Fidelity Decay as an Efficient Indicator of Quantum Chaos
Joseph Emerson
Recent work has connected the type of fidelity decay in perturbed quantum models to the presence of chaos in the associated classical models. We demonstrate that a system's rate of fidelity decay under repeated perturbations may be measured efficiently on a quantum information processor, and analyze the conditions under which this indicator is a reliable probe of quantum chaos and related statistical properties of the unperturbed system. The type and rate of the decay are not dependent on the eigenvalue statistics of the unperturbed system, but depend on the system's eigenvector statistics in the eigenbasis of the perturbation operator. For random eigenvector statistics the decay is exponential with a rate fixed precisely by the variance of the perturbation's energy spectrum. Hence, even classically regular models can exhibit an exponential fidelity decay under generic quantum perturbations. These results clarify which perturbations can distinguish classically regular and chaotic quantum systems.
cue.tex
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\title{Fidelity Decay as an Efficient Indicator of Quantum Chaos}
%\title {Sensitivity to Perturbations for Random Unitary Maps}

\author{Joseph Emerson}
\email[Corresponding author: ]{jemerson@mit.edu}
\affiliation{Department of Nuclear Engineering}
\author{Yaakov S.~Weinstein} 
\affiliation{Department of Nuclear Engineering}
\author{Seth Lloyd} 
\affiliation{Department of Mechanical Engineering \\
Massachusetts Institute of Technology, 
Cambridge, Massachusetts, 02139} 
\author{D.G. Cory}
\affiliation{Department of Nuclear Engineering}

\date{\today}

\begin{abstract}
Recent work has connected the type of fidelity decay 
in perturbed quantum models to the presence of chaos 
in the associated classical models.  We demonstrate that a 
system's rate of fidelity decay under repeated perturbations 
may be measured efficiently on a 
quantum information processor, and analyze the conditions under 
which this indicator is a reliable probe of 
quantum chaos and related statistical properties of the unperturbed system.  
The type and rate of the decay are not dependent on the eigenvalue statistics of the unperturbed system, 
but depend on the system's eigenvector statistics in the  
eigenbasis of the perturbation operator. For random 
eigenvector statistics the decay is exponential with a rate fixed precisely by 
the variance of the perturbation's energy spectrum. 
Hence, even classically regular models can exhibit an 
exponential fidelity decay under generic quantum perturbations.
These results clarify which perturbations can distinguish 
classically regular and chaotic quantum systems.
\end{abstract}

\pacs{05.45.Mt, 03.67.Lx}

\maketitle

Over the last two decades a great deal of insight has been achieved 
regarding the manifestations of chaos and complexity in 
quantum systems. We are interested in the problem of identifying 
such signatures in the context of quantum simulation on 
a quantum information processor (QIP). It is known that 
a QIP enables efficient simulation of the dynamics of a 
wide class of quantum systems \cite{Lloyd,ZalkaWeisner}. 
In the case of quantum chaos models, quantum simulation 
provides an exponential speedup over direct classical 
simulation \cite{Schack,Shep,Casati}.  
Recently, the quantum baker's map has been implemented 
using a nuclear magnetic resonance QIP \cite{Yaakov}. 
%As a consequence it is important to determine 
These developments highlight the importance of devising efficient QIP 
methods for the measurement of quantum chaos signatures 
and related properties of complex quantum systems.

Perhaps the most established signature of quantum chaos  
is given by the (nearly) universal correspondence 
between the eigenvalue \cite{BT77BGS84} and eigenvector 
\cite{IZR87KMH88HZ90} statistics 
of quantized classically chaotic systems and those of  
the canonical ensembles of random matrix theory (RMT). 
%The relevant matrix ensemble is fixed 
%uniquely by the invariant measure under the system's symmetries. 
%In contrast, the spectral fluctuations of generic classically integrable 
%systems deviate significantly from the RMT predictions and 
%are well described by Poissonian statistics.  
%The conjecture for this signature, developed in the 
%work of Berry and Tabor \cite{BT}, and Bohigas, Giannoni 
%and Schmidt (BGS) \cite{BGS}, builds upon the original idea of Wigner, who 
%surmised that the eigenspectra of sufficiently 
%complex quantum systems might exhibit the 
%the same spectral fluctuations as those derived from RMT \cite{Wigner}. 
%In the context of quantum simulation, 
%On a QIP the eigenvalues of a unitary operator may be 
%measured using the phase estimation algorithm of 
%Abrams and Lloyd \cite{AL97}. 
%However, in order to determine the eigenstructure and extract 
%information about the local spectral fluctuations an exponential 
%amount of information is required.  
%Consequently any direct measurement of spectral statistics 
%appears to be inefficient. % also in the QIP framework. 
% method becomes impractical precisely when 
%simulation on a QIP device offers any advantage over direct 
%diagonalization on a classical computer. 
%More rece an earlier observation due to 
Unfortunately direct detection of these spectral signatures  
is algorithmically inefficient by any known technique. 
However, following the original observation of Peres \cite{Peres}, 
some recent work has demonstrated that, under 
sufficiently strong perturbation, 
the fidelity exhibits a characteristic exponential inthe case of 
classical chaotic systems
\cite{JP01Cucchietti01,Jacquod01,Casati01,Prosen02}. 
Below, we demonstrate that this charecteristic decay 
may be measured by an efficient algorithim, and analyze 
in detail how the observation of this decay may be 
applied as an indicator of canonical RMT 
statistics (quantum chaos)  in the {\em unperturbed} system. 

%For perturbation strength above a prescribed 
%non-perturbative border, previous work suggests that 
%the fidelity decay is given by a characteristics exponential 
%in the case of classically chaotic systems 
%\cite{JP01Cucchietti01,Jacquod01,Casati01,Prosen02}.  
By first considering RMT models we show that 
Wigner-Dyson fluctuations in the system eigenvalue spectrum 
are not necessary to produce this characteristic 
decay. More importantly, 
%rather, the 
%crucial determinants of exponential fidelity decay and the associated rate are 
we find that the canonical RMT statistics for the system 
eigenvector components, expressed  in the eigenbasis of the perturbation 
operator, are sufficient to produce the characteristic exponential decay. 
These observations are checked in the case of a dynamical model, 
where we demonstrate that an 
exponential fidelity decay can arise not only for a classically chaotic 
system, but also for classically regular system 
under a generic choice of perturbation. 
%this characteristic exponential decay can still
%arise for certain types of perturbation specified below.  
%that this distinction generically fails for perturbations in a 
%non-classical eigenbasis. 
Our analysis implies specific restrictions that must be imposed on 
the choice of applied perturbation operator in order to extract 
useful information about the statistical signatures 
of quantum chaos in the unperturbed system. 
%Moreover, in these cases the exponential rate may be predicted accurately 
%if the unperturbed system eigenvectors are random in the eigenbasis 
%of the perturbation operator.

