Subnostr

(completely theoretical and unfinished)

|ψ₁⟩ ⊗ |ψ₂⟩ ⊗ |ψ₃⟩ ⊗ |ψ₄⟩ = (α₁|00⟩ + β₁|01⟩ + γ₁|10⟩ + δ₁|11⟩) ⊗ (α₂|00⟩ + β₂|01⟩ + γ₂|10⟩ + δ₂|11⟩) ⊗ (α₃|00⟩ + β₃|01⟩ + γ₃|10⟩ + δ₃|11⟩) ⊗ (α₄|00⟩ + β₄|01⟩ + γ₄|10⟩ + δ₄|11⟩)

---------------------------------------------------------------------------

|ϕ⟩ = α|0⟩ + β|1⟩

|ψ⟩ = γ|0⟩ + δ|1⟩

|ξ⟩ = ε|0⟩ + ζ|1⟩

|η⟩ = η|0⟩ + θ|1⟩

|ϕ₁⟩ ⊗ |ψ₂⟩ ⊗ |ξ₃⟩ ⊗ |η₄⟩ = (α₁|00⟩ + β₁|01⟩ + γ₁|10⟩ + δ₁|11⟩) ⊗ (α₂|00⟩ + β₂|01⟩ + γ₂|10⟩ + δ₂|11⟩) ⊗ (α₃|00⟩ + β₃|01⟩ + γ₃|10⟩ + δ₃|11⟩) ⊗ (α₄|00⟩ + β₄|01⟩ + γ₄|10⟩ + δ₄|11⟩)

|ϕ⟩ = α|00⟩ + β|01⟩ + γ|10⟩ + δ|11⟩

α|+⟩ + β|-⟩ + γ|+⟩ + δ|-⟩

α|-⟩ + β|+⟩ + γ|-⟩ + δ|+⟩

α|-⟩ + β|-⟩ + γ|+⟩ + δ|+⟩

α|+⟩ + β|+⟩ + γ|-⟩ + δ|-⟩

|ψ⟩ = α'|00⟩ + β'|01⟩ + γ'|10⟩ + δ'|11⟩

α'|+⟩ + β'|-⟩ + γ'|+⟩ + δ'|-⟩

α'|-⟩ + β'|+⟩ + γ'|-⟩ + δ'|+⟩

α'|-⟩ + β'|-⟩ + γ'|+⟩ + δ'|+⟩

α'|+⟩ + β'|+⟩ + γ'|-⟩ + δ'|-⟩

|ξ⟩ = α''|00⟩ + β''|01⟩ + γ''|10⟩ + δ''|11⟩

α''|+⟩ + β''|-⟩ + γ''|+⟩ + δ''|-⟩

α''|-⟩ + β''|+⟩ + γ''|-⟩ + δ''|+⟩

α''|-⟩ - β''|-⟩ + γ''|+⟩ + δ''|+⟩

α''|+⟩ + β''|+⟩ + γ''|-⟩ + δ''|-⟩

|η⟩ = α'''|00⟩ + β'''|01⟩ + γ'''|10⟩ + δ'''|11⟩

α'''|+⟩ + β'''|-⟩ + γ'''|+⟩ + δ'''|-⟩

α'''|-⟩ + β'''|+⟩ + γ'''|-⟩ + δ'''|+⟩

α'''|-⟩ + β'''|-⟩ + γ'''|+⟩ + δ'''|+⟩

α'''|+⟩ + β'''|+⟩ + γ'''|-⟩ + δ'''|-⟩

|ϕ₁⟩ ⊗ |ψ₂⟩ ⊗ |ξ₃⟩ ⊗ |η₄⟩ = (α₁|00⟩ + β₁|01⟩ + γ₁|10⟩ + δ₁|11⟩

α|+⟩ + β|-⟩ + γ|+⟩ + δ|-⟩ )

⊗

(α₂|00⟩ + β₂|01⟩ + γ₂|10⟩ + δ₂|11⟩

α'|+⟩ + β'|-⟩ + γ'|+⟩ + δ'|-⟩ )

⊗

(α₃|00⟩ + β₃|01⟩ + γ₃|10⟩ + δ₃|11⟩

α'|+⟩ + β'|-⟩ + γ'|+⟩ + δ'|-⟩ )

⊗

(α₄|00⟩ + β₄|01⟩ + γ₄|10⟩ + δ₄|11⟩

α'''|+⟩ + β'''|-⟩ + γ'''|+⟩ + δ'''|-⟩ )

set 1:

