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arXiv:2604.25807

Proof of the Error Scaling for Universally Robust Dynamical Decoupling Sequences

Domenico D'Alessandro, Phattharaporn Singkanipa, Daniel Lidar

quant-phcs.ITSubmitted Apr 28, 2026• Announced Apr 29, 2026• 0 citations
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Abstract

Universally robust dynamical decoupling (UR$n$) sequences were proposed to compensate pulse imperfections arising from arbitrary experimental parameters while achieving high-order error suppression with only a linear increase in the number of pulses. Although their performance was supported by analytical arguments, numerical simulations, and experiments, a complete mathematical proof of the claimed order of error compensation has been absent. In this work, we present a rigorous proof for UR$n$ DD sequences with even $n$. Using a series expansion of a quantity whose modulus is the fidelity $F$, we derive necessary and sufficient conditions for the cancellation of its coefficients up to, but not including, order $n$. The UR$n$ phase prescription satisfies these conditions, and therefore $1-F=O(ε^n)$. Our results establish the UR$n$ construction on firm analytical grounds and clarify the structure responsible for its high-order robustness.

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Universally robust dynamical decoupling (URnnn) sequences were proposed to compensate pulse imperfections arising from arbitrary experimental parameters while achieving high-order error suppression with only a linear increase in the number of pulses. Although their performance was supported by analytical arguments, numerical simulations, and experiments, a complete mathematical proof of the claimed order of error compensation has been absent. In this work, we present a rigorous proof for URnnn DD sequences with even nnn. Using a series expansion of a quantity whose modulus is the fidelity FFF, we derive necessary and sufficient conditions for the cancellation of its coefficients up to, but not including, order nnn. The URnnn phase prescription satisfies these conditions, and therefore 1−F=O(εn)1-F=O(ε^n)1−F=O(εn). Our results establish the URnnn construction on firm analytical grounds and clarify the structure responsible for its high-order robustness.
We introduce universally robust sequences for dynamical decoupling, which simultaneously compensate pulse imperfections and the detrimental effect of a dephasing environment to an arbitrary order, work with any pulse shape, and improve performance for any initial condition. Moreover, the number of pulses in a sequence grows only linearly with the order of error compensation. Our sequences outperform the state-of-the-art robust sequences for dynamical decoupling. Beyond the theoretical proposal, we also present convincing experimental data for dynamical decoupling of atomic coherences in a solid-state optical memory.
Dynamical decoupling (DD) is perhaps the simplest and least resource-intensive error-suppression strategy for improving quantum computer performance. Here we report on a large-scale survey of the performance of 60 different DD sequences from ten families, including basic as well as advanced sequences with high-order error cancelation properties and built-in robustness. The survey is performed using three different superconducting-qubit IBMQ devices, with the goal of assessing the relative performance of the different sequences in the setting of arbitrary quantum state preservation. We find that the high-order universally robust (UR) and quadratic DD (QDD) sequences generally outperform all other sequences across devices and pulse-interval settings. Surprisingly, we find that DD performance for basic sequences such as the Carr-Purcell-Meiboom-Gill and XY4 sequences can be made to nearly match that of UR and QDD sequences by optimizing the pulse interval, with the optimal interval being substantially larger than the minimum interval possible on each device.
We prove the universality of the generalized QDDN1N2{}_{{N}_{1}{N}_{2}}N1​N2​​ (quadratic dynamical decoupling) pulse sequence for near-optimal suppression of general single-qubit decoherence. Earlier work showed numerically that this dynamical decoupling sequence, which consists of an inner Uhrig DD (UDD) and outer UDD sequence using N1{N}_{1}N1​ and N2{N}_{2}N2​ pulses, respectively, can eliminate decoherence to O(TN)O({T}^{N})O(TN) using O(N2)O({N}^{2})O(N2) unequally spaced ``ideal'' (zero-width) pulses, where TTT is the total evolution time and N=N1=N2N={N}_{1}={N}_{2}N=N1​=N2​. A proof of the universality of QDD has been given for even N1{N}_{1}N1​. Here we give a general universality proof of QDD for arbitrary N1{N}_{1}N1​ and N2{N}_{2}N2​. As in earlier proofs, our result holds for arbitrary bounded environments. Furthermore, we explore the single-axis (polarization) error suppression abilities of the inner and outer UDD sequences. We analyze both the single-axis QDD performance and how the overall performance of QDD depends on the single-axis errors. We identify various performance effects related to the parities and relative magnitudes of N1{N}_{1}N1​ and N2{N}_{2}N2​. We prove that using QDDN1N2{}_{{N}_{1}{N}_{2}}N1​N2​​ decoherence can always be eliminated to O(TminN1,N2)O({T}^{\mathrm{min}{{N}_{1},{N}_{2}}})O(TminN1​,N2​).
Active protection of quantum states is an essential prerequisite for the implementation of quantum computing. Dynamical decoupling (DD) is a promising approach that applies sequences of control pulses to the system in order to reduce the adverse effect of system-environment interactions. Since every hardware device has finite precision, the errors of the DD control pulses can themselves destroy the stored information rather than protect it. We experimentally compare the performance of different DD sequences in the presence of an environment that was chosen such that all relevant DD sequences can equally suppress its effect on the system. Under these conditions, the remaining decay of the qubits under DD allows us to compare very precisely the robustness of the different DD sequences with respect to imperfections of the control pulses.
Abstract Dynamical decoupling (DD), usually implemented by sophisticated sequences of instantaneous control pulses, is a well-established quantum control technique for quantum information and quantum sensing. In practice, the pulses are inevitably imperfect with many systematic errors that may influence the performances of DD. In particular, Rabi error and detuning are primary systemic errors arising from finite pulse duration, incorrect time control, and frequency instability. Here, we propose a phase-modulated DD with staggered global phases for the basic units of the pulse sequences to suppress these systemic errors. By varying the global phases appended to the pulses in the dynamical decoupling unit alternatively with 0 or π, our protocol can significantly reduce the influences of Rabi error and detuning. Our protocol is general and can be combined with the most existing DD sequences such as universal DD, knill DD, XY, etc. As an example, we further apply our method to quantum lock-in detection for measuring time-dependent alternating signals. Our study paves the way for a simple and feasible way to realize robust dynamical decoupling sequences, which can be applicable for various quantum sensing scenarios.
Dynamical decoupling (DD) refers to a well-established family of methods for error mitigation, comprising pulse sequences aimed at averaging away slowly evolving noise in quantum systems. Here we revisit the question of its efficacy in the presence of noisy pulses in scenarios important for quantum devices today: pulses with gate control errors, and the computational setting where DD is used to reduce noise in every computational gate. We focus on the well-known schemes of periodic (or universal) DD and its extension, concatenated DD, for scaling up its power. The qualitative conclusions from our analysis of these two schemes nevertheless apply to other DD approaches. In the presence of noisy pulses, DD does not always mitigate errors. It does so only when the added noise from the imperfect DD pulses does not outweigh the increased ability in averaging away the original background noise. We present break-even conditions that delineate when DD is useful, and further find that there is a limit in the performance of concatenated DD, specifically in how far one can concatenate the DD pulse sequences before the added noise no longer offers any further benefit in error mitigation.