%In this approach 
The aim of this Letter is to characterize certain static 
properties of a unitary map, $U$, by observing 
the rate of divergence between a fiducial Hilbert 
space vector evolved under 
this map, $|\psi_u(n) \rangle =  U^n |\psi(0) \rangle$, 
and the same initial vector evolved under this map but 
subject also to a sequence of small 
perturbations $| \psi_p(n) \rangle = (U_p U)^n |\psi(0) \rangle$, where 
$U_p = \exp(-i V \delta)$ is some unspecified perturbation operator 
and $n$ denotes the number of iterations. 
%A convenient indicator of this stability is 
The fidelity,  
\begin{equation}
\label{eqn:fidelity}
%F(n) & = & \langle \psi_u(n) | \psi_p(n) \rangle, \\
O(n)  =  | \langle \psi_u(n) | \psi_p(n) \rangle |^2, 
\end{equation}
provides a natural indicator of this divergence. 
%e stability of $U$ against these perturbations.   
%It is clear that the decay of $O(n)$ may be measured directly in QIP 
%by performing state tomography 
%on both $|\psi_p(n) \rangle$ and $|\psi_u(n) \rangle$, though 
%this inefficeint method is unnecessary. 
%%inefficient since the number of components 
%%of either state grows exponentially with the number of qubits. 
The value of $O(n)$ may be determined by an 
efficient algorithm on a QIP as follows. 
%by storing its value in a single additional qubit 
%In the case of the overlap, however, we can 
%efficiently record the time-dependent value of the overlap in 
%the state of a single ancilla bit.   
%To see this we first observe that $$, where $psi_f $
We start by preparing the fiducial state 
$| \psi(0) \rangle = U_o | 0 \rangle^{\otimes n_q}$, where $n_q$ is the number of 
qubits required to span the system's Hilbert space $N = 2^{n_q}$. 
As shown below, the choice of fiducial state is not critical and 
the computational basis states, which are simplest to implement, 
are a convenient set. 
%the desired initial state is first prepared,   
%$ | \psi(0) \rangle = U_o | 0 \rangle $. 
%We show below that initial computational basis states 
%produce the same characteristic fidelity decay as the initial 
%eigenstates of the map $U$.  
%but initial eigenstates are also of interest  
%and a random eigenstate may be prepared efficiently
%using the algorithm in Ref.~\cite{AL97}. 
After applying the sequence $(U^\dagger)^{n}(U_p U)^n$, 
the system register contains a final state $| \psi(n) \rangle$.  
The circuit implementation of $U$ requires only Poly($n_q$) operations 
for the simulation of a  wide class of quantum 
systems to arbitrary accuracy \cite{Lloyd,ZalkaWeisner,Shep,Schack,Casati}. 
%is either local \cite{Lloyd} or the Hamiltonian 
%involves a sum of functions that 
%may be evaluated efficiently and stored in  
%an additional work computational basis 
%For some chaotic quantum maps 
%this additional computational basis is not necessary \cite{Schack,Casati}. 
Here we observe that $ | \langle \psi_p(n) | \psi_u(n) \rangle| = 
| \langle \psi(n) | \psi(0) \rangle|  =  
| \langle \psi_f | 0 \rangle |$,   
where the state $| \psi_f \rangle$ is obtained by time-reversing the 
initial state preparation $ |\psi_f \rangle = U_o^\dagger | \psi(n) \rangle$.  
The magnitude of $O(n)$ is then determined from sampling 
the population of the state $| 0\rangle $. 
%This is followed by the application of a C$^0$NOT operation on the 
%ancilla bit conditioned on the  
%$| 0 \rangle^{\otimes n_q}$ component of the system state.   
%After tracing over the system, the ancilla state has the 
%diagonal components $ \rho_a =  
%O(n) |0\rangle\langle 0| + (1- O(n)) |1\rangle \langle 1| $ 
%as well as irrelevant off-diagonal terms.  
%Thus mreasurment of O(n) requires is obtained from 
%tomography on a single 
%In the case of an ensemble NMR quantum 
%information processor the ancilla bit is unnecessary and the net 
%polarization of the pseudo-pure 
%state $| \psi_f(2n) \rangle$ contains the overlap information. 
%This C$^0$NOT operation requires Poly($n_q$) gates. 
The entire algorithm therefore scales as Poly($n_q$).
%, as does the number 
%of trials required for sampling $O(n)$ provided 
%the required time-scale of observation of $O(n)$ does not scale  
%exponentially with $n_q$ and 
%the decay rate of $O(n)$ 
%does not increase exponentially with $n_q$. 

