Analytically quantifying cat scar enforced discrete time crystalline orders*

We construct an analytical perturbation theory to quantitatively characterize rare Schrodinger's cat eigenstates, dubbed cat scars, in a Floquet system. The theory allows for computing the analytical scaling relations for their associated discrete time crystal (DTC) dynamics, including DTC amplitude, Fock space localization, and DTC lifetime scaling. In particular, we show that a plethora of inhomogeneous cat eigenstate configurations can be precisely engineered using a minimal number of different two-qubit gates for interactions. Thus, our theory allows for deterministically enhancing the resource usage efficiency when engineering cat scar enforced DTCs in current noisy-intermediate-scale-quantum devices. Further, the analytical framework sheds light on several subtle issues. They include the conditions to avoid Floquet many-body resonances in systems with strong Ising interactions, and the practical methods to verify genuine cat eigenstate enforced many-body DTCs. Relevant experiments on observing unconventional cat state dynamics in a cat-scarred DTC will also be discussed.