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Current noisy quantum computers have multiple types of errors, which can occur in the state preparation, measurement/readout, and gate operation, as well as intrinsic decoherence and relaxation. Partly motivated by the booming of intermediate-scale quantum processors, measurement and gate errors have been recently extensively studied, and several methods of mitigating them have been proposed and formulated in software packages (e.g., in IBM Qiskit). Despite this, the state preparation error and the procedure to quantify it have not yet been standardized, as state preparation and measurement errors are usually considered not directly separable. Inspired by a recent work of Laflamme, Lin, and Mor \cite{laflamme2022algorithmic}, we propose a simple and resource-efficient approach to quantify separately the state preparation and readout error rates. With these two errors separately quantified, we also propose methods to mitigate them separately, especially mitigating state preparation errors with linear (with the number of qubits) complexity. As a result of the separate mitigation, we show that the fidelity of the outcome can be improved by an order of magnitude compared to the standard measurement error mitigation scheme. We also show that the quantification and mitigation scheme is resilient against gate noise and can be immediately applied to current noisy quantum computers. To demonstrate this, we present results from cloud experiments on IBM's superconducting quantum computers. The results indicate that the state preparation error rate is also an important metric for qubit metrology that can be efficiently obtained. 

We propose a Floquet time crystal model that responds in arbitrary multiples of the driving period. Such an 𝑛-tuple discrete time crystal is theoretically constructed by permutations in a disordered spin chain and is well suited for experimental implementations. Transitions between these time crystals with different periods give rise to a novel phase of matter that we call subspace-thermal discrete time crystals, where states within subspaces of definite charges are fully thermalized at an early time. However, the whole system still robustly responds to the periodic driving subharmonically, with a period being the greatest common divisor of the original two periods. Existing theoretical analysis from many-body localization cannot be used to understand the rigidity of such subspace-thermal time crystal phases. To resolve this, we develop a theoretical framework for the robustness of DTCs from the perspective of the robust 2⁢𝜋/𝑛 quasi-energy gap. Its robustness is rigorously proved if the system satisfies a certain condition where the mixing length, defined by the Hamming distance of the symmetry charges, does not exceed a global threshold. Although whether the condition is satisfied in generic disordered systems is unclear, the rigorous proof for DTC properties applies beyond the models considered here and extends to other existing DTCs realized by kicking disordered MBL systems, where the condition is automatically satisfied and conventional MBL-DTCs can be regarded as a special case of the subspace-thermal DTC with the subspace dimension being one, thus offering a systematic way to construct discrete time crystal models. We also introduce the notion of DTC charges that allow us to probe observables that spontaneously break the time-translation symmetry in both regular discrete time crystals and subspace-thermal discrete time crystals. Moreover, our discrete time crystal models can be generalized to systems with higher spin magnitudes or qudits, as well as to higher spatial dimensions.