⚡ Pluto.jl ⚡

Problem 1. Consider the unary gates (2 x 2 unitary matrices)

$N = (\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}) H = \frac{1}{\sqrt{2}} (\begin{matrix} 1 & 1 \\ 1 & - 1 \end{matrix})$

$V = (\begin{matrix} 1 & 0 \\ 0 & e^{\frac{i π}{2}} \end{matrix}) W = (\begin{matrix} 1 & 0 \\ 0 & e^{\frac{i π}{4}} \end{matrix})$

and the normalized state

$| ψ ⟩ = \frac{1}{\sqrt{2}} (\begin{matrix} 1 \\ 1 \end{matrix})$

calculate $U | ψ ⟩$ where $U = N \cdot H \cdot V \cdot W$ and also calculate the epectation value $⟨ ψ | U | ψ ⟩$ .

Note that $U$ is the Pauli-X gate, $H$ the Hadamard gate, $V$ phase gate, and $W$ $π / 8$ gate.

2.5 ms

Solution 1

8.0 μs

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using SymPy

7.8 s

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begin
    𝑖,π = sympy.symbols("𝑖 π")
    sqrthalf = 1/sympy.sqrt(2)
    e = sympy.exp
    ⋅ = *
end;

174 ms

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begin
    N = sympy.Matrix([0 1;
         1 0])
    
    H = sqrthalf * sympy.Matrix([1 1;
                1 -1])
    
    V = sympy.Matrix([1 0;
         0 e(𝑖⋅π / 2)])
    
    W = sympy.Matrix([1 0;
         0 e(𝑖⋅π / 4)])
end;

1.8 s

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begin
    ψ = sqrthalf * sympy.Matrix([1;1])
    ψᵀ = ψ'
end;

72.4 ms

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U = N ⋅ H ⋅ V ⋅ W;

15.5 ms

$U = [\begin{matrix} \frac{\sqrt{2}}{2} & - \frac{\sqrt{2} e^{\frac{3 π 𝑖}{4}}}{2} \\ \frac{\sqrt{2}}{2} & \frac{\sqrt{2} e^{\frac{3 π 𝑖}{4}}}{2} \end{matrix}]$

296 ms

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ψₜ = U⋅ψ;

7.6 ms

$U | ψ ⟩ = [\begin{matrix} \frac{1}{2} - \frac{e^{\frac{3 π 𝑖}{4}}}{2} \\ \frac{e^{\frac{3 π 𝑖}{4}}}{2} + \frac{1}{2} \end{matrix}]$

23.8 ms

The expectation $⟨ ψ | U | ψ ⟩$ is then:

6.1 μs

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𝙴 = ψᵀ ⋅ U ⋅ ψ;

110 ms

$𝙴 = \frac{\sqrt{2}}{2}$

16.0 ms