Dynamical decoupling of a qubit with always-on control fields

Abstract

We consider dynamical decoupling schemes in which the qubit is continuously manipulated by a control field at all times. Building on the theory of the Uhrig dynamical decoupling (UDD) sequence and its connections to Chebyshev polynomials, we derive a method of always-on control by expressing the UDD control field as a Fourier series. We then truncate this series and numerically optimize the series coefficients for decoupling, constructing the Chebyshev and Fourier expansion sequence. This approach generates a bounded, continuous control field. We simulate the decoupling effectiveness of our sequence versus a continuous version of UDD for a qubit coupled to fullyquantum and semi-classical dephasing baths and find comparable performance. We derive filter functions for continuous-control decoupling sequences, and we assess how robust such sequences are to noise on control fields. The methods we employ provide a variety of tools to analyze continuous-control dynamical decoupling sequences. 4. Decoupling a qubit from a quantum bath 13 5. Filter function analysis 16 6. Decoupling a qubit from Born-Markov classical baths 18 7. Conclusion 20 Acknowledgments 20 Appendix A. Proof that Fourier series components do not affect 'sine'-like DD constraints 20 Appendix B. Derivation of filter function for continuous sequences 22 References 23simple solution to this problem: by applying a determined sequence of control pulses, one can significantly enhance the lifetime of a qubit. Early work in DD sought to suppress decoherence by applying a periodic sequence of instantaneous 'bang-bang' pulses, which periodically flip the state of a qubit and undo the coupling to the bath [12][13][14][15]. This work was extended to Hamiltonian engineering by using decoupling sequences to selectively enable coupling Hamiltonians [16,17]. Subsequently, attention turned to using Eulerian graphs to design sequences using bounded control operations that were robust to many types of systematic control errors [18]. This led to the notion of dynamically-corrected gates [19], which combine techniques from DD and composite pulse sequences from NMR to produce error-suppressing quantum gates.The idealized 'bang-bang' control pulse is instantaneous, like the Dirac delta function. However, real physical pulses can only approximate 'bang-bang' control, because such idealized control would require infinite power. The implications of a real, continuous-time function can be significant. Most DD sequences are designed to correct noise errors between pulses, but offer no intrinsic protection to errors during pulses. Early studies of always-on DD arose in the problem of dipolar decoupling in solid-state NMR [20], culminating from substantial work in extending pulsed decoupling techniques to 'soft', or low-power, pulses. A modern consideration of this problem in the quantum information context is found in [21]. More recently, optimal control approaches to customizing continuous, bounded controls to decouple a specific but arbitrary noise bath were considered in [22][2...

Dynamically corrected gates from geometric space curves

“…These techniques have been extended to more recent contexts such as quantum information processing, where NMR sequences such as Hahn echo and CPMG have proven effective at preserving the information stored in qubits [12][13][14][15][16][17][18][19]. These newer applications have also driven the search for additional sequences that can more efficiently extend qubit lifetimes [20][21][22][23][24].…”

General solution to inhomogeneous dephasing and smooth pulse dynamical decoupling

“…Spin echo and related multipulse sequences [20][21][22][23][24][25][26][27] have also been widely employed to mitigate other types of decoherence such as environmental noise fluctuations [28][29][30]. Much work has been done to extend dynamical decoupling to not only preserve the state of the system, but to also cancel errors while performing operations on the system (dynamical gate correction) [31][32][33][34][35][36][37][38][39][40][41][42][43].Although these dynamical decoupling methods have been broadly successful, there are many systems, especially in the context of quantum information technologies, where they exhibit substantial drawbacks. This is because the highly idealized pulse waveforms needed, i.e., δ-functions or square pulses, can be challenging to generate in systems where the dynamics occurs on nanosecond timescales, pushing the limits of current waveform generators, for which minimal achievable rise times are on the scale of 100 ps-1 ns.…”

“…In addition, building control sequences from a very restricted set of pulse shapes leaves few tunable parameters and leads to unnecessarily long sequences that may compete with other decoherence or loss mechanisms that become important on longer timescales. Smooth pulses can be generated numerically using optimal control techniques such as GRAPE [44,45] or using continuous analogs of composite sequences such as CORPSE [27,34], however such methods yield locally optimal pulses that often exhibit complicated shapes, and obtaining globally optimal quantum controls necessitates the use of analytical methods.…”