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arXiv:2610.24039

Designing Robust Quantum Gates

Yuanxing Hao, Marin Bukov

quant-phSubmitted Oct 16, 2026• Announced Oct 19, 2026• 0 citations
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Abstract

For given control channels, quantum control theory determines whether quantum gates can be realized, but not their resilience to errors. We address this problem by establishing existence criteria for robust gate implementation in $d$-level systems. Our criteria reveal that implementability of all gates is not enough: a universal family of robust gates exists if and only if the driven Hamiltonians share no common symmetry. When these criteria hold, we can make gates resilient to any combination of static perturbations, uniform Hamiltonian rescaling, perturbations at a finite set of selected frequencies, and perturbations with arbitrary selected waveforms. We can impose robustness conditions broadly with limited information about errors, e.g., against any static perturbations; we can also tailor these conditions to the platform, e.g., against environment-induced noise with dominant spectral peaks, requiring no spectral details beyond the peak frequencies. We numerically construct explicit protocols for robust gates, reducing average gate infidelity by approximately two to seven orders of magnitude under Hamiltonian perturbations and environment-induced noise. Together, our existence criteria and numerical constructions provide practical guidance for evaluating the potential of quantum platforms and designing robust quantum gates.

Comments
47 pages, 10 figures, 3 tables. The data associated with this manuscript version is available under DOI:10.5281/zenodo.23339159

