Geometrical Formalism for Dynamically Corrected Gates in Multiqubit Systems

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 Schrodinger 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 multi-dimensional Euclidean space. Cancellation of noise errors to leading order corresponds to closure of the curve, a condition that can be satisfied without solving the Schrodinger 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 examples by using the geometric method to design dynamically corrected gates for a class of two-qubit Hamiltonians that is relevant for both superconducting transmon qubits and semiconductor spin qubits. We propose this geometric formalism as a general technique for pulse-induced error suppression in quantum computing gate operations.