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5.2 Case 2: N=15, a=11

Figure 10 5.2 Case 2: N=15, a=11

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~~ NEO Lo _ _ 1° °F _ _ | os _ _ | oo” _ _ | 4 _ _ | ei _ _ | _ _ 1° * NU 4 If state |1) is put in, the output will be a superposition of all states from |0) to |7) with the same amplitude. But the phase difference between any two adjacent states is eiz/4. So the frequency ot he output is 1. If state |2) is put in, the amplitude of all possible states is also the same, but the yhase difference between any two adjacent states doubles, namely, the output has a frequency of 2. f the input is a superposition state, say |1) and |2), then the output is the superposition of the wave yf frequency 1 and the wave of frequency 2. See Figure 1 (4-qubit OFT)
~~ NEO Lo _ _ 1° °F _ _ | os _ _ | oo” _ _ | 4 _ _ | ei _ _ | _ _ 1° * NU 4 If state |1) is put in, the output will be a superposition of all states from |0) to |7) with the same amplitude. But the phase difference between any two adjacent states is eiz/4. So the frequency ot he output is 1. If state |2) is put in, the amplitude of all possible states is also the same, but the yhase difference between any two adjacent states doubles, namely, the output has a frequency of 2. f the input is a superposition state, say |1) and |2), then the output is the superposition of the wave yf frequency 1 and the wave of frequency 2. See Figure 1 (4-qubit OFT)
3. Realization of QFT and IQFT on IBMQ 21 Decion of OFT [641
3. Realization of QFT and IQFT on IBMQ 21 Decion of OFT [641
Figure 5 Result of 4-qubit QFT 3.2 Implementation of QFT and IQFT [8]
Figure 5 Result of 4-qubit QFT 3.2 Implementation of QFT and IQFT [8]
In summary, Uf gate selects the x that make f(x) =a certain z. They are the very x that are put in the IQFT. Note that x’s are separated by a constant value p, and the IQFT detects the periods of the input states. Accordingly, the measurement of the output will reflect the periodicity of the input by showing yet another periodicity. We can utilize the period of the measurement in the output to deduce the period of the input.
In summary, Uf gate selects the x that make f(x) =a certain z. They are the very x that are put in the IQFT. Note that x’s are separated by a constant value p, and the IQFT detects the periods of the input states. Accordingly, the measurement of the output will reflect the periodicity of the input by showing yet another periodicity. We can utilize the period of the measurement in the output to deduce the period of the input.
Figure 8 Procedure of Shor's Algorithm
Figure 8 Procedure of Shor's Algorithm
The circuit for Shor’s algorithm is designed as Figure 9. Qubits 0-2 form register A; qubits 3-6 form register B.
The circuit for Shor’s algorithm is designed as Figure 9. Qubits 0-2 form register A; qubits 3-6 form register B.
Related topics:
MathematicsPhysicsDiscrete LogarithmThermodynamics: An Engineering Approach
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