Spin swap vs. double occupancy in quantum gates
T. A. Kaplan and C. Piermarocchi
We propose an approach to realize quantum gates with electron spins localized in a semiconductor that uses double occupancy to advantage. With a fast (non-adiabatic) time control of the tunnelling, the probability of double occupancy is first increased and then brought back exactly to zero. The quantum phase built in this process can be exploited to realize fast quantum operations. We illustrate the idea focusing on the half-swap operation, which is the key two-qubit operation needed to build a CNOT gate.
spinswapcarlo5.tex
\documentclass[twocolumn,showpacs,prl]{revtex4}
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\usepackage{amsmath,amsthm,amscd}
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\usepackage{graphicx} \usepackage{epsfig} \begin{document}
\title{Spin swap vs. double occupancy in quantum gates}
\author{T. A. Kaplan and C. Piermarocchi}
\affiliation{Department of Physics and Astronomy and Institute for
Quantum Sciences, \\ Michigan State University, East Lansing, Michigan
48824}

\begin{abstract} We propose an approach to realize quantum
gates with electron spins localized in a semiconductor that uses double occupancy to
advantage. With a fast (non-adiabatic) time control of the tunnelling, the probability
of double occupancy is first increased and then brought back exactly to zero. The
quantum phase built in this process can be exploited to realize fast quantum
operations. We illustrate the idea focusing on the half-swap operation, which is the
key two-qubit operation needed to build a CNOT gate.  \end{abstract}

\pacs{75.10.Pq}

\date{\today}
\maketitle

Many solid state proposals for quantum computation make use of the
spin of an electron localized in a quantum dot~\cite{friesen03,
loss98} or by a donor ion~\cite{vrijen00} as a qubit. In these schemes
the overlap between the electron wave functions in neighboring dots is
controlled by gates, and the exchange interaction provides the
mechanism for the realization of conditional operations on two
qubits. Among the requirements for such proposals is that one wants no
or small double occupancy, i.e. small probability to have the two
electrons in the same dot or donor. In fact, the state space
$\mathcal{S}$ for 2 qubits has to be 4-dimensional and the mixing with
the states of double occupancy brings the physical system out of the
computational space.  Hu and Das Sarma~\cite{hu} observed that the
double occupancy will in principle degrade any implementation of
quantum gates, and noted the intrinsic conflict of trying to minimize
this double occupancy while making the overlap large enough to realize
a sizeable two-qubit operation, like e.g. a spin swap. The problem was
further analyzed in Ref.~\cite{schliemann}, where it was shown that
using an adiabatic control in the quantum dots case, the double
occupancy was small enough so that the degradation can be made
negligible, consistently with the requirement of large enough
overlap. The operation critical to quantum computing is the
\emph{half-swap} operation (also written as $\sqrt{SWAP}$),
i.e. having the state going half of the way to the complete swap, and
we focus on this process to illustrate the approach.

We show here that \emph{a half-swap operation without degradation can be realized by
designing a time control where the probability of double occupancy first increases and
then is brought back exactly to zero. In this cycle the quantum state in $\mathcal{S}$
picks up a phase that is exploited in completing the process.} As a consequence, the
swap can be much faster than that in the case where double occupancy is kept small at
all times. We will consider explicitly two cases where the control of the tunneling
amplitude follows in time a square and a hyperbolic secant pulse shape. The conditions
for a perfect $\sqrt{SWAP}$, i. e. a fast half-swap gate with double occupancy
identically zero, can be given in these two cases by simple analytic rules.  The
results are exact within the single band model considered, the validity of which will
be discussed below for realistic systems.

%The adiabatic
%condition is not needed in the scheme and square pulses as well as
%other pulse shapes can be used.

The dynamics during the half swap can be described by using two spatial orbitals,
which we call $w_a$ and $w_b$ ($w$ for Wannier functions), orthonormal, and localized
respectively around the two sites $a$ and $b$~\cite{hu, schliemann, burkard}. The
$w$'s are linear combinations of non-orthogonal ``atomic" functions whose overlap is
$\Lambda$. This implies a 4-state one-electron basis, and therefore a 6-dimensional
complete vector space for 2 electrons.~\cite{slater} The Hamiltonian $H$ contains the
kinetic and potential energies, plus a magnetic field interacting with the orbital
magnetic moment. The quantum control is realized by changing the profile of the
potential energy, therefore modifying the tunnelling amplitude between the two sites.
We will not consider a magnetic field term, since it is irrelevant in the proposed
scheme. The 2-electron basis used is
\begin{subequations}
\label{5}
\begin{eqnarray}
|\Phi_1\rangle&=&2^{-1/2}(c^\dagger_{a\uparrow}c^\dagger_{b\downarrow}
-c^\dagger_{a\downarrow}c^\dagger_{b\uparrow})|0\rangle  \\
|\Phi_2\rangle&=&2^{-1/2}(c^\dagger_{a\uparrow}c^\dagger_{a\downarrow}
+c^\dagger_{b\uparrow}c^\dagger_{b\downarrow})|0\rangle \\
|\Phi_3\rangle
&=&2^{-1/2}(c^\dagger_{a\uparrow}c^\dagger_{a\downarrow}
-c^\dagger_{b\uparrow}c^\dagger_{b\downarrow})|0\rangle \\
|\Phi_4\rangle
&=&c^\dagger_{a\uparrow}c^\dagger_{b\uparrow}|0\rangle \\
|\Phi_5\rangle
&=&2^{-1/2}(c^\dagger_{a\uparrow}c^\dagger_{b\downarrow}
+c^\dagger_{a\downarrow}c^\dagger_{b\uparrow})|0\rangle \\
|\Phi_6\rangle&=&c^\dagger_{a\downarrow}c^\dagger_{b\downarrow}|0\rangle~,
\end{eqnarray}
\end{subequations}
where $c^\dagger_{\nu\sigma}$ creates an electron in the orbital $w_{\nu}$ with spin
$\sigma$.  Physically we speak of $\Phi_{\nu}$ with $\nu =1,4,5,6$ as having 1
electron near each site, while for $\nu=2,3$, the states have 2 electrons on one site,
and are called doubly-occupied states. Assuming inversion symmetry, the time-dependent
Hamiltonian $H_{ij}(t)\equiv \langle\Phi_i|H(t)|\Phi_j\rangle$ reduces by symmetry to
four $1 \times 1$'s (the triplet $H_{44}$, $H_{55}$, and $H_{66}$, and the {\it
ungerade} singlet $H_{33}$), plus a $2 \times 2 $ which yields the remaining two
(\emph{gerade}) singlets.  We assume that before applying the control pulse, at time
$t=-\infty$, the wave function is $|\Psi(-\infty)\rangle
=c^\dagger_{a\uparrow}c^\dagger_{b\downarrow}|0\rangle
=2^{-1/2}(|\Phi_1\rangle+|\Phi_5\rangle)~.$ Given the form of the initial state and
the symmetry properties discussed above, $|\Phi_3 \rangle$, $|\Phi_4\rangle$ and
$|\Phi_6\rangle$ do not enter in the dynamics and we calculate
$|\Psi(t)\rangle=a(t)|\Phi_1\rangle+b(t)|\Phi_2\rangle +c(t) |\Phi_5\rangle $ by the
(time-dependent) Schr\"{o}dinger equation
\begin{equation} i\frac{d}{dt}\left[
\begin{array}{c} a(t)\\ b(t)\\ c(t) \end{array}\right]^T=\left[
\begin{array}{ccc} 0 & \frac{\Omega(t)}{2}& 0 \\ \frac{\Omega(t)}{2}&
\Delta &0 \\ 0 & 0& J_P(t) \end{array} \right] \left[ \begin{array}{c} a(t)\\ b(t)\\
c(t) \end{array}\right] \end{equation} where $\hbar\Omega(t)=2 H_{12}(t)$,
$\hbar\Delta=H_{22}-H_{11}$ and $\hbar J_P(t)=H_{55}(t)-H_{11}(t)$. Notice that
$\Omega(t)$ is first order in the overlap $\Lambda$, while $\Delta$ depends only on
the on-site and long-range Coulomb interaction: $\hbar\Delta =\langle
w_a,w_a|v|w_a,w_a\rangle - \langle w_a,w_b|v|w_a,w_b\rangle $, where $v$ is the
Coulomb interaction between the two electrons. This term is of order $\Lambda^0$, and
therefore only weakly dependent on the tunnelling amplitude. So we assume $\Delta$
constant during the gate operation. $\hbar J_P(t)=-2 \langle w_a,w_b|v|w_b,w_a\rangle$
is the potential exchange~\cite{anderson}, which is O($\Lambda^2)$. The hopping term
$\hbar \Omega(t)=4\langle w_a|h_1|w_b\rangle+4\langle w_a,w_b|v|w_a,w_a\rangle$, where
$h_1$ is the single particle kinetic and potential energy, is responsible for the
kinetic exchange. Both $J_P(t)$ and $\Omega(t)$ are controlled by the tunnelling
amplitude through their dependence on the overlap $\Lambda$. We remark that these
parameters simplify in the Hubbard model to $J_P(t)=0$ and $\hbar \Delta=U$. In this
case the hopping amplitude $\Omega(t)$ is the only mechanism for the exchange
interaction ($J_K \cong -\hbar^2\Omega(t)^2/2U$ for small $\Lambda$, the kinetic
exchange~\cite{anderson}). The presence of the potential exchange $J_P(t)$ is not
essential in the realization of the proposed scheme, but we keep it in order to prove
the reliability of the scheme in the most general case.

It is now simple to calculate the quantities we want to monitor. One is the
expectation value of $s_z=n_{a\uparrow}-n_{a\downarrow}$, the z-component of spin at
site $a$. ($n_c$ are occupation numbers.) It is sufficient to consider just one site
since the total z-component of spin is conserved and so must remain 0. Using the
property $s_z|\Phi_1\rangle=|\Phi_5\rangle$, $s_z|\Phi_5\rangle=|\Phi_1\rangle$, and
$s_z|\Phi_2\rangle=0$, we find
\begin{equation}
\sigma_z(t)\equiv \langle\Psi(t)|s_z|\Psi(t)\rangle=2 Re[a(t)c^*(t)]~. \label{sigmaz}
\end{equation}
We also want the probability $P_d$ of double occupancy, i.e. the
probability of finding the system in the state $\Phi_2$. This is
\begin{equation}
P_d(t)=|\langle\Phi_2|\Psi(t)\rangle|^2= |b(t)|^2 \label{12}~.
\end{equation}
In order to have the perfect $\sqrt{SWAP}$ at the end of the pulse,
i.e. at $t=+\infty$, we need to design the control in such a way that
the two conditions
\begin{subequations}
\label{c1c2}
\begin{eqnarray}
P_d(+\infty)=0 \label{c2} \\
\sigma_z(+\infty)=0 \label{c1}
\end{eqnarray}
\end{subequations}
are verified. One can see that Eq.~(\ref{c1c2}) implies
$|\Psi(+\infty)\rangle=2^{-1/2}(c^\dagger_{a\uparrow}c^\dagger_{b\downarrow} \pm i
c^\dagger_{a\downarrow}c^\dagger_{b\uparrow})|0\rangle$ (Bell entangled states). In
the usual approach a weaker constraint than the one in Eq.~(\ref{c2}) is used (i.e.
$P_d(+\infty)\ll 1$) satisfied by slow adiabatic switching and small mixing with
doubly occupied states. This condition is implicit when we describe the system of two
localized electrons using the (effective) Heisenberg Hamiltonian. Our approach
consists in finding first the general conditions on the control to satisfy exactly
Eq.~(\ref{c2}), which does not imply necessarily adiabaticity and small average mixing
with the doubly occupied states, and then adjust the remaining control parameters to
satisfy Eq.~(\ref{c1}).

The time dependence of $c(t)$ can be integrated directly from the
Schroedinger equation and we obtain
\begin{equation}
c(+\infty)=2^{-1/2} \exp(-i\int_{-\infty}^{+\infty}J_P(t)~dt)~.
\end{equation}
The dynamics of $a(t)$ and $b(t)$ can be mapped to the evolution of a pseudospin in
the presence of a time dependent effective magnetic field. The condition in
Eq.~(\ref{c2}) means that the pseudospin makes a closed loop in the Bloch sphere,
while the condition in Eq.~(\ref{c1}) ensures that the phase picked up by the
pseudospin evolution during such a closed loop compensates the phase due to the
potential exchange for a perfect $\sqrt{SWAP}$.  The analytic solution for the
evolution of a spin in a time dependent magnetic field can be found only in particular
cases~\cite{ishk00}, and we will discuss here two of them: the Rabi
solution~\cite{allen}, corresponding to a square pulse shape, and the Rosen and Zener
solution~\cite{rosen32} corresponding to a hyperbolic secant pulse shape.

