Quantum-mechanical computers and uncomputability
Abstract
The time evolution operator for any quantum-mechanical computer is diagonalizable, but to obtain the diagonal decomposition of a program state of the computer is as hard as actually performing the computation corresponding to the program. In particular, if a quantum-mechanical system is capable of universal computation, then the diagonal decomposition of program states is uncomputable. As a result, in a universe in which local variables support universal computation, a quantum-mechanical theory for that universe that supplies its spectrum cannot supply the spectral decomposition of the computational variables. A ``theory of everything'' can be simultaneously correct and fundamentally incomplete.
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A wide variety of physical systems-ranging from nearest neighbor Ising models to quantum electrodynamics and conformal field theories-are known to be computationally universal.
“Although computational universality might at first seem to be a stringent demand on a physical system, a wide variety of physical systems - ranging from nearest neighbor Ising models 52 to quantum electrodynamics 84 and conformal field theories 86 - are known to be computationally universal 51 -53 , 55 -65 .”
From Ultimate physical limits to computation (2000) · cites this paper as [51]
Cites
Quantum theory, the Church–Turing principle and the universal quantum computer
Deutsch, D. (1985) · Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences · doi:10.1098/rspa.1985.0070 · Cited by 4,597 · see citing works
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Quantum computation and Shor's factoring algorithm
Ekert, A., & Jozsa, R. (1996) · Reviews of Modern Physics · doi:10.1103/revmodphys.68.733 · Cited by 1,354 · see citing works
Ultimