Table 8 Number of projection measurements required by the proposed scheme to obtain an arbitrary process matrix element for a multi-particle system

Diagonal Elements \({\chi }_{J,K}^{M,N}(M=N,J=K)\)

1

Requires only a single parameter setting. The diagonal element can be directly obtained from one projection probability.

Off-diagonal Elements (satisfying the mathematical form \({\chi }_{J,K}^{M,N}(M=N,J\,\ne \,K)\) or \({\chi }_{J,K}^{M,N}(M\,\ne \,N,J=K)\))

4

Requires four different parameter settings. The real and imaginary parts of the element are obtained via a linear combination of the four corresponding projection probabilities.

Off-diagonal Elements (satisfying the mathematical form \({\chi }_{J,K}^{M,N}(M\,\ne \,N,J\,\ne \,K)\))

16

Requires sixteen different circuit parameter settings. The real and imaginary parts are determined through a linear combination of the sixteen corresponding projection probabilities.