The Rabi solution describes the precession of the pseudospin in a constant effective
magnetic field. A square pulse in the tunnelling amplitude keeps this effective field
on for a time $T$, and then turns it off. We can write the hopping term as a constant
$\Omega(t)=\Omega$, and consider the time evolution given by the Rabi solution from a
time $t=0$ to $t=T$ (just the solution for $t$-independent $H$). In a similar way the
potential exchange term $J_P(t)=J_P$ for $0 \le t \le T$ and zero otherwise. The
probability of double occupancy at the end of the pulse is given by:
\begin{equation}
P_d(T)=|b(T)|^2=\frac{\Omega^2/2}{\Omega^2+\Delta^2}
\sin^2\left(\frac{\sqrt{\Omega^2+\Delta^2}}{2}T\right)~.
\end{equation}
Therefore the condition in Eq.~(\ref{c2}) is satisfied by fixing the
length of the pulse to
\begin{equation}
T_n= \frac{2 n \pi}{\sqrt{\Omega^2+\Delta^2}}~.
\end{equation}
The integer $n$ identifies the number of full precessions of the
pseudospin during the pulse.  We find from the Rabi solution that $
a(T_n)=\exp (-i\Delta T_n /2)/\sqrt{2}$. Using this solution in
Eq.~(\ref{sigmaz}), the condition in Eq. (\ref{c1}) for the perfect
$\sqrt{SWAP}$ is rewritten as
\begin{equation}
\cos n\pi\left(\frac{\Delta-2J_P}{\sqrt{\Omega^2+\Delta^2}} \right)=0. \label{genR}
\end{equation}
This can be recast in terms of the separation between the triplet and
the lower singlet eigenvalues $J$, and the separation between the two
\emph{gerade} singlets $\Delta E$ as
\begin{equation}
\frac{2 n J}{\Delta E}=\frac{2k+1}{2}~,
\label{condR}
\end{equation}
where $k$ is any integer.
%The state at the end of the pulse is found to be
%\begin{equation}
%\Psi(t =+\infty)=2^{-1/2}(c^\dagger_{a\uparrow}c^\dagger_{b\downarrow}
%+ (i)^{2k+1} c^\dagger_{a\downarrow}c^\dagger_{b\uparrow})|0>,
%\label{15}
%\end{equation}
%which is a maximally entangled state.

\begin{figure}
%\hspace{-1.5 cm}
\centering \includegraphics[height=8. cm, angle=270]{Fig1spinswapcarlo4.eps}
\caption{$\sigma_z(t)$ and $P_d(t)$ for two electrons confined by
 hydrogenic potentials. Infinite tunnelling barriers are switched off
 during the time interval $0<t<T$. Time $t$ is in units of
 $\hbar$/Ry$^*$. (a) The separation between the sites is $4.53$ au*. At
 the end of the pulse we have a half swap with a small but nonzero
 $P_d$. (b) The separation is $2.38$ au* and the energy levels satisfy
 the condition in Eq.~(\ref{condR}). The half swap is realized in a
 shorter time and with $P_d=0$ identically at the end of the pulse
 (see inset).}
\label{fig:sigma}
\end{figure}

To illustrate the idea we show in Fig.~\ref{fig:sigma} the evolution of $\sigma(t)$
and $P_d(t)$ in the square pulse case calculated assuming a hydrogenic confinement
potential and localized functions of the Slater form, which models well the case of
electrons confined in shallow donors. We want to stress that the general approach we
are suggesting is independent of this particular choice. In fact, for any reasonable
form of the confinement potential it is possible to modify the ratio in
Eq.~(\ref{condR}) by changing the maximum tunnelling amplitude or the separation
between the dots/impurities. When the separation is large, so the overlap
%$\int d^3r|w_a(\mathbf{r})w_b(\mathbf{r})|$
is small (or the tunnelling barrier is large), there is a small admixture of
doubly-occupied states at any time. This is the region where the low-lying states are
described accurately by the Heisenberg Hamiltonian $2J\mathbf{S}_a\cdot\mathbf{S}_b$.
The spins here are associated with the sites; e.g. $S_a^+=c^\dagger_{a\uparrow}
c_{a\downarrow}$~\cite{anderson}. This case is considered in Fig.~\ref{fig:sigma}~(a),
obtained with a separation between the localized electrons of 4.53 au$^*$. We indicate
by au$^*$ and Ry$^*$ the atomic units scaled by the electron effective mass and the
static dielectric constant. Time is in units of $\hbar$/Ry$^*$. For a typical GaAs
based system 1 Ry$^*$ $\sim$ 5 meV, 1 au$^*$ $ \sim $ 80 \AA, and $\hbar/Ry* \sim $
0.1 ps.  We see that at the end of the pulse we satisfy the condition for the half
swap $\sigma_z=0$ with a small double occupancy $P_d \sim 0.03 $ (see inset in
Fig.~1~(a)). The new approach is illustrated in Fig.~\ref{fig:sigma}~(b) corresponding
to a separation of 2.38 au$^*$. At this separation the energy levels satisfy the
condition in Eq.~(\ref{condR}) yielding a perfect half swap with both $\sigma_z=0$ and
$P_d=0$ identically. Fig.~\ref{fig:sigma}~(b) shows a marked increase in swap speed
and a much higher probability of double occupancy during the gate compared to
Fig.~\ref{fig:sigma}~(a). The case in Fig.~\ref{fig:sigma}~(b) corresponds to $n=1$,
higher $n$ will give a perfect half swap in longer times.

