otimes |\Psi_{\mathrm{trial}}(0)\rangle$, implying that the norm of the trial state is $\langle \Psi_{\mathrm{trial}}(0)|\Psi_{\mathrm{trial}}(0)\rangle =1$. Moreover, we assume that this trial state can be constructed using a number of gates scaling like $\mathrm{poly}(n)$. This is possible in principle because it is an arbitrary state where the coefficient $\alpha_{k}$ can take any value. One possible way to implement the trial state is the utilization of the technique described in Refs. \cite{Zalka08011998,kaye2004quantum,grover2002creating}, which allows for the efficient initialization of a certain class of function. Another approach is given in Ref. \cite{PhysRevLett.91.257902} where an approximation of the eigenstate is obtained by an efficient grid refinement. The trial state can also be a thermal random state as in the \textit{deterministic quantum computation with one quantum bit} (DQC1) model of computation \cite{PhysRevLett.81.5672}. The minimal requirement is that the overlap with the desired spectral components should not be exponentially small, as discussed further in the next section.

For the second step, we use the fact that the filtering of the trial function given in Eq. \eqref{eq:eigen_state} can be approximated by a quadrature formula of the form
\begin{align}
\label{eq:eigen_state_quad}
|\Psi_{\rho}\rangle &=  \sum_{i=0}^{N_{t}} B_{i}|\Psi_{\rm trial}(t_{i})\rangle + O(\Delta t^{q}) ,\\
B_{i} &:= \frac{u_{i} w_{\rho}(t_{i})e^{iE_{\rho}t_{i}}}{N_{t}} ,
\end{align}
where $N_{t}$ is the number of time steps, $t_{i} := i\Delta t$ is the time where the integrand is evaluated and $(u_{i})_{i=0,\cdots,N_{t}}$ are coefficients defined by the quadrature rule \footnote{For example, for the trapezoidal rule, we have $u_{0} = u_{N_{t}} = 1/2$ and $(u_{i})_{i=1,\cdots,N_{t}-1}=1$.}: they are chosen such that the order of accuracy is at least $(\Delta t)^{q} \approx \epsilon$ to ensure that this numerical error does not dominate over the error attributable to the time evolution approximation. The sum in Eq. \eqref{eq:eigen_state_quad} is then computed by alternating a gate $\hat{B}_{i}$ with a controlled gate c-$U_{i+1}$. The latter evolves the trial function by one time step while $\hat{B}_{i}$ actually performs the sum. The quantum circuit associated to this algorithm is displayed in Fig. \ref{fig:init_ff}.

As mentioned earlier, it is assumed throughout that the time evolution implemented by the gates c-$U_{i+1}$ is efficient, i.e. that it yields an accurate approximation of Eq. \eqref{eq:evol_op} in $\mathrm{poly}(n) $ number of gates. This property of the time-dependent solver is very important for the global efficiency of the initialization algorithm.

The operator $\hat{B}_{i}$ is a non-unitary operator applied on the added qubit $|c \rangle$ and is given in the computational basis by
\begin{align}
\label{eq:gate_B}
\hat{B}_{i} :=  \frac{1}{ \sqrt{1+ \frac{|B_{i}|^{2}}{2} + |B_{i}|\sqrt{1+\frac{|B_{i}|^{2}}{4}}}}
\begin{bmatrix}
1 & 0 \\
B_{i} & 1
\end{bmatrix},
\end{align}
where the prefactor changes the normalization and guarantees that the operator $\hat{B}_{i}$ can be \textit{literally realized} \cite{Blass2015}. It can be checked that after $N_{t}$ iterations, the quantum register will be in the state
\begin{align}
\label{eq:iterations}
|c\rangle \otimes |\psi_{n} \rangle& = \frac{|0\rangle \otimes |\Psi_{\mathrm{trial}}(T)\rangle +  |1\rangle \otimes |\Psi_{\rho}\rangle}
{N}, \\
N &:= \sqrt{\langle \Psi_{\mathrm{trial}}(T)|\Psi_{\mathrm{trial}}(T)\rangle + \langle \Psi_{\rho} |\Psi_{\rho}\rangle}, \\
&= \sqrt{1 + \langle \Psi_{\rho} |\Psi_{\rho}\rangle},
\end{align}
where the last equality is obtained because unitary operations are used to evolve the trial state and thus, do not change its norm.

Performing a projective measurement $|1\rangle \langle 1|$ (see Fig. \ref{fig:init_ff}), one obtains that $|c\rangle \otimes |\psi_{n} \rangle \mapsto |1\rangle \otimes |\Psi_{\rho} \rangle$. This state
can then be used as an initial condition for the time-dependent simulation of a physical system. The probability of success of the projective measurement is related to the probability to be in the eigenstate, i.e. $P_{\rho} := \langle \Psi_{\rho} | \Psi_{\rho}\rangle/N^{2}$.

\begin{figure*}
\includegraphics[width=0.7\textwidth]{feitfleck}
\caption{Circuit diagram for the quantum implementation of the spectral filtering method. The gate $U_{i}$ advances the trial solution by $\Delta t$. The gate $B_{i}$ is a non-unitary operation. It is defined in Eq. \eqref{eq:gate_B} and its quantum circuit is displayed in Fig. \eqref{fig:nonunitary}. The gate $|\Psi_{\rm trial}\rangle$ initializes the quantum register to an arbitrary state. The projective measurement operator implements the projective measurement $|1\rangle \langle 1|$ that collapses the register to the wanted wave function. Finally, the gate $\hat{U}$ performs the quantum simulation by evolving in time the initial state constructed by the filtering algorithm. }
\label{fig:init_ff}
\end{figure*}

The main challenge of this strategy is the implementation of the non-unitary operation $\hat{B}_{i}$. Such operations and their quantum gate decomposition have already been considered \cite{Terashima2005,mezzacapo2015quantum,Blass2015,Childs:2012:HSU:2481569.2481570,PhysRevLett.114.090502} and a similar strategy is utilized here. First, the matrix $\hat{B}_{i}$ is re-written as
$\hat{B}_{i} = U_{i} \Sigma_{i} V^{\dagger}_{i}$,
obtained from the usual singular value decomposition. This yields two unitary matrices $U_{i}, V^{\dagger}_{i}$ and the diagonal matrix $\Sigma_{i} = \mathrm{diag}(1,s_{i})$. The matrices $U_{i}, V^{\dagger}_{i}$ can be evaluated explicitly and are given in Appendix \ref{app:svd}. The second singular value is
\begin{align}
s_{i} &= \sqrt{ \frac{1+ \frac{|B_{i}|^{2}}{2} - |B_{i}|\sqrt{1+\frac{|B_{i}|^{2}}{4}}}{1+ \frac{|B_{i}|^{2}}{2} + |B_{i}|\sqrt{1+\frac{|B_{i}|^{2}}{4}}}} .
\end{align}
The value of the prefactor in Eq. \eqref{eq:gate_B} was chosen such that the first singular value is ``1'' while the second obey $|s_{i}| \leq 1$, in accordance with the exact realization theorem \cite{Blass2015}. Then, the probability interpretation is preserved and the operator $\Sigma_{i}$ can be literally realized with one ancilla qubit, a unitary transformation and a projective measurement. This procedure is now detailed.