%The conjecture that the decay rate of the fidelity might 
%provide an indication of quantum chaos 
%was first proposed by Peres \cite{Peres}. 
%Within first order perturbation theory the eigenstates of 
%$(U_p U)$ are well-approximated by those of $U$ and one has 
%$O(n) \simeq \left| {\textstyle \sum_m} |c_m |^2 \exp( - i \delta V_{mm} n ) 
%\right|^2$,  
%where the distribution of coefficients 
%$c_m = \langle m | \psi(0) \rangle$ is fixed by the 
%initial state and the matrix elements $V_{mn}$ refer to 
%the ordered eigenbasis of $U$.  
%(We have chosen units such that $\hbar =1 $.) 
%If only a few eigenstates are populated by $| \psi(0) \rangle$ 
%then $O(n)$ remains close to unity with small oscillations. 
%For initial states that are well distributed over the 
%eigenbasis of $U$, a parabolic decay 
%$O(n) \simeq 1 - \delta^2 (\Delta V)^2 n^2 
%+ O(\delta^4 (\Delta V)^4 n^4 )$  is assured 
%{\em irrespective} of the system properties 
%over a short time-scale determined by  $(\delta \Delta V n)^2 \ll 1$, where 
%$(\Delta V)^2 = \sum_m |c_m |^2 (V_{mm})^2 - (\sum | c_m^2| V_{mm} )^2$.  
%%provided $\sum | c_m^2| V_{mm} \simeq 0$. 
%However, for chaotic systems RMT arguments may be 
%applied \cite{Peres,Tomsovic02}, 
%%in which case the $V_{mm}$
%%are expected to be Gaussian distributed, 
%%and the fter this charac
%leading to a Gaussian decay over times 
%much longer than the initial parabolic decay.
%%\begin{equation}
%%O(n) \simeq \exp( - \delta^2 \overline{ V_{mm}^2} n^2 ).
%%\end{equation}
%%where $\sigma^2 = \delta^2 \overline{ V_{mm}^2} $ stands for a 
%%typical perturbation strength. 
Recently, Jacquod and coworkers \cite{Jacquod01} observed that 
for $|\psi(0)\rangle 
= |v_o\rangle$, an initial eigenstate of $U$ 
with eigenphase $\phi_o$, the fidelity
relates to the local density of states 
(LDOS), $\eta(\phi_o-\phi_m') =  |\langle v_o | v_m' \rangle|^2$,  
via Fourier transform, 
\begin{equation}
\label{eqn:FT}
O(n) = | \sum_m \eta(\phi_o-\phi_m') \exp(-i (\phi_o -\phi_m') n) |^2. 
%=  \langle v_m | v_m' \rangle  , 
\end{equation}
In the above $|v_m'\rangle$ are eigenstates of the perturbed map 
$U_p U | v_m' \rangle = \exp(-i\phi_m') | v_m' \rangle $.
%In this case $O(n)$ is just the survival probability 
%of the initial state $|v_o\rangle$. %\cite{WC01}. 
These observations locate the decay of $O(n)$
in an existing theoretical framework. 
In the non-perturbative regime $\sigma / \Delta \agt 1$, 
where $\sigma^2 = \delta^2 \overline{ V_{mn}^2}$ denotes  
a typical off-diagonal matrix element
and $\Delta$ is the average level spacing,  
previous work suggests that when the perturbed system is complex 
the LDOS is typically Lorentzian \cite{Wigner,FCIC96}, 
\begin{equation}
\label{eqn:LDOS}
\eta(\phi_o - \phi_m') \propto \frac{\Gamma}{ (\phi_o - \phi_m')^2 + (\Gamma/2)^2 }
\end{equation}
with width $\Gamma = 2 \pi \sigma^2 / \Delta$ determined by 
the Fermi golden rule (FGR). %For complex systems 
%One therefore expects a transition from Gaussian to 
From (\ref{eqn:FT}) and (\ref{eqn:LDOS}) one expects 
the exponential decay, 
\begin{equation}
\label{eqn:fgr}
O(n) \simeq \exp(-\Gamma n). 
\end{equation} 
%for perturbation strength above the non-perturbative border 
%and not larger than the bandwidth of $V_{mn}$ 
%In the case of generic chaotic systems 
%strength that is fixed from 
%RMT considerations. 
%(above the non-perturbative border) 
%We should also mention  
%that for classically chaotic $U$, initial states,  
%and perturbation $V$ amenable 
%to a semi-classical analysis, under increasing 
%perturbation strength the rate of the exponential can eventually saturate 
%at the classical Lyapunov 
%exponent \cite{JP01Cucchietti01,Jacquod01,Casati01}.  
The onset of the exponential decay (\ref{eqn:fgr}) 
has been confirmed recently in a few  
classically chaotic models \cite{Jacquod01,Casati01}, 
though the rate is not always given by the golden 
rule \cite{Wisniacki02,JP01Cucchietti01}. 
However, in the case of integrable $U$ the situation is less clear  
since under some perturbations 
%a non-exponential decay has been observed and conjectured 
%to hold rather generally \cite{Prosen02}, 
%under some perturbations 
%clear that 
%for many-body non-interacting systems under a two-body perturbations 
%important to recall that under certain perturbations 
the LDOS is known to take on a Lorentzian shape \cite{FCIC96,BGI98}.
%leading again to the FGR decay (\ref{eqn:fgr}). 

Below, we examine 
which statistical properties of the {\em unperturbed} system 
lead to the FGR decay, and how this characteristic decay  depends on the 
properties of the perturbation operator.  
This approach is motivated by the context of QIP simulation, in which 
the eigenbasis of this perturbation 
may be mapped onto an arbitrary basis of 
the simulated system $U$ \cite{Cory99}. 
Our choice of perturbation eigenvalue structure 
is motivated from the 
perspective of quantum control studies. 
Specifically, we consider  
\begin{equation}
\label{eqn:cbr}
%U_p = \Pi_{j=1}^{n_q} \exp(- i \delta {\bf {\hat n}} \cdot {\bf \sigma}^j / 2 ),
U_p = \Pi_{j=1}^{n_q} \exp(- i \delta \sigma_z^j / 2 ),
\end{equation}
%where $\bf{\sigma}$ are the usual Pauli matrices, $\bf{\hat{n}}$ a unit vector, 
where $\sigma_z$ is the usual Pauli matrix
and (\ref{eqn:cbr}) therefore corresponds to a 
collective rotation of all the qubits by an angle $\delta$. 
%Further motivation for studying 
%The perturbation (\ref{eqn:cbr}) is also relevant 
Eq.~(\ref{eqn:cbr}) is a model of coherent far-field 
errors \cite{Fortunato}, and for this type of error model 
a better understanding of the fidelity decay is a 
subject of intrinsic interest. 
%about a ``z-axis''  associated with the computational basis basis. 
%In the quantum simulation context, the efficient implementation  
%of $U$ requires mapping one of the  
%physical coordinates of the system 
%onto the computational basis \cite{ZalkaWeisner,Shep,Cory}. 
%A perturbation operator which is diagonal in 
%computational basis is therefore diagonal in the 
%eigenbasis of a canonical coordinate of the system. 
%When the perturbation which is 
%diagonal in this basis, enabling the connection with 
%RMT for chaotic systems. 

As a first test of the LDOS/FGR framework in the case of the 
qubit perturbation (\ref{eqn:cbr})  
we evaluate the fidelity decay for a map $U=U_{\text{CUE}}$ 
drawn from the circular unitary ensemble (CUE). 
These matrices form an established model for 
classically fully chaotic time-periodic systems 
(without additional symmetries) since these 
systems (almost always) exhibit the same characteristic 
(Wigner-Dyson) spectral fluctuations and  eigenvector statistics 
as the CUE \cite{Haake01}. 
%This ensemble consists of unitary matrices 
%drawn uniformly with respect to the invariant (Haar) measure \cite{Haake01}. 
%If global classical chaos is present, spectral fluctuations of the 
%quantum Floquet map mimick those of CUE. 
%On the other hand, for generic classically 
%integrable systems the Floquet eigenphases (quasi-energies) are 
%distributed uniformly on the unit circle and the spectral 
%fluctuations obey Poissonian statistics.
%%$P(S) \propto \exp(-S) $. 
%%, the most celebrated statistic being the 
%%distribution of nearest-neigbor eigenphase spacings, which for 
%%CUE reads $P(S) \propto S^2 \exp(- \pi S^2)$. 
%and compared with the FGR prediction Eq. \ref{eqn:fgr}. 
The randomness of the system eigenvectors 
enables a system-independent estimate of the rate $\Gamma$ of the FGR decay.  
%determined using (\ref{eqn:Vmn}).
Since the components  of random eigenstates 
are distributed uniformly over 
the basis states and uncorrelated 
with the distribution of eigenvalues, % of the associated operators,  
%For such random eigenstates 
% of the perturbation operator,  
%Since the eigenstates of a CUE map are random by construction, 
%the rate of the FGR decay may be determined using (\ref{eqn:Vmn}).
the second moment of the matrix 
elements $V_{mn}$ may be directly evaluated, 
\begin{equation}
\label{eqn:Vmn}
\overline{V_{mn}^2} = \overline{\lambda^2}/ N  
\end{equation}
assuming $\overline{\lambda}=0$ and where 
$\overline{\lambda^2} = N^{-1} \sum_{i=1}^N \lambda_i^2$
denotes the variance of the eigenvalues of $V$.  
%and $N=2^{n_q}$ is the dimension of the Hilbert space.  
As a result, the rate of the FGR decay is determined by 
%may be predicted 
%when the system is classically chaotic,  
%from knowledge of the control parameter $\delta$ and 
the eigenvalues of the perturbation, 
\begin{equation}
\label{eqn:Gamma}
	\Gamma = \delta^2 \overline{\lambda^2}, 
\end{equation}
where we have used $\Delta = 2 \pi /N$. 
For the qubit perturbation (\ref{eqn:cbr}) the variance of 
the eigenvalues has a simple form,
\begin{equation}
\label{eqn:bin}
	\overline{\lambda^2 } = \frac{1}{N}
%{\textstyle 
\sum_{k=0}^{n_q} 
\left(\frac{2k - n_q}{2} \right)^2  C^{n_q}_{k}, 
\end{equation}
where %the sum is over even $\lambda_k$ and 
the $C^{n_q}_k$ are binomial coefficients. 
%For the qubit perturbation (\ref{eqn:cbr}),  
%the collective qubit rotation (\ref{eqn:cbr}) and choosing 
Using our RMT estimate (\ref{eqn:Gamma}), for $n_q=10$, the rate is, 
\begin{equation}
\label{eqn:Gammacbr}
	\Gamma = 2.50 \; \delta^2.
\end{equation}
%where we have used (\ref{eqn:bin}). 