Set 2:

|ϕ₁⟩ ⊗ |ψ₂⟩ ⊗ |ξ₃⟩ ⊗ |η₄⟩ = (α₁-|00⟩ + β₁-|01⟩ + γ₁-|10⟩ + δ₁-|11⟩) ⊗ (α₂-|00⟩ + β₂-|01⟩ + γ₂-|10⟩ + δ₂-|11⟩) ⊗ (α₃-|00⟩ + β₃-|01⟩ + γ₃-|10⟩ + δ₃-|11⟩) ⊗ (α₄-|00⟩ + β₄-|01⟩ + γ₄-|10⟩ + δ₄-|11⟩)

= α₁α₂α₃α₄|0000⟩

α₁α₂α₃β₄-|0001⟩

α₁α₂α₃γ₄|0010⟩

α₁α₂α₃δ₄-|0011⟩

α₁α₂β₃α₄|0100⟩

α₁α₂β₃β₄-|0101⟩

α₁α₂β₃γ₄|0110⟩

α₁α₂β₃δ₄-|0111⟩

α₁α₂γ₃α₄|1000⟩

α₁α₂γ₃β₄-|1001⟩

α₁α₂γ₃γ₄|1010⟩

α₁α₂γ₃δ₄-|1011⟩

α₁α₂δ₃α₄|1100⟩

α₁α₂δ₃β₄-|1101⟩

α₁α₂δ₃γ₄|1110⟩

α₁α₂δ₃δ₄-|1111⟩

+ ... +

β₁β₂β₃δ₄|1101⟩

β₁β₂γ₃α₄-|1010⟩

β₁β₂γ₃β₄|1011⟩

β₁β₂γ₃γ₄-|1100⟩

β₁β₂γ₃δ₄|1101⟩

β₁β₂δ₃α₄-|1110⟩

+... +

β₁β₂δ₃α₄|1111⟩

------------------------------------------------------------------

define the phase shift gate S as the following unitary matrix:

S = [1, 0; 0, i]

Step 1: Apply the phase shift gate S to the qubit in the state |01⟩

After applying the phase shift gate, the equation becomes:

1': |ϕ₁⟩ ⊗ |ψ₂⟩ ⊗ |ξ₃⟩ ⊗ |η₄⟩ = (α₁|00⟩ + i*β₁|01⟩ + γ₁|10⟩ + δ₁|11⟩) ⊗ (α₂|00⟩ + β₂|01⟩ + γ₂|10⟩ + δ₂|11⟩) ⊗ (α₃|00⟩ + β₃|01⟩ + γ₃|10⟩ + δ₃|11⟩) ⊗ (α₄|00⟩ + β₄|01⟩ + γ₄|10⟩ + δ₄|11⟩)

After applying the phase shift gate, the equation becomes:

2': |ϕ₁⟩ ⊗ |ψ₂⟩ ⊗ |ξ₃⟩ ⊗ |η₄⟩ = (α₁-|00⟩ + e^(iπ/2)β₁-|01⟩ + γ₁-|10⟩ + δ₁-|11⟩) ⊗ (α₂-|00⟩ + β₂-|01⟩ + γ₂-|10⟩ + δ₂-|11⟩) ⊗ (α₃-|00⟩ + β₃-|01⟩ + γ₃-|10⟩ + δ₃-|11⟩) ⊗ (α₄-|00⟩ + β₄-|01⟩ + γ₄-|10⟩ + δ₄-|11⟩)

|ϕ₁⟩ ⊗ |ψ₂⟩ ⊗ |ξ₃⟩ ⊗ |η₄⟩ = (α₁|00⟩ + β₁-|01⟩ + γ₁|10⟩ + δ₁-|11⟩) ⊗ (α₂|00⟩ + β₂-|01⟩ + γ₂|10⟩ + δ₂-|11⟩) ⊗ (α₃|00⟩ + β₃-|01⟩ + γ₃|10⟩ + δ₃-|11⟩) ⊗ (α₄|00⟩ + β₄-|01⟩ + γ₄-|10⟩ + δ₄|11⟩)

