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Physical Review Letters

Minimum Energy Requirements for Quantum Computation

Julio R. Gea-Banacloche

Published Nov 4, 2002• 74 citations
Publisher

Abstract

A lower bound on the amount of energy needed to carry out an elementary logical operation on a quantum computer, with a given accuracy and in a given time, is derived. The bound arises from the requirement that the controls used to manipulate the qubits, which ultimately are themselves quan-tum mechanical systems, must nonetheless be classical to a sufficiently good approximation; it is expected to hold under a wide variety of conditions, and independently of the nature of the physical systems used to encode the qubits. This could have important consequences for very large-scale quantum computations.

DOI
10.1103/physrevlett.89.217901

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A lower bound on the amount of energy needed to carry out an elementary logical operation on a quantum computer, with a given accuracy and in a given time, is derived. The bound arises from the requirement that the controls used to manipulate the qubits, which ultimately are themselves quan-tum mechanical systems, must nonetheless be classical to a sufficiently good approximation; it is expected to hold under a wide variety of conditions, and independently of the nature of the physical systems used to encode the qubits. This could have important consequences for very large-scale quantum computations.
A lower bound on the amount of energy needed to carry out an elementary logical operation on a qubit system, with a given accuracy and in a given time, has been recently postulated. This paper is an attempt to formalize this bound and explore the conditions under which it may be expected to hold. For a specific, important case (namely, when the control system is a quantized electromagnetic field) it is shown how one can extend this result to a generally stronger constraint on the minimum energy density required, per pulse.
Abstract While recent breakthroughs in quantum computing promise the nascence of the quantum information age, quantum states remain delicate to control. Moreover, the required energy budget for large scale quantum applications has only sparely been considered. Addressing either of these issues necessitates a careful study of the most energetically efficient implementation of elementary quantum operations. In the present analysis, we show that this optimal control problem can be solved within the powerful framework of quantum speed limits. To this end, we derive state-independent lower bounds on the energetic cost, from which we find the universally optimal implementation of unitary quantum gates, for both single and N -qubit operations.
Landauer’s principle asserts that any computation has an unavoidable energy cost that grows proportionally to its degree of logical irreversibility. But even a logically reversible operation, when run on a physical processor that operates on different energy levels, requires energy. Here we quantify this energy requirement, providing upper and lower bounds that coincide up to a constant factor. We derive these bounds from a general quantum resource-theoretic argument, which implies that the initial resource requirement for implementing a unitary operation within an error ε grows like 1/√ε times the amount of resource generated by the operation. Applying these results to quantum circuits, we find that their energy requirement can, by an appropriate design, be made independent of their time complexity.
Landauer's principle asserts that any computation has an unavoidable energy cost that grows proportionally to its degree of logical irreversibility.But even a logically reversible operation, when run on a physical processor that operates on different energy levels, requires energy.Here we quantify this energy requirement, providing upper and lower bounds that coincide up to a constant factor.We derive these bounds from a general quantum resource-theoretic argument, which implies that the initial resource requirement for implementing a unitary operation within an error ϵ grows like 1= ffiffi ffi ϵ p times the amount of resource generated by the operation.Applying these results to quantum circuits, we find that their energy requirement can, by an appropriate design, be made independent of their time complexity.
One version of the energy-time uncertainty principle states that the minimum time ${T}_{\ensuremath{\perp}}$ for a quantum system to evolve from a given state to any orthogonal state is $h/(4\mathrm{\ensuremath{\Delta}}E)$, where $\mathrm{\ensuremath{\Delta}}E$ is the energy uncertainty. A related bound called the Margolus-Levitin theorem states that ${T}_{\ensuremath{\perp}}\ensuremath{\ge}h/(2\ensuremath{\langle}E\ensuremath{\rangle})$, where $\ensuremath{\langle}E\ensuremath{\rangle}$ is the expectation value of energy and the ground energy is taken to be zero. Many subsequent works have interpreted ${T}_{\ensuremath{\perp}}$ 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 $\ensuremath{\langle}E\ensuremath{\rangle}$ and $\mathrm{\ensuremath{\Delta}}E$ 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.