Editing Glossary
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===Unitary Operations=== | ===Unitary Operations=== | ||
*<math>\text{X (NOT gate)}</math>: <math>X|0\rangle\,\to\,\ |1\rangle,\quad X|1\rangle\,\to\,\ |0\rangle,\quad X|+\rangle\,\to\,\ |+\rangle,\quad *X|-\rangle\,\to\,\ -|-\rangle</math> | *<math>\text{X (NOT gate)}</math>: <math>X|0\rangle\,\to\,\ |1\rangle,\quad X|1\rangle\,\to\,\ |0\rangle,\quad X|+\rangle\,\to\,\ |+\rangle,\quad *X|-\rangle\,\to\,\ -|-\rangle</math> | ||
*<math>\text{Z}</math>: <math>Z|+\rangle \,\to\,\ |-\rangle,\quad Z|-\rangle \,\to\,\ |+\rangle,\quad Z|0\rangle \,\to\,\ |0\rangle,\quad Z|1\rangle \,\to\,\ -|1\rangle </math></br> | *<math>\text{Z (Phase gate)}</math>: <math>Z|+\rangle \,\to\,\ |-\rangle,\quad Z|-\rangle \,\to\,\ |+\rangle,\quad Z|0\rangle \,\to\,\ |0\rangle,\quad Z|1\rangle \,\to\,\ -|1\rangle </math></br> | ||
Thus, <math>|0\rangle </math>, <math>|1\rangle </math> are eigenstates of Z gate and <math>|+\rangle</math>, <math>|-\rangle </math> are eigenstates of X gate. | Thus, <math>|0\rangle </math>, <math>|1\rangle </math> are eigenstates of Z gate and <math>|+\rangle</math>, <math>|-\rangle </math> are eigenstates of X gate. | ||
*<math>\text{H (Hadamard gate)}</math>: <math>H|0\rangle \,\to\,\ |+\rangle </math> or <math>H|1\rangle \,\to\,\ |-\rangle </math> | *<math>\text{H (Hadamard gate)}</math>: <math>H|0\rangle \,\to\,\ |+\rangle </math> or <math>H|1\rangle \,\to\,\ |-\rangle </math> | ||
<math> | <math>X= | ||
X= | |||
\left[ {\begin{array}{cc} | \left[ {\begin{array}{cc} | ||
0 & 1 \\ | 0 & 1 \\ | ||
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*<math>\text{Controlled-U(CU)}</math>: uses two inputs, control qubit and target qubit. It operates U on the second(target) qubit only when the first (source) qubit is 1. C-U gates are used to produce entangled states, when the target qubit is <math>|+\rangle</math> and control qubit is not an eigenstate of U. In the given equation 'i' denotes the source qubit and 'j', the target qubit. Following are two important C-U gates. | *<math>\text{Controlled-U(CU)}</math>: uses two inputs, control qubit and target qubit. It operates U on the second(target) qubit only when the first (source) qubit is 1. C-U gates are used to produce entangled states, when the target qubit is <math>|+\rangle</math> and control qubit is not an eigenstate of U. In the given equation 'i' denotes the source qubit and 'j', the target qubit. Following are two important C-U gates. | ||
<math> | <math> | ||
\text{Controlled-NOT( | \text{Controlled-NOT(CX or CNOT): }CX_{ij}|+\rangle_i|0\rangle_j\,\to\,\ \frac{1}{\sqrt{2}} (|0_i0_j\rangle+|1_i1_j\rangle)</math></br> | ||
<math>\text{Controlled- | <math>\text{Controlled-Phase(CZ): }CZ_{ij}|+\rangle_i|+\rangle_j\,\to\,\ \frac{1}{\sqrt{2}} (|0_i+_j\rangle+|1_i-_j\rangle)</math> | ||
The commutation relations for the above gates are as follows:</br> | The commutation relations for the above gates are as follows:</br> | ||
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*'''Pauli Gates(U):''' Single Qubit Gates I (Identity), X, Y, Z. All the gates in this set follow <math>U^2=I</math> | *'''Pauli Gates(U):''' Single Qubit Gates I (Identity), X, Y, Z. All the gates in this set follow <math>U^2=I</math> | ||
*'''Clifford Gates(C):''' Pauli Gates, Phase Gate, C-NOT. This set of gates can be simulated on classical computer. All the gates in this set follow CU=U'C, where U and U' are two different Pauli gates depending on C | *'''Clifford Gates(C):''' Pauli Gates, Phase Gate, C-NOT. This set of gates can be simulated on classical computer. All the gates in this set follow CU=U'C, where U and U' are two different Pauli gates depending on C | ||
*'''Universal Set of gates:''' This set consists of all Clifford gates and one Non-Clifford gate (T gate). If a model can realise Universal Set of gates, it can imlement any quantum computation efficiently. T gates follow <math>UT=P^aU'T</math>, where P is the phase gate and U, U' are any two Pauli gates depending on | *'''Universal Set of gates:''' This set consists of all Clifford gates and one Non-Clifford gate (T gate). If a model can realise Universal Set of gates, it can imlement any quantum computation efficiently. T gates follow <math>UT=P^aU'T</math>, where P is the phase gate and U, U' are any two Pauli gates depending on C. Parameter <math>a\epsilon{0,1}</math> is obtained from U, such that <math>P^0=I</math>, <math>P^1=P</math>.</br> | ||
To summarize, if <math>C^1=</math>P, <math>C^2=</math>C, <math>C^3=</math>T, then <math>C^{k}=\{U:UQU=C^{k-1}|Q\ | To summarize, if <math>C^1=</math>P, <math>C^2=</math>C, <math>C^3=</math>T, then <math>C^{k}=\{U:UQU=C^{k-1}|Q\epsilon C^1\}</math> | ||
===Magic States=== | ===Magic States=== | ||
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# ''For each column j ≡ 3 (mod 8) and each odd row i, apply the operator c-Z on qubits (i,j) and (i + 1,j) and also on qubits (i,j + 2) and (i + 1,j + 2).'' | # ''For each column j ≡ 3 (mod 8) and each odd row i, apply the operator c-Z on qubits (i,j) and (i + 1,j) and also on qubits (i,j + 2) and (i + 1,j + 2).'' | ||
# ''For each column j ≡ 7 (mod 8) and each even row i, apply the operator c-Z on qubits (i,j) and (i + 1,j) and also on qubits (i,j + 2) and (i + 1,j + 2).'' | # ''For each column j ≡ 7 (mod 8) and each even row i, apply the operator c-Z on qubits (i,j) and (i + 1,j) and also on qubits (i,j + 2) and (i + 1,j + 2).'' | ||
====Flow Construction-Determinism==== | ====Flow Construction-Determinism==== | ||
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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"> | <div id="swap"> | ||
[[File:SWAP_test_figure.png |center|thumb|500px|Figure | [[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. | 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> | 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. | 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> |