After applying the phase shift gate, the equation becomes:

|ϕ₁⟩ ⊗ |ψ₂⟩ ⊗ |ξ₃⟩ ⊗ |η₄⟩ = (α₁|00⟩ + e^(iπ/2)β₁-|01⟩ + γ₁|10⟩ + e^(iπ/2)δ₁-|11⟩) ⊗ (α₂|00⟩ + e^(iπ/2)β₂-|01⟩ + γ₂|10⟩ + e^(iπ/2)δ₂-|11⟩) ⊗ (α₃|00⟩ + e^(iπ/2)β₃-|01⟩ + γ₃|10⟩ + e^(iπ/2)δ₃-|11⟩) ⊗ (α₄|00⟩ + e^(iπ/2)β₄-|01⟩ + γ₄|10⟩ + δ₄|11⟩)

|ϕ₁⟩ ⊗ |ψ₂⟩ ⊗ |ξ₃⟩ ⊗ |η₄⟩ = (α₁-|00⟩ + β₁|01⟩ + γ₁-|10⟩ + δ₁|11⟩) ⊗ (α₂-|00⟩ + β₂|01⟩ + γ₂-|10⟩ + δ₂|11⟩) ⊗ (α₃-|00⟩ + β₃|01⟩ + γ₃-|10⟩ + δ₃|11⟩) ⊗ (α₄-|00⟩ + β₄|01⟩ + γ₄-|10⟩ + δ₄|11⟩)

After applying the phase shift gate, the equation becomes:

---------------------------------------------------------------------

Here, i is the imaginary unit and θ is the phase shift angle. The prime notation denotes the state after the phase shift gate operation.

----------------------------------------------------------------------

let's say we have these four quantum states:

|ϕ⟩ = α|0⟩ + β|1⟩

|ψ⟩ = γ|0⟩ + δ|1⟩

|ξ⟩ = ε|0⟩ + ζ|1⟩

|η⟩ = η|0⟩ + θ|1⟩

We can represent these states in the computational basis as:

|ϕ⟩ = α|00⟩ + β|01⟩

|ψ⟩ = γ|10⟩ + δ|11⟩

|ξ⟩ = ε|00⟩ + ζ|01⟩

|η⟩ = η|10⟩ + θ|11⟩

The tensor product of these states in the computational basis would be:

|ϕ⟩ ⊗ |ψ⟩ ⊗ |ξ⟩ ⊗ |η⟩ = (α|00⟩ + β|01⟩) ⊗ (γ|10⟩ + δ|11⟩) ⊗ (ε|00⟩ + ζ|01⟩) ⊗ (η|10⟩ + θ|11⟩) = αγεη|0000⟩ + αγζθ|0001⟩ + βγ

In the computational basis, we can express these states as:

|ϕ⟩ = α|00⟩ + β|01⟩ + γ|10⟩ + δ|11⟩

|ψ⟩ = α'|00⟩ + β'|01⟩ + γ'|10⟩ + δ'|11⟩

|ξ⟩ = α''|00⟩ + β''|01⟩ + γ''|10⟩ + δ''|11⟩

|η⟩ = α'''|00⟩ + β'''|01⟩ + γ'''|10⟩ + δ'''|11⟩

In the Hadamard basis, we can express these states as:

|ϕ⟩ = α|+⟩ + β|-⟩

|ψ⟩ = α'|+⟩ + β'|-⟩

|ξ⟩ = α''|+⟩ + β''|-⟩

|η⟩ = α'''|+⟩ + β'''|-⟩

--------------------------------------------------------------------

1': |ϕ₁⟩ ⊗ |ψ₂⟩ ⊗ |ξ₃⟩ ⊗ |η₄⟩ = (α₁|00⟩ + i*β₁|01⟩ + γ₁|10⟩ + δ₁|11⟩) ⊗ (α₂|00⟩ + β₂|01⟩ + γ₂|10⟩ + δ₂|11⟩) ⊗ (α₃|00⟩ + β₃|01⟩ + γ₃|10⟩ + δ₃|11⟩) ⊗ (α₄|00⟩ + β₄|01⟩ + γ₄|10⟩ + δ₄|11⟩) / sqrt(4)

3':|ϕ₁⟩ ⊗ |ψ₂⟩ ⊗ |ξ₃⟩ ⊗ |η₄⟩ = (α₁|00⟩ + e^(iπ/2)β₁-|01⟩ + γ₁|10⟩ + e^(iπ/2)δ₁-|11⟩) ⊗ (α₂|00⟩ + e^(iπ/2)β₂-|01⟩ + γ₂|10⟩ + e^(iπ/2)δ₂-|11⟩) ⊗ (α₃|00⟩ + e^(iπ/2)β₃-|01⟩ + γ₃|10⟩ + e^(iπ/2)δ₃-|11⟩) ⊗ (α₄|00⟩ + e^(iπ/2)β₄-|01⟩ + γ₄|10⟩ + δ₄|11⟩) / sqrt(4)