Introducing another ancilla in the state $|0\rangle$, the matrix $\Sigma_{i}$ can be decomposed into the following transformations (up to a normalization):
\begin{align}
& |0\rangle \otimes V_{i}^{\dagger} \left(|0\rangle \otimes |\Psi_{\mathrm{trial}}(i\Delta t)\rangle + |1\rangle \otimes |\Psi^{(i-1)}_{\rho}\rangle \right)= \nonumber \\
& |0\rangle \otimes \left(|0\rangle \otimes |\Psi_{0}^{(i)} \rangle + |1\rangle \otimes |\Psi_{1}^{(i)}\rangle \right) \\
\label{eq:map_nonunit}
&\mapsto   |00 \rangle  \otimes |\Psi_{0}^{(i)}\rangle + \left( s_{i} |01\rangle + \sqrt{1-s_{i}^{2}} |11\rangle \right) \otimes |\Psi_{1}^{(i)}\rangle \\
\label{eq:map_meas}
&\mapsto |0\rangle \otimes \left( |0 \rangle \otimes |\Psi_{0}^{(i)} \rangle + s_{i} |1\rangle \otimes |\Psi_{1}^{(i)}\rangle  \right) \\
\label{eq:map_last}
&\mapsto |0\rangle \otimes U_{i} \left( |0 \rangle \otimes |\Psi_{0}^{(i)} \rangle + s_{i} |1\rangle \otimes |\Psi_{1}^{(i)}\rangle  \right) = \nonumber \\
&\quad \; \; |0\rangle \otimes \left(|0\rangle \otimes |\Psi_{\mathrm{trial}}(i\Delta t)\rangle + |1\rangle \otimes |\Psi^{(i)}_{\rho}\rangle \right),
\end{align}
where $|\Psi^{(j)}_{\rho}\rangle$ is the partial sum (for $i=0,\cdots,j$) in Eq. \eqref{eq:eigen_state_quad} and where $|\Psi_{0,1}^{(i)}\rangle$ are linear combinations of $|\Psi_{\mathrm{trial}}(i\Delta t)\rangle$ and $|\Psi^{(i-1)}_{\rho}\rangle$ obtained by applying $V_{i}^{\dagger}$ on the qubit $|c\rangle$.
The first mapping (Eq. \eqref{eq:map_nonunit}) is performed by a unitary operation in the subspace of the ancilla and the $|c\rangle$ qubits corresponding to a controlled inverse rotation where the rotation angle $\theta_{i}$ is $\cos(\theta_{i})= s_{i}$, controlled by the qubit $|c\rangle$. In turn, Eq. \eqref{eq:map_meas} is obtained by a non-unitary projective measurement $|0\rangle \langle 0|$ on the second ancilla qubit while Eq. \eqref{eq:map_last} is achieved by applying the operator $U_{i}$ on the first ancilla qubit $|c \rangle$. The corresponding circuit diagram is displayed in Fig. \ref{fig:nonunitary}. Going from Eq. \eqref{eq:map_nonunit} to Eq. \eqref{eq:map_meas}, the normalization changes from $N_{i-1}^{2}$ to $N_{i}^{2}$, where
\begin{align}
N_{i} := \sqrt{ 1 + \langle \Psi^{(i)}_{\rho}|\Psi^{(i)}_{\rho}\rangle }.
\end{align}

\begin{figure}
\includegraphics[width=0.5\textwidth]{nonunitary}
\caption{Circuit diagram for the implementation of the nonunitary operation. The upper qubit is an ancilla qubit prepared in the state $|0\rangle$. The measurement operator implements the projective measurement $|0\rangle \langle 0|$. If the measurement yields the state $|1\rangle$, the calculation has to be redone from the beginning.}
\label{fig:nonunitary}
\end{figure}

This algorithm is nondeterministic: every time the projective measurement is performed, the value of the ancilla qubit is verified. If it is in the state $|0\rangle$, this is a ``success'' and the rest of the algorithm can follow because the non-unitary operation has been performed properly. Otherwise, if it is in the state $|1 \rangle$, this is a ``failure'' and the initialization phase has to be reworked from the beginning. Each measurement has a definite success and failure probability. From Eq. \eqref{eq:map_nonunit}, the probability of failure of one projective measurement at step $i=0,\cdots,N_{t}$ is given by
\begin{align}
\label{eq:p_failure}
p_{\mathrm{failure}}^{(i)} = \frac{\langle \Psi_{1}^{(i)}|\Psi_{1}^{(i)} \rangle  (1-s_{i}^{2})}{N_{i-1}^{2}}.
\end{align}
The failure probability is maximized when $p_{1}^{(i)}:=\langle \Psi_{1}^{(i)}|\Psi_{1}^{(i)} \rangle/N_{i-1}$ is maximal. It is demonstrated in Appendix \ref{app:bound_p} that the maximization over the norm of $\Psi^{(i-1)}_{\rho}$ is given by
\begin{align}
\max_{  \left\lVert \Psi^{(i-1)}_{\rho} \right\lVert} p_{1}^{(i)}= \frac{1}{1+(F_{i}^{-})^{2}},
\end{align}
where $F_{i}^{-}$ is defined in Eq. \eqref{eq:F} and depends on $B_{i}$.

The success probability, on the other hand, is $p_{\mathrm{success}}^{(i)} = 1-p_{\mathrm{failure}}^{(i)}$. Maximizing $p_{1}^{(i)}$, a first inequality can be written as
\begin{align}
p_{\mathrm{success}}^{(i)
&\geq 1- \left(\max_{  \left\lVert \Psi^{(i-1)}_{\rho} \right\lVert} p_{1}^{(i)} \right)(1-s_{i}^{2}).
\end{align}
The right-hand side of the last expression is a monotonically decreasing function of $|B_{i}|$. As a consequence, the success probability is minimized when $|B_{i}|$ is maximized. As can be readily verified, $\max|B_{i}| = 1/N_{t}$ because the other terms have values in the interval $|w_{\rho}|,|u_{i}|,|e^{iEt}| \in [0,1]$.
Using this result, the success probability after $N_{t}+1$ applications of operator $\hat{B}_{i}$, as required by the algorithm and denoted by $P_{\mathrm{success}}$, is bounded by
\begin{align}
P_{\mathrm{success}}
&\geq  \left[ \frac{1+ \frac{1}{4N_{t}^{2}} - \frac{1}{2N_{t}}\sqrt{1+\frac{1}{4N_{t}^{2}}}}{1+ \frac{1}{4N_{t}^{2}} + \frac{1}{2N_{t}}\sqrt{1+\frac{1}{4N_{t}^{2}}}} \right]^{N_{t}+1}
\end{align}
For a large number of iterations, this becomes
\begin{align}
\label{eq:prob_success}
P_{\mathrm{success}} &\geq  \frac{1}{e} \left[ 1 - \frac{1}{N_{t}} \right]  + O(N_{t}^{-2}),
\end{align}
where $e \approx 2.7183$ is Euler's number.