%We note that this perturbation 
%is not a semiclassically ``reasonable'' 
%operator \cite{Peres} in the sense that it  
%does not have a proper classical limit, {\em i.e.}\ 
%all elements $V_{mn} \rightarrow 0$ as $N \rightarrow \infty$.


%Numerical values for saved CUE matrix: 
%RMS values is $(\overline{V_{mn}^2})^{1/2} = 0.0494$ 
%and average magnitude is $\overline{|V_{mn}|} = 0.0438$ 
%RMS values is $(\overline{V_{mm}^2})^{1/2} = 0.0444$ 
%and average magnitude is $\overline{|V_{mm}|} = 0.0350$ 
%RMS eigenvalue is 1.5811 and average eigenvalue magnitude is 1.2305. 
%Note that $rmseig/sqrt(N) = 0.0494$ and $typeig/sqrt(N)$ = 0.0385.
%Numerical values for second CUE matrix: 
%RMS values is $(\overline{V_{mn}^2})^{1/2} = 0.0494$ 
%and average magnitude is $\overline{|V_{mn}|} = 0.0438$ 
%RMS values is $(\overline{V_{mm}^2})^{1/2} = 0.0452$ 
%and average magnitude is $\overline{|V_{mm}|} = 0.0363$ 
%RMS eigenvalue is $1.5811$ and average eigenvalue magnitude is $1.2305$. 
% for the rest of this Letter. 
%The perturbation also represents collective (or far-field) noise, 
%which arises. Since 


%$\overline{V_{mn}^2} \simeq \overline{\lambda^2}/ N = (0.0494)^2$.  
%random basis are given by Eq. \ref{eqn:Vmn}.  
%where $\overline{\lambda^2} \simeq (1.66)^2$ and $N = 1024$. 
%The FGR decay (\ref{eqn:fgr})
%has been conjectured to hold not just for 
%initial eigenstates, but also for more general 
%initial states \cite{Jacquod01}. 

While a CUE map may be generated on a QIP using the 
gate decomposition devised in Ref.~\cite{Zyc}, 
for our numerical study we construct $U=U_{\text{CUE}}$ 
directly from the eigenvectors of a random Hermitian matrix. 
Since computational basis states are easiest to prepare 
in the QIP setting, we consider the fidelity decay 
for both single computational basis states and averages over 
50 such states.  The behaviour of the fidelity decay 
for a matrix typical of CUE is displayed in Fig.~\ref{fig:cue}.  
The three perturbation values displayed in the figure 
are chosen near the onset of the non-perturbative regime ($\delta > 0.1 $) 
and it is evident that the fidelity decay even for 
{\em individual} computational basis 
states exhibits FGR decay (\ref{eqn:fgr}) at the expected rate.
% in expected rate (\ref{eqn:Gamma}).
%$\Gamma \simeq 2.5 \delta^2$.  
%The decay rate for this type of initial state is in 
%excellent agreement 
%spite of the coherent averaging over system eigenstates. 
%The overlap decay is well approximated by the exponential 
%In either case t
The FGR decay persists for a 
time-scale $\Gamma^{-1}\log(N)$ 
until saturation at a time-average that decreases 
as $1/N$. 
%Our numerical studies also confirm that the decay rate 
%for CUE maps remains in agreement with  $\Gamma = 2.5 \delta^2$ 
%until saturation at the stroboscopic limit. 
%Therefore the CUE models reproduce the transition from 
%Gaussian to Golden rule decay that has been 
%observed in a few of the model quantum chaos systems, but, 
%are incapable of reproducing system specific effects such as 

\begin{figure}
%\includegraphics{Fig1}
\epsfig{file=Fig1.eps, height = 5.8cm, width = 8cm}
%{\epsfxsize=2.75in\centerline{\epsffile{Fig1.eps}}}
%{\epsfxsize=2.75in\centerline{\epsffile{CUEp1p2p4ups.eps}}}
%{\epsfxsize=3in\centerline{\epsffile{CUEexp.eps}}}
%\vspace{.1cm}
%\vspace{.25cm}
\caption{\label{fig:cue} Fidelity decay for $U_{\text{CUE}}$ 
averaged over 50 computational basis states (dash lines) 
%and an initial computational basis state $|0\rangle^{\otimes n_q}$ (solid lines)
is in excellent agreement with the golden rule decay 
(\ref{eqn:fgr}) 
and the RMT rate (\ref{eqn:Gammacbr}) 
%$\Gamma =  2.5 \delta^2$ 
for $\delta = (0.1,0.2,0.4)$ (solid lines top to bottom). 
Chain lines show the fidelity decay for two typical 
computational basis states in the case $\delta = 0.1$.}
%The decay of the  state $|0\rangle^{\otimes n_q}$ is 
%accurately described by the FGR in spite of the 
%coherence effects  system eigenstates.}
%with  numerically determined $B = $ where $\gamma^2 = B \delta^2$. 
%Theoretically expect $B = \overline{V_{mn}^2} = 0.04^2 $. }
\end{figure}

%\begin{figure}
%%{\epsfxsize=3in\centerline{\epsffile{ExpGUE.ps}}}
%\vspace{.25cm}
%\caption{ This figure demonstrates that random states 
%for GUE exhibit Fermi golden rule decay. Solid line is  
%$\Gamma = A \delta^2$ with $A = 2.4$ determined from CUE figure.
%Theoretically expect $A = {\overline V_{mn}^2} N = 0.04*0.04 1024 = 1.64 $. }
%\end{figure}

