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Walsh Codes for Multi-Qubit Decoupling with Fluctuating Ising Crosstalk

Francesco Vicari

Published Oct 26, 2026• 0 citations
Publisher

Abstract

Digital dynamical decoupling (DD) of an n-qubit register assigns each qubit a Walsh switching function. Static pairwise ZZ crosstalk is removed to first order by any set of distinct indices (orthogonal arrays). When Ising couplings fluctuate, the products of pair switching functions must also be filtered. Nested sequences achieve this but scale exponentially. We treat the index selection under a fixed slot budget (~4n) as a code-design problem. The infidelity is additive: a spectral cost per single-qubit function plus one per pair product. This cost ranks fidelities with Spearman ρ = 0.97–0.98. With static couplings, the maximal-sequency assignment reaches the computed optimum. With fluctuating couplings, the maximal-sequency sequence collapses (F = 0.302 vs 0.700 at n = 10, 64 slots). The optimum moves to a fast subspace closed under XOR. At fixed total time the code outperforms nested Uhrig decoupling (n = 5: F = 0.929 with 221 pulses versus 0.576 with 1023). In a 6-qubit density-matrix check with T1 and finite pulses it is the best design in all four conditions tested, with paired differences over the natural Hadamard order all above zero. Scripts and data: doi:10.5281/zenodo.23661132.

