Supplementary Information: Difference between revisions

Graph states The above operation can also be viewed as a graph state with two nodes and one edge. The qubit 1 is measured in a rotated basis HZ(θ), thus leaving qubit 2 in desired state and Pauli Correction Xs1HZ(θ1)|ψi, where s1 is the measurement outcome of qubit 1.<br/>

Now, suppose we need to operate the state with two unitary gates Z(θ1) and Z(θ2). This can be done by taking the output state of Z(θ1) gate as the input state of Z(θ2) gate and then repeating gate teleportation for this setup, as described above. Thus, following the same pattern for graph states we have now three nodes (two measurement qubits for two operators and one output qubit) with two edges, entangled as one dimensional chain(See Figure 4).<br/>

   

The measurement on qubit 1 will operate Xs1HZ(θ1)|ψi⊗I on qubits 2 and 3. If qubit 2 when measured in the given basis yields outcome s2, qubit 3 results in the following state Xs2HZ(θ2)Xs1HZ(θ1)|ψi. Using the relation we shift all the Pauli corrections to one end i.e. qubit 3 becomes Xs2Zs1HZ(±θ2)HZ(θ1)|ψi{equation missing}(Zs1H = HXs1). This method of computation requires sequential measurement of states i.e. all the states should not be measured simultaneously. As outcome of qubit 1 can be used to choose sign of ±θ2. This technique is also known as adaptive measurement. With each measurement, the qubits before the one measured at present have been destroyed by measurement. It is a feed-forward mechanism, hence known as one way quantum computation.<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/>

   

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  '''Figure 6: Measurement Pattern from Circuit Model''''''

Although cluster states are universal for MBQC, yet we need to tailor these to the specific computation by performing some computational (Z) basis measurements. If we were to use this principle for blind quantum computing, Client would have to reveal information about the structure of the underlying graph state. Thus, for the UBQC protocol, we introduce a new family of states called the Brickwork states which are universal for X − Y plane measurements and thus do not require the initial computational basis measurements. It was later shown that the Z-basis measurements can be dropped for cluster states and hence cluster states are also universal in X-Y measurements.

   

'''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]).