dorsal/arxiv
View SchemaOn the Interpretation of Energy as the Rate of Quantum Computation
| Authors | Michael P. Frank |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0409056 |
| URL | https://arxiv.org/abs/quant-ph/0409056 |
| DOI | 10.1007/s11128-005-7851-5 |
| Journal | Quantum Information Processing 4(4):283-334, Oct. 2005 |
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.
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"abstract": "Over the last few decades, developments in the physical limits of computing\nand quantum computing have increasingly taught us that it can be helpful to\nthink about physics itself in computational terms. For example, work over the\nlast decade has shown that the energy of a quantum system limits the rate at\nwhich it can perform significant computational operations, and suggests that we\nmight validly interpret energy as in fact being the speed at which a physical\nsystem is \"computing,\" in some appropriate sense of the word. In this paper, we\nexplore the precise nature of this connection. Elementary results in quantum\ntheory show that the Hamiltonian energy of any quantum system corresponds\nexactly to the angular velocity of state-vector rotation (defined in a certain\nnatural way) in Hilbert space, and also to the rate at which the state-vector\u0027s\ncomponents (in any basis) sweep out area in the complex plane. The total angle\ntraversed (or area swept out) corresponds to the action of the Hamiltonian\noperator along the trajectory, and we can also consider it to be a measure of\nthe \"amount of computational effort exerted\" by the system, or effort for\nshort. For any specific quantum or classical computational operation, we can\n(at least in principle) calculate its difficulty, defined as the minimum effort\nrequired to perform that operation on a worst-case input state, and this in\nturn determines the minimum time required for quantum systems to carry out that\noperation on worst-case input states of a given energy. As examples, we\ncalculate the difficulty of some basic 1-bit and n-bit quantum and classical\noperations in an simple unconstrained scenario.",
"arxiv_id": "quant-ph/0409056",
"authors": [
"Michael P. Frank"
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"doi": "10.1007/s11128-005-7851-5",
"journal_ref": "Quantum Information Processing 4(4):283-334, Oct. 2005",
"title": "On the Interpretation of Energy as the Rate of Quantum Computation",
"url": "https://arxiv.org/abs/quant-ph/0409056"
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