The use of square pulses gives a direct illustration of the new approach. However, for
practical purposes, a continuous time dependence of the pulse is desirable. Using the
picture of the pseudospin in an effective magnetic field, we need to design in the
general case a $\mathbf{B}_{eff}(t)$ that generates closed loops in the Bloch sphere.
Once this first condition is realized, the condition on the control parameters for a
perfect half swap can be found accordingly. This is possible for the hopping amplitude
of the form $\Omega(t)=\Omega/\cosh\left( \pi t/T\right)$. Then
$\int_{-\infty}^{+\infty}\Omega(t)~dt=\Omega T$, the same as for the square pulse. The
probability of double occupancy at the end of the pulse (at $t=+\infty$) can be
obtained from the Rosen and Zener result~\cite{rosen32}
\begin{equation}
P_d=\frac{\sin^2\Omega T/2}{2 \cosh^2\Delta T/2}.
\label{rzres}
\end{equation}
In the adiabatic approach the condition $\Delta T\gg 1$ in the denominator makes this
probability small.  Our approach takes advantage of the numerator in
Eq.~({\ref{rzres}), which implies that for
\begin{equation}
T_n= \frac{2 n \pi}{\Omega}
\end{equation}
we will have no double occupancy at the end of the pulse. In order to
determine the phase of $a(+\infty)$, we use the transformation
$2z=\tanh(\pi t/T)+1$ to map the infinite time interval to $[0,1]$,
and the equation for the coefficient $a(z)$ reads
\begin{equation}
(z^2-z)a^{\prime \prime}+\left(\frac{1}{2}+\frac{i\Delta T}{2 \pi}-z\right)a^{\prime}
+ \left(\frac{\Omega T}{2\pi}\right)^2 a=0~.
\label{hyper}
\end{equation}
This is the hypergeometric equation and in general we obtain a
solution in terms of a linear combination of hypergeometric
functions. However, it is interesting to notice that if the condition
for no double occupancy is met, then $\left(\Omega
T/2\pi\right)=n$, and we will fall in the degenerate case of the
hypergeometric equation. As discussed in Ref.~\cite{erdelyi53}, the
solution in this case consists of a polynomial of degree $n$. We find
that the solution to Eq.~(\ref{hyper}) with the initial condition
$a(0)=1$ and with $|a(1)|^2=1$ can be written explicitly as
$a_n(z)=n!P^{\alpha_n,\alpha_n^*}_{n}(1-2z)/(\alpha_n+1)_n ~,$ where
$P^{\beta,\gamma}_{n}(x)$ are Jacobi polynomials, $\alpha_n= -1/2 + i n
\Delta/\Omega$, and
$(x)_n=\Gamma\left(x+n\right)/\Gamma\left(x\right)$.  The condition in
Eq.~(\ref{c1}) can then be written in a compact form using the
hypergeometric function (which in this case terminates after $n$ terms
in the series) as
%\begin{equation}
%Re[P^{-1/2+ i n \Delta/\Omega,-1/2- i n \Delta/\Omega}(-1)/(1/2+ i n \Delta/\Omega)_n e^{i 4 n J_D/\Omega}]=0~,
%\end{equation}
\begin{equation}
Re[F(-n,n,1/2+i n \Delta/\Omega,1) e^{i 4 n J_P/\Omega}]=0~,
\label{condRZ}
\end{equation}
which is the parallel of Eq.~(\ref{genR}) for this continuous pulse shape. The fact
that the potential exchange $J_P(t)$ is O($\Lambda^2)$ led us to take $J_P(t)=J_P
\mbox{sech}^2(t/T)$ in Eq.~(\ref{condRZ}), although the solvability and essential
aspects of the solution are not limited to this. We show in Fig.~\ref{fig2} the
realization of a perfect half swap in the hydrogenic case. At the separation of about
2.36 au* the matrix elements of the Hamiltonian satisfy the condition in
Eq.~({\ref{condRZ}) and a perfect half swap can be realized. The case in
Fig.~\ref{fig2} corresponds to $n=1$ and we have checked that for higher $n$ we can
always find at least one separation to satisfy Eq.~(\ref{condRZ}). The inset shows the
exponential converging of $P_d$ and $\sigma_z$ toward the desired value. We remark
that in GaAs based systems this translates in a half swap operated with an accuracy of
$10^{-10}$ in a time of the order of 3 ps.
\begin{figure}
%\hspace{-2.5 cm}
\centering \includegraphics[scale=0.3]{Fig2spinswapcarlo4.eps} \caption{$\sigma_z(t)$
and $P_d(t)$ for two electrons confined by hydrogenic potentials with barriers
controlled using a hyperbolic secant pulse shape.  The separation is 2.36$\cdots$
au$^*$. Time $t$ is in units of $\hbar$/Ry$^*$. The inset shows the Log of $\sigma(t)$
and $P_d(t)$ at longer times.} \label{fig2}
\end{figure}

We now consider likely sources of error and other difficulties to be
expected in a real implementation of the ideas proposed. We pose the
question, if $\Omega$ and $\Delta$ deviate slightly from values needed
for a perfect half-swap, how much do $\sigma(+\infty)$ and
$P_d(+\infty)$ differ from their ideal values of zero? Using the
analytic solutions we proved that if the deviation of the parameters
is O($\epsilon$) then the deviation $\sigma(+\infty)=$O$(\epsilon)$
and $P_d(+\infty)=$O$(\epsilon^2)$.
%, i.e. the control errors do not
%propagate with an exponential or singular behavior.
Higher intra-dot excitations will complicate our considerations. If they are large
compared to $\hbar \Delta$, however, their effect will be small. To get an estimate of
the requirements, consider these excitations for the infinite square well, side length
$L$. Then $\delta E=3\pi^2\hbar^2/2m^*L^2=6\pi^2(\text{au*}/L)^2 \text{Ry*}$, is the
first excitation energy, and $\hbar \Delta\approx 2 e^2/(\epsilon
L)=2(\text{au*}/L)\text{Ry*}$, so $\delta E/\hbar \Delta\approx 3\pi^2(\text{au*}/L)$.
For two GaAs quantum dots of lateral size $L=100 \AA$, this gives $\delta
E/\hbar\Delta\approx 27$ and $\hbar \Delta\approx 2\text{Ry*}$, satisfying the
requirement $\delta E/\hbar \Delta>>1$. Finally, we remark that in order to take
advantage of the proposed scheme the time resolution needed for controlling the gate
is related to the intra-dot Coulomb repulsion $\hbar \Delta$. In principle, a
subpicosecond switching time for a pulse gate is not out of the
question~\cite{tessmer}.  By increasing the dot size $L$ while maintaining a large
$\delta E/\hbar\Delta $ it is possible to increase the resolution time $\Delta^{-1}$.
We estimate that this condition can be well satisfied for large ($\sim$ 300-400 \AA)
GaAs quantum dots in such a way to limit the resolution to the picosecond range, which
is within state-of-the-art capabilities.