The success probability $P_{\mathrm{success}}$ relates to the implementation of the non-unitary operations that filter the trial function. As a consequence, the mean number of realizations required to implement the filtering part of the algorithm is approximately  $  e $. This is low enough that the efficiency of the global algorithm, including the time evolution, is not deteriorated by the nondeterministic nature of the spectral filtering method. However, the total success probability of the algorithm is
\begin{align}
P_{\mathrm{total}} &= P_{\rho}P_{\mathrm{success}}, \\
& = \frac{\langle \psi_{\rho}| \psi_{\rho}\rangle}{N^{2}e} + O(N_{t}^{-1}).
\end{align}
Consequently, the total success probability of the whole procedure depends on both the probability of being in the eigenstate $P_{\rho}$ and the probability of success of the non-unitary operation. The total success probability and the accuracy of the method are analyzed in more detail in the next section.

\section{Accuracy, complexity analysis and resource requirements}
\label{sec:complexity_an}

In this section, some properties of the filtering technique are studied quantitatively and compared to the ALT. In particular, estimates for the accuracy and for the resource requirements are obtained in terms of the filter properties.

\subsection{Accuracy of the filtering method}

The accuracy of the filtering method is evaluated in Appendix \ref{app:accuracy}. If the suppression of the filter is large enough and if the trial function is real, the error is bounded by
\begin{align}
\label{eq:err_est_text}
\epsilon \leq \max \left[ CN_{t}^{-q}, \epsilon_{\mathrm{f}} \right].
\end{align}
The first argument on the right-hand side of Eq. \eqref{eq:err_est_text} is the error of the time evolution discretization scheme, which is $O((\Delta t)^{q})$. The value of the constant $C$ depends on the Hamiltonian considered while $q$ depends on the order of accuracy of the time evolution and on the smoothness of the Hamiltonian \cite{1751-8121-43-6-065203}. The second argument defined by
\begin{align}
\epsilon_{\mathrm{f}} := \frac{S^{2}(1-A)}{A|L(0)|^{2}},
\end{align}
is the error of the filtering procedure, where $S$ is the minimum suppression of the filter (defined in Eq. \eqref{eq:supp_fact}), $A:=|\langle \varphi_{\rho}| \Psi_{\mathrm{trial}}(0) \rangle|^{2} $ is the overlap of the trial function with the desired eigenstate and $|L(0)| \in [0,1]$ is the coherent gain of the filter. Typical values for the coherent gain lie in the interval $0.3 \lesssim |L(0)| \lesssim 0.8$, with a value of  $|L(0)|=1$ for the rectangular window function \cite{harris1978use,heinzel2002spectrum}. Therefore,  the accuracy of the filtering is controlled by the filter properties (the suppression and the coherent gain) and by the initial trial state.

The value of the suppression factor $S$ depends on the filter chosen. One popular choice is the Hann function \cite{Feit1982412}. However, there now exist high-performance filters where the side lobe suppression levels can reach up to -248~dB \cite{heinzel2002spectrum}, reducing the power spectral density by a factor of $\approx 10^{25}$ outside the range of interest and relative to the central frequency. As a comparison, a rectangular window function, as in the ALT, reaches -13.3~dB while the Hann window function has a suppression factor of -31.5~dB. Choosing a filter with a large suppression could potentially lead to a very accurate eigenstate estimation.

\subsection{Performance of the method}

The high suppression factors described in the last section are possible only when the filter resolves the eigenenergies, when the energy width of the line shape is smaller than the energy interval between the wanted eigenenergy and its closest neighbor  as $\Delta E_{\mathrm{window}} < \Delta E_{\rho}:= \min|E_{\rho} - E_{\rho \pm 1}|$. Here, $\Delta E_{\mathrm{window}}$ is the bandwidth defined from the center of the filter $E_{\rho}$ to the minimum between the main peak and the first side lobe. The resolution of the filter is then related to the total time of the calculation as $\Delta E_{\mathrm{window}} =W\pi/T$, where $W \in \mathbb{R}^{+}$ is the line shape width. The rectangular filter has the best resolution with $W_{\mathrm{rectangular}}=1$. For other filters, with higher suppression strength, typical values are $2.0 \lesssim W \lesssim 6.0$. Therefore, having a higher suppression strength usually entails more time iterations to resolve the eigenstates.

The number of iterations $N_{t}$ required is determined by fixing $\Delta t$ and $T$ to obtain the desired bandwidth and resolution. First, the size of the time step $\Delta t$ should guarantee that the time-dependent evolution reaches convergence: this is achieved when the largest frequency component is resolved and when the numerical scheme is stable (as an example, for explicit time integration scheme in real space, a necessary condition for stability is the Courant-Friedrichs-Lewy condition \cite{leveque2002finite}). In addition, the value of $\Delta t$ should make sure that the error due to the time evolution is at least as small as the wanted accuracy. Finally, because the energy range of the filter is $[- \pi/\Delta t, \pi/\Delta t]$, yielding a bandwidth of $B:=2\pi/\Delta t$, the time step has to be set to give an energy range that accommodates all of the desired spectral components. Second, as mentioned earlier, the final time should be large enough such that eigenenergies are resolved. These considerations yield the following condition on the number of time steps:
\begin{align}
N_{t} > \max \left[\frac{ W B}{2(\Delta E_{\rho}) }, \left(\frac{C}{\epsilon} \right)^{\frac{1}{q}} \right].
\end{align}
This condition ensures that the filter resolves the eigenstate and that the desired accuracy is achieved.
Therefore, if a high resolution is required, the computation time is proportional to the width of the filter and inversely proportional to the energy interval between eigenvalues. For a given Hamiltonian, the number of time steps required is then $N_{t} = \mathrm{poly}(n)$, as long as the eigenstates are not exponentially close together, obeying $(\Delta E_{\rho})^{-1} = \mathrm{poly}(n)$. For many cases of interest, this condition is fulfilled and our initialization technique becomes exponentially faster than any classical algorithm. For well-separated eigenenergies, the number of time steps is rather determined by the accuracy of the time evolution. Then, the width of the filter is not critical to the performance of the method.

