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====Flow Construction-Determinism==== | ====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 | | 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_{2x1}</math>.<br/> | ||
<math>C_{2x1} = CZ_{ij} |\psi_{ii} |+_{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 {|+ | If one measures qubit i in <math>\{|+_i\rangle,|−_i\rangle\}</math> basis and gets outcome s, qubit j reduces to,<br/> | ||
= (a + b)| | <math>= (a + b)|0_i\rang;e + (a − b)|1_i\rangle, if s=0</math><br/> | ||
= (a − b)| | <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 | | 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/> | ||
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 |+ | *''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/> | *''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/> | *''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/> | ||