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#Easy to compute: The mapping <math>f : k\rightarrow |f_k\rangle</math> is easy to compute by a quantum polynomial-time algorithm. | #Easy to compute: The mapping <math>f : k\rightarrow |f_k\rangle</math> is easy to compute by a quantum polynomial-time algorithm. | ||
#Hard to invert: Given <math>|f_k\rangle</math>, it is impossible to invert k by virtue of fundamental quantum information theory. | #Hard to invert: Given <math>|f_k\rangle</math>, it is impossible to invert k by virtue of fundamental quantum information theory. | ||
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<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 <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/> | 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/> | ||
===SWAP test=== | |||
<div id="swap"> | |||
[[File:SWAP_test_figure.png |center|thumb|500px|Figure 8: Gate Teleporation for Multiple Single Qubit Gates]]</div> | |||
SWAP helps to compare two states <math>|\psi\rangle</math> and <math>|\psi'\rangle</math>. An ancilla qubit is prepared here in the state <math>\frac{|0\rangle + |1\rangle}{2}</math> and a controlled swap test is performed on two states <math>|\psi\rangle</math> and <math>|\psi'\rangle</math>. | |||
If <math>|\psi\rangle</math> = <math>|\psi'\rangle</math>, then the ancilla qubit, after performing a Hadamard operation, yields <math>|0\rangle</math> when measurement is applied in computational basis. SWAP test is passed here. | |||
If <math>|\psi\langle|\psi'\rangle \leq \delta</math> the ancilla qubit, after performing the necessary Hadamard Gates, upon measurement passes the test with probability <math>\frac{1+\delta^2}{2}</math> | |||
and fails the test with probability <math>\frac{1-\delta^2}{2}</math>. Hence, the SWAP test always passes for the same inputs and sometimes fails if they are different. By repeating the SWAP test, its efficiency can be amplified. | |||
==References== | ==References== | ||
<div style='text-align: right;'>''*contributed by Shraddha Singh''</div> | <div style='text-align: right;'>''*contributed by Shraddha Singh''</div> | ||