For quasigenerate states exponentially close to each other, when $\Delta E_{\rho} \sim 2^{-n}$, the quantum efficiency is absent because resolving these states necessitates an exponential number of time steps. One example is the  Coulomb-like system for which the eigenvalues display an accumulation point at $E=0$. Resolving the highly excited states in the vicinity of $E=0$ requires an exponential number of operations because of their quasi-degenerate nature.

Also, this technique could cease to be efficient for many-body systems, in the limit of a large number of strongly correlated particles \cite{schuch2009computational}. In this case, the required accuracy decreases exponentially with the number of particles \cite{RevModPhys.71.1253}, demanding an exponential calculation time.  As a matter of fact, there exist classes of local Hamiltonian for which the eigenenergies are QMA-hard to obtain \cite{doi:10.1137/S0097539704445226} and therefore, are likely to involve an exponential number of operations. Nevertheless, although our method is not valid for all QMA-hard problems, it may still be useful in practical applications, such as quantum chemistry simulations \cite{kassal2011simulating}. These limitations are also shared by the ALT.

The average number of operations $\bar{N}$ required to initialize the wave function using this quantum algorithm scales like $\bar{N} = e N_{t} \mathrm{poly}(n)/P_{\rho}$. This can be compared to the general states initialization methods described at the beginning, which scale like $O(2^{n})$. In the limit of a large number $n$ of qubits, the quantum spectral filtering method has an exponential gain of performance, if $P_{\rho}$ is not exponentially small. This probability is now discussed in more details. As long as  $\epsilon_{\mathrm{f}} \ll 1$, the probability can be estimated as
\begin{align}
P_{\rho} = \frac{A|L(0)|^{2}}{1+A|L(0)|^{2}} + O(\epsilon_{\mathrm{f}}),
\end{align}
and thus, varies in the interval $P_{\rho}\in [0,1/2]$, according to the value of the overlap and the coherent gain. Similar to the accuracy, this probability depends on the coherent gain of the filter and on the trial state overlap. As mentioned earlier,  typical values of the coherent gain for most filters are $\sim 10^{-1}$, reducing the performance of the algorithm when compared to the rectangular filter. This probability is also controlled by the overlap, confirming that the trial state choice is highly important for the computational cost.

\subsection{Comparison with the ALT}

Using the estimates derived above, it is now possible to make a comparison with the ALT.
The computational complexity of the filtering algorithm is similar to the ALT: the number of operations in the ALT scales like $ \bar{N}^{(\mathrm{ALT})} = M \mathrm{poly}(n)/P_{\rho}^{(\mathrm{ALT})}$, where $M = N_{t}$, the number of iterations, determines the resolution and is similar to our number of time steps. However, the success probability is different and is given by $P_{\rho}^{(\mathrm{ALT})} = A + O(\epsilon_{\mathrm{f}})$ \cite{PhysRevLett.91.257902}. Combining the preceding results, it is possible to compare the resource requirements for both methods. We get
\begin{align}
\frac{\bar{N}}{\bar{N}^{(\mathrm{ALT})}} < \frac{2e}{|L(0)|^{2}} \frac{N_{t}}{N_{t}^{(\mathrm{ALT})}}.
\end{align}
For applications where the resolution is critical, this becomes $\bar{N}/\bar{N}^{(\mathrm{ALT})} < 2eW/|L(0)|^{2}$ and the performance is dictated by the coherent gain and the width of the filter. However, if the accuracy determines the number of time steps, we have $\bar{N}/\bar{N}^{(\mathrm{ALT})} < 2e/|L(0)|^{2}$ and only the coherent gain is important for the performance.

In terms of the accuracy, the comparison between the two methods yields
\begin{align}
\frac{\epsilon}{\epsilon^{(\mathrm{ALT})}} = \frac{S^{2}}{S^{2}_{\mathrm{rectangular}}|L(0)|^{2}},
\end{align}
where it is assumed that $CN_{t}^{-q} < \epsilon_{\mathrm{f}},\epsilon_{\mathrm{f}}^{(\mathrm{ALT)}}$. Therefore, the accuracy of our filtering technique can potentially be superior to the ALT.

However, there is clearly a tradeoff between accuracy and performance, that depends on the type of filter and the technique used. In particular, the ALT is more efficient by a constant factor that depends on the properties of the filter. On the other hand, our technique can potentially yield more accurate results if a larger suppression factor is considered.

Moreover, our technique does not require a large number of ancilla qubits, a useful property given that the number of qubits is limited to $\approx 14$ in actual quantum computer registers \cite{PhysRevLett.106.130506,ladd2010quantum,0034-4885-74-10-104401}.
A typical quantum simulation in real or momentum space for a non-relativistic single particle system, for example, requires at least $\approx$ 256-512 lattice points in 1D to reach convergence, making for a quantum register having more than $\approx$ 8-9 qubits to store the wave function. For the ALT method, the register for the quantum Fourier transform requires approximately $\approx$ 6-7 additional qubits \cite{PhysRevLett.83.5162}. In addition, a certain number of qubits should be reserved for error correction and possibly, for additional work space. Therefore, using ALT, the resource requirement for this type of quantum simulation is at the limit and probably above the resources available. With our algorithm, only two ancilla qubits are necessary, for any energy resolution. This reduction of the number of qubits could allow for the simulation of a certain type of systems using actual quantum computers, whereas they would be out of reach using the ALT. This of course, assumes that the coherence time of the quantum device can accommodate the larger computational cost of our method.

\section{Specific example: the harmonic oscillator}
\label{sec:harm_osci}

In this section, some features of the filtering method are illustrated in a simple example: the quantum harmonic oscillator. The Hamiltonian, expressed in the oscillator natural units where the energy is in units of $\hbar \omega$, lengths are in units of $\sqrt{\hbar/(m\omega)}$, $\omega$ is the oscillator frequency and $m$ the particle mass, is given by
\begin{align}
\hat{H} = \frac{\hat{p}^{2}}{2} + \frac{1}{2}\hat{x}^{2}.
\end{align}
The ground state of this system is now generated on a classical computer by the filtering method combined with techniques borrowed from real space quantum simulation. In particular, the 1D time-dependent Schr\"{o}dinger equation is solved by using the pseudospectral second order split operator scheme described in Appendix \ref{app:td_sh}. The latter could be implemented efficiently on a quantum computer by using the quantum Fourier transform \cite{OPPROP:PR877,Strini2008}.

The filtering is applied to a trial state given by
\begin{align}
\Psi_{\mathrm{trial}}(0,x) =
\begin{cases}
\cos^{2} \left( \frac{\pi x}{2\ell} \right), & x\in[-\ell,\ell]\\
0,& \mathrm{else}
\end{cases}
\end{align}
where $\ell$ is the wave function width. This trial state can be generated efficiently with $\mathrm{poly}(n)$ quantum gates using the technique given in Refs. \cite{Zalka08011998,kaye2004quantum,grover2002creating} because its square can be integrated analytically.