We next consider $O(n)$ for the Gaussian unitary ensemble 
(GUE) in order to clarify the relationship between the FGR decay 
and the distinct statistical features of RMT that represent 
signatures of quantum chaos. 
The GUE consists of Hermitian matrices with independent elements 
drawn randomly with respect to the unique unitarily-invariant 
measure \cite{Haake01}.  GUE forms the relevant RMT model 
for the important class of chaotic or complex 
autonomous Hamiltonian systems 
that are unrestricted by any additional symmetries.  
%$ \mu({\text d} H) = \Pi_{i<j} \; {\text Re}\{H_{ij}\} \Pi_{i<j} \; 
%{\text Im}\{H_{ij}\}$. 
We may examine the sensitivity to perturbations for the GUE 
by constructing the  
unitary operator $U_{\text{GUE}} = \exp(-i H_{\text{GUE}} \tau )$, 
where $\tau$ is a time-delay between perturbations. We consider the same 
%$H$ is drawn from GUE and 
perturbation %(a collective rotation of the qubits) 
as for the CUE case.  For sufficiently small $\tau$ the 
propagator approaches identity and the overlap decay 
is dominated by the perturbation operator, 
%\begin{equation}
$O(n) = |\langle \exp( - i \delta V) \rangle|^2 + O(\delta^2 \tau^2 n^2)$.  
%\end{equation}
%irrespective of the Hamiltonian $H$.   
This behaviour is demonstrated in Fig.~\ref{fig:gue} for $\tau = 0.001$ 
and $\tau=0.01$ and with $\delta = 0.3$.
For larger values of $\tau$ the fidelity decay under $U_{\text{GUE}}$ 
obeys the FGR with the RMT rate (\ref{eqn:Gammacbr}). 
%an exponential decay rate that is the same as that 
%obtained for $U_{\text{CUE}}$. 


\begin{figure}
\epsfig{file=Fig2.eps, height = 5.8cm, width = 8cm}
%\includegraphics{Fig2}
%{\epsfxsize=2.75in\centerline{\epsffile{Fig2.eps}}}
%{\epsfxsize=2.75in\centerline{\epsffile{GUEtausupsp3wPoisstau100.eps}}}
%\vspace{.1cm}
%\vspace{.25cm}
\caption{ \label{fig:gue} Fidelity decay for $U_{\text{GUE}}$ averaged 
over 50 computational basis states % $|0\rangle^{\otimes n_q}$ 
with $\delta = 0.3$  for 
$\tau = 0.001$ (dashed line),  
$\tau = 0.01$ (dotted line) 
and $\tau = 0.1$ and $100$ (chain lines), compared to 
the FGR/RMT prediction (solid line). Inset: Spacing distribution 
for $U_{\text{GUE}}$ for $\tau =100$ compared to 
Poissonian (solid line) and 
Wigner-Dyson (dashed line) distributions.} 
%with $\Gamma = 2.5 \delta^2$. } 
%numerical determined $A = 2.4$ where $\Gamma = A \delta^2$. 
%Theoretically expect $A = 2 \pi {\overline V_{mn}^2} / Delta = 0.04*0.04 1024 = 1.64 $. }
\end{figure}



The important point is that for sufficiently large 
$\tau$ the eigenphases of $U_{\text{GUE}}$ become 
spread pseudo-randomly in the interval $[0,2\pi)$. 
%This feature of the 
%eigenspectrum holds irrespective of the choice of $H$, 
%and is guaranteed if, 
%\begin{equation}
%(E_{\text max} - E_{\text min} ) \tau \gg 2 \pi, 
%\end{equation}
%where $E_{\text min}$ is the smallest eigenvalue of $H$.
Under these conditions the 
eigenphases of the map $U_{\text{GUE}}$ exhibit the 
Poissonian spectral fluctuations that are characteristic 
of classically integrable (time-periodic) systems. 
We checked the nearest-neighbor spacing distribution 
of $U_{\text{GUE}}$ and 
found that for $\tau = 100$ the statistics are in excellent 
agreement with the Poissonian distribution $P(s) \propto \exp(-s)$ 
(see inset to Fig.~\ref{fig:gue}). 
%Wigner-Dyson $P(s) \propto s^2 \exp(-4s^2/pi) $ statistics
However, the eigenvectors of $U_{\text{GUE}}$ 
are random (by construction) and 
independent of $\tau$  (for finite $\tau$). 
From these observations it is clear that the presence of 
Wigner-Dyson spectral 
fluctuations in the implemented $U$, which comprises  
the only basis-independent criterion of quantum chaos, 
%are not necessary for an FGR decay of the overlap.  
%spectral statistics of a unitary operator 
is not actually necessary for the onset of exponential 
(FGR) decay at the rate (\ref{eqn:Gamma}). 
This suggests that it is the RMT statistics 
of the eigenvectors of $U$ 
that lead to the FGR decay with the RMT rate.  

%(and exhibit the appropriate random statistics).
%, that leads to 
%a connection with the FGR decay and excellent 
%agreement with the prediction (\ref{eqn:Vmn}).   

We next consider the fidelity decay for the quantum kicked 
top, which is an exemplary dynamical model of quantum 
chaos \cite{HKS87,Haake01}. 
%In light of our earlier observations 
%under the qubit perturbation described above. 
The kicked top is a unitary map $U_{\text{QKT}} = \exp(-i \pi J_y /2) 
\exp(- i k J_z^2 /j)$ acting on the Hilbert space of dimension 
$N = 2j+1$ associated with an 
irreducible representation of the angular momentum operator $\vec{J}$. 
% (though most choices will be lead to inefficient implementations). 
In previous fidelity decay and LDOS studies the choice of 
perturbation has usually been tied to a physical 
coordinate of the system $U$. 
%has typically been constructed by varying a 
%parameter in the Hamiltonian, applying a small (stroboscopic) 
%Galilean displacement, or deforming the walls of a billiard. 
%In the case of multi-body (complex) 
%systems the perturbation is usually associated with   
%two-body interactions in the position coordinates. 
% or. As a result of this approach 
%A common feature of these approaches is that 
%implementation of both the perturbation operator and the system 
%operator onto the computational basis. 
%As a check we first 
We first follow this convention and identify 
the eigenbasis of the perturbation (\ref{eqn:cbr}) 
with the eigenbasis $| m_j \rangle$  of the system coordinate  
$J_z$ (where $m_j = \{j, \dots, -j\} $). 
%, which is  
%with the computational basis 
%states $| m_N \rangle$, where $m_N = \{0, \dots, N-1\} $. 
In Fig.~\ref{fig:qkt} we compare the fidelity decay 
for the chaotic and regular regimes of the kicked top 
for averages over 50 initial computational basis states.  
The fidelity decay for the chaotic top ($k=12$) 
is well described by the FGR prediction (\ref{eqn:fgr}) 
and the RMT rate (\ref{eqn:Gammacbr}), 
whereas the regular top  ($k = 1$) 
shows a slower {\em non-exponential} decay rate. 
Similarly, if we associate the perturbation eigenbasis 
with the basis of the $J_y$ coordinate, 
% and plot the decay results in Fig.~3. 
the fidelity decay for the chaotic top remains 
in agreement with the RMT rate and the regular top 
again exhibits non-exponential decay, though in this case with a 
{\em faster} decay than the RMT rate (\ref{eqn:Gammacbr}). 
%We should also mention 
%recent evidence that the {\em averaged} decay of $O(n)$ 
%is given by a power-law for integrable systems \cite{Jacquod02b}. 
%It is worth noting that for a chaotic systems 
%the magnitude of the fluctuations should match those 
%prescribed by RMT considerations, 
%although measurement of these fluctuations would 
%be inefficent in the QIP context.
%In the context of QIP simulation, however, 
%universal control enables an arbitrary choice of 
%perturbation operator.  
%Our goal here is to analyze the relationship
%between the choice of perturbation eigenbasis, 
%the spectral statistics of $U_{\text{QKT}}$, 
%and the observation of FGR decay. 
However, we now demonstrate that an exponential decay at the RMT rate 
arises even for the regular kicked top when the qubit 
perturbation (\ref{eqn:cbr}) is diagonal in 
a generic basis relative to the eigenbasis of $U_{\text{QKT}}$. 
Specifically, we leave the perturbation 
eigenvalue spectrum unchanged but set 
\begin{equation}
\label{eqn:rcbr}
U_p = T \left[ \Pi_j \exp(-i\delta \sigma_z^j /2 ) \right] T^{-1}, 
\end{equation}
where $T$ is drawn from CUE. As demonstrated in the inset 
to Fig.~\ref{fig:qkt}, under this type of perturbation the fidelity 
decay  for the regular top is indistinguishable from that of the  
chaotic top and is very accurately 
described by the FGR at the RMT rate (\ref{eqn:Gammacbr}). 

