Fig. 3: Implementations of the Deutsch-Jozsa algorithm and quantum phase estimation algorithm over two remote network nodes.

a Quantum circuit for implementing the Deutsch-Jozsa algorithm. x is a single query bit, and y is an auxiliary bit, F is the Fourier transform, F⁻¹ is the inverse Fourier transform. The box U_f represents a unitary operation specific to each of the functions f. b, c The results represent the case of identity (ID) and NOT operations, which belong to the constant function, so the register measurement result collapses to \(\left\vert H\right\rangle\). d, e The results represent the case of the CNOT and zero-CNOT (ZCNOT) operations, which belong to the balance function, so the register measurement result collapses to \(\left\vert V\right\rangle\). f Quantum circuit for the k-th iteration of the iterative phase estimation algorithm (IPEA). The algorithm is iterated m times to get an m-bit \(\tilde{\varphi }\), which is the approximation to the phase of the eigenstate φ. Z_φ represents the phase correction as obtained from previous iterations. g–j Quantum phase estimation results with m = 3. The corresponding eigenvalues are phase shifts that can be perfectly represented by three binary values with φ = 0, π/2, 5π/4, 3π/2. The blue part represents a phase estimation result of 0, while the green part represents a phase estimation result of 1. The error bars are one standard deviation.