On the Interpretation of Energy as the Rate of Quantum Computation

On the Interpretation of Energy as the Rate of Quantum Computation Over the last few decades, developments in the physical limits of computing and quantum computing have increasingly taught us that it can be helpful to think about physics itself in computational terms. For example, work over the last decade has shown that the energy of a quantum system limits the rate at which it can perform significant computational operations, and suggests that we might validly interpret energy as in fact being the speed at which a physical system is “computing,” in some appropriate sense of the word. In this paper, we explore the precise nature of this connection. Elementary results in quantum theory show that the Hamiltonian energy of any quantum system corresponds exactly to the angular velocity of state-vector rotation (defined in a certain natural way) in Hilbert space, and also to the rate at which the state-vector’s components (in any basis) sweep out area in the complex plane. The total angle traversed (or area swept out) corresponds to the action of the Hamiltonian operator along the trajectory, and we can also consider it to be a measure of the “amount of computational effort exerted” by the system, or effort for short. For any specific quantum or classical computational operation, we can (at least in principle) calculate its difficulty, defined as the minimum effort required to perform that operation on a worst-case input state, and this in turn determines the minimum time required for quantum systems to carry out that operation on worst-case input states of a given energy. As examples, we calculate the difficulty of some basic 1-bit and n-bit quantum and classical operations in an simple unconstrained scenario http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Quantum Information Processing Springer Journals

On the Interpretation of Energy as the Rate of Quantum Computation

On the Interpretation of Energy as the Rate of Quantum Computation

Quantum Information Processing, Vol. 4, No. 4, October 2005 (© 2005) DOI: 10.1007/s11128-005-7851-5 On the Interpretation of Energy as the Rate of Quantum Computation Michael P. Frank Received January 3, 2005; accepted July 12, 2005 Over the last few decades, developments in the physical limits of computing and quantum computing have increasingly taught us that it can be helpful to think about physics itself in computational terms. For example, work over the last decade has shown that the energy of a quantum system limits the rate at which it can perform significant computational operations, and suggests that we might validly interpret energy as in fact being the speed at which a physical system is “computing,” in some appropriate sense of the word. In this paper, we explore the precise nature of this connection. Elementary results in quantum theory show that the Hamilto- nian energy of any quantum system corresponds exactly to the angular velocity of state-vector rotation (defined in a certain natural way) in Hilbert space, and also to the rate at which the state-vector’s components (in any basis) sweep out area in the complex plane. The total angle traversed (or area swept out) corresponds to the action of the Hamiltonian operator along the trajectory, and we can also consider it to be a measure of the “amount of computational effort exerted” by the system, or effort for short. For any specific quantum or classical computational operation, we can (at least in principle) calculate its difficulty, defined as the mini- mum effort required to perform that operation on a worst-case input state, and this in turn determines the minimum time required for quantum systems to carry out that operation on worst-case input states of a given energy. As examples, we calcu- late the difficulty of some basic 1-bit and n-bit quantum and classical operations in an simple unconstrained scenario. KEY WORDS: Time evolution operator;...
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Publisher
Springer Journals
Copyright
Copyright © 2005 by Springer Science+Business Media, Inc.
Subject
Physics; Quantum Information Technology, Spintronics; Quantum Computing; Data Structures, Cryptology and Information Theory; Quantum Physics; Mathematical Physics
ISSN
1570-0755
eISSN
1573-1332
D.O.I.
10.1007/s11128-005-7851-5
Publisher site
See Article on Publisher Site

Abstract

Over the last few decades, developments in the physical limits of computing and quantum computing have increasingly taught us that it can be helpful to think about physics itself in computational terms. For example, work over the last decade has shown that the energy of a quantum system limits the rate at which it can perform significant computational operations, and suggests that we might validly interpret energy as in fact being the speed at which a physical system is “computing,” in some appropriate sense of the word. In this paper, we explore the precise nature of this connection. Elementary results in quantum theory show that the Hamiltonian energy of any quantum system corresponds exactly to the angular velocity of state-vector rotation (defined in a certain natural way) in Hilbert space, and also to the rate at which the state-vector’s components (in any basis) sweep out area in the complex plane. The total angle traversed (or area swept out) corresponds to the action of the Hamiltonian operator along the trajectory, and we can also consider it to be a measure of the “amount of computational effort exerted” by the system, or effort for short. For any specific quantum or classical computational operation, we can (at least in principle) calculate its difficulty, defined as the minimum effort required to perform that operation on a worst-case input state, and this in turn determines the minimum time required for quantum systems to carry out that operation on worst-case input states of a given energy. As examples, we calculate the difficulty of some basic 1-bit and n-bit quantum and classical operations in an simple unconstrained scenario

Journal

Quantum Information ProcessingSpringer Journals

Published: Oct 1, 2005

References

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