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quantum, computing, geometric, phase, berry, spacetime, fine structure, memory, curvature, computer, bit, iteration, nonlinear
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%keywords: quantum, computing, geometric, phase, berry, spacetime, fine structure, memory, curvature, computer, bit, iteration, nonlinear
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\begin{document} About Geometric Phases Serving as Memory in Quantum Computing
\begin{center}
{\Large\bf Spacetime Memory: Phase-Locked Geometric Phases}\\
%//{\Large\bf The Spacetime Memory of Geometric Phases and Quantum Computing}\\
\vspace{.4in} {B{\sc ernd}~B{\sc inder}\footnote{\it email:
\href{mailto:binder@quanics.com} {binder@quanics.com}, {\it
Weildorferstr.22, 88682 Salem-Neufrach, Germany \copyright 2002}}}
\\{\tiny Date: 29.8.-1.9.2002}
\end{center}
\footnotesize
%====================================================================
%\begin{abstract}\noindent
Spacetime memory is defined with a holonomic approach to
information processing, where multi-state stability is introduced
by a non-linear phase-locked loop. Geometric phases serve as the
carrier of physical information and geometric memory (of
orientation) given by a path integral measure of curvature that is
periodically refreshed. Regarding the resulting spin-orbit
coupling and gauge field, the geometric nature of spacetime memory
suggests to assign intrinsic computational properties to the
electromagnetic field.
\\\\
%\end{abstract}
%====================================================================
\tiny PACS 02.10.De, 03.65.Bz, 03.65.Vf, 03.67.Lx \\
%\small All pictures can be found in \cite{online1}

\normalsize
\begin{multicols}{2}

Recently, geometric phases \cite{berry} are getting considerable
attention in quantum computing \cite{ZanRas}. In this paper we
shall evaluate basic couplings of non-abelian geometric phases
(holonomies) and try to find out what is necessary to setup,
couple, and process spacetime memory. Geometric phases are subject
of concepts in differential-geometry and topology \cite{NAK}
associated with {\em non-abelian} groups, i.e. $U(N)$ \cite{WIZE}.
The first successful implementations via NMR have been reported
\cite{JonesN}, so it is likely that a quantum computer will be
operated in the near future by the non-abelian Berry connection of
a quantum computational bundle. To reveal the role of geometric
phases, one has to discuss the necessities in information
\begin{itemize}
  \item storage, transfer, and processing.
\end{itemize}
In \cite{ZanRas} it has been shown how quantum information can be
encoded in an eigenspace of a degenerate Hamiltonian $H$ such,
that one can in principle achieve the full quantum computational
power by using holonomies only. For an introduction to geometric
phases see i.e. \cite{Batterman}.

%#############################################################
\subsection*{Information storage}
%#############################################################

