Abstract
The greatest challenge in achieving the high level of control needed for future technologies based on coherent quantum systems is the decoherence induced by the environment. Here, we present an analytical approach that yields explicit constraints on the driving field which are necessary and sufficient to ensure that the leading-order noise-induced errors in a qubit’s evolution cancel exactly. We derive constraints for two of the most common types of noise that arise in qubits: slow fluctuations of the qubit energy splitting and fluctuations in the driving field itself. By theoretically recasting a phase in the qubit’s wavefunction as a topological winding number, we can satisfy the noise-cancelation conditions by adjusting driving field parameters without altering the target state or quantum evolution. We demonstrate our method by constructing robust quantum gates for two types of spin qubit: phosphorous donors in silicon and nitrogen-vacancy centers in diamond.
Highlights
To the time-dependent Schrödinger equation are known[24], a fact which leads to proposals involving sequences of idealized control pulses, such as delta functions or square waveforms, that are often neither optimal nor implemented in real experimental setups. Replacing such waveforms with smoother shapes such as Gaussians[21] can make them easier to generate in real systems, but the fact remains that using a preselected pulse shape repeatedly provides few tunable parameters and leads to unnecessarily long control sequences which may be impractical depending on the physical system in question
We present a general solution to the quantum control problem for a two-level system
We do not preselect a particular waveform to serve as the building block of composite sequences, but rather develop a formalism that systematically generates optimal waveforms
Summary
To the time-dependent Schrödinger equation are known[24], a fact which leads to proposals involving sequences of idealized control pulses, such as delta functions or square waveforms, that are often neither optimal nor implemented in real experimental setups. We obtain a general solution to this problem by deriving a set of constraints which any robust control field must obey and showing how these can be solved systematically. We can construct noise-resistant driving fields by requiring these variations to vanish at the final time t = tf; this imposes constraints on χ0(t), the solutions to which can be input into Eq (2) to obtain forms of Ω 0(t) that implement robust control.
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