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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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===Bloch Sphere=== | ===Bloch Sphere=== | ||
In quantum mechanics, the Bloch sphere is a geometrical representation of the pure state space of a two-level quantum mechanical system (qubit). The Bloch sphere is a unit 2-sphere, with antipodal points corresponding to a pair of mutually orthogonal state vectors. The north and south poles of the Bloch sphere are typically chosen to correspond to the standard basis vectors <math>|0\rangle</math>, <math>|1\rangle</math> respectively, which in turn might correspond e.g. to the spin-up and spin-down states of an electron. This choice is arbitrary, however. The points on the surface of the sphere correspond to the pure states of the system, whereas the interior points correspond to the mixed states. | In quantum mechanics, the Bloch sphere is a geometrical representation of the pure state space of a two-level quantum mechanical system (qubit). The Bloch sphere is a unit 2-sphere, with antipodal points corresponding to a pair of mutually orthogonal state vectors. The north and south poles of the Bloch sphere are typically chosen to correspond to the standard basis vectors <math>|0\rangle</math>, <math>|1\rangle</math> respectively, which in turn might correspond e.g. to the spin-up and spin-down states of an electron. This choice is arbitrary, however. The points on the surface of the sphere correspond to the pure states of the system, whereas the interior points correspond to the mixed states. | ||
===Quantum One Way Function=== | ===Quantum One Way Function=== | ||
Based on the fundamental principles of quantum mechanics, QOWF was proposed by Gottesman and Chuang [https://arxiv.org/abs/quant-ph/0105032] and its definition is presented as follows.</br> | Based on the fundamental principles of quantum mechanics, QOWF was proposed by Gottesman and Chuang [https://arxiv.org/abs/quant-ph/0105032] and its definition is presented as follows.</br> | ||
'''Definition 1''' Let k, <math>|f_k\rangle</math> be classical bits string of length <math>L_1</math>, quantum state of <math>L_2</math> qubits, respectively. A function <math>f : k\rightarrow |f_k\rangle</math>, where <math>|f_k\rangle</math> satisfies <math>\langle f_k|f_{k'}\rangle\le\delta < 1</math> for <math>k\ne k'</math>, is called a QOWF under physical mechanics if | '''Definition 1''' Let k, <math>|f_k\rangle</math> be classical bits string of length <math>L_1</math>, quantum state of <math>L_2</math> qubits, respectively. A function <math>f : k\rightarrow |f_k\rangle</math>, where <math>|f_k\rangle</math> satisfies <math>\langle f_k|f_{k'}\rangle\le\delta < 1</math> for <math>k\ne k'</math>, is called a QOWF under physical mechanics if | ||
#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. | ||
===Gate Teleportation=== | ===Gate Teleportation=== | ||
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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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<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/> | ||
==References== | ==References== | ||
<div style='text-align: right;'>''*contributed by Shraddha Singh''</div> | <div style='text-align: right;'>''*contributed by Shraddha Singh''</div> |