What characterizes memory? A memory has a time scale much longer
than the time scale of information processing. This is a basic
requirement, since computation is useless if the input is
forgotten while waiting for the output. These two timescales are
natural requirements of geometric phases: in a physical situation
the long time scale is given by the path of a vector signal on a
curved manifold, i.e. the orbital period, the short time scale
defines the vector signal, i.e. it's spinning period. The
information has to be `imprinted' in a spacetime structure, where
the topology is implemented by the proper spacetime manifold of
the signal path. The information coded in the geometric phase is
given by a path integral measure of curvature that modulates the
vector, on $S^2$ by a typical conic precession. In the holonomic
approach, information is encoded in a degenerate eigenspace of a
parametric family of Hamiltonians and manipulated by the
associated holonomic gates. Important for the realization of a
memory unit, the non-adiabatic generalization of \cite{ahanan}
defines a geometric phase factor for any cyclic evolution of a
quantum system. Consider a $T$-periodic cyclic vector
$\ket{\psi(\tau)}$ that evolves on a closed path ${\cal C}$
according to
\begin{equation}
\ket{\psi(T)}= e^{i\varphi(T)}\;\ket{\psi(0)} \labeqn{sv02},
\end{equation}
where the total phase $\varphi(T)$ acquired by the cyclic vector
can naturally be decomposed into a geometric $\varphi_{g}(T)$ and
dynamical phase $\varphi_{d}(T)$
\begin{equation}
\varphi(T) = \varphi_{g}(T) + \varphi_{d}(T) \labeqn{sv03}.
\end{equation}
The dynamical phase for one loop $t\in [0;T]$ is with the
Schr\"odinger equation given by
\begin{equation}
\varphi_{d}(T) = - {1 \over \hbar}\int\limits_0^T
\bra{{\psi}(\tau)} H(\tau) \ket{{\psi}(\tau)}d\tau
 \labeqn{sv05}.
\end{equation}
The Berry phase or geometric phase depends not on the explicit
time dependence of the trajectory and is for one loop given by
\begin{equation}
\varphi_{g}(T) = i \oint_{\cal C} \bra{{\psi}(\tau)} {d
}\ket{{{\psi}(\tau)}}  \labeqn{sv04}.
\end{equation}
The `parallel transported' spin vector will come back after every
loop with a directional change $\varphi_{g}(T)$ equal to the
curvature enclosed by the path $\cal C$. On the unit sphere the
curvature increment is proportional to the area increment that can
be a spherical triangle with area given by
\begin{eqnarray}
  d\Omega := [1-\cos \theta(\tau)] d\varphi(\tau),
\end{eqnarray}
the total area enclosed by the closed orbit (loop) is equal to
\begin{eqnarray}
 \Omega = \oint_{\cal C} d\Omega :=
 \int_0^T d\tau [1-\cos \theta(\tau)] \dot \varphi(\tau).
\end{eqnarray}
The Berry phase $\varphi_{g}(T) = J \Omega $ and the total phase
are proportional to spin $J$. In the standard case of precession
on the sphere
\begin{equation}
  \varphi_{g}(T) = 2\pi J (1-\cos\theta), \quad \varphi(T)= 2\pi J  \labeqn{sv08},
\end{equation}
where $\theta$ is the vertex cone semiangle, $\varphi_{d}(T)= 2\pi
J \cos\theta$. For the two level system, the geometric phase is
equal to half of the solid angle subtended by the area in the
Bloch sphere enclosed by the closed evolution loop of the
eigenstate. With $n$ parameters $\lambda_\mu(t)$, $\mu=1,2,...,n$
that span a closed curve $\cal C$ in the $T$-periodic parameter
space $\lambda_\mu(0)=\lambda_\mu(T)$, the Berry phase may be
represented in terms of the `gauge potential' $A$ with connection
matrix
\begin{equation}
(A_{\mu})^{\alpha\beta}:= \langle\psi^\alpha(\lambda)|
\,{\partial}/{\partial\lambda^\mu}\,
|\psi_{}^\beta(\lambda)\rangle \label{conn}
\end{equation}
where $A =\sum_\mu A_{\mu}\,d\lambda_\mu,$ and
\begin{equation}
\varphi_{g}(T)  =  \oint_{\cal C} A  =  \int\limits_{\cal S_{\cal
C}} F , \quad F = dA \labeqn{berry-geometric}.
\end{equation}
$A$ is the non-abelian gauge potential that can be regarded as a
winding number density and allows for parallel transport of
vectors over $\cal S_{\cal C}$, an arbitrary surface in the
parameter space bounded by the contour $\cal C$. For more details
regarding monopoles and Wilson loops on the lattice in non-abelian
gauge theories, see e.g. \cite{Zakharov}.


%#############################################################
\subsection*{Information transfer}
%#############################################################

What characterizes information channels? Our geometric memory will
have quantum nature: it is periodically refreshed or regenerated
with quantum memory loss and information transfer given by the
correspondent phase-frequency modulation. Degeneracy plays a
crucial role in quantum computing and allows to transfer the phase
states between energetically equivalent sub-systems. The dimension
$d$ of the manifold $U(N)$ reaches its minimum for $d=1$, in this
extreme case $N=1$ denotes the maximally degenerate case
\cite{ZanRas}. In this case (the gauge group $U(1)$ has the
topology of a circle on which the homotopy classes of closed
curves are labelled by their winding or loop numbers) the
wave-function in the $M$-fold degenerate case transforms as
\begin{equation}
\psi \rightarrow e^{\pm i M \Lambda } \psi,
 \labeqn{trans01}
\end{equation}
where one unit corresponds to the phase sub-interval $[0,2\pi/M]$.
Information transfer of $M$ quantum information units per unit
cycle (adiabatic loop) at dynamical phase evolution frequency
$\omega_{M}$ could be realized by a coupling energy $\triangle E$
sponsored by a carrier with energy $E$ where
\begin{equation}
\triangle E = E - M \hbar \omega_{M} \labeqn{trans02},
\end{equation}
where $\triangle E \ll M\hbar \omega_{M}$. In electromagnetism $M$
is the (magnetic) quantum number quantizing charge taking integral
values $\pm 0, \pm 1, \pm 2, ...$ \cite{Dirac-31}. It is quite
often that the relevant systems provide for the required discrete
symmetries and large degenerate eigenspaces, i.e. rotational
invariance, see \eqn{trans01}.