In conclusion, we have shown that by non-adiabatic control it is possible to take
advantage of the double occupied states to make fast and accurate half swap
operations.  The idea can be applied to many systems currently under investigation for
experimental realization of quantum computing. Even in systems where the mixing with
double occupancy is small, the simple rules given here can be used as an error
correction method. Possible generalizations to more than two spins~\cite{lidar04} are
under consideration. It is also of interest to consider more general pulse shapes,
particularly with the object of seeing what forms of $\mathbf{B}_{eff}(t)$ will allow
the pseudospin to return to its starting value (i.e. corresponding to $P_d=0$).

%The scheme proposed can be extended to
%other two qubit operations, like e.g. a full swap. An open question is
%related to the extension of the idea to more than two
%electrons\cite{lidar}, we plan to explore the non-adiabatic
%control of many electronic spin localized in semiconductors.

\begin{acknowledgements}
We thank S. H. Tessmer, M. I. Dykman, and S. D. Mahanti for
enlightening discussions. C.P. acknowledges support by NSF under
contract DMR-0312491.
\end{acknowledgements}


\begin{thebibliography}{99}
\bibitem{friesen03} M. Friesen, P. Rugheimer, D. E. Savage,
M. G. Lagally, D. W. van der Weide, R. Joint, and M. A. Eriksson,
Phys. Rev. B {76}, 121301 (2003).
 \bibitem{loss98} D. Loss and D. P. DiVincenzo, Phys. Rev. A {\bf 57},
 120 (1998).
\bibitem{vrijen00} R. Vrijen, E. Yablonovitch, K. Wang, H. W. Jiang,
A. Balandin, V. Roychowdhury, T. Mor, and D. DiVincenzo, Phys. Rev. A
{\bf 62}, 122306 (2000).
 \bibitem{hu} X. Hu and S. Das Sarma, Phys. Rev. A \textbf{61}, 062301
 (2000).
\bibitem{schliemann} J. Schliemann, D. Loss and A. H. MacDonald,
Phys. Rev. B \textbf{63}, 085311 (2001).
\bibitem{burkard} G. Burkard, D. Loss and D. P. DiVincenzo,
Phys. Rev. B\textbf{59}, 2070 (1999).
\bibitem{slater} J. C. Slater, \emph{Quantum Theory of Molecules and
Solids} Vol.1, Chap.4 McGraw-Hill, New York (1963).
\bibitem{anderson} P. W. Anderson, Solid State Physics \textbf{14},99
(1963).
\bibitem{ishk00} A. M. Ishkhanyan, J. Phys. A {\bf 33}, 5539 (2000).
\bibitem{allen} L. Allen and J. H. Eberli, \emph{Optical Resonance and
Two-Level Atoms}, Wiley, New York 1975.
\bibitem{rosen32} N. Rosen and C. Zener, Phys. Rev. {\bf 40} 502
(1932).
\bibitem{erdelyi53} {\it Higher Trascendental Functions}, A. Erd\'elyi
Editor, McGraw-Hill, New York (1953).
\bibitem{tessmer} S. Tessmer, private communication.
\bibitem{lidar04} D. A. Lidar and A. Mizel, cond-mat/0401081 (2004).

%\bibitem{shi} Yu Shi, quant-ph/0205069(2002); A. Mizel, private comm.
%\bibitem{werner89} R. F. Werner, Phys. Rev. A {\bf 40} 4277 (1989).
%\bibitem{kaplan} T. A. Kaplan, Bull. Am. Phys. Soc., March APS meeting (2004).
%\bibitem{zanardipc} P. Zanardi, private comm.
%\bibitem{zanardi01} P.Zanardi, Phys. Rev. Lett. {\bf 87} 077901 (2001).
%\bibitem{barnum03} H. Burnam et al., quant-ph/0305023 (2003).
\end{thebibliography}
\end{document}

%, namely, (i)
%errors in the fixing the parameters, (ii) errors due to the neglect of
%higher intra-dot states, (iii) possible difficulties in achieving the
%faster switching times.


 put $\Delta=\Delta_0+c_1\epsilon$ and
$\Omega=\Omega_0+c_2\epsilon$ into the analytic solutions for $a(t)$
and $b(t)$, where $\epsilon$ is small and $c_1$ and $c_2$ are of order
unity. we find $\sigma(+\infty)=$O$(\epsilon)$ and
$P_d(+\infty)=$O$(\epsilon^2)$. Thus the deviation of the wave
function from being in $\mathcal{S}$ is O$(\epsilon^2)$, while its
deviation from the wanted state is O$(\epsilon)$.

There are two situations where this degradation is not important:

(i) For a large $\Gamma=U/J$, which corresponds to a small tunneling coupling $H_{12}$. This is the region where the low-lying states are described accurately
by the Heisenberg Hamiltonian $-2J\mathbf{S}_a\cdot\mathbf{S}_b$. The spins here are associated with the sites; e.g. $S_a^+=c^\dagger_{a\uparrow}
c_{a\downarrow}$~\cite{anderson}. From~(\ref{7}) we can see that for large $\Gamma$
\begin{eqnarray}
 \nu_1 &\cong& -\frac{H_{12}}{H_{22}-H_{11}}\nonumber\\
 \nu_2 &\cong& \frac{H_{22}-H_{11}}{H_{12}}.\label{13}
\end{eqnarray}
Thus the amplitude of $\cos(U t)$ in $\sigma(t)$ is much smaller than that of $\cos(J t)$, allowing the achievement of almost a complete swap. In this
case, the probability of double occupancy remains small during the gate operation.
(ii) It can be seen from Eqs. \ref{11} and \ref{12} that a perfect complete swap occurs whenever $J=k\omega_0,U=l\omega_0,$
where $k,l$ are odd integers. The complete-swap time $t_0$ is then given by $\omega_0t_0=\pi$. Furthermore, if $k+l$ is a multiple of four, then at $t_0/2$ one will get a perfect half swap. The state at time $t_0/2$ is found to be
\begin{eqnarray}
\Psi(t_0/2)&=&2^{-1/2}(i^k\Phi_1+\Phi_2)\label{14}\\&=&2^{-1/2}e^{\pm i\pi/4}
(c^\dagger_{a\uparrow}c^\dagger_{b\downarrow}
\mp i c^\dagger_{a\downarrow}c^\dagger_{b\uparrow})|0>,
\label{15}
\end{eqnarray}
upper or lower signs for $(k-1)/2$ even or odd, respectively.
We also note that in the case where $k+l$ is not multiple of four we still obtain states with $P_d=0$ at certain times. For instance, for $k=1$ and $l=5$ $P_d=0$ at $t_0/3$, and $2t_0/3$. The corresponding states do map to states in $\mathcal{S}$, and the associated gates can be seen as a $\sqrt[3]{SWAP}$ and $\sqrt[3]{SWAP^2}$, respectively.


