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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. |
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| ===Density Matrices===
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| ===Fidelity===
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| ===Superposition===
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| ===Entanglement===
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| ===Measurement===
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| ===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). |
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| ====Flow Construction-Determinism====
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| 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/>
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| <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/>
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| If one measures qubit i in <math>\{|+_i\rangle,|−_i\rangle\}</math> basis and gets outcome s, qubit j reduces to,<br/>
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| <math>= (a + b)|0_i\rang;e + (a − b)|1_i\rangle, if s=0</math><br/>
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| <math>= (a − b)|0_i\rangle + (a + b)|1_i\rangle, if s=1</math><br/>
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| 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/>
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| 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/>
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| 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/>
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| *''Preparation'' prepares all input qubits in the required state, generally represented as <math>|+\theta_i\rangle</math> = where
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| *''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/>
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| *''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/>
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| *''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/>
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| 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/>
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| t α s α t s (−1)sα+tπ [Mi ] = Mi Z X = Mi{equation missing}π<br/>
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| We shall denote measurement in X-basis ({equation missing} and Y-basis ({equation missing}<br/>
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| Commutation relations:<br/>
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| EijXis = XisZisEij (EX)<br/>
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| EijXjs = XjsZjsEij (EX)<br/>
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| EijZit = ZitEij (EZ){equation missing} <br/>
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| = (EZ){equation missing} <br/>
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| t α s r<br/>
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| [Mi ] Xi = t α s+r[Mi ]{equation missing}(MX)<br/>
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| MixXis = Mix (MX){equation missing} <br/>
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| = (MZ){equation missing} <br/>
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| 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/>
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| Outcomes: s1,s2,s3,s4<br/>
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| Circuit Operation: .<br/>
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| EX =⇒{equation missing} <br/>
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| EX =⇒{equation missing} <br/>
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| MX =⇒{equation missing} <br/>
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| X4s3Z4s2Z1s2M3xM2xE13E234{equation missing} <br/>
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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 {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/>
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