Fast quantum computation at arbitrarily low energy

TL;DR

This paper challenges the notion that energy alone limits quantum computational speed by presenting Hamiltonians where clock speed can grow arbitrarily fast, emphasizing the need for additional physical constraints.

Contribution

It introduces local time-independent Hamiltonians with unbounded computational speed, showing energy constraints are insufficient to set speed limits.

Findings

Hamiltonians with arbitrarily high clock speeds relative to energy.

Energy considerations alone do not impose upper bounds on quantum computation speed.

Additional physical limits are necessary to bound computational speed.

Abstract

One version of the energy-time uncertainty principle states that the minimum time $T_{\perp}$ for a quantum system to evolve from a given state to any orthogonal state is $h/(4 \Delta E)$ where $\Delta E$ is the energy uncertainty. A related bound called the Margolus-Levitin theorem states that $T_{\perp} \geq h/(2 E)$ where E is the expectation value of energy and the ground energy is taken to be zero. Many subsequent works have interpreted $T_{\perp}$ as defining a minimal time for an elementary computational operation and correspondingly a fundamental limit on clock speed determined by a system's energy. Here we present local time-independent Hamiltonians in which computational clock speed becomes arbitrarily large relative to E and $\Delta E$ as the number of computational steps goes to infinity. We argue that energy considerations alone are not sufficient to obtain an upper bound on computational speed, and that additional physical assumptions such as limits to information density and information transmission speed are necessary to obtain such a bound.