The states
$\Phi_i, i=3-6$ are directly eigenstates, while $\Phi_1$ and $\Phi_2$ participate in the $2\times2$. Clearly the only matrix elements connecting the initial state with these
basis states are with $\Phi_1$ and $\Phi_5$. However, because $\Phi_2$ gets mixed with $\Phi_1$ in the eigenstates, the expansion~(\ref{4}) contains $\Phi_1,\Phi_2$ and $\Phi_5$, or, alternately, it contains $\Psi_1,\Psi_2$ and $\Psi_5$, where $\Psi_1,\Psi_2$ originate in the $2 \times 2$ and are given by
\begin{equation}

\Psi_i=\frac{\Phi_1+\nu_i\Phi_2}{1+\nu_i^2},\ i=1,2,\label{6}
\end{equation}
where
\begin{equation}
\nu_{
\begin{subarray}{c}
1 \\
2
\end{subarray}}
=\frac{H_{22}-H_{11}\mp\sqrt{(H_{22}-H_{11})^2+4H_{12}^2}}{2H_{12}},\label{7}
\end{equation}
with $H_{ij}\equiv (\Phi_i,H\Phi_j)$.
The resulting wave function is
\begin{eqnarray}
\Psi(t)&=&\frac{exp{(-iE_1t/\hbar)}}{\sqrt{2(1+\nu_1^2})}\Psi_1
+\frac{exp{(-iE_2t/\hbar)}}{\sqrt{2(1+\nu_2^2})}\Psi_2\nonumber\\
& &+\frac{exp{(-iE_5t/\hbar)}}{\sqrt{2}}\Psi_5.\label{9}
\end{eqnarray}
Thus the ungerade singlet $\Psi_3\equiv\Phi_3$ never enters into play. It can be proved that the highest as well as the lowest
singlets originates in the $2 \times 2$, while the energy of the ungerade singlet is always between them.  With the help of
\begin{eqnarray}
\frac{1}{1+\nu_1^2}+\frac{1}{1+\nu_2^2}&=&1\nonumber\\
\frac{\nu_1}{1+\nu_1^2}+\frac{\nu_2}{1+\nu_2^2}&=&0,\label{10}
\end{eqnarray}
and of Eq.~(\ref{6}), one has an easy check that the state in Eq.~(\ref{9})
reduces to the correct $\Psi(0)$, namely
$(\Phi_1+\Phi_5)/\sqrt{2}$.



In terms of the basis set in Eq. \ref{5} the initial state can be written as
\begin{equation}
|\Psi(-\infty)\rangle=.
\end{equation}

Here, $E_n, \Psi_n$ are the eigenstates of $H$ and $A_n$ are the coefficients of the $\Psi_n$ in the expansion of $\Psi(0)$, as evident from~(\ref{4}).
We assume that the quantum control is realized using square pulses and the $H$ is constant during the operation.



We will prove that a perfect $\sqrt{SWAP}$ can be realized when the splitting $J$ of two lowest singlet and triplet states, and the separation $U$ between the lower triplet and the upper gerade singlet, satisfy the condition
\begin{equation}
J/k=U/l~,
\end{equation}
where $k$ and $l$ are odd integers whose sum is a multiple of 4.


giving a half swap time $t_0=J \pi/ 2 k$.







We assume that at time $t=0$ the wave function is
\begin{equation}
\Psi(0)=c^\dagger_{a\uparrow}c^\dagger_{b\downarrow}|0>,\label{3}
\end{equation}
and we calculate the wave function at time $t$ by the (time-dependent)
Schr\"{o}dinger equation
\begin{eqnarray}
\Psi(t)&=&e^{-iHt/\hbar}\Psi(0)\nonumber\\
&=&\sum A_n e^{-iE_nt/\hbar}\Psi_n\label{4}.
\end{eqnarray}
Here, $E_n, \Psi_n$ are the eigenstates of $H$ and $A_n$ are the coefficients of the $\Psi_n$ in the expansion of $\Psi(0)$, as evident from~(\ref{4}).
We assume that the quantum control is realized using square pulses and the $H$ is constant during the operation. The eigenvectors are determined from diagonalization of a $6\times6$ matrix, but which can be reduced by symmetry to four $1\times1$'s (including the triplet and the ungerade singlet), plus a $2\times2$ which yields the remaining two singlets.






It is interesting to note that for arbitrary (incommensurate) $J$ and
$E_S$, $\Psi(t)$ never repeats. This is reflected in the behavior of
$\sigma(t)$; the fact that it does not repeat, and in particular does not achieve the value -1, represents a degradation of the spin swap process. This is because in general the oscillation of the double occupancy and the dynamics of the spins are not synchronized.






