Data used in XOR experiment

The input state can be represented by a 6-dimensional complex vector

$$\psi_{in}={\left\langle c_{01}, c_{11},c_{02}, c_{12},c_{0,out}, c_{1,out}\right\rangle}^t.$$

In this case, the unitary operator $U$ can be expressed in a matrix form:

$$U=e^{-iHt\over \hbar}$$

where $H$ is an Hermitian matrix. After the system is let go for free evolution, the actual time point $t$ at which the measurement is performed can be absorbed into $H$. This is also the case for the minus sign and $\hbar$. We then write the exponent of $e$ still as $H$. The reader should bear in mind that $H$ is a short-hand form of ( $-Hamiltonian\cdot t/\hbar$). The end state of the system $\psi_{out}$ can be then expressed as a matrix-vector multiplication:

$${\psi}_{out}=U{\psi}_{in}$$

The simulation data used in Section 6.2 are summarized as follows.

$$Re[H]=\left( \matrix{ -0.1400 & -0.03995 & 0.008367 & -0.2734... ...4 & -0.8315 & -0.8313 & 0.3009 & 0.03478 & 0.06618 \cr }\right)$$

$$Im[H]=\left( \matrix{ 0 & -0.2129 & -0.1333 & -0.1391 & 0.312... ....6552 & -0.4363 & -0.3461 & 0.5109 & -0.01860 & 0 \cr } \right)$$

$$Re[U]=\left( \matrix{ 0.5320 & 0.5283 & 0.06375 & 0.09863 & ... ...321 & 0.3306 & -0.3866 & 0.0002425 & -0.0001584 \cr } \right)$$

$$Im[U]=\left( \matrix{ -0.1011 & -0.1126 & -0.1866 & -0.09938 ... ...747 & -0.3728 & 0.3161 & 0.0002185 & -0.0001794 \cr } \right)$$

The following can be easily checked:

$$U=e^{iH}$$

$$U^\dag ={(e^{iH})}^\dag = {e^{-iH^\dag }}=e^{-iH}$$

since $H$ is Hermitian ($H^\dag =H$). So we have:

$$U^\dag U=e^{-iH}e^{iH}=e^0=I$$

and

$$UU^\dag =e^{iH}e^{-iH}=e^0=I$$

where $I$ is an identity matrix of dimension 6. So $U^\dag =U^{-1}$.

When a phase-mapping function is applied to the AND-training data (or the OR-training data) before submitting it to the unitary transformation (trained for XOR without phase-mapping), the system can perform AND-operation (OR-operation). The phase-mapping function can be written in a matrix form (see Section 6.2) and calculated using standard minimization procedure (we use the random walk method). The phase-mapping matrice used in Section 6.2 for the AND (OR) training data are:

$$\Phi_{\wedge}=\left( {\matrix{ -10.16 & 3.557 & 6.005 & -8.74... ... & 3.088 \cr -2.400 & -7.193 & -3.493 & -3.156 \cr }} \right)$$

$$\Phi_{\vee}=\left( {\matrix{ 1.289 & -7.746 & 8.088 & 1.335 \... ...& 3.588 \cr -8.610 & -0.02628 & -5.716 & 2.173 \cr }} \right)$$