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== Measurement Based Quantum Computation (MBQC)== | == Measurement Based Quantum Computation (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. | 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 [[#xref:Figure 1|Supplementary Information|label=Figure 1] for circuit.<br/> | ||
[[File:One Bit Teleportation.jpg|right|thumb|1000px|One Bit Teleportation]](H ⊗ I)(CZ12)|ψi1 |+i2 <br/> | [[File:One Bit Teleportation.jpg|right|label=Figure 1|thumb|1000px|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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= |mi ⊗ XmH |ψi<br/> | = |mi ⊗ XmH |ψi<br/> | ||
Similarly if we have the input state rotated by a Z(θ) gate the circuit would look like Figure 2. As the rotation gate Z(θ) commutes with Controlled-Phase gate. Hence, Figure 3 is justified.<br/> | Similarly if we have the input state rotated by a Z(θ) gate the circuit would look like Figure 2. As the rotation gate Z(θ) commutes with Controlled-Phase gate. Hence, Figure 3 is justified.<br/> | ||
<div><ul> | <div><ul> | ||
<li style="display: inline-block;"> [[File:Modified Input.jpg|thumb|widths=500px|Modified Input]]</li> | <li style="display: inline-block;"> [[File:Modified Input.jpg|thumb|widths=500px|Modified Input]]</li> | ||
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[[File:Cluster State.jpg|center|thumb|500px|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/> | [[File:Cluster State.jpg|center|thumb|500px|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/> | ||
<div><ul> | <div><ul> | ||
<li style="display: inline-block;"> [[File:Circuit Diagram to implement C-NOT.jpg| | <li style="display: inline-block;"> [[File:Circuit Diagram to implement C-NOT.jpg|frame|500px|Circuit Diagram to implement C-NOT]] </li> | ||
<li style="display: inline-block;"> [[File:Graph State Pattern for C-NOT.jpg| | <li style="display: inline-block;"> [[File:Graph State Pattern for C-NOT.jpg|frame|500px|Graph State Pattern for C-NOT]] </li> | ||
< | <center><caption>Measurement Pattern from Circuit Model</caption></center> | ||
</ul></div><br/> | |||
Figure 6b shows implementation of the first Hadamard gate on the second input state as measurement M2 on qubit 2. Then C-Z gate is implemented by the entangled qubits 3 and 1 in the graph state. Qubit 3 is entangled to another qubit 4 to record the output while measurement M3 on qubit 3 implements the second Hadamard gate. Finally, the states to which qubits (1) and (4) are reduced to determine the output states of the two input qubits after C-NOT gate operation. It is evident that one needs to remove certain nodes from the cluster state in order to implement the above shown graph state. This can be done by Z-basis measurements. Such measurements would leave the remaining qubits in the cluster state with extra Pauli corrections. This can be explained as follows. Consider a 2-dimesional graph state {1,2}. If qubit 1 is to be eliminated, we operate C-Z with 2 as target and 1 as control.<br/> | Figure 6b shows implementation of the first Hadamard gate on the second input state as measurement M2 on qubit 2. Then C-Z gate is implemented by the entangled qubits 3 and 1 in the graph state. Qubit 3 is entangled to another qubit 4 to record the output while measurement M3 on qubit 3 implements the second Hadamard gate. Finally, the states to which qubits (1) and (4) are reduced to determine the output states of the two input qubits after C-NOT gate operation. It is evident that one needs to remove certain nodes from the cluster state in order to implement the above shown graph state. This can be done by Z-basis measurements. Such measurements would leave the remaining qubits in the cluster state with extra Pauli corrections. This can be explained as follows. Consider a 2-dimesional graph state {1,2}. If qubit 1 is to be eliminated, we operate C-Z with 2 as target and 1 as control.<br/> | ||
CZ12 |+i1 |+i2 = |0i1 |+i2 + |1i1 |−i2 ,<br/> | CZ12 |+i1 |+i2 = |0i1 |+i2 + |1i1 |−i2 ,<br/> | ||