The domain has a length of $L=40$ (all quantities are in the quantum oscillator natural units) and is discretized by 1024 lattice points. The trial state width is set to $\ell = 10$ and is evolved to a final time of $T=100$ using $N_{t} = 8192$ time steps. The power spectrum of the trial function is computed by using Eq. \eqref{eq:auto_corr} and the results are displayed in Fig. \ref{fig:power_spec}. It can be verified that the energy of the highest peaks in the power  spectrum are positioned at energies
\begin{align}
E = E_{m} = m + \frac{1}{2}, \;\; m =1,3,5,\cdots,
\end{align}
in agreement with the analytical solution of the harmonic oscillator. Components with $m$ even do not appear in the spectrum because they are antisymmetric while the trial wave function is symmetric.

The trial state is then filtered using Eq. \eqref{eq:eigen_state}. The simulation parameters are the same as above. Two window functions are considered: a rectangular window as in the ALT and the Hann window, given by
\begin{align}
w(t) = \frac{1}{2}\left[ 1 - \cos\left(\frac{2\pi t}{T} \right) \right].
\end{align}
The resulting power spectrum of the filtered ground state (with $E_{0} = 1/2$) is also displayed in Fig. \ref{fig:power_spec}. For both window types, the power spectrum is sharply peaked around the ground state energy. However, the Hann window function suppresses the unwanted modes by up to five orders of magnitude more than the rectangular window function, conferring a distinct advantage of our apodization technique because it yields more accurate states. This is confirmed by computing the numerical error $\epsilon = \left\lVert \psi_{\mathrm{filtered}} - \psi_{\mathrm{exact}} \right\lVert$. We find that $\epsilon_{\mathrm{Hann}} \approx 2.42 \times 10^{-8}$ while $\epsilon^{(\mathrm{ALT})} = \epsilon_{\mathrm{rect}} \approx 1.77 \times 10^{-5}$.

\begin{figure}
\includegraphics[width=0.5\textwidth]{power_spectrum}
\caption{Power spectra for the ground state generated from the filtering method using the constant or Hann window function.  The spectrum is also given for the trial state, prior to filtering.}
\label{fig:power_spec}
\end{figure}

The calculation considered here could be performed on a 12 qubit register: 10 qubits are used to store the wave function and two ancilla qubits are utilized in the initialization phase, as discussed previously. In comparison, for the same energy resolution, the ALT would require approximately 23 qubits: 10 qubits for the wave function and $\approx 13$ qubits for the quantum Fourier transform. Given that the state-of-the-art quantum registers have $\approx 14$ qubits \cite{PhysRevLett.106.130506}, reducing the number of ancilla qubits is still an important issue for the success of quantum simulations. Therefore, our quantum filtering technique is an interesting alternative from this standpoint.

The success probability, on the other hand, favors the ALT. The overlap of the trial state with the desired state yields a success probability of $P^{(\mathrm{ALT})}_{\mathrm{total}} \approx 0.45$. For our filtering technique, this probability is evaluated explicitly using Eqs. \eqref{eq:p1_app}, \eqref{eq:p_failure} and the expression of $P_{\rho}$. We find that $P_{\mathrm{total}} \approx 0.061$. Therefore, our filtering scheme demands more computation time than for ALT, demonstrating the tradeoff between accuracy, computation time and the number of qubits required.

Interestingly, the accuracy of our filtering scheme is approximately the same as for ALT using a number of time steps given by $N_{t} = 1600$, using the same value for other parameters. In this case, the accuracy is reduced to $\epsilon_{\mathrm{Hann}} \approx 1.66 \times 10^{-5}$ while the change in the probability of success is negligible. Therefore, one obtains that $\frac{\bar{N}}{\bar{N}^{(\mathrm{ALT})}} \approx 1.44$, showing that the performance overhead can be relatively unimportant in some cases for a fixed accuracy, although the number of required qubits is reduced.

\section{Conclusion}
\label{sec:conclusion}

In conclusion, the algorithm we are presenting allows for an efficient initialization of a quantum register to a state with a specified energy range under some specific conditions: the overlap of the trial state with the eigenstate and the energy difference between eigenstates should not be exponentially small. Our algorithm improves upon the ALT on two main aspects: the inclusion of a window function allows for more accurate eigenstates (the accuracy depends on the choice of filter) and the number of ancilla qubit is limited to two, for any energy resolution. In the ALT, this energy resolution dictates the number of qubits. However, our technique generally requires more quantum operations than the ALT and thus, there is a tradeoff between accuracy, computation time and resource requirements. These features were demonstrated in the simple example of the quantum harmonic potential system.

Such a procedure is important for dynamical quantum simulations on digital quantum computers where the goal is to simulate the time-dependent behavior of the wave function and to determine scattering process probabilities where the initial state is one specific eigenstate of some static Hamiltonian. These are important in many fields such as atomic, molecular, optical and condensed matter physics.

Furthermore, our algorithm can be applied to many physical systems in any basis (real space, momentum space, etc) as long as an efficient time evolution algorithm is available and the conditions given above are fulfilled. Other explicit implementations for the solution of quantum equations will be given elsewhere.

Finally, it may be interesting to apply the novel concept of qubitization to our filtering method. This new technique promises optimal query complexity for the computation of a large class of unitary and non-unitary operators \cite{low2016hamiltonian}. This will be investigated further in the future.


\section{Introduction}

Rotating magnetospheres, with relatively strong
magnetic fields, surrounding active galactic nuclei (AGN), are believed to be the regions where cosmic rays might originate {\bf \cite{rieger1,zev}}. This particles are highly relativistic and one of the open problems in modern astrophysics is how do they attain  such high energies.

Several mechanism have been suggested to explain acceleration of
leptons or hadrons to ultra high energies. In particular, in the so-called Fermi class
(and its modifications) of processes \cite{fermi,bell1,bell2}, the acceleration occurs in
relatively strong magnetic fields in the magnetospheres of astrophysical
objects. The Fermi processes, it turns out, are efficient only if the particles are already relativistic. For overall efficiency, therefore,  some kind of a pre-acceleration is desirable \cite{rieger1}.

In this paper, however, we explore the Langmuir-Landau-Centrifugal Drive (LLCD) mechanism that makes use of the plasma collective phenomena manifested though the Langmuir wave that will serve as a conduit for transferring gravitational energy in the rotating magnetosphere of an AGN to particle kinetic /thermal energy. Although LLCD has been discussed and developed in several papers \cite{review,sgrA,screp1,zev}, we will summarise its salient qualitative features in the introduction.