\begin{figure}
\epsfig{file=Fig3.eps, height = 5.8cm, width = 8cm}
%\includegraphics{Fig3}
%{\epsfxsize=3in\centerline{\epsffile{QKTZYp1p3k12k1wins.eps}}}
%{\epsfxsize=3in\centerline{\epsffile{QKTp1p3exp.eps}}}
\caption{ \label{fig:qkt} 
Decay of $O(n)$ for the kicked top in chaotic regime ($k=12$) 
averaged over 50 computational basis states, with   
the perturbation eigenbasis mapped to the eigenbases of 
$J_z$ (dash lines) and $J_y$ (chain lines) compared to 
%is in excellent agreement with 
the FGR decay (solid lines) at the RMT rate (\ref{eqn:Gammacbr}) for 
$\delta = (0.1,0.3)$ (top to bottom).  
%with $\Gamma = 2.5 \delta^2$ 
Lines with circles and squares are for 
the regular kicked top ($k=1$) with the perturbation 
eigenbasis tied to the $J_z$ and $J_y$ 
coordinate bases respectively (for $\delta = 0.1$).  
Inset: Average fidelity decay for regular kicked top ($k=1$),   
when the qubit perturbation is 
in a random eigenbasis (\ref{eqn:rcbr})
and with $\delta = (0.1,0.3)$ (dashed lines),   
compared to the FGR/RMT rate (solid lines).}
\end{figure}

The sensitive dependence of the type of fidelity decay on the 
eigenbasis of the applied perturbation 
suggests a close connection with the basis-dependence of 
the eigenvector statistics of classically 
regular quantum models.  
Expressed in a generic quantum basis,  
the eigenvectors of any quantized classical system $U$ 
will have randomly (Gaussian) distributed components.  
%(i.e., matching the states  
%drawn uniformly from the invariant distribution on Hilbert space) 
In contrast, {\em in the eigenbases of the system coordinates} 
the components of classically chaotic and integrable systems 
are known to be different \cite{IZR87KMH88HZ90}, 
with the former Gaussian distributed 
and the latter exhibiting substantial deviation from 
the canonical Gaussian distribution. 
%For classically chaotic systems 
%the components of the eigenstates exhibit the 
%statistics of random states 
%in the bases of the system coordinates. 
%In contrast, the components of the eigenstates 
%for integrable systems should never be randomly 
%distributed {\em in the eigenbases of the system coordinates}. 
%However, for the perturbation (\ref{}), the eigenvectors of the 
%unperturbed system are random in the generic eigenbasis. 
%Therefore under these conditions the rate of the fidelity decay 
%should deviate significantly from the RMT prediction (\ref{eqn:Gamma}). 
%On the other hand, eigenvectors will appear random in a random 
In light of this connection, in the case of quantized classical models 
we infer that exponential (non-exponential) fidelity decay 
%at the FGR rate 
can be correlated with the presence (absence) of characteristic RMT  
spectral fluctuations in the unperturbed system 
{\em provided that} the applied 
perturbation commutes with a system coordinate.  

%How then can regular and chaotic systems be reliably distinguished 
%if even regular systems can lead to an exponential fidelity decay? 
%However, an experimental observation of fidelity decay 
%may still be applied to distinguish 
%between chaotic and regular systems provided 
%The answer is that the choice of 
%perturbation is suitably restricted. 
%. though the full set of useful perturbations 
%probably includes perturbations that commute with any  
%``classical'' operator, {\it e.g}., 
%those constructed from simple functions of the classical coordinates,  
%as defined more rigourously in Refs.~\cite{JaffePeres}. 

In summary, we have shown that the fidelity decay 
may be measured efficiently on a QIP. 
We then examined 
which statistical properties of the unperturbed system  
determine the type and rate of the decay. 
In the case of random unitary and Hermitian matrices, 
as well as a classically chaotic dynamical model, we have 
shown that %under non-classical perturbations   
the fidelity decays exponentially with a characteristic rate 
given precisely by the variance of the perturbation's 
eigenspectrum. The occurrence of 
the exponential decay is not directly dependent on the 
Wigner-Dyson fluctuations of the unperturbed spectrum, but does  
depend sensitively on the RMT statistics of the system 
eigenvectors in the eigenbasis of the applied perturbation. 
Hence, the fidelity decay for both classically regular 
and chaotic dynamical systems is given by the FGR under all but a 
small subset of unitary perturbation operators. 
%This is a striking demonstration that only a small subset 
%of all possible perturbations can reliably distinguish  
%regular from chaotic statistics in the unperturbed system 
%and requires some explanation. 
%However, in the case of a dynamical model, 
%the eigenvectors are expected to be random in the bases of the 
%system coordinates only if the unperturbed system 
%is classically chaotic. 
%Consequently, 
In the case of classical models, we conclude that the fidelity decay 
provides a reliable indicator of RMT statistics (quantum chaos) in the 
unperturbed system only when the applied perturbation is restricted 
to the subset of perturbations that commute with a classical coordinate.
%or such classical models we conclude that the fidelity decay 
%rovides a reliable 
%ndicator of RMT statistics (quantum chaos) in the 
%nperturbed system if the applied perturbation is restricted 
%o the subset of perturbations that commute with a system coordinate. 
%In the more general case of a map $U$ without classical analogue, 
%the fidelity decay can reveal only basis-dependent 
%information tied to the eigenbasis of the perturbation. 

