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*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 |+i 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. | *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 |+i 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. | ||
*Controlled-NOT(CX): | *Controlled-NOT(CX): | ||
*Controlled-Phase(CZ): | *Controlled-Phase(CZ): <br/> | ||
The commutation relations for the above gates are as follows: | The commutation relations for the above gates are as follows: | ||
XH = HZ, XZ = −ZX<br/> | XH = HZ, XZ = −ZX<br/> | ||
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MBQC is a formalism used for quantum computation by operating only single qubit measurements on a fixed set of entangled states, also known as graph states. Graph states denote any graph where each node represents a quantum state, and the edges denote entanglement between any two vertices. The measurement on successive layers of qubits is decided by previous measurement outcomes. Outcomes of last qubit layer gives the result of concerned computation. Following, we elucidate in detail certain concepts necessary to understand the working of MBQC. | MBQC is a formalism used for quantum computation by operating only single qubit measurements on a fixed set of entangled states, also known as graph states. Graph states denote any graph where each node represents a quantum state, and the edges denote entanglement between any two vertices. The measurement on successive layers of qubits is decided by previous measurement outcomes. Outcomes of last qubit layer gives the result of concerned computation. Following, we elucidate in detail certain concepts necessary to understand the working of MBQC. | ||
Gate Teleportation The idea comes from one-qubit teleporation. This means that one can transfer an unknown qubit |ψi without actually sending it via a quantum channel. The underlying equations explain the notion. See Figure 1 for circuit.<br/> | Gate Teleportation The idea comes from one-qubit teleporation. This means that one can transfer an unknown qubit |ψi without actually sending it via a quantum channel. The underlying equations explain the notion. See Figure 1 for circuit.<br/> | ||
[[File:One Bit Teleportation.jpg|right|caption=One Bit Teleportation]](H ⊗ I)(CZ12)|ψi1 |+i2 <br/> | [[File:One Bit Teleportation.jpg|right|caption="One Bit Teleportation"]](H ⊗ I)(CZ12)|ψi1 |+i2 <br/> | ||
= (H ⊗ I)(CZ12)(a|0i1 + b|1i1)|+i2<br/> | = (H ⊗ I)(CZ12)(a|0i1 + b|1i1)|+i2<br/> | ||
= (H ⊗ I)(a|0i1 |+i2 + b|1i1 |−i2)<br/> | = (H ⊗ I)(a|0i1 |+i2 + b|1i1 |−i2)<br/> | ||
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Cluster States In case of multi-qubit quatum circuits, one needs a 2-dimensional graph state. Cluster State is a square lattice used as substrate for such computation. All the nodes are in |+i entangled by C-Z indicated by the edges. It is known to be universal i.e. it can simulate any quatum gate.<br/> | Cluster States In case of multi-qubit quatum circuits, one needs a 2-dimensional graph state. Cluster State is a square lattice used as substrate for such computation. All the nodes are in |+i entangled by C-Z indicated by the edges. It is known to be universal i.e. it can simulate any quatum gate.<br/> | ||
[[File:Cluster State.jpg|caption=Cluster State]] | [[File:Cluster State.jpg|right|caption=Cluster State]]Each row would thus represent the teleporation of starting qubit in that row horizontally. On the other hand, vertical edges indicate different input qubits linked with multi-qubit gates (same as circuit model). For example, see Figure 6 to understand the conversion from circuit model to graph state model. As the computation relation follows X = HZH, thus, Figure 6a represents Circuit diagram for C-NOT gate in terms of C-Z gate and Single Qubit Gate H.<br/> | ||
Each row would thus represent the teleporation of starting qubit in that row horizontally. On the other hand, vertical edges indicate different input qubits linked with multi-qubit gates (same as circuit model). For example, see Figure 6 to understand the conversion from circuit model to graph state model. As the computation relation follows X = HZH, thus, Figure 6a represents Circuit diagram for C-NOT gate in terms of C-Z gate and Single Qubit Gate H.<br/> | |||
[[File: Circuit Diagram to implement C-NOT|left|caption=Circuit Diagram to implement C-NOT]][[File:Graph State Pattern for C-NOT.jpg|right|caption=Graph State Pattern for C-NOT]] | [[File: Circuit Diagram to implement C-NOT|left|caption=Circuit Diagram to implement C-NOT]][[File:Graph State Pattern for C-NOT.jpg|right|caption=Graph State Pattern for C-NOT]] | ||
'''Figure 6: Measurement Pattern from Circuit Model'''''' | '''Figure 6: Measurement Pattern from Circuit Model'''''' | ||