a) What is QFT6 of 1/√2(|0>+|3>)?
b) What is QFT6 of 1/√2(|1>+|4>)?
c) What is QFT6 of 1/√3(|0>+|2>+|4>)?
d) What is QFT6 of 1/√3(|1>+|3>+|5>)?
Answer like:
a) |0> = ; |1> = ; |2> = ; |3> = ; |4> = ; |5> = ...
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Anyone for b, c, d?
b) What is QFT6 of 1/�ã2(|1>+|4>)?
c) What is QFT6 of 1/�ã3(|0>+|2>+|4>)?
d) What is QFT6 of 1/�ã3(|1>+|3>+|5>)?
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To find the quantum Fourier transform (QFT6) of the given states, we'll go through the following steps:
Step 1: Express the given states in binary form.
Step 2: Apply the QFT gate to each binary state.
Step 3: Express the resulting states in ket notation.
a) QFT6 of 1/√2(|0>+|3>):
Step 1: The binary representation of |0> is 000000, and |3> is 000011.
Step 2: Apply the QFT gate to each of the binary states:
- QFT6(|0>) = 1/√2 (|0> + |1> + |2> + |3> + |4> + |5> + |6> + |7>)
- QFT6(|3>) = 1/√2 (|0> + i|1> - |2> - i|3> + |4> + i|5> - |6> - i|7>)
Step 3: Express the resulting states in ket notation:
a) QFT6 of 1/√2(|0>+|3>) = (1/√2) (|0> + |1> + |2> + |3> + |4> + |5> + |6> + |7>)
b) QFT6 of 1/√2(|1>+|4>):
Step 1: The binary representation of |1> is 000001, and |4> is 000100.
Step 2: Apply the QFT gate to each of the binary states:
- QFT6(|1>) = 1/√2 (|0> + i|1> - |2> - i|3> + |4> + i|5> - |6> - i|7>)
- QFT6(|4>) = 1/√2 (|0> + |1> + |2> + |3> - |4> - |5> - |6> - |7>)
Step 3: Express the resulting states in ket notation:
b) QFT6 of 1/√2(|1>+|4>) = (1/√2) (|0> + i|1> - |2> - i|3> + |4> + |5> + |6> + |7>)
c) QFT6 of 1/√3(|0>+|2>+|4>):
Step 1: The binary representation of |0> is 000000, |2> is 000010, and |4> is 000100.
Step 2: Apply the QFT gate to each of the binary states:
- QFT6(|0>) = 1/√2 (|0> + |1> + |2> + |3> + |4> + |5> + |6> + |7>)
- QFT6(|2>) = 1/√2 (|0> + |1> - |2> - |3> + |4> + |5> - |6> - |7>)
- QFT6(|4>) = 1/√2 (|0> + |1> + |2> + |3> - |4> - |5> - |6> - |7>)
Step 3: Express the resulting states in ket notation:
c) QFT6 of 1/√3(|0>+|2>+|4>) = (1/√3) (|0> + |1> + |2> - i|3> + |4> + |5> - |6> - i|7>)
d) QFT6 of 1/√3(|1>+|3>+|5>):
Step 1: The binary representation of |1> is 000001, |3> is 000011, and |5> is 000101.
Step 2: Apply the QFT gate to each of the binary states:
- QFT6(|1>) = 1/√2 (|0> + i|1> - |2> - i|3> + |4> + i|5> - |6> - i|7>)
- QFT6(|3>) = 1/√2 (|0> + |1> + |2> + |3> - |4> - |5> - |6> - |7>)
- QFT6(|5>) = 1/√2 (|0> - |1> + |2> - |3> + |4> - |5> + |6> - |7>)
Step 3: Express the resulting states in ket notation:
d) QFT6 of 1/√3(|1>+|3>+|5>) = (1/√3) (|0> + |1> + |2> + |3> - |4> - |5> + |6> + |7>)
Please note that the calculations above are based on the standard definition of the QFT6 gate.