%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
I would like to add some remarks concerning the question of \textbf{fermion
entanglement}. The historical definition of an entangled state is, for two particles,
one that is a product $\phi(\mathbf{r}_1,s_1)\psi(\mathbf{r}_2,s_2), \mathbf{r}_i,s_i$
being the position and spin coordinates of particle $i$. Clearly there exist no
functions $\phi,\psi$ such that the product is antisymmetric on permutation of the
particles. Thus, by this definition, fermions are always entangled. More recently, a
different definition has been used, e.g. ~\cite{schliemann, shi}, namely a single
Slater determinant is non-entangled. I find this definition unacceptable because it
states that a typical EPR entangled state~\cite{einstein, bohm} is non-entangled. For
example, the molecular orbital ground state for H$_2$ is a single determinant:
\begin{equation}
C^\dagger_\uparrow
C^\dagger_\downarrow|0>=u(\mathbf{r}_1)u(\mathbf{r}_2\chi_0,\label{12}
\end{equation}
where $u=(w_a+w_b)/\sqrt{2}$ (the bonding molecular orbital), and $C^\dagger_\sigma$
creates an electron in $u$ with spin $\sigma$. As I said, the presence of the singlet
makes this a typical EPR entangled state (it could be the initial wave function, which
if the nuclei were removed, would then spread, the two electrons flying off in
opposite directions having the important famously mysterious quantum correlations). So
this definition of entanglement is orthogonal to the historic idea of entanglement,
and therefore unacceptable. But the original definition, which says fermions are
always entangled presents difficulty to attempts to define \emph{degree} of
entanglement, since there is no state with zero entanglement. Nevertheless, there are
special circumstances where that idea can be rescued. In particular, for the hydrogen
molecule ar large $R$, we saw that because the only states that enter into the
low-energy properties are the ground singlet and triplet, all the relevant physics can
be accounted for in terms of the spins $\mathbf{S}_i$ acting in the 4-dimensional spin
space. There is a $1\leftrightarrow 1$ mapping between the full fermion wave functions
and the spin functions. The latter can be considered as localized (even though there
is important non-zero interaction between them), so that non-entangled states in this
language, e.g. $\alpha(1)\beta{2}$, are not only permissible, but must be available.
This concept is well-known to generalize to N sites and N electrons. But, I emphasize,
this mapping can not always be made, e.g. when $R$ in H$_2$ is not large.

In the calculations above concerning spin swaps, it wasn't necessary to think about
entanglement, but it was necessary to represent the wave functions as proper fermion
states, at least for general $R$. So I conclude that in general fermions are
entangled, and can only be considered as possibly non-entangled under very special
conditions, such as where the mapping just discussed is possible.



When the overlap is small it is known that there is only a small admixture of doubly-occupied states, e.g. $c_{a_\uparrow}^\dagger c_{a\downarrow}^\dagger|0>$, in the ground state singlet. ($c^\dagger_{a\sigma}$ is the creation operator for an electron in $w_a$ with spin $\sigma$, etc., and $|0>$ is the electron vacuum.) But
this admixture can become large at smaller $R$.





The calculation is straightforward. The 2 dots are modelled by the hydrogen molecule
with the electron mass replaced by the effective mass $m^*$, the Coulomb interactions
scaled by the dielectric constant $\epsilon$, and the hydrogen 1s functions scaled to
the size of the dots; we also omit the (c-number) proton-proton interaction. Thus the
Schr\"{o}dinger equation is
\begin{equation}
[-\sum_{i=1}^2\nabla_i^2+2/r_{12}-\sum_{i,j=1}^2\label{1}
2/(\mathbf{r}_i-\mathbf{R}_j)]\Psi=E\Psi,
\end{equation}
where the energies $E$ and lengths are respectively in units of effective Rydbergs
$(Ry^*)$ and effective Bohr $(au^*)$: $Ry^*=\frac{m^*}{m\epsilon^2}Ry;

au^*=\frac{\epsilon m}{m^*}au.$
For numbers we will use the parameters for GsAs: $m/m^*=0.067$ and $\epsilon=13.1$, so
that $Ry^*=5.31 meV$ and $au^*=97\AA$. The localized states in the dots are of the form
\begin{equation}
\sqrt{\alpha^3/\pi}exp{(-\alpha r)}\label{2}
\end{equation}
 where $1/\alpha$ is the characteristic
length scale of the wave function in the dot measured in $au*$. Thus if we take the
dot diameter as 100$\AA$, this translates to about $\alpha=1$ (defining the edge of
the dot as the point where the charge density is $1/e$ of its value at the center). We
will use this in the numerical results to be presented.



This was carried out by Slater
~\cite{slater}, (although he worked in the (6-dimensional) molecular orbital basis,
which of course gives the same results as the calculation working in the Wannier
function basis). The Wannier functions were taken as linear combinations of the scaled hydrogen 1s functions~(\ref{2}), centered on the two sites.



The eigenvalues are shown in Figure 1 (in scaled atomic units: $Ry*$ and $au*$); these agree with
Slater's~\cite{slater}, but are carried to a bit larger $R$'s.
\begin{figure}[h]
\centering\includegraphics[height=3.0in]{figDotEnergies.eps}
 \caption{Energy eigenvalues for the 1-band model of H$_2$.}\label{fig:energies}
\end{figure}
They are labelled as the triplet $^3E$ and the three singlets $^1E_i$.

To understand the result to be obtained, it is worthwhile to explain some details of
the wave function determination. %[\hsize\textwidth\columnwidth\hsize\csname@twocolumnfalse\endcsname
In the last column we provide the case $\Gamma=3$, which occurs at $R=$ 2.38au$^*$, which gives $P_d=0$ identically for the half swap. We call the process a \emph{perfect half-swap} when $P_d=0$ for the half-swap. The figure also shows the marked increase in swap speed for this case as compared to the other one with smaller double occupancy during the gate.


 In this case there is a small, rapidly fluctuating double-occupancy probability  and we notice that at the . The rapid fluctuations can be removed by adiabatically reducing the tunnel barrier~\cite{schliemann}. Where the maximum or average double-occupancy is small, and there is very small degradation, $\sigma(t)$ reaching zero with a small probability double occupancy.


We will also generalize the idea to a larger class of gates able to generate unitary transformations in the space $\mathcal{S}$, even if during the operation the system has a large probability of being out of it. These gates represent additional tools in the control of the two-qubit space.

