Fast quantum computation at arbitrarily low energy

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 $T_{\perp }\ge h/(2E)$ 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.

Added ref 36. Published version

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• Holographic complexity and thermodynamics of AdS black holes
• Holographic complexity for black branes with momentum relaxation
• Hamiltonian simulation for low-energy states with optimal time dependence
• Holographic complexity: braneworld gravity versus the Lloyd bound
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• Constraints on physical computers in holographic spacetimes
• Holographic complexity of rotating black holes
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