Since in the AGN magnetospheres magnetic fields are of the order of $10^{1-4}$G, the charged particle flows will follow co-rotating magnetic field lines (frozen-in condition), experiencing very strong relativistic centrifugal
force in the light cylinder (LC) zone (area where the linear velocity of rotation exactly equals
the speed of light) \cite{gold}. If the plasma number density is hight enough to provide the screening of longitudinal (along the magnetic field lines) electric field up to the LC area, the centrifugal mechanism becomes very efficient \cite{rieger2}. The strong relativistic differential centrifugal force leads to several consequences, for instance, the direct acceleration of electrons to very high energies (Lorentz factors of the order of $10^7$) \cite{rieger1,osm7}. Although the LLCD mechanism has been shown to accelerate particles to even higher energies, we in this paper emphasize its possible role in heating of the magnetosphere.

The first step in the LLCD process is the parametric excitation of Langmuir waves through charge separation caused by differential rotation of different charge species (electrons and protons ) endowed with different Lorentz factors. This process, examined for millisecond pulsars \cite{langm2} and AGN \cite{langm1}, proves to be very effective in pumping enormous energy (from rotation, which in turn is gravitationally driven) into what could be called centrifugally driven electrostatic waves.

In the second stage of LLCD, the electrostatic waves Landau damp and transfer energy from
centrifugally amplified modes to particle kinetic energy. The efficacy of this wave-particle energy transfer has been demonstrated for millisecond and newly born pulsars \cite{screp1,screp2}; it was shown that LLCD could energize particles to the tune of $10^{18}$eV.

In the study of the role of LLCD in the AGN magnetospheres \cite{zev}, and SgrA$^{\star}$
(located in the centre of the Milky Way) \cite{sgrA}, we found that the parametrically amplified
Langmuir waves, before undergoing standard Landau damp, go through another intermediate boosting via Langmuir collapse. One learns from the classical work by \cite{zakharov}, that the Langmuir turbulence in the non-linear regime creates unstably excited caverns or the low-density regions. As a result, the high
frequency pressure pulls the particles from the cavern, which in turn, provokes highly unstable
collapse leading to fast amplification of the electrostatic field. The corresponding problem has been studied for three dimensional geometry by \cite{galeev} considering spectra of Langmuir turbulence. The termination by means of the Landau damping of the collapse process has been numerically studied by \cite{degtiarev}. This last stage is the most important one in energy pumping process, because the whole energy of Langmuir waves transfers to particle kinetic energies. Applying this process to AGN \cite{zev} and the central black hole of our galaxy \cite{sgrA}, it has been found that protons might achieve extremely high energies of the order of $ZeV$ (in AGNs) and $PeV$ (in SgrA$^{\star}$).

A principal consequence of  Langmuir collapse might be not only the particle acceleration in a particular direction but also the efficient heating of the sustaining plasma.  It is this heating that is posited here to be the cause of the observed intrinsic brightness (high temperatures ~$10^{6-9}$K) of some AGNs \cite{agn}.

The paper is organized   as follows: in Sec.2, the essential formulation of LLCD is recalled, in Sec. 3, the theoretical model is applied to AGN heating problem, and in Sec. 4, our principal results are summarized.

\section{The theoretical model of LLCD}

The plasma under consideration lies in the magnetosphere of a supermassive black hole of typical mass
$M =10^{8-9}\times M_{\odot}$ ($M_{\odot}\approx 2\times 10^{33}$g is the solar mass), and corresponding angular velocity of rotation
\begin{equation}
\label{rotat} \Omega\approx\frac{a c^3}{GM}\approx
10^{-3}\frac{a}{M_8}rad/s^2,
\end{equation}
where $c$ is the speed of light,
$G\approx 6.67\times 10^{-8}$dyne-cm$^2$/g$^2$ is the gravitational constant and
$M_8\equiv M/(10^8M_{\odot})$ and $0<a\leq 1$ are respectively dimensionless mass and a
dimensionless parameter measuring the rate of rotation.

If the plasma in the near vicinity of a supermassive black hole is anchored
by a magnetic field strong enough to maintain frozen-in condition, i.e, the  charged particles
will follow the field lines. In our model, we assume that the AGN radiation energy density and magnetic field energy density are of the same orders of magnitude (approximate equipartition). Then, on the light cylinder surface, where the relativistic effects of rotation are the most efficient, the strength of magnetic field is given by \cite{osm7},
\begin{equation}
\label{mag} B\approx\sqrt{\frac{2L}{R_{lc}^2c}}\approx
870\times a\times L_{45}^{1/2}\times M_8 \;G,
\end{equation}
where $R_{lc}=c/\Omega$ is
the light cylinder radius and $L_{45}=L/\left(10^{45}erg/s\right)$ is the dimensionless
luminosity. As it has been explained in detail by \cite{rieger1}, the charged particles co-rotate with the field lines until the Lorentz factor reaches its maximum value, $\gamma_{max}\sim\left(eB/(2m_p\Omega c)\right)^{2/3}\sim 1.2\times 10^7L^{1/3}_{45}$ (here $m_p$ is the proton's mass). Due accretion, the nearby area of AGN contains soft thermal photons which, by means of the IC scattering, might affect the process of acceleration. IC operates primarily on electrons while it is strongly suppressed for protons (by a factor ~$10^{-13}$) and does not limit the maximum energy protons may acquire. In the framework of our approach  the magnetic field lines almost up to the LC are supposed to be approximately straight \footnote{In the rotating magnetosphere this means that the particles in the rotating frame of reference move along straight trajectories. However, it has been shown that on the LC itself the field lines are swept back, lagging behind the rotation \cite{curve} but up to the mentioned zone the field lines are quasi straight.}

The original content of the LLCD mechanism, the  parametric generation of centrifugally driven Langmuir waves,  is contained in the set of linearized fluid equations (in Fourier space) coupled to the Poisson equation \cite{screp1,zev,screp2}:
 \begin{equation}
\label{eul3} \frac{\partial p_{_{\beta}}}{\partial
t}+ik\upsilon_{_{\beta0}}p_{_{\beta}}=
\upsilon_{_{\beta0}}\Omega^2r_{_{\beta}}p_{_{\beta}}+\frac{e_{_{\beta}}}{m_{_{\beta}}}E,
\end{equation}
\begin{equation}
\label{cont1} \frac{\partial n_{_{\beta}}}{\partial
t}+ik\upsilon_{_{\beta0}}n_{_{\beta}}, +
ikn_{_{\beta0}}\upsilon_{_{\beta}}=0
\end{equation}
\begin{equation}
\label{pois1} ikE=4\pi\sum_{_{\beta}}n_{_{\beta0}}e_{_{\beta}},
\end{equation}
where Eq. (\ref{eul3}) is the equation of motion in the presence of a centrifugal force, Eq. (\ref{cont1}) represents the
continuity equation, and Eq. (\ref{pois1}) is the Poisson equation for the electric field E. In the preceding equations,
 ${\beta}$ is the species index (either electrons or protons), $p_{_{\beta}}$ is the first order dimensionless momentum ($p_{_{\beta}}\rightarrow
p_{_{\beta}}/m_{_{\beta}}$), $k$ represents the wave number of the excited mode, $\upsilon_{_{\beta0}}(t) \approx
c\cos\left(\Omega t + \phi_{_{\beta}}\right)$ is the zeroth order
velocity and $r_{_{\beta}}(t) \approx
\frac{c}{\Omega}\sin\left(\Omega t + \phi_{_{\beta}}\right)$ is the
radial coordinate \cite{zev}, $e_{_{\beta}}$ is the particle's
charge and $n_{_{\beta}}$ and $n_{_{\beta0}}$ are, respectively the perturbed and
unperturbed Fourier components of the number density.

