Suppose we have a quantum circuit that takes the input |0> and outputs |+>, and also takes the input |1> and outputs −|−>. If we input √2*i/√3|+> + 1√3|−>, what does the circuit output?
In the form a│0> + b│1>
To determine what the circuit outputs when the input is √2*i/√3|+> + 1/√3|−>, we can apply the circuit gate-by-gate to the input state.
Let's break down the given input state:
√2*i/√3|+> + 1/√3|−>
To simplify the calculations, let's express |+> and |−> in terms of the computational basis states |0> and |1>:
√2*i/√3(|0> + |1>) + 1/√3(|0> − |1>)
Expanding this expression further:
√2*i/√3|0> + √2*i/√3|1> + 1/√3|0> - 1/√3|1>
Grouping the terms:
(√2*i/√3 + 1/√3)|0> + (√2*i/√3 - 1/√3)|1>
Now, let's apply the circuit to this input state.
Since the circuit takes |0> to |+> and |1> to −|−>, we can substitute these values:
(|+> + |−>)/√2|0> + (|+> − |−>)/√2|1>
Expanding further:
(|+>/√2 + |−>/√2)|0> + (|+>/√2 - |−>/√2)|1>
Simplifying:
1/√2(|+> + |−>)|0> + 1/√2(|+> - |−>)|1>
Finally, expressing |+> and |−> in terms of the computational basis states:
1/√2 (|0> + |1>)|0> + 1/√2 (|0> - |1>)|1>
Distributing the coefficients:
1/√2 |0>|0> + 1/√2 |1>|0> + 1/√2 |0>|1> - 1/√2 |1>|1>
Combining terms:
(1/√2 |0> + 1/√2 |1>)|0> + (1/√2 |0> - 1/√2 |1>)|1>
Simplifying:
(1/√2)|+>|0> + (1/√2)|−>|1>
So, the output of the circuit when the input is √2*i/√3|+> + 1/√3|−> is (1/√2)|+>|0> + (1/√2)|−>|1>, which can be written in the desired form as 1/√2|0> + 1/√2|1>.