The second column in Fig.~\ref{fig:sigma} corresponds to $\Gamma=5$, which occurs for a separation of 2.93 au*. There is a very large $P_d$ at the half-swap point, but it does give states with $P_d=0$ at 1/3 and 2/3 of the complete swap time. These states do map to states in $\mathcal{S}$, and the corresponding quantum gates might play a convenient role in the quantum control of the two qubit space. We notice that in such a case there would be a complete swap at time $\pi/J$, despite an appreciable double occupancy during the evolution. Furthermore, the frequency governing the swap, $J$, is much greater than in the case analyzed in the first column, so the swap would be much faster.

The form of the wave function in Eq.~(\ref{15}) tempts one to say that $\Psi(t_0/2)$ is maximally \emph{entangled}. And this would follow if we were to use a definition of an entangled state for fermions used by some~\cite{schliemann,shi}, namely a quantum state is entangled if can not be written as a single Slater determinant $D$. Unfortunately this definition is unacceptable because it would label as non-entangled some states agreed to be entangled in terms of the historical definition of entanglement~\cite{werner89},  i.e. an entangled state is a state that cannot be rewritten as a product of single-particle states. An example is provided by the molecular orbital singlet approximate ground state for H$_2$, which is a single $D$ with doubly occupied bonding orbital and clearly entangled in terms of this historical definition~\cite{kaplan}.  However, the concept of entanglement is currently under revision~\cite{zanardipc,zanardi01,barnum03}, and it is clear that in order to be meaningful it has always to be related to a set of available measurements and it depends on the physical regime, i.e. the energy scale at which these measurements are performed. In our system,  if the physical regime allows only  the 4 states (each $D$'s) $c^\dagger_{a\sigma}c^\dagger_{b\sigma'}|0>, \sigma,\sigma'=\uparrow,\downarrow$, as in~(\ref{15}), these clearly \emph{map to} the 4-state spin space $\mathcal{S}$, which states do not require any notion of antisymmetrization. Working within $\mathcal{S}$,  and assuming that the available measurements are described by single spin operators, we can have non-entangled states, and we can talk of the mapped state in Eq.~(\ref{15}) as being entangled. In these terms, the states in Eq.~(\ref{15}) are recognized as two of the four Bell states and they span the space $(S_z=0)$.

\begin{eqnarray}
\Psi(t =+\infty)&=&2^{-1/2}(i^k\Phi_1+\Phi_2)\label{14}\\&=&2^{-1/2}e^{\pm i\pi/4}
(c^\dagger_{a\uparrow}c^\dagger_{b\downarrow}
\mp i c^\dagger_{a\downarrow}c^\dagger_{b\uparrow})|0>,
\label{15}
\end{eqnarray}
and is of the Bell form.

This results gives directly the phase at the end of the pulse ~\cite{hypern}. In the $n=1$ case which corresponds to the shorter $T$, we find explicitly
\begin{equation}
a_1(z)= \frac{-i + 2 i z + \Delta T / \pi}{ - i + \Delta T / \pi}
\end{equation}
And the condition for the perfect $\sqrt{SWAP}$ can be rewritten in the form
\begin{equation}
\frac{\Delta}{\Omega_0}=\frac{1-\sin\Lambda}{2 \cos \Lambda}~,
\end{equation} where $\Lambda$ is the contribution due to the Dirac exchange. We notice that in the Hubbard case $\Lambda=0$, and this condition simplify to $\Delta=\Omega_0/2$. All the other solutions with $n>1$ corresponds in general to a higher $\Delta/\Omega_0$ and are in principle more accessible form an experimental standpoint. We show in Fig.~\ref{zener} the realization of perfect $\sqrt{SWAP}$ using the smoother pulse shape. The cases $n=1$  and $n=3$ are shown. The case $n=3$ has more than one solution.

cond-mat0402674.extracted_bib
\begin{thebibliography}{99}
\bibitem{friesen03} M. Friesen, P. Rugheimer, D. E. Savage,
M. G. Lagally, D. W. van der Weide, R. Joint, and M. A. Eriksson,
Phys. Rev. B {76}, 121301 (2003).
 \bibitem{loss98} D. Loss and D. P. DiVincenzo, Phys. Rev. A {\bf 57},
 120 (1998).
\bibitem{vrijen00} R. Vrijen, E. Yablonovitch, K. Wang, H. W. Jiang,
A. Balandin, V. Roychowdhury, T. Mor, and D. DiVincenzo, Phys. Rev. A
{\bf 62}, 122306 (2000).
 \bibitem{hu} X. Hu and S. Das Sarma, Phys. Rev. A \textbf{61}, 062301
 (2000).
\bibitem{schliemann} J. Schliemann, D. Loss and A. H. MacDonald,
Phys. Rev. B \textbf{63}, 085311 (2001).
\bibitem{burkard} G. Burkard, D. Loss and D. P. DiVincenzo,
Phys. Rev. B\textbf{59}, 2070 (1999).
\bibitem{slater} J. C. Slater, \emph{Quantum Theory of Molecules and
Solids} Vol.1, Chap.4 McGraw-Hill, New York (1963).
\bibitem{anderson} P. W. Anderson, Solid State Physics \textbf{14},99
(1963).
\bibitem{ishk00} A. M. Ishkhanyan, J. Phys. A {\bf 33}, 5539 (2000).
\bibitem{allen} L. Allen and J. H. Eberli, \emph{Optical Resonance and
Two-Level Atoms}, Wiley, New York 1975.
\bibitem{rosen32} N. Rosen and C. Zener, Phys. Rev. {\bf 40} 502
(1932).
\bibitem{erdelyi53} {\it Higher Trascendental Functions}, A. Erd\'elyi
Editor, McGraw-Hill, New York (1953).
\bibitem{tessmer} S. Tessmer, private communication.
\bibitem{lidar04} D. A. Lidar and A. Mizel, cond-mat/0401081 (2004).

%\bibitem{shi} Yu Shi, quant-ph/0205069(2002); A. Mizel, private comm.
%\bibitem{werner89} R. F. Werner, Phys. Rev. A {\bf 40} 4277 (1989).
%\bibitem{kaplan} T. A. Kaplan, Bull. Am. Phys. Soc., March APS meeting (2004).
%\bibitem{zanardipc} P. Zanardi, private comm.
%\bibitem{zanardi01} P.Zanardi, Phys. Rev. Lett. {\bf 87} 077901 (2001).
%\bibitem{barnum03} H. Burnam et al., quant-ph/0305023 (2003).
\end{thebibliography}