The first term on the righthand side of Eq. (\ref{eul3}) represents the relativistic analogue of the centrifugal force. Since the centrifugal  force differentiates between different species (electrons and protons), the resulting charge separation excites the Langmuir instability.

Notice that due to the time dependence of the centrifugal force, the preceding system cannot be "mode analyzed" by the standard techniques. However, we can extract almost as much information from the system as in a typical system, by the method developed in \cite{screp2}. The ansatz
\begin{equation}
\label{ansatz}
n_{\beta}=N_{\beta}e^{-\frac{iV_{\beta}k}{\Omega}\sin\left(\Omega t
+ \phi_{\beta}\right)},
\end{equation}
converts the aforementioned set of governing equations to a pair of coupled equations (time dependence is left only in $\chi$)
\begin{equation}
\label{ME1} \frac{d^2N_p}{dt^2}+{\omega_p}^2 N_p= -{\omega_p}^2 N_e
e^{i \chi},
\end{equation}
\begin{equation}
\label{ME2} \frac{d^2N_e}{dt^2}+{\omega_e}^2 N_e= -{\omega_e}^2 N_p
e^{-i \chi},
\end{equation}
where $\omega_{e,p}\equiv\sqrt{4\pi e^2n_{e,p}/m_{e,p}\gamma_{e,p}^3}$ is the relativistic
plasma frequency of the corresponding specie, $\gamma_{e,p}$ is the Lorentz factor, $\chi = b\cos\left(\Omega t+\phi_{+}\right)$, $b =
\frac{2ck}{\Omega}\sin\phi_{-}$ and $2\phi_{\pm} = \phi_p\pm\phi_e$.

A quasi "dispersion relation" for the electrostatic wave is derived after a formal Fourier transform (For the detailed derivation please see the paper by \cite{screp2} of Eqs. (\ref{ME1},\ref{ME2})

\begin{equation}
\label{disp} \omega^2 -\omega_e^2 - \omega_p^2  J_0^2(b)= \omega_p^2
\sum_{\mu} J_{\mu}^{2}(b) \frac{\omega^2}{(\omega-\mu\Omega)^2},
\end{equation}
where $J_{\mu}(x)$ is the Bessel function. For resonant modes, $\omega = \mu\Omega+\Delta=\omega_r+\Delta$, $\Delta\ll \omega_r$, we find that the effective dispersion relation is contained in
 \begin{equation}
 \label{disp1}
 \Delta^3=\frac{\omega_r {\omega_p}^2 {J_{\mu_{r}}(b)}^2}{2},
 \end{equation}
 that is readily solved to yield the instability growth rate
\begin{equation}
 \label{grow1}
 \Gamma= \frac{\sqrt3}{2}\left (\frac{\omega_e {\omega_p}^2}{2}\right)^{\frac{1}{3}}
 {J_{\mu}(b)}^{\frac{2}{3}}.
\end{equation}
where $\mu = \omega_e/\Omega$.

\begin{figure}
  \resizebox{\hsize}{!}{\includegraphics[angle=0]{fig01.eps}}
  \caption{Behaviour of $T$ (solid line) and $T_0$ (dashed line) with respect to the
  bolometric luminosity of AGN.
The set of parameters is: $a = 1$, $\epsilon = 0.1$, $\kappa = 0.5$, $M_8 = 1$, $\gamma_1 = 7\times 10^3$, $\gamma_2 =  10^4$ and $R_0\approx 0.01$pc.}\label{fig1}
\end{figure}

\section{Discusion}

Let us now examine the relevance of this instability as an agent for energy transfer in the AGN context.
Since this instability is driven by differential rotation between the electron and ion fluids, it falls in the general class of two stream instabilities. We now show that
for the magnetospheric plasma, consisting of relativistic electrons and protons, the instability growth rates are "large" implying that  Langmuir wave generation  is very efficient.

From Eq. (\ref{disp1}), one can straightforwardly show that for electrons with Lorentz factors $\gamma_1 = 7\times 10^3$, and protons having the same value $\gamma_2=10^4$, the timescale of energy pumping from rotation
to electrostatic waves, $\tau\sim 1/\Gamma$, varies in the range $600-2000$sec. On the other hand, the kinematic timescale (escape time-scale) of co-rotating particles (electrons and protons) equals $P/4=\pi/(2\Omega)\approx 1.5\times 10^4$sec. As it is evident, the instability timescale is much less than the kinematic timescale, indicating extremely high efficiency of centrifugally driven electrostatic waves.

These relatively high amplitude Langmuir waves will, now, induce high frequency
pressure pushing the particles out from the perturbed zone  \cite{zakharov}. In the resulting flow density
areas (caverns), the penetrating waves will amplify pressure, which
in turn, augments the process of pulling out of the particles resulting in the Langmuir collapse.

We assume that the kinetic and potential energies of plasmons inside the caverns are of the same orders
of magnitude \cite{arcimovich}
\begin{equation}
\label{k} k^2\lambda_D^2\sim\frac{\mid\delta n\mid}{n_0},
\end{equation}
where $\lambda_D\equiv \sqrt{k_{B}T_0/(4\pi n_0e^2)}$ is the
Debye length-scale, $k_{B}\approx 1.38\times 10^{-16}$ erg K$^{-1}$
is the Boltzmann constant and $T_0$ is the temperature provided by the accretion process. Since the perturbation density is much less than the unperturbed density, $\delta n\ll n_0$, energy of plasmons is almost constant
\begin{equation}
\label{E2a} \int {d\bf r}\mid E\mid^2 = const.
\end{equation}

It is useful to recognize that  $k$ defines the system length-scale ($k\sim l^{-1}$).  Equation (\ref{E2a}), then, implies that the electrostatic energy density behaves as $E^2\propto l^{-q}$ where $q$ denotes the dimensionality of the process. Coupled with Eq.(\ref{k}), we find that the high frequency
pressure, $P_{hf}\approx -E^2\delta n/(24\pi
k^2\lambda_D^2n_0)\propto E^2\propto l^{-q}$   \cite{arcimovich}, suppresses
the thermal pressure, $P_{th} = k_BT_0\delta n\propto\delta n\propto k^2\propto l^{-2}$ only for three dimensional
geometry. Inside the magnetosphere, however, the effective geometry is one dimensional ($q=1$) since the particles are in the frozen-in condition and follow the field lines. In this region,  thus, the collapse
is impossible ($P_{hf}\sim l^{-1}$ and $P_{th}\sim l^{-2}$);  it can be realized only in the outer regions of the magnetospheres, for distances exceeding the LC radius.

