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===Quantum States=== | ===Quantum States=== | ||
*<math>|+\rangle=\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle),\quad |-\rangle=\frac{1}{\sqrt{2}}(|0\rangle-|1\rangle)</math></br> | *<math>|+\rangle=\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle),\quad |-\rangle=\frac{1}{\sqrt{2}}(|0\rangle-|1\rangle)</math></br> | ||
* | *Bell/ EPR pairs: | ||
* | *GHZ States: | ||
*W States: | |||
===Unitary Operations=== | ===Unitary Operations=== | ||
* | *X (NOT gate): <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> | ||
* | *Z (Phase gate): <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. | ||
* | *H (Hadamard gate): <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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\end{array} }\right]\quad | \end{array} }\right]\quad | ||
</math> | </math> | ||
* | *Controlled-U(CU): 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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===Universal Resource=== | ===Universal Resource=== | ||
A set of <math>|+_\theta\rangle </math>states on which applying Clifford operations is enough for universal quantum computation. | A set of <math>|+_\theta\rangle </math>states on which applying Clifford operations is enough for universal quantum computation. | ||
===Density Matrices=== | |||
===Fidelity=== | ===Fidelity=== | ||
===Superposition=== | |||
=== | |||
===Entanglement=== | |||
===Measurement=== | |||
===Gate Teleportation=== | ===Gate Teleportation=== | ||
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<div id="7"> | <div id="7"> | ||
[[File:Brickwork State.jpg|center|thumb|500px|Figure 7: Brickwork State]]</div> | [[File:Brickwork State.jpg|center|thumb|500px|Figure 7: Brickwork State]]</div> | ||
'''Definition 1 | '''Definition 1''' A brickwork state Gn×m, where m ≡ 5 (mod 8), is an entangled state of n × m qubits constructed as follows (see also Figure 7): | ||
# | # Prepare all qubits in state |+i and assign to each qubit an index (i,j), i being a column (i ∈ [n]) and j being a row (j ∈ [m]). | ||
# | # For each row, apply the operator c-Z on qubits (i,j) and (i,j + 1) where 1 ≤ j ≤ m − 1. | ||
# | # 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). | ||
==== | ====Flow Construction-Determinism==== | ||
Measurement outcomes of qubits is not certain, hence it renders MBQC a non-deterministic model. This can still be rectified by invoking Pauli corrections based on the previous outcomes, as evident from above. For example, to implement Hadamard gate on input state <math>|\psi_i\rangle = a|0_i\rangle + b|1_i\rangle</math>, we consider the case of a two qubit graph state <math>C_{2*1}</math>.<br/> | |||
<math>C_{2x1} = CZ_{ij} |\psi_{ii}\rangle |+_{ij}\rangle = a|00_i\rangle + a|01_i\rangle + b|10_i\rangle − b|11_i\rangle</math><br/> | |||
If one measures qubit i in <math>\{|+_i\rangle,|−_i\rangle\}</math> basis and gets outcome s, qubit j reduces to,<br/> | |||
<math>= (a + b)|0_i\rang;e + (a − b)|1_i\rangle, if s=0</math><br/> | |||
<math>= (a − b)|0_i\rangle + (a + b)|1_i\rangle, if s=1</math><br/> | |||
As, the two possible output states are different, it shows this method is non-deterministic. One could end up with any of the two states and there is no certainty. Nevertheless, we observe that if X gate is operated on the second qubit after measurement if outcome is 1, both the equations would be same and hence, obtained state is H |\psi_i\rangle.<br/> | |||
Measurement outcomes of qubits is not certain, hence it renders MBQC a non-deterministic model. This can still be rectified by invoking Pauli corrections based on the previous outcomes, as evident from above. For example, to implement Hadamard gate on input state <math>|\ | |||
If one measures qubit i in <math>\{|+\rangle,| | |||
<math>=(a | |||
As, the two possible output states are different, it shows this method is non-deterministic. One could end up with any of the two states and there is no certainty. Nevertheless, we observe that if X gate is operated on the second qubit after measurement if outcome is 1, both the equations would be same and hence, obtained state is | |||
