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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Energy requirements for quantum computation

Julio R. Gea-Banacloche

Published May 9, 2003• 2 citations
Publisher

Abstract

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.

DOI
10.1117/12.488787

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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.
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.
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.
Quantum measurements are an essential ingredient in numerous protocols and quantum technologies.As a characteristic trait, quantum measurements are typically employed a vast number of times to allow for higher accuracy, better control or just increased performance of quantum devices.In this thesis, we pursue a resource-theoretic approach to quantum measurements, aiming at the quantification of their energetic cost.While there exists a plethora of literature on this topic already, we develop a general framework that allows us to unify several of these previous results and to improve them systematically.We highlight an issue common to most of these previous works, where a notion of quantum measurement is employed which is not detailed enough to be applicable to several realistic and important applications for quantum measurement.More concretely, previous work typically involves a state transformation concept, similar to thermodynamic considerations, where only a specific initial and final state of the measured system are specified to characterise the process.For real applications however, quantum measurement devices are typically required to perform the measurement for all possible initial states of the measured system.In this thesis we analyse this requirement and obtain as a consequence much stronger energy constraints on quantum measurements than previously known.For particular forms of measurement, namely dephasing channels, projective quantum instruments, and projection valued measures, we even determine the exact energy costs in contrast to the common lower bound statements in previous literature.Remarkably, the exact cost of projective quantum instruments constitutes the strongest energy constraint on quantum measurement known so far.We demonstrate the implications of these results in two important applications, where quantum measurements play a significant role, namely quantum Zeno stabilisation and quantum error correction.The exact energetic requirements we compute for these two protocols show that any restriction on the amount of available energy immediately places fundamental constraints on the performance of these protocols.
We show that if an electromagnetic energy pulse in a multimode coherent state with average photon number n‾\overline{n}n is used to carry out the same quantum logical operation on a set of NNN atoms, either simultaneously or sequentially, the overall error probability in the worst-case scenario (i.e., maximized over all the possible initial atomic states) scales as N2∕n‾{N}^{2}∕\overline{n}N2∕n. This means that in order to keep the error probability bounded by NϵNϵNϵ, with $ϵ\ensuremath{\sim}1∕\overline{n}$, one needs to use Nn‾N\overline{n}Nn photons or, equivalently, NNN separate ``minimum-energy'' pulses: in this sense the pulses cannot, in general, be shared. The origin of this phenomenon is found in atom-field entanglement. These results may have important consequences for quantum logic and, in particular, for large-scale quantum computation.