Zakharov  showed that, for three dimensional geometry, $q=3$,
the driven electrostatic field and the corresponding length-scale of the cavern behave as
\cite{zakharov}
\begin{equation}
\label{E2} \mid E\mid\approx \mid E_0\mid\frac{t_0}{t_0-t}
\end{equation}
\begin{equation}
\label{l} l\approx l_0\left(\frac{t_0}{t_0-t}\right)^{-2/3},
\end{equation}
where $t_0$ is the time when the electric field fully collapses, and $E_0\approx 4\pi ne\Delta r\exp\left(\Gamma P/4\right)$ is the electrostatic field before the collapse starts (the initial field amplified by means of the electrostatic instability) and $\Delta r\approx
R_{lc}/(2\gamma_{_{max}})$ represents a length scale in the LC zone where the
process of energy pumping occurs \cite{zev} and we have taking into account that the escape time of particles equals $P/4$ \cite{MR}. It is clear from the preceding equations, that
in due course of time, the length-scale of the cavern goes to zero, whereas the electrostatic field
asymptotically increases.

This process is terminated by means of Landau damping, when $l$ reaches the dissipation
length-scale, $l_d\approx 2\pi\lambda_D$, \cite{arcimovich}. From Eqs. (\ref{E2},\ref{l}) one can
show that the electrostatic field will be boosted by the factor $\left(\Delta r/l_d\right)^{3/2}$. Correspondingly, a certain fraction, $\kappa$, of the electrical energy converts to heat, increasing the temperature of the ambient plasma,
\begin{equation}
\label{temp1} 2\kappa\pi R_{lc}l_dH\frac{E^2}{8\pi}\approx\frac{4\sigma}{c}T^4\pi R_{0}^2H,
\end{equation}
where (in the framework of the equipartition approach) it has been assumed that the energy pumped by means of the collapse nearby the LC zone in a thin
layer, $l_d$ (being of the order of $(3-5)\times 10^3$cm), is uniformly distributed in a cylindrical area with typical radius, $R_{0}$; the energy is emitted away as a black body radiation. Here we assume that the typical heated radius  is of the order of $0.01$pc (an approximate outer radius of the accretion disc for the given mass of the BH \cite{carroll}, and $H$ represents the height of the corresponding cylinder. From Eq. (\ref{temp1}) one can, straightforwardly, obtain the increased temperature of the AGN ambient
\begin{equation}
\label{temp} T\approx\left(\frac{c\kappa l_d R_{lc}E^2}{16\pi\sigma R_{0}^2}\right)^{1/4}.
\end{equation}
The proposed heating mechanism is efficient, if the corresponding temperature exceeds that of the
accretion disk temperature, which is estimated as \cite{carroll}
\begin{equation}
\label{T0} T_0\approx\left(\frac{3GM\dot{M}}{8\pi\sigma R^3}\right)^{1/4}\left(\frac{R}{R_{lc}}\right)^{3/4}
\left(1-\sqrt{\frac{R}{R_{lc}}}\right)^{1/4},
\end{equation}
where $R\equiv 2GM/c^2$ is the Schwarzschild radius of the supermassive black hole, $\dot{M}=L/(\epsilon c^2)$ is the accretion rate and $\epsilon\leq 1$ describes efficiency of the accretion process.

In Fig. 1 we show the dependance of $T_0$ and $T$ on the bolometric luminosity of AGN. The set of parameters is: $a = 0.1$, $\epsilon = 0.1$, $\kappa = 0.5$, $M_8 = 1$, $\gamma_1 = 7\times 10^3$, $\gamma_2 =  10^4$, and $R_0 \approx 0.01$pc.
As is evident from the plots, the centrifugally driven heating mechanism becomes efficient for highly luminous AGN. In particular, for AGN with $L>>3\times 10^{44}$erg/s the temperature becomes much higher than the initial value ($T_0\sim 10^5$K) acquired in the accretion processes. On the other hand, one can straightforwardly check that the heating process is not sensitive with the rotation rate, which is a direct result of the power $1/4$ (see Eq. (\ref{temp})). Here we assumed the equipartition distribution of energy: half of the pumped energy goes to acceleration and half - to the heating process.

The heating mechanism considered in the present work is so efficient that it can heat the system to blackbody radiation temperatures~$10^{9}$K, which have been observed in the $X$-ray corona of AGNs \cite{agn}.

It is worth noting that during the acceleration the particles might lose energy by means of the several cooling mechanisms. This problem is considered in detail by \cite{zev}.The synchrotron process almost from the very beginning of motion becomes insignificant because the corresponding radiation is so efficient that the particles very soon transit to the ground Landau level, follow the magnetic field lines and do not emit any more in the synchrotron regime.

The curvature radiation does not impose any significant constraints as well. In particular, on the last stage of LLCD the particles achieve such high energies that the magnetic field becomes dependent on plasma energy density, the particles move along straight trajectories and as a result the curvature radiation is terminated.

The inverse Compton (IC) scattering for the considered energies occurs in the Klein-Nishina regime, when the corresponding timescale is a continuously increasing function of the proton energy \cite{blumenthal}, therefore this mechanism does not limit maximum achievable energies.

Another mechanism, which potentially might limit the process of proton acceleration is the photo-pion energy losses. The corresponding timescale is a continuously decreasing function of energy and for the extremely high values $10^{19-20}$eV is of the order of $10^7$sec. On the other hand, the timescale of energy pumping into Langmuir waves is by several orders of magnitude less than the aforementioned value. The collapse in turn, since characterised by blowing up (see Eq. (\ref{E2}-\ref{l})) is even more efficient than any other mechanisms. Therefore, the photo-pion cooling does not impose any constraints on proton acceleration and consequently on the heating process.

\section{Summary}

By examining the LLCD mechanism we have considered the system of equations composed
by the Euler equation, continuity equation and the Poisson equation respectively. It has been shown
that by means of the centrifugal force, the Langmuir waves parametrically amplify. As a result,
the excited electrostatic waves efficiently pump energy from rotation.

Amplification of electrostatic field on the next stage is provided by means of the Langmuir collapse, which terminates on the Debye length-scales, resulting in the heating of the surrounding medium. It has been shown that for luminosities greater than $2\times 10^{45}$erg/s the mentioned process can provide temperatures
in the following interval $10^{6-9}$K.

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