Quantum control and fault-tolerant quantum computing (FTQC) are two of the cornerstones on which the hope of realizing a large-scale quantum computer is pinned, yet only preliminary steps have been taken towards formalizing the interplay between them. Here we explore this interplay using the powerful strategy of dynamical decoupling (DD), and show how it can be seamlessly and optimally integrated with FTQC. To this end we show how to find the optimal decoupling generator set (DGS) for various subspaces relevant to FTQC, and how to simultaneously decouple them. We focus on stabilizer codes, which represent the largest contribution to the size of the DGS, showing that the intuitive choice comprising the stabilizers and logical operators of the code is in fact optimal, i.e., minimizes a natural cost function associated with the length of DD sequences. Our work brings hybrid DD-FTQC schemes, and their potentially considerable advantages, closer to realization.
Quantum error correction/detection (QEC/QED) and dynamical decoupling (DD) are tools for protecting quantum information. A natural goal is to combine them to outperform either approach alone. Such a benefit is not automatic: physical DD can conflict with encoded subspace, and QEC performance is governed by the errors that survive decoding, not just those DD suppresses. We analyze a hybrid memory cycle where DD is implemented logically (LDD) using normalizer elements of an
[[n,k,d]] stabilizer code, followed by a round of syndrome measurement and recovery (or, in the detection setting, postselection on a trivial syndrome). In an effective Pauli model with physical error probability
p, LDD suppression factor
pDD, and recovery imperfection rate
pQEC (or
pQED), we derive closed-form entanglement-fidelity expressions for QEC-only, QED-only, LDD-only, physical DD, and the hybrid LDD+QEC protocol. The formulas are expressed via a small set of code-dependent weight enumerator polynomials, making the role of the decoder and the LDD group explicit. For ideal recovery LDD+QEC outperforms QEC-only iff the conditional fraction of uncorrectable Pauli errors is larger in the LDD-suppressed sector than in the unsuppressed sector. In the low-noise regime, a sufficient design rule guaranteeing hybrid advantage is that LDD suppresses at least one minimum-weight uncorrectable Pauli error for the chosen recovery map. We show how stabilizer-equivalent choices of LDD generators can enforce this condition. We supplement our analysis with numerical results for the
[[7,1,3]] Steane code and a
[[13,1,3]] code, mapping regions of hybrid-protocol advantage in parameter space beyond the small-
p regime. Our work illustrates the need for co-design of the code, decoder, and logical decoupling group, and clarifies the conditions under which the hybrid LDD+QEC protocol is advantageous.
Quantum computation (QC) relies on the ability to implement high-fidelity quantum gate operations and successfully preserve quantum state coherence. One of the most challenging obstacles for reliable QC is overcoming the inevitable interaction between a quantum system and its environment. Unwanted interactions result in decoherence processes that cause quantum states to deviate from a desired evolution, consequently leading to computational errors and loss of coherence. Dynamical decoupling (DD) is one such method, which seeks to attenuate the effects of decoherence by applying strong and expeditious control pulses solely to the system. Provided the pulses are applied over a time duration sufficiently shorter than the correlation time associated with the environment dynamics, DD effectively averages out undesirable interactions and preserves quantum states with a low probability of error, or fidelity loss. In this study various aspects of this approach are studied from sequence construction to applications of DD to protecting QC. First, a comprehensive examination of the error suppression properties of a near-optimal DD approach is given to understand the relationship between error suppression capabilities and the number of required DD control pulses in the case of ideal, instantaneous pulses. While such considerations are instructive for examining DD efficiency, i.e., performance vs the number of control pulses, high-fidelity DD in realizable systems is difficult to achieve due to intrinsic pulse imperfections which further contribute to decoherence. As a second consideration, it is shown how one can overcome this hurdle and achieve robustness and recover high-fidelity DD in the presence of faulty control pulses using Genetic Algorithm optimization and sequence symmetrization. Thirdly, to illustrate the implementation of DD in conjunction with QC, the utilization of DD and quantum error correction codes (QECCs) as a protection method for adiabatic quantum computing (AQC) is discussed. A performance comparison between two deterministic DD schemes is given, where preference towards one particular method is found due to sequence structure and procedure from which higher accuracy sequences are generated.
We study how dynamical decoupling (DD) pulse sequences can improve the reliability of quantum computers. We prove upper bounds on the accuracy of DD-protected quantum gates and derive sufficient conditions for DD-protected gates to outperform unprotected gates. Under suitable conditions, fault-tolerant quantum circuits constructed from DD-protected gates can tolerate stronger noise and have a lower overhead cost than fault-tolerant circuits constructed from unprotected gates. Our accuracy estimates depend on the dynamics of the bath that couples to the quantum computer and can be expressed either in terms of the operator norm of the bath's Hamiltonian or in terms of the power spectrum of bath correlations; we explain in particular how the performance of recursively generated concatenated pulse sequences can be analyzed from either viewpoint. Our results apply to Hamiltonian noise models with limited spatial correlations.
Constructing high-fidelity control pulses that are robust to control and system/environment fluctuations is a crucial objective for quantum information processing (QIP). We combine dynamical decoupling (DD) with optimal control (OC) to identify control pulses that achieve this objective numerically. Previous DD work has shown that general errors up to (but not including) third order can be removed from {pi}- and {pi}/2-pulses without concatenation. By systematically integrating DD and OC, we are able to increase pulse fidelity beyond this limit. Our hybrid method of quantum control incorporates a newly-developed algorithm for robust OC, providing a nested DD-OC approach to generate robust controls. Motivated by solid-state QIP, we also incorporate relevant experimental constraints into this DD-OC formalism. To demonstrate the advantage of our approach, the resulting quantum controls are compared to previous DD results in open and uncertain model systems.
Dynamic quantum circuits integrate unitary evolution with mid-circuit measurement and feedforward, enabling conditional operations essential for efficient quantum algorithms and foundational for fault-tolerant quantum computation. However, such operations introduce measurement-induced errors and control constraints that are not addressed by conventional error-suppression techniques. Here, we introduce an empirical learning framework that optimizes dynamical decoupling (DD) sequences for dynamic circuits at the level of circuit subintervals and qubit subregisters. Applying empirically learned DD sequences, we achieve a three-fold reduction in average dynamic circuit error rates as measured via randomized benchmarking. We apply the learned strategies to the dynamic circuit implementation of the quantum Fourier transform with measurement (QFT+M), demonstrating nontrivial process fidelity on connected chains of up to 20 qubits. Applying the resulting enhancement, we perform a high signal-to-noise QFT immediately following the preparation of a 10-qubit entangled state. Our results demonstrate that empirically optimized DD systematically outperforms theoretically derived sequences for dynamic circuits, establishing it as an efficient approach for error suppression in dynamic quantum circuits, with direct relevance to applications requiring measurement and feedback such as quantum error correction.