%#############################################################
\subsection*{Information processing}
%#############################################################

What characterizes information processing? Computation requires
that memory is multi-stable and coupled to the quantum state
transfer, where the time evolution of a quantum sub-system can be
controlled by the state of another sub-system. The computational
dynamics is obtained by switching on and off by a set of gate
Hamiltonians that generate a small set of basic paths given by
unitary transformation on the quantum state-space. Multi-state
stability can be introduced by non-linear behavior. The well known
example of a simple flip-flop-type feedback process (a bi-state or
half spin configuration) can be realized with a geometric phase
that is driven by it's own precession dynamics. Let the precession
cone vertex semiangle $\theta$ of \eqn{sv08} realize a bi-stable
flip-flop configuration characterized by two states:
\begin{itemize}
  \item $M_+>0$, $0<\theta_+<\pi$
  \item $M_-<0$, $\pi<\theta_-<2\pi$
\end{itemize}
that can be stabilized by the iteration
%\begin{equation}
%{M_\pm \theta_\pm }  = \pi \cos \theta\({\theta_\pm}\)
%\labeqn{proc01}.
%\end{equation}
\begin{eqnarray}
\theta_{\pm,i+1} = \frac{\pi \cos\theta_{\pm,i}}{M_\pm}
 \labeqn{spinorbit0},
\end{eqnarray} and converges after a few steps to a special
$\theta_\pm$-value. A fast convergence is crucial for the
performance of the spacetime computer. The coupling can be
interpreted as a navigational iteration on the closed path, where
one iteration step requires to exchange one bit of information
between the orbital system partners controlling each other in a
center-of mass system. The bit is the sign of phase that gets lost
in the cosine-function on $S^2$ independent of the resulting
coupling shift. Therefore, a virtual coupling bit-stream can be
assigned to the closed orbital path on $S^2$. The choice of the
form \eqn{spinorbit0} is adjusted to the energy transfer relation
\eqn{trans02} with $M \rightarrow M_\pm$ and $\theta \rightarrow
\theta_\pm$. This has the following background: the phase
evolution can be divided into the two parts of geometric and
dynamic phase evolution, where the geometric evolution can be
assigned to a precession frequency $\omega_{p}$ with ratio
adjusted to the sponsored energy
\begin{equation}
{\omega_{p} \over \gamma \omega} = {\triangle E \over E} =
{\varphi_{g}(T) \over 2\pi J }
 \labeqn{phase02g}
\end{equation}
including relativistic correction $\gamma$. The dynamical phase
evolution corresponds to the cyclic frequency $\omega_{M_\pm}$ and
characterizes the general expression for spin-rotation coupling
observed in the laboratory frame which can be assumed to be on
$S^2$
\begin{equation}
  M_\pm {\omega_{M_\pm} \over \omega} = \pm 1 \mp {\omega_{p}  \over \gamma
  \omega}=\cos\theta_{\pm}
\labeqn{phase02}
\end{equation}
in accordance with \eqn{sv08}. \\ Omitting the $\pm$ polarity the
relative dynamical coupling constant $\alpha(M)$ can be defined by
the ratio dynamical phase evolution frequency $\omega_{M}$ divided
by the carrier (Compton) frequency $\omega$ driven by particle
spin $J$
\begin{equation}
 {\alpha(M) } ={J \triangle\varphi_{d}(T) \over \varphi(T)}= {J \omega_{M} \over \omega}
\labeqn{phase02d},
\end{equation}
where the coupling is proportional to the evolution of the
dynamical part $\triangle \varphi_{d}(T)/\varphi(T)$. With
\eqn{phase02} in \eqn{phase02d}
\begin{equation}
  {\varphi_{g}(T) \over 2\pi J }
  = 1-{M \alpha  \over J} \labeqn{sv091},
\end{equation}
the dynamical part of \eqn{sv091} provides for
\begin{equation}
{M \alpha }  = J \cos \({\theta}\) \labeqn{spinorbit11}.
\end{equation}
Comparing \eqn{phase02} - \eqn{spinorbit11} the precession cone
vertex angle $2{\theta}$ is linearly related to the dynamic
spin-orbit interaction and given by the feedback coupling relation
of the most trivial kind
\begin{equation}
 {\theta} = \pi \alpha
 \labeqn{sv07b}.
\end{equation}
\eqn{sv07b} has a simple geometric interpretation: spin-orbit
coupling modelled by a `rolling cone' representing a vector state
or signal. Rotated once, the cone will change its orbital
orientation by a special angle $2\pi/M$, rotated $M$-times in the
quantum case, the cone will return to the initial position with
integral $M$ (providing for single-valuedness). If the base of the
cone has radius $\theta/\pi$, the side length is
$M\theta/\pi=\cos(\theta)$. As shown in \cite{alpha137MN}, for $M
= 137$ and vitual photon vector coupling with $J=1$ the coupling
constant $\alpha \approx 1/137.03600941164$ fits within error
range to a neutral and theory independent determination of the
Sommerfeld fine structure constant. This suggests to assign
intrinsic computational entities and capabilities to the
electromagnetic field.