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For given control channels, quantum control theory determines whether quantum gates can be realized, but not their resilience to errors. We address this problem by establishing existence criteria for robust gate implementation in ddd-level systems. Our criteria reveal that implementability of all gates is not enough: a universal family of robust gates exists if and only if the driven Hamiltonians share no common symmetry. When these criteria hold, we can make gates resilient to any combination of static perturbations, uniform Hamiltonian rescaling, perturbations at a finite set of selected frequencies, and perturbations with arbitrary selected waveforms. We can impose robustness conditions broadly with limited information about errors, e.g., against any static perturbations; we can also tailor these conditions to the platform, e.g., against environment-induced noise with dominant spectral peaks, requiring no spectral details beyond the peak frequencies. We numerically construct explicit protocols for robust gates, reducing average gate infidelity by approximately two to seven orders of magnitude under Hamiltonian perturbations and environment-induced noise. Together, our existence criteria and numerical constructions provide practical guidance for evaluating the potential of quantum platforms and designing robust quantum gates.
Abstract A protocol for implementing a universal set of quantum gates, which are safeguarded from noise through geometric, dephasing‐free, and dynamical‐decoupling approaches is proposed. With properly designed Hamiltonians, the desired interactions are realized directly within the Schrödinger picture rather than relying on the interaction picture. This design confers two advantages. One is that the system's evolution remains continuously protected by the noise‐mitigation strategies at every stage, eliminating the need to revert to the Schrödinger picture. The other one is that the dynamical decoupling technique effectively averages out small residual free terms in the system's Hamiltonian, thus reducing the stringent requirements for precise control of system parameters to avoid errors. Furthermore, numerical results are presented to support our protocol by finding gate fidelities in the absence (presence) of noises or errors, without or with dynamical decoupling technique.
We derive a universal performance limit for coherent quantum control in the presence of modeled and unmodeled uncertainties. For any target unitary WWW that is implementable in the absence of error, we prove that the worst-case (and hence the average) gate fidelity obeys the lower bound $F \ge \Flb\bigl(\tf \Omeff\bigr)$, where $\tf$ is the gate duration and $\Omeff$ is a single frequency-like measure that aggregates \emph{all} bounded uncertainty sources, e.g., coherent control imperfections, unknown couplings, and residual environment interactions, without assuming an initially factorizable system-bath state or a completely positive map. The bound is obtained by combining an interaction-picture averaging method with a Bellman-Gronwall inequality and holds for any finite-norm Hamiltonian decomposition. Hence it applies equally to qubits, multi-level qudits, and ancilla-assisted operations. Because $\Flb$ depends only on the dimensionless product $\tf\Omeff$, it yields a device-independent metric that certifies whether a given hardware platform can, in principle, reach a specified fault-tolerance threshold, and also sets a quantitative target for robust-control synthesis and system identification.
Geometric quantum gates are often expected to be more resilient than dynamical gates against certain types of error, which would make them ideal for robust quantum computing.However, this is still up for debate due to seemingly conflicting results in the literature.Here we use dynamical invariant theory in conjunction with filter functions in order to analytically characterize the noise sensitivity of an arbitrary quantum gate.For any control Hamiltonian that produces a geometric gate, we find that, under certain common conditions, one can construct another control Hamiltonian that produces an equivalent dynamical gate with identical noise sensitivity (as characterized by the filter function).Our result holds for a Hilbert space of arbitrary dimensions, but we illustrate our result by examining experimentally relevant singlequbit scenarios and providing explicit examples of equivalent geometric and dynamical gates.
Quantum computers promise to solve computational problems that are inaccessible to classical machines. The reliable execution of large-scale quantum algorithms requires robust operations in the presence of noise. Quantum error correction addresses this challenge by encoding quantum information redundantly across several physical qubits, such that errors can be detected and corrected during a computation. However, no single error-correcting code intrinsically supports an inherently robust, fault-tolerant implementation of all gates needed for universal quantum computing. Moreover, most modern approaches to both quantum error correction and the implementation of universal logical operations on encoded qubits rely on mid-circuit measurements and feed-forward control. These types of operations are technologically demanding, often slow, and prone to high error rates, and they limit the overall success probability of quantum algorithms. In this thesis, we develop fault-tolerant protocols and circuits for quantum error correction and universal logical computation, with an emphasis on implementations compatible with current hardware systems. We first focus on fault-tolerant quantum error correction on small codes. By comparing different protocols, we identify trade-offs between compact circuit constructions and their robustness to noise. We then assess their practical performance by comparing numerical simulations of noisy circuits with experimental data. Beyond the preservation of encoded information, we study how universal logic can be implemented in a fault-tolerant manner on encoded qubits. A central approach considered in this thesis is code switching, in which logical information is transferred between two distinct error-correcting codes that support complementary sets of inherently fault-tolerant logical gates. In combination, these codes give access to a complete universal fault-tolerant logical gate set. In this thesis, we determine conditions that allow us to construct fault-tolerant protocols for code switching. We adapt these protocols to the capabilities of current hardware, thereby enabling the first experimental realization of universal fault-tolerant logic by means of code switching. However, the overall performance of these schemes is limited by slow and noisy mid-circuit measurements. To address this bottleneck, we construct a suite of measurement-free fault-tolerant universal operations that do not require any measurements during the execution of an algorithm. In this framework, error correction and logical operations are implemented coherently as part of the quantum circuit, without mid-circuit measurements or in-sequence classical control. We introduce measurement-free code-switching schemes that can be scaled to arbitrary code distances through concatenation, and which thereby provide, in principle, arbitrarily strong error suppression. We further tailor our approach to the capabilities of current trapped-ion quantum processors. New circuit constructions support the first experimental implementation of a logical quantum algorithm without mid-circuit measurements, demonstrated through Grover's search on three encoded qubits. Together, the results of this thesis expand the toolbox for early fault-tolerant quantum computation. They show how protocols for quantum error correction and universal logical gates can be designed without mid-circuit measurements with realistic hardware constraints in mind, opening new pathways towards universal quantum logic.
Coherent control errors, for which ideal Hamiltonians are perturbed by unknown multiplicative noise terms, are a major obstacle for reliable quantum computing. In this paper we present a framework for analyzing the robustness of quantum algorithms against coherent control errors using Lipschitz bounds. We derive worst-case fidelity bounds which show that the resilience against coherent control errors is mainly influenced by the norms of the Hamiltonians generating the individual gates. These bounds are explicitly computable even for large circuits and they can be used to guarantee fault tolerance via threshold theorems. Moreover, we apply our theoretical framework to derive a guideline for robust quantum algorithm design and transpilation, which amounts to reducing the norms of the Hamiltonians. Using the three-qubit quantum Fourier transform as an example application, we demonstrate that this guideline targets robustness more effectively than existing ones based on circuit depth or gate count. Furthermore, we apply our framework to study the effect of parameter regularization in variational quantum algorithms. The practicality of the theoretical results is demonstrated via implementations in simulation and on a quantum computer.
Geometric quantum gates are conjectured to be more resilient than dynamical gates against certain types of error, which makes them ideal for robust quantum computing. However, there are conflicting claims within the literature about the validity of that robustness conjecture. Here we use dynamical invariant theory in conjunction with filter functions in order to analytically characterize the noise sensitivity of an arbitrary quantum gate. Under certain conditions, we find that there exists a transformation of the Hamiltonian that leaves invariant the final gate and noise sensitivity (as characterized by the filter function) while changing the phase from geometric to dynamical. Our result holds for a Hilbert space of arbitrary dimensions, but we illustrate our result by examining experimentally relevant single-qubit scenarios and providing explicit constructions of such a transformation.