Fig. 1: The proposed protocol for the realization of QSP on a noisy quantum computer.

We choose Hamiltonian simulation as the application. We start with a necessary preprocessing step (A) that maps the input parameters to an effective Hamiltonian \(\tilde{H}\) and an effective simulation time \(\tilde{t}\). In step (B), \(\tilde{H}\) is embedded in a unitary operator. By classically optimizing/compiling a circuit \({{{\mathcal{W}}}}\) this step produces a compressed version of a block-encoding circuit. Next, in the operator-function design (C), we approximate the real-time evolution function, e^−ixt, by a polynomial f(x) of degree d. While increasing the degree leads to a more accurate polynomial approximation, the computation suffers from larger noise effects. This is due to the growing depth of the QSP circuit, consisting of \({{{\mathcal{O}}}}(d)\) primitive gates. By accounting for the error rate p_TQ of two-qubit gates, we heuristically estimate the optimal degree yielding the smallest combined error. The processing step (D) finally realizes QSP using the compressed block-encoding circuit \({{{\mathcal{W}}}}\) and the designed polynomial f(x). Upon postselection on the ancilla’s measurement outcomes, we obtain an approximation to the desired real-time evolution e^−iHt. An error mitigation scheme based on the error rate p_TQ further reduces the effect of noise on the output.