%#############################################################
\subsection*{Conclusion}
%#############################################################

In the holonomic approach to information processing geometric
phases serve as the carrier of physical information. In this case
geometric phases are the primordial memory of orientation given by
a path integral measure of curvature on $S^2=SU(2)/U(1)$, where
the coupling of intrinsic spin with rotation reveals the quantum
of rotational inertia $\equiv$ memory $\equiv$ angular momentum
quantum $\hbar$. The system carries pair-creation energy $E$ and
coupling energy $\triangle E$ stabilized by a phase-locked
feedback loop, a periodical refreshment including precession as a
form of phase-frequency modulation. The non-linear iteration
converges quickly and provides for a fast and flip-flop-type
situation: the prototype spacetime imprint of a polar binary
system. A virtual coupling bit-stream can be assigned to the
navigational iteration on the closed path, since one non-linear
iteration step requires to exchange one bit of information.
Regarding the polar coupling constant and the magnetic monopole
topology \cite{berry,Dirac-31}, it can be proposed that natural
memories optimized by iterative phase relationships can be found
everywhere. The non-linear phase-locked feedback mechanism
provides for an hidden and very fast bit-stream running at a nice
bandwidth $\approx \omega_M$. In such a natural high-performance
computer, fine structure as a pure number would divide hardware
from software. Additional details can be found in
\cite{alpha137MN}, \cite{alpha137top}, and \cite{online1}.

%\newpage
\footnotesize
%............................................................#
\begin{thebibliography}{40}

\bibitem{berry} M. V. Berry, Proc. Roy. Soc. Lond. A {\bf 392},
45 (1984).

\bibitem{ZanRas} P. Zanardi, M. Rasetti, Phys. Lett.
{\bf A264}, 94 (1999). quant-ph/9904011

\bibitem{NAK} M. Nakahara, {\em Geometry, Topology and Physics}, IOP Publishing Ltd., 1990

\bibitem{WIZE} F. Wilczek and A. Zee., Phys. Rev. Lett. {\bf {52}}, 2111
(1984).

\bibitem{JonesN} J.A. Jones, V. Vedral, A. Ekert, G. Castagnoli, Nature {\bf 403}, 869 (2000).

\bibitem{Batterman} R. Batterman, `Falling Cats, Parallel Parking, and Polarized
Light' (2002);
\\\href{http://philsci-archive.pitt.edu/documents/disk0/00/00/05/83/index.html}{PITT-PHIL-SCI00000583}.

\bibitem{ahanan} Y. Aharonov, J. Anandan, `Phase Change During a
Cyclic Quantum Evolution', Phys. Rev. Lett. {\bf 58}, 1593 (1987).

\bibitem{Zakharov} F.V. Gubarev, V.I. Zakharov, Int. J. Mod.
Phys. {\bf A17}, 157 (2002);
\\\href{http://xxx.lanl.gov/abs/hep-th/0004012}{hep-th/0004012}.

\bibitem{Dirac-31} P. A. M.  Dirac, Proc. Roy. Soc. London A {\bf
133}, 60 (1931).

\bibitem{alpha137MN} B. Binder, `Berry's Phase and Fine Structure' (2002);
\\\href{http://philsci-archive.pitt.edu/documents/disk0/00/00/06/82/index.html}{PITT-PHIL-SCI00000682}.

\bibitem{alpha137top} B. Binder, `Geometric Phase Locked in Fine Structure' (2002);
\\\href{http://philsci-archive.pitt.edu/documents/disk0/00/00/07/74/index.html}{PITT-PHIL-SCI00000774}.

\bibitem{online1}{B. Binder, alpha simulation;}\\
\href{http://www.quanics.com/spinorbit.html}{http://www.quanics.com/spinorbit.html}.

\end{thebibliography}
\end{multicols}

\end{document}
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