In NMR experiments and quantum computation, many pulse (quantum gate) sequences called the composite pulses, were developed to suppress one of two dominant errors; a pulse length error and an off-resonance error. We describe, in this paper, a general prescription to design a single-qubit concatenated composite pulse (CCCP) that is robust against two types of errors simultaneously. To this end, we introduce a new property, which is satisfied by some composite pulses and is sufficient to obtain a CCCP. Then we introduce a general method to design CCCPs with shorter execution time and less number of pulses.
In NMR experiments and quantum computation, many pulse (quantum gate) sequences called the composite pulses, were developed to suppress one of two dominant errors; a pulse length error and an off-resonance error. We describe, in this paper, a general prescription to design a single-qubit concatenated composite pulse (CCCP) that is robust against two types of errors simultaneously. To this end, we introduce a new property, which is satisfied by some composite pulses and is sufficient to obtain a CCCP. Then we introduce a general method to design CCCPs with shorter execution time and less number of pulses.
We study the performance of composite pulses in the presence of time-varying control noise on a single qubit. These protocols, originally devised only to correct for static, systematic errors, are shown to be robust to time-dependent non-Markovian noise in the control field up to frequencies as high as $\ensuremath{\sim}10%$ of the Rabi frequency. Our study combines a generalized filter-function approach with asymptotic dc-limit calculations to give a simple analytic framework for error analysis applied to a number of composite-pulse sequences relevant to nuclear magnetic resonance as well as quantum information experiments. Results include examination of recently introduced concatenated composite pulses and dynamically corrected gates, demonstrating equivalent first-order suppression of time-dependent fluctuations in amplitude and/or detuning, as appropriate for the sequence in question. Our analytic results agree well with numerical simulations for realistic
1/f noise spectra with a roll-off to
1/f2, providing independent validation of our theoretical insights.
Residual-error preservation (REP) provides a sufficient rule for constructing concatenated composite pulses that compensate multiple systematic errors. We show that local REP is not necessary: deviations from REP can cancel across the full sequence. Using first-order error generators, we derive a necessary and sufficient condition for concatenation to retain the robustness of an outer sequence. A simple sufficient condition is that all inner pulses rescale the corresponding elementary error generators by a common real factor, not necessarily unity. We illustrate this principle using pulse-length and off-resonance errors, two systematic error models commonly considered in magnetic resonance. Short CORPSE and SCROFULOUS each compensate one error without preserving the other locally. Combining these inner pulses with suitable equal-rotation-angle outer sequences yields simultaneous first-order compensation of both errors, extending concatenated composite-pulse design beyond local REP.