We are grateful to K. Zyczkowski for suggesting the relevance 
of the eigenvector statistics, Ph.~Jacquod, E.M.~Fortunato, 
C.~Ramanathan, and T.F.~Havel for 
helpful discussions, and DARPA and the NSF for financial support. 
%\vspace{0.5in}

\begin{thebibliography}{99}
\bibitem{Lloyd}
S.~Lloyd, Science {\bf 273}, 1073 (1996).  
\bibitem{ZalkaWeisner}
C.~Zalka, Proc.~Roy.~Soc.~London, Ser.~A {\bf 454}, 313 (1998); 
%\bibitem{Weisner}
S.~Weisner, quant-ph/9603028 (1996). 
\bibitem{Shep}
B.~Georgeot and D.L.~Shepelyansky, Phys. Rev. Lett. {\bf 86}, 2890 (2001).
\bibitem{Schack}
R.~Schack, Phys.~Rev.~A {\bf 57}, 1634 (1998). 
\bibitem{Casati}
G.~Benenti, G.~Casati, S.~Montangero, and D.L.~Shepelyansky, 
Phys.~Rev.~Lett.~{\bf 87}, 227901 (2001).
%quant-ph/0107036.
\bibitem{Yaakov}
Y.S.~Weinstein, S.~Lloyd, 
J.~Emerson, and D.~Cory, Phys. Rev. Lett. in press (2002).
\bibitem{BT77BGS84}
M.V.~Berry and M.~Tabor, Proc. Roy. Soc. Lond. {\bf A356}, 375 (1977);
%\bibitem{BGS84}
O. Bohigas, M.J. Giannoni, C. Schmit, Phys. Rev Lett. {\bf 52}, 1 (1984). 
\bibitem{IZR87KMH88HZ90}
F.M.~Izrailev, Phys.~Lett.~{\bf 125A}, 250 (1987); 
M. Kus, J. Mostowski, and F. Haake, J. Phys. A: 
Math. Gen. {\bf 21}, L1073 (1988); 
F. Haake and K. Zyczkowski, Phys. Rev. A {\bf A42}, 1013 (1990). 
\bibitem{Peres}
A. Peres, Phys. Rev. A {\bf 30}, 1610 (1984). 
%{\it Quantum Theory: Concepts and Methods}, Kluwer Academic Publishers (1995).
%\bibitem{L+L}
%A.J. Lichtenberg and M.A. Lieberman {\it Regular and Chaotic Dynamics} 
%(Springer Verlag, 1992). 
%\bibitem{B1}
%M. V. Berry, {\it Quantum Chaology} Proc. R. Soc. London Ser. A {\bf 413}, 
%183 (1987).
%\bibitem{B2}
%M. V. Berry in {\it New Trends in Nuclear Collective Dynamics}, edited by Y. Abe, H. Horiuchi, and K Matsuyanagi (Springer, Berlin, 1992), p. 183.
%\bibitem{AL97}
%D. Abrams and S. Lloyd, Phys. Rev. Lett. {\bf 83}, 5162 (1997).
%{\it Stability of Quantum Motion and Correlation Decay}, 
%FIDELITY DECAY: Exponential to Gaussian (semiclassical and RMT)
%\bibitem{Tomsovic02}
%N. Cerruti and S. Tomsovic, Phys. Rev. Lett. {\bf 88}, 054103 (2002).
%FIDELITY DECAY: Gaussian to Lyapunov (numerical)
\bibitem{JP01Cucchietti01}
R.A. Jalabert and H.M. Pastawski, Phys. Rev. Lett. {\bf 86}, 2490 (2001); 
F. Cucchietti, C.H. Lewenkopf, E.R. Mucciolo, H. Pastawski
and R.O. Vallejos, nlin.CD/0112015.
\bibitem{Jacquod01}
Ph. Jacquod, P.G. Silvestrov, C.W.J. Beenakker, Phys. Rev. E {\bf 64}, 
055203 (2001).
\bibitem{Casati01}
G. Benenti and G. Casati, quant-ph/0112060. 
\bibitem{Prosen02}
T. Prosen and M. Znidaric, J. Phys. A {\bf 35}, 1455 (2002).
%Fidelity vs LDOS: correlation effects and semiclassical effects:
%\bibitem{WC01}
%D. Wisniacki and D. Cohen, quant-ph/0111125.  
\bibitem{Wigner}
E.P. Wigner, Ann. Math. {\bf 62}, 548 (1955); {\bf 65}, 203 (1957).
%DISC of LDOS; universality of Lorentzian: transition FROM SEMICIRCLE TO BW 
%FOR SPARSE WRBM:
\bibitem{FCIC96}
Y.V. Fyodorov, O.A. Chubykalo, F.M. Izrailev, and G. Casati, Phys.~Rev.~
Lett.~{\bf 76}, 1603 (1996).
%LDOS FOR BRM: localization and BW form.
%\bibitem{Jacquod95}
%Ph. Jacquod and D.L. Shepelyansky, Phys.~Rev.~Lett.~{\bf 75}, 3501 (1995).
%FIDELITY DECAY: Exponential to Lyapunov (sawtooth)
%FIDELITY DECAY: Exponential to Lyapunov (semiclassical: Bunomivich )
\bibitem{Wisniacki02}
D. Wisniacki, E. Vergini, H. Pastawski, and F. Cucchietti, nlin.CD/0111051.
%LDOS IS BW ONLY WITHIN NARROW BAND FOR REAL SYSTEM
\bibitem{BGI98}
F. Borgonovi, I Guarneri, and F.M. Izrailev, Phys. Rev. E {\bf 57}, 5291 (1998).
%FIDELITY DECAY: Gaussian to Lyapunov (semiclassical and RMT)
\bibitem{Cory99}
C.H. Tseng, S. Somaroo, Y. Sharf, E. Knill, R. Laflamme, T.F. Havel, and 
D. Cory, Phys. Rev. A {\bf 61}, 012302 (1999).
\bibitem{Fortunato}
L. Viola, E. Fortunato, M. Pravia, E. Knill, R. Laflamme, and D. Cory, 
Science {\bf 293}, 2059 (2001).  
\bibitem{Haake01}
F. Haake, {\it Quantum Signatures of Chaos} (Springer, New York, 2001).
%\bibitem{Nielson} 
%M. Neilson and I. Chuang {\it Quantum Computation  
%and Quantum Information}, Cambridge Univ. Press (2000). 
%\bibitem{Jacquod02b}
%Ph. Jacquod, I. Adagideli, C.W.J. Beenakker, quant-ph/0206160 (2002).
\bibitem{Zyc}
K. Zyczkowski and M. Kus, J. Phys. A {\bf 27}, 4235 (1994). 
\bibitem{HKS87}
F.~Haake, M.~Kus, and R.~Scharf, Z.~Phys.~B.~{\bf 65}, 381 (1987).
%\bibitem{JaffePeres}
%L.G.~Jaffe, Rev.~Mod.~Phys.~ {\bf 54}, 407 (1982);
%A.~Peres, Phys.~ Rev.~ A {\bf 30}, 504 (1984).
\end{thebibliography}

