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The last second equation is implied from the fact that for <math>M_i^x</math>, <math>x=0=-0</math>. Thus, X gate has no effect on measurement in X-basis for the given states. Using above notations we express the equation for circuit model of C-NOT from Figure 6 with two inputs and two outputs: Qubits: 1,2,3,4<br/> | The last second equation is implied from the fact that for <math>M_i^x</math>, <math>x=0=-0</math>. Thus, X gate has no effect on measurement in X-basis for the given states. Using above notations we express the equation for circuit model of C-NOT from Figure 6 with two inputs and two outputs: Qubits: 1,2,3,4<br/> | ||
Outcomes: <math>s_1, s_2, s_3, s_4</math><br/> | Outcomes: <math>s_1, s_2, s_3, s_4</math><br/> | ||
Circuit Operation: <math>X^{s_3}_4M_3^xE_{34}E_{13}X_3^{s_2}M_2^xE_{23}</math><br/><div style='text-align: center;'> | Circuit Operation: <math>X^{s_3}_4M_3^xE_{34}E_{13}X_3^{s_2}M_2^xE_{23}</math><br/> | ||
<math>X^{s_3}_4M_3^xE_{34}\mathbf{E_{13}X_3^{s_2}}M_2^xE_{23} | <div style='text-align: center;'> | ||
<math>X^{s_3}_4Z_1^{s_2}M_3^x\mathbf{E_{34}X_3^{s_2}}M_2^xE_{13}E_{23} | <math>X^{s_3}_4M_3^xE_{34}\mathbf{E_{13}X_3^{s_2}}M_2^xE_{23}\overset{\text{EX}}{\implies}</math><br/> | ||
<math>X^{s_3}_4Z_4^{s_2}Z_1^{s_2}\mathbf{M_3^xX_3^{s_2}}M_2^xE_{13}E_{23}E_{34} | <math>X^{s_3}_4Z_1^{s_2}M_3^x\mathbf{E_{34}X_3^{s_2}}M_2^xE_{13}E_{23}\overset{\text{EX}}{\implies}</math><br/> | ||
<math>X^{s_3}_4Z_4^{s_2}Z_1^{s_2}\mathbf{M_3^xX_3^{s_2}}M_2^xE_{13}E_{23}E_{34}\overset{\text{MX}}{\implies}</math><br/> | |||
<math>X^{s_3}_4Z_4^{s_2}Z_1^{s_2}M_3^xM_2^xE_{13}E_{234}</math><br/></div> | <math>X^{s_3}_4Z_4^{s_2}Z_1^{s_2}M_3^xM_2^xE_{13}E_{234}</math><br/></div> | ||
Hence, we obtain a measurement pattern to implement C-NOT gate with a T-shaped graph state with three qubits entangled chain {2,3,4} and 1 entangled to 3. X dependency sets for qubit 1:{ | Hence, we obtain a measurement pattern to implement C-NOT gate with a T-shaped graph state with three qubits entangled chain <math>\{2,3,4\}</math> and 1 entangled to 3. X dependency sets for qubit <math>1:\{s_3\}</math>, <math>2:\phi</math>, <math>3:\phi</math>, <math>4:\phi</math>. Z dependency sets for qubit <math>1:\{s_2\}</math>, <math>2:\phi</math>, <math>3:\phi</math>, <math>4:\{s_2\}</math>. The measurements are independent of any outcome so they can all be performed in parallel. In the end, Pauli corrections are performed as such. Parity (modulo 2 sum) of all the previous outcomes in the dependency set is calculated for each qubit (i), for X (<math>s^X_i=s_1\oplus s_2\oplus...</math>) and Z (<math>s^Z_i=s_1\oplus s_2\oplus...</math>), separately. Thus, <math>X^{s^X_i}Z^{s^Z_i}</math> is operated on qubit i. <br/> | ||
==References== | ==References== | ||
<div style='text-align: right;'>''*contributed by Shraddha Singh''</div> | <div style='text-align: right;'>''*contributed by Shraddha Singh''</div> | ||