How fast can a quantum gate be? Exact speed limits from geometry

TL;DR

This paper derives fundamental speed limits for quantum gate operations based on geometric and spectral constraints, revealing variability across different gates and providing a geometric framework for understanding quantum speed limits.

Contribution

The authors establish a general, tight quantum speed limit for any unitary evolution with bounded spectral width, applying it to standard quantum gates and introducing a geometric control formalism.

Findings

QSL varies significantly across different quantum gates.

Time-optimal gates correspond to helices in the geometric formalism.

The slowest operator determines the minimal gate time.

Abstract

The speed of quantum evolution is limited under finite energy resources. While most quantum speed limits (QSLs) are formulated in terms of quantum states, they can be extended to the evolution operator itself, and thus impose fundamental limits on how quickly logical gate operations can be implemented on a quantum computer. Here, we derive a general, tight QSL that holds for any unitary evolution under the constraint that the spectral width of the Hamiltonian is bounded. We apply this result to obtain QSLs for several standard quantum gates, including Hadamard, CNOT, and Toffoli gates, finding that the QSL can vary significantly across different gates, including ones with the same entangling power. These findings can be understood geometrically using the Space Curve Quantum Control formalism, which maps unitary evolution to space curves in Euclidean space. In this formalism, the problem of finding QSLs is recast as the problem of finding minimal-length curves obeying a curvature bound. We find that time-optimal gates map to helices of varying dimensions, and that QSLs can be understood from the perspective of a bottleneck principle in which the operator that evolves the slowest governs the minimal gate time.