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arXiv:2105.02882 · PRX Quantum

Conditions for Equivalent Noise Sensitivity of Geometric and Dynamical Quantum Gates

Ralph Kenneth Colmenar, Utkan Güngördü, J. P. Kestner

Published Jul 19, 2022• 12 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/prxquantum.3.030310

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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.
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.
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.
We derive an integral expression for the filter-transfer function of an arbitrary one-qubit gate through the use of dynamical invariant theory and Hamiltonian reverse engineering. We use this result to define a cost function which can be efficiently optimized to produce one-qubit control pulses that are robust against specified frequency bands of the noise power spectral density. We demonstrate the utility of our result by generating optimal control pulses that are designed to suppress broadband detuning and pulse amplitude noise. We report an order of magnitude improvement in gate fidelity in comparison with known composite pulse sequences. More broadly, we also use the same theoretical framework to prove the robustness of nonadiabatic geometric quantum gates under specific error models and control constraints.
We derive an integral expression for the filter-transfer function of an arbitrary one-qubit gate through the use of dynamical invariant theory and Hamiltonian reverse engineering. We use this result to define a cost functional which can be efficiently optimized to produce one-qubit control pulses that are robust against specified frequency bands of the noise power spectral density. We demonstrate the utility of our result by generating optimal control pulses that are designed to suppress broadband detuning and pulse amplitude noise. We report an order of magnitude improvement in gate fidelity in comparison with known composite pulse sequences. More broadly, we also use the same theoretical framework to prove the robustness of nonadiabatic geometric quantum gates under specific error models and control constraints.
The ability to perform gates in multiqubit systems that are robust to noise is of crucial importance for the advancement of quantum information technologies.However, finding control pulses that cancel noise while performing a gate is made difficult by the intractability of the time-dependent Schrödinger equation, especially in multiqubit systems.Here, we show that this issue can be sidestepped by using a formalism in which the cumulative error during a gate is represented geometrically as a curve in a multidimensional Euclidean space.Cancelation of noise errors to leading order corresponds to closure of the curve, a condition that can be satisfied without solving the Schrödinger equation.We develop and uncover general properties of this geometric formalism, and derive a recursion relation that maps control fields to curvatures for Hamiltonians of arbitrary dimension.We demonstrate the utility of the formalism by employing it to design pulses that simultaneously correct against both noise errors and crosstalk for two qubits coupled by an Ising interaction.We give examples both of a single-qubit rotation and a two-qubit maximally entangling gate.The results obtained in this example are relevant to both superconducting transmon qubits and semiconductor quantum-dot spin qubits.We propose this geometric formalism as a general technique for pulse-induced error suppression in quantum computing gate operations.
The realistic application of geometric quantum computation is crucially dependent on an unproved robustness conjecture, claiming that geometric quantum gates are more resilient against random noise than dynamic gates. We propose a suitable model that allows a direct and fair comparison between geometrical and dynamical operations. In the presence of stochastic control errors we find that the maximum of gate fidelity corresponds to quantum gates with a vanishing dynamical phase. This is a clear evidence for the robustness of nonadiabatic geometric quantum computation. The predictions here presented can be experimentally tested in almost all of the already existing quantum computer candidates.