13
ρ (θ)̸=ρ . Following the same approach as before, we prepare the quantum states:
s s
1.ρ =U |0⟩⟨0|⊗nU†,
s s s
2.ρ =U |0⟩⟨0|⊗nU†,
s′ s′ s′
3.ρ =(I−2ρ )ρ (I−2ρ )
ss′ s′ s s′
=ρ −2ρ ρ −2ρ ρ +4ρ ρ ρ , (A4)
s s′ s s s′ s′ s s′
4.ρ (θ)=G (θ)ρ G (θ)†
s′s s s′ s
=(I−ρ (θ))ρ (I−ρ (θ))
s s′ s
=ρ −ρ ρ (θ)−ρ (θ)ρ +ρ (θ)ρ ρ (θ).
s′ s′ s s s′ s s′ s
From the above equation, we observe that rotating the quantum gate G by an arbitrary angle θ ̸=kπ,k =0,1,2···
0
changes the originally linearly dependent quantum states, prepared using G , into linearly independent ones.
0
By using similar methods, it is possible to prepare the remaining d(d−1) quantum states and ensure that they
are linearly independent. This allows us to construct a complete d2-dimensional linear space for performing a full
tomography of the PTM under study.
Appendix B: More Numerical Simulation
For the Hadamard Test procedure, we incorporate noise simulation into its measurement process by injecting
randomnumbersdrawnfromaGaussiandistributionwithastandarddeviationof0.01. Thegraphicalrepresentation
below illustrates the outcomes prior to and subsequent to the integration of these random numbers:
FIG. 6. This figure presents numerical results for Tr{ρ2} and Tr{ρ3}, taking into account noise, using the Hadamard Test
method. Subfigure(a)displaysthenumericalresultsforTr{ρ2},andsubfigure(b)showsthenumericalresultsforTr{ρ3}. The
black lines represent the results after adding random numbers, while the red lines represent the results before adding random
gates.
ForTomography,weintroducenoisesimulationbyaddingrandomnumberstoitsPTMelementsaswellasg-matrix
elements. Since the singular values of the g-matrix are small, we choose random numbers sampled from a Gaussian
distributionwithastandarddeviationof0.0001tosimulatethenoise. Theresultsbeforeandafteraddingtherandom
numbers are shown in the graph below: