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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).'' | ||
====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 <math>|\psi\rangle = a|0\rangle + b|1\rangle</math>, we consider the case of a two qubit graph state <math>\mathcal{C}_{2\text{x}1}</math>.<br/> | |||
<div style='text-align: center;'><math>\mathcal{C}_{2\text{x}1}= CZ_{ij} |\psi_{i}\rangle |+_{i}\rangle = a|00\rangle + a|01\rangle + b|10\rangle -b|11\rangle</math><br/></div> | |||
If one measures qubit i in <math>\{|+\rangle,|-\rangle\}</math> basis and gets outcome s, qubit j reduces to,<br/> | |||
<div style='text-align: center;'><math>=(a+b)|0\rangle+(a-b)|1\rangle, \text{if s=0}</math></br> | |||
<math>=(a-b)|0\rangle+(a+b)|1\rangle, \text{if s=1}</math></div> | |||
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 <math>H|\psi\rangle</math>.<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/> | |||
*'''Preparation''' prepares all input qubits in the required state, generally represented as <math>|+_\theta=\frac{1}{\sqrt{2}}(|0\rangle+e^{i\theta}|1\rangle)</math> where <math>\theta\epsilon[0,2\pi]</math>. | |||
*'''Entanglement''' entangles all the qubits according to the required graph state. This operation is denoted by <math>E_{ij}</math>, 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 <math>M^\alpha_i</math>: the qubit ’i’ would be measured in <math>\{|+_\alpha\rangle,|-_\alpha\rangle\}</math> basis i.e. if the state is <math>\frac{1}{\sqrt{2}}(|0\rangle+e^{i\alpha}|1\rangle)</math> one gets outcome 0 and if the state is <math>\frac{1}{\sqrt{2}}(|0\rangle-e^{i\alpha}|1\rangle)</math>, the outcome is 1.<br/> | |||
*'''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/> | |||
The effect of X gate on a measurement angle (<math>\alpha</math>) in X-Y plane is to change its sign and Z gate is to add a phase <math>\pi</math>.<br/> | |||
<div style='text-align: center;'><math> ^t[M^\alpha_i]^s=M_i^\alpha Z^t X^s=M_i^{(-1)^s\alpha+t\pi}</math></div> | |||
We shall denote measurement in X-basis (<math>M_i^0</math>) as <math>M_i^x</math> and Y-basis (<math>M_i^\frac{\pi}{2}</math>) as <math>M_i^y</math><br/> | |||
Commutation relations:<br/><div style='text-align: center;'> | |||
<math>E_{ij}X_i^s=X_i^sZ_i^sE_{ij}\quad\quad (EX)</math><br/> | |||
<math>E_{ij}X_j^s=X_j^sZ_j^sE_{ij}\quad\quad(EX)</math><br/> | |||
<math>E_{ij}Z_i^t=Z_i^tE_{ij}\quad\quad(EZ)</math><br/> | |||
<math>E_{ij}Z_j^t=Z_j^tE_{ij}\quad\quad(EZ)</math><br/> | |||
<math>^t[M^\alpha_i]^sX^r_i=^t[M^\alpha_i]^{s+r}\quad\quad(MX)</math><br/> | |||
<math>M_i^xX_i^s=M_i^x\quad\quad(MX)</math><br/> | |||
<math>^t[M^\alpha_i]^sZ^r_i=^{t+r}[M^\alpha_i]^s\quad\quad(MZ)</math></div> | |||
The last second equation is implied from the fact that for <math>M_i^x</math>, <math>x=0=-0</math>. 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/> | |||
Outcomes: <math>s_1, s_2, s_3, s_4</math><br/> | |||
Circuit Operation: <math>X^{s_3}_4M_3^xE_{34}E_{13}X_3^{s_2}M_2^xE_{23}</math><br/><div style='text-align: center;'> | |||
<math>X^{s_3}_4M_3^xE_{34}\mathbf{E_{13}X_3^{s_2}}M_2^xE_{23}&\overset{\text{EX}}{\implies}</math><br/> | |||
<math>X^{s_3}_4Z_1^{s_2}M_3^x\mathbf{E_{34}X_3^{s_2}}M_2^xE_{13}E_{23}&\overset{\text{EX}}{\implies}</math><br/> | |||
<math>X^{s_3}_4Z_4^{s_2}Z_1^{s_2}\mathbf{M_3^xX_3^{s_2}}M_2^xE_{13}E_{23}E_{34}&\overset{\text{MX}}{\implies}</math><br/> | |||
<math>X^{s_3}_4Z_4^{s_2}Z_1^{s_2}M_3^xM_2^xE_{13}E_{234}</math><br/></div> | |||
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/> | |||
==References== | |||
<div style='text-align: right;'>''*contributed by Shraddha Singh''</div> | |||