Scaling of entangling-gate errors in large ion crystals

Abstract Trapped-ion has shown great advantages in building quantum computers. While high fidelity entangling-gate has been realized for a few ions, how to maintain the high fidelity for large scale trapped-ions remains an open problem. Here, we present an analysis of arbitrary scale ion chains and focus on motional-related errors, reported as one of the leading error sources in state-of-the-art experiments. We theoretically analyze two-qubit entangling-gate infidelity in a large ion crystal under the phase insensitive configuration. To verify our result, we develop an efficient numerical simulation algorithm that avoids exponential increases in of the Hilbert space dimension. For the motional heating error, we derive a much tighter bound of gate infidelity than previously estimated <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>N</mml:mi> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:mi>τ</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> , and we give an intuitive understanding from the trajectories in the phase space of motional modes. Our discoveries may inspire the scheme of pulse design against incoherent errors and shed light on the way toward constructing scalable quantum computers with large ion crystals.