Optimal digital dynamical decoupling for general decoherence via Walsh modulation

Abstract

We provide a general framework for constructing digital dynamical decoupling sequences based on Walsh modulation -applicable to arbitrary qubit decoherence scenarios. By establishing equivalence between decoupling design based on Walsh functions and on concatenated projections, we identify a family of optimal Walsh sequences, which can be exponentially more efficient, in terms of the required total pulse number, for fixed cancellation order, than known digital sequences based on concatenated design. Optimal sequences for a given cancellation order are highly non-uniquetheir performance depending sensitively on the control path. We provide an analytic upper bound to the achievable decoupling error, and show how sequences within the optimal Walsh family can substantially outperform concatenated decoupling, while respecting realistic timing constraints. We validate these conclusions by numerically computing the average fidelity in a toy model capturing the essential feature of hyperfine-induced decoherence in a quantum dot.PACS numbers: 03.65. Yz,03.67.Pp,89.70.+c Dynamical decoupling (DD) techniques, based on open-loop quantum control, provide an effective strategy to reduce decoherence from temporally correlated noise processes in realistic quantum information processing platforms [1,2]. In its simplest form, DD coherently averages out the unwanted system-environment interaction through the application of tailored sequences of (ideally, instantaneous) pulses, whose net action on the system translates, in the frequency domain, into a high-pass noise filter [3][4][5][6]. To date, the most efficient DD schemes known for generic error models -notably, Uhrig DD [7] and quadratic DD [8] for pure dephasing and general decoherence on a single qubit -involve pulse sequences with irrational pulse timing [9]. However, consideration of practical constraints highlights crucial advantages of digital DD, whereby all pulse separations are integer multiples of an experimentally restricted minimum time interval. Irrationally-timed DD sequences have been found to be more sensitive to both the form of the spectral cutoff and to inevitable pulse errors [10][11][12], while being less amenable to the additional compensation steps (e.g., via phase-shifts or composite pulses) that are needed to mitigate these errors for arbitrary input states [13][14][15][16]. Even in situations where pulse imperfections are unimportant, digital DD sequences are highly compatible with hardware constraints stemming from digital sequencing circuitry and clocking, which makes them attractive in terms of minimizing sequencing complexity, as ultimately demanded for large-scale implementations.Control modulation based on Walsh functions [17], has been proposed as a unifying approach for generating digital-efficient protocols, for both dynamically corrected quantum storage and gates [18][19][20][21]. Walsh DD (WDD) has been shown to naturally incorporate existing digital sequences as special instances (including concatenated DD for both single-and multi-axis...

Digital noise spectroscopy with a quantum sensor

“…Equivalently, the binary values of the mth Walsh modulation function are given by the mth row of the N × N Walsh matrix W, such that w m (t/τ ) = W[m, j], where t ∈ [jτ, (j + 1)τ ). Walsh functions are intrinsically compatible with the available quantum control hardware, supporting microwave or optical pulses and time discretization [27], and have been used to design optimal DD or coherence protection in the presence of both dephasing [22] and general multi-axis noise [23].…”

“…The protocols which have been most commonly employed to date, inspired by classical signal processing, perform spectral reconstruction in the frequency domain by measuring the decoherence of a qubit sensor subjected to sequences of spin-flip pulses [5,7,21], also known as dynamical decoupling (DD) sequences because of their noise filtering capability [22,23]. Whereas these protocols claim a simple intuitive formalism, the exact digital filters they produce are a product of trigonometric functions-effectively converting the reconstruction problem into a nonlinear inversion problem that necessitates numerical deconvolution algorithms [19].…”

Quantum control methods for robust entanglement of trapped ions

“…The PDD protocols can be made compatible with classical digital circuitry by building the pulse timings as multiples of a clock cycle [52][53][54]. The timings are encoded by Walsh functions, which are a series of square pulses with values {−1, +1}, where π-pulses are applied during zero-crossings.…”

Quantum control methods for robust entanglement of trapped ions

“…The PDD protocols can be made compatible with classical digital circuitry by building the pulse timings as multiples of a clock cycle [48][49][50]. The timings are encoded by Walsh functions, which are a series of square pulses with values {−1, +1}, where π-pulses are applied during zero-crossings.…”