Fig. 1: Numerical analysis on error-crafted synthesis for the Pauli constraint.
a Diamond distance \({d}_{\diamond }(\hat{{\mathcal{U}}},{{\mathcal{U}}}_{{\rm{synth}}})\) between the target single-qubit Haar random unitary \(\hat{{\mathcal{U}}}\) and the synthesized channel \({{\mathcal{U}}}_{{\rm{synth}}}:={\sum }_{j}{p}_{j}{{\mathcal{U}}}_{j}\). The black real line indicates the results from unitary synthesis by the Ross-Selinger algorithm, while the filled dots are results from mixed synthesis with shift factors of c = 2, 3, 5, 7. When we increase the number of synthesized unitaries by a factor of R = 3 (see main text for detailed definition) for c = 7, shown by unfilled triangles, the remnant synthesis error is suppressed below the bound in Eq. (5) and approaches the value of ϵ2, which matches the upper bound for the non-constrained case as in Eq. (1). The plots are averaged over 200 random instances of target unitaries. b Surface plot of failure rate pfail of mixed synthesis under the Pauli constraint, averaged over 200 instances for ϵ = 10−4. Scaling with c and R are shown in (c), (d), where the number of instances is 10,000 and 2000, respectively.