DOI
10.5281/zenodo.23661599

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Digital dynamical decoupling (DD) of an n-qubit register assigns each qubit a Walsh switching function. Static pairwise ZZ crosstalk is removed to first order by any set of distinct indices (orthogonal arrays). When Ising couplings fluctuate, the products of pair switching functions must also be filtered. Nested sequences achieve this but scale exponentially. We treat the index selection under a fixed slot budget (~4n) as a code-design problem. The infidelity is additive: a spectral cost per single-qubit function plus one per pair product. This cost ranks fidelities with Spearman ρ = 0.97–0.98. With static couplings, the maximal-sequency assignment reaches the computed optimum. With fluctuating couplings, the maximal-sequency sequence collapses (F = 0.302 vs 0.700 at n = 10, 64 slots). The optimum moves to a fast subspace closed under XOR. At fixed total time the code outperforms nested Uhrig decoupling (n = 5: F = 0.929 with 221 pulses versus 0.576 with 1023). In a 6-qubit density-matrix check with T1 and finite pulses it is the best design in all four conditions tested, with paired differences over the natural Hadamard order all above zero. Scripts and data: doi:10.5281/zenodo.23661132.
Theoretically understanding and experimentally characterizing and modifying the underlying Hamiltonian of a quantum system is of utmost importance in achieving high-fidelity quantum gates for quantum computing. In this work, we explore the use of dynamical decoupling (DD) in characterizing and suppressing undesired two-qubit couplings as well as the underlying single-qubit decoherence, both significant hurdles to achieving precise quantum control and realizing quantum computing on many hardware prototypes. Through discrete search of DD sequences, we identify sequences that protect against decoherence and selectively target unwanted two-qubit interactions of general form. On a transmon-qubit-based superconducting quantum device, we identify separate white and 1/f1/f1/f noise components underlying the single-qubit decoherence and a static ZZ coupling between pairs of qubits. A family of syncopated DD sequences is found and their efficiency is demonstrated in two-qubit benchmarking experiments. The syncopated decoupling technique significantly boosts performance in a realistic algorithmic quantum circuit.
Abstract Dynamical decoupling (DD) is one of the simplest error suppression methods, aiming to enhance the coherence of qubits in open quantum systems. Moreover, DD has demonstrated effectiveness in reducing coherent crosstalk, one major error source in near-term quantum hardware, which manifests from two types of interactions. Static crosstalk exists in various hardware platforms, including superconductor and semiconductor qubits, by virtue of always-on qubit-qubit coupling. Additionally, driven crosstalk may occur as an unwanted drive term due to leakage from driven gates on other qubits. Here we explore a novel staggered DD protocol tailored for multi-qubit systems that suppresses the decoherence error and both types of coherent crosstalk. We develop two experimental setups—an ‘idle–idle’ experiment in which two pairs of qubits undergo free evolution simultaneously and a ‘driven-idle’ experiment in which one pair is continuously driven during the free evolution of the other pair. These experiments are performed on an IBM Quantum superconducting processor and demonstrate the significant impact of the staggered DD protocol in suppressing both types of coherent crosstalk. When compared to the standard DD sequences from state-of-the-art methodologies with the application of X 2 sequences, our staggered DD protocol enhances circuit fidelity by 19.7% and 8.5%, respectively, in addressing these two crosstalk types.
We introduce a high-order dynamical decoupling (DD) scheme for arbitrary bounded system-bath interactions in the weak-coupling regime. Given any decoupling group G\mathcal GG that averages the interaction to zero, our construction {guarantees the existence of pulse sequences with at most (∣G∣−1)K(|\mathcal G|-1)K(∣G∣−1)K pulses}, while canceling all error terms linear in the system-bath coupling strength up to order KKK in the total evolution time. As a corollary, for an nnn-qubit system with kkk-local system-bath interactions, we obtain an O(nk−1K)\mathcal{O}(n^{k-1}K)O(nk−1K)-pulse sequence, a significant improvement over existing schemes with O(exp⁡(n))\mathcal{O}(\exp(n))O(exp(n)) pulses (for k=O(1)k=\mathcal{O}(1)k=O(1)). The construction is obtained via a mapping to the continuous necklace-splitting problem, which asks how to cut a multi-colored interval into pieces that give each party the same share of every color. We provide explicit pulse sequences for suppressing general single-qubit decoherence, prove that the pulse count is asymptotically optimal, and verify the predicted error scaling in numerical simulations. For the same number of pulses, we observe that our sequences outperform the state-of-the-art Quadratic DD in the weak-coupling regime. We also construct explicit high-order sequences for suppressing representative 2-local noise models. Finally, the same construction extends to suppress slow, time-dependent classical noise and to filter-function design.
We model repetitive quantum error correction (QEC) with the single-error-correcting five-qubit code on a network of individually controlled qubits with always-on Ising couplings. We use our previously designed universal set of quantum gates based on sequences of shaped decoupling pulses. In addition to being accurate quantum gates, the sequences also provide dynamical decoupling (DD) of low-frequency phase noise. The simulation involves integrating the unitary dynamics of six qubits over the duration of tens of thousands of control pulses, using classical stochastic phase noise as a source of decoherence. The combined DD and QEC protocol dramatically improves the coherence, with the QEC alone being responsible for more than an order of magnitude infidelity reduction.
Dynamical decoupling (DD) emerges as a promising tool to suppress decoherence of quantum systems by eliminating the system-environment coupling. We investigate its power in combating pure dephasing of a qubit under random classical noise. Interestingly, the decoherence rate can be viewed as the interplay between noise spectrum and a controllable filter function. More specifically, the filter function is determined by the DD sequence we employed. Here we take Uhrig DD (UDD) as a benchmark sequence to illustrate the filtering process. We find that UDD sequence works as a perfect dynamical filter for low-frequency noise. For spectrum with hard cutoff, UDD increases the performance order by order with linearly increased number of pulses. However, for noise spectrum with soft cutoff, the required pulse number grows exponentially.
Aperiodic dynamical decoupling (DD) sequences of π pulses are of great interest to decoherence control and have been recently extended from single-qubit to two-qubit systems. If the environmental noise power spectrum is made available, then one may further optimize aperiodic DD sequences to reach higher efficiency of decoherence suppression than known universal schemes. This possibility is investigated in this work for the protection of two-qubit states, using an exactly solvable pure dephasing model including both local and nonlocal noise. The performance of optimized DD sequences in protecting two-qubit states is compared with that achieved by nested Uhrig's DD (nested-UDD) sequences, for several different types of noise spectrum. Except for the cases with noise spectrum decaying slowly in the high-frequency regime, optimized DD sequences with tens of control pulses can perform orders of magnitude better than that of nested-UDD. A two-qubit system with highly unbalanced local noise is also examined to shed more light on a recent experiment. Possible experiments that may be motivated by this work are discussed.