Flow construction is the semanticism to construct the measurement pattern for a given graph state. It gives the Pauli corrections for a particular qubit in the graph state, called X and Z dependencies. In simpler words, it records the effect of shifting all Pauli X and Z corrections to the left of (after performing) Entanglement and Entanglement operations, respectively, for each qubit. The result is recorded as sets of X, Z corrections required for each site. These sets depend on the graph state and not the computation or measurement results. Hence, it can be computed while choosing the graph state for a required computation. Following we illustrate how to construct such models with ease using Measurement Calculus to invoke determinism in One-Way Quantum Computation(MBQC).<br/> | Flow construction is the semanticism to construct the measurement pattern for a given graph state. It gives the Pauli corrections for a particular qubit in the graph state, called X and Z dependencies. In simpler words, it records the effect of shifting all Pauli X and Z corrections to the left of (after performing) Entanglement and Entanglement operations, respectively, for each qubit. The result is recorded as sets of X, Z corrections required for each site. These sets depend on the graph state and not the computation or measurement results. Hence, it can be computed while choosing the graph state for a required computation. Following we illustrate how to construct such models with ease using Measurement Calculus to invoke determinism in One-Way Quantum Computation(MBQC).<br/> | ||
Choose a unitary gate for the circuit and hence, its measurement pattern. In order to implement this pattern on a graph state, there are four basic steps: Preparation, Entanglement, Measurement, Corrections.<br/> | Choose a unitary gate for the circuit and hence, its measurement pattern. In order to implement this pattern on a graph state, there are four basic steps: Preparation, Entanglement, Measurement, Corrections.<br/> | ||
* | *''Preparation'' prepares all input qubits in the required state, generally represented as <math>|+\theta_i\rangle</math> = where | ||
* | *''Entanglement'' entangles all the qubits according to the required graph state. This operation is denoted by Eij, where C-Z is operated with i as control qubit and j as target qubit.<br/> | ||
* | *''Measurement'' assigns measurement angle in X-Y plane for each qubit. Measurement operator is notated as Miα: the qubit ’i’ would be measured in {|+αi,|−αi} basis i.e. if the state is ) one gets outcome 0 and if the state is ), the outcome is 1.<br/> | ||
* | *''Correction'' calculates all Pauli corrections to be applied on a given qubit in the pattern. The set of such parameters are called Dependencies for X and Z operators individually. To calculate all the Pauli Corrections on a given qubit, one needs to take into account the measurement outcomes of previous qubits as well as commutation relations. Both affect the Pauli corrections for a given qubit. Below is a formalism to explain the process with an example.<br/> | ||
The effect of X gate on a measurement angle ( | The effect of X gate on a measurement angle (α) in X-Y plane is to change its sign and Z gate is to add a phase π.<br/> | ||
t α s α t s (−1)sα+tπ [Mi ] = Mi Z X = Mi{equation missing}π<br/> | |||
We shall denote measurement in X-basis ( | We shall denote measurement in X-basis ({equation missing} and Y-basis ({equation missing}<br/> | ||
Commutation relations:<br/ | Commutation relations:<br/> | ||
EijXis = XisZisEij (EX)<br/> | |||
EijXjs = XjsZjsEij (EX)<br/> | |||
EijZit = ZitEij (EZ){equation missing} <br/> | |||
= (EZ){equation missing} <br/> | |||
t α s r<br/> | |||
[Mi ] Xi = t α s+r[Mi ]{equation missing}(MX)<br/> | |||
MixXis = Mix (MX){equation missing} <br/> | |||
The last second equation is implied from the fact that for | = (MZ){equation missing} <br/> | ||
Outcomes: | The last second equation is implied from the fact that for , x=0=-0. 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/> | ||
Circuit Operation: | Outcomes: s1,s2,s3,s4<br/> | ||
Circuit Operation: .<br/> | |||
EX =⇒{equation missing} <br/> | |||
EX =⇒{equation missing} <br/> | |||
MX =⇒{equation missing} <br/> | |||
X4s3Z4s2Z1s2M3xM2xE13E234{equation missing} <br/> | |||
Hence, we obtain a measurement pattern to implement C-NOT gate with a T-shaped graph state with three qubits entangled chain | 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:{s3}, 2:φ, 3:φ, 4:φ. Z dependency sets for qubit 1:{s2}, 2:φ, 3:φ, 4:{s2}. 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{equation missing} (i), for X (sXi = s1 ⊕ s2 ⊕ ...) and Z (sZi = s1 ⊕ s2 ⊕ ...), separately. Thus, is operated on qubit i.{equation missing} <br/> | ||
===Fault Tolerance=== | |||
===Quantum Error Correction=== | |||
*Quantum Error Correcting Codes (QECCs) | |||
*Stabilizer Codes | |||
=== | ===Topological Error Correction=== | ||
== | ===[[Adversarial Definitions]]=== | ||