\end{document}



quant-ph0207099.extracted_bib
\begin{thebibliography}{99}
\bibitem{Lloyd}
S.~Lloyd, Science {\bf 273}, 1073 (1996).  
\bibitem{ZalkaWeisner}
C.~Zalka, Proc.~Roy.~Soc.~London, Ser.~A {\bf 454}, 313 (1998); 
%\bibitem{Weisner}
S.~Weisner, quant-ph/9603028 (1996). 
\bibitem{Shep}
B.~Georgeot and D.L.~Shepelyansky, Phys. Rev. Lett. {\bf 86}, 2890 (2001).
\bibitem{Schack}
R.~Schack, Phys.~Rev.~A {\bf 57}, 1634 (1998). 
\bibitem{Casati}
G.~Benenti, G.~Casati, S.~Montangero, and D.L.~Shepelyansky, 
Phys.~Rev.~Lett.~{\bf 87}, 227901 (2001).
%quant-ph/0107036.
\bibitem{Yaakov}
Y.S.~Weinstein, S.~Lloyd, 
J.~Emerson, and D.~Cory, Phys. Rev. Lett. in press (2002).
\bibitem{BT77BGS84}
M.V.~Berry and M.~Tabor, Proc. Roy. Soc. Lond. {\bf A356}, 375 (1977);
%\bibitem{BGS84}
O. Bohigas, M.J. Giannoni, C. Schmit, Phys. Rev Lett. {\bf 52}, 1 (1984). 
\bibitem{IZR87KMH88HZ90}
F.M.~Izrailev, Phys.~Lett.~{\bf 125A}, 250 (1987); 
M. Kus, J. Mostowski, and F. Haake, J. Phys. A: 
Math. Gen. {\bf 21}, L1073 (1988); 
F. Haake and K. Zyczkowski, Phys. Rev. A {\bf A42}, 1013 (1990). 
\bibitem{Peres}
A. Peres, Phys. Rev. A {\bf 30}, 1610 (1984). 
%{\it Quantum Theory: Concepts and Methods}, Kluwer Academic Publishers (1995).
%\bibitem{L+L}
%A.J. Lichtenberg and M.A. Lieberman {\it Regular and Chaotic Dynamics} 
%(Springer Verlag, 1992). 
%\bibitem{B1}
%M. V. Berry, {\it Quantum Chaology} Proc. R. Soc. London Ser. A {\bf 413}, 
%183 (1987).
%\bibitem{B2}
%M. V. Berry in {\it New Trends in Nuclear Collective Dynamics}, edited by Y. Abe, H. Horiuchi, and K Matsuyanagi (Springer, Berlin, 1992), p. 183.
%\bibitem{AL97}
%D. Abrams and S. Lloyd, Phys. Rev. Lett. {\bf 83}, 5162 (1997).
%{\it Stability of Quantum Motion and Correlation Decay}, 
%FIDELITY DECAY: Exponential to Gaussian (semiclassical and RMT)
%\bibitem{Tomsovic02}
%N. Cerruti and S. Tomsovic, Phys. Rev. Lett. {\bf 88}, 054103 (2002).
%FIDELITY DECAY: Gaussian to Lyapunov (numerical)
\bibitem{JP01Cucchietti01}
R.A. Jalabert and H.M. Pastawski, Phys. Rev. Lett. {\bf 86}, 2490 (2001); 
F. Cucchietti, C.H. Lewenkopf, E.R. Mucciolo, H. Pastawski
and R.O. Vallejos, nlin.CD/0112015.
\bibitem{Jacquod01}
Ph. Jacquod, P.G. Silvestrov, C.W.J. Beenakker, Phys. Rev. E {\bf 64}, 
055203 (2001).
\bibitem{Casati01}
G. Benenti and G. Casati, quant-ph/0112060. 
\bibitem{Prosen02}
T. Prosen and M. Znidaric, J. Phys. A {\bf 35}, 1455 (2002).
%Fidelity vs LDOS: correlation effects and semiclassical effects:
%\bibitem{WC01}
%D. Wisniacki and D. Cohen, quant-ph/0111125.  
\bibitem{Wigner}
E.P. Wigner, Ann. Math. {\bf 62}, 548 (1955); {\bf 65}, 203 (1957).
%DISC of LDOS; universality of Lorentzian: transition FROM SEMICIRCLE TO BW 
%FOR SPARSE WRBM:
\bibitem{FCIC96}
Y.V. Fyodorov, O.A. Chubykalo, F.M. Izrailev, and G. Casati, Phys.~Rev.~
Lett.~{\bf 76}, 1603 (1996).
%LDOS FOR BRM: localization and BW form.
%\bibitem{Jacquod95}
%Ph. Jacquod and D.L. Shepelyansky, Phys.~Rev.~Lett.~{\bf 75}, 3501 (1995).
%FIDELITY DECAY: Exponential to Lyapunov (sawtooth)
%FIDELITY DECAY: Exponential to Lyapunov (semiclassical: Bunomivich )
\bibitem{Wisniacki02}
D. Wisniacki, E. Vergini, H. Pastawski, and F. Cucchietti, nlin.CD/0111051.
%LDOS IS BW ONLY WITHIN NARROW BAND FOR REAL SYSTEM
\bibitem{BGI98}
F. Borgonovi, I Guarneri, and F.M. Izrailev, Phys. Rev. E {\bf 57}, 5291 (1998).
%FIDELITY DECAY: Gaussian to Lyapunov (semiclassical and RMT)
\bibitem{Cory99}
C.H. Tseng, S. Somaroo, Y. Sharf, E. Knill, R. Laflamme, T.F. Havel, and 
D. Cory, Phys. Rev. A {\bf 61}, 012302 (1999).
\bibitem{Fortunato}
L. Viola, E. Fortunato, M. Pravia, E. Knill, R. Laflamme, and D. Cory, 
Science {\bf 293}, 2059 (2001).  
\bibitem{Haake01}
F. Haake, {\it Quantum Signatures of Chaos} (Springer, New York, 2001).
%\bibitem{Nielson} 
%M. Neilson and I. Chuang {\it Quantum Computation  
%and Quantum Information}, Cambridge Univ. Press (2000). 
%\bibitem{Jacquod02b}
%Ph. Jacquod, I. Adagideli, C.W.J. Beenakker, quant-ph/0206160 (2002).
\bibitem{Zyc}
K. Zyczkowski and M. Kus, J. Phys. A {\bf 27}, 4235 (1994). 
\bibitem{HKS87}
F.~Haake, M.~Kus, and R.~Scharf, Z.~Phys.~B.~{\bf 65}, 381 (1987).
%\bibitem{JaffePeres}
%L.G.~Jaffe, Rev.~Mod.~Phys.~ {\bf 54}, 407 (1982);
%A.~Peres, Phys.~ Rev.~ A {\bf 30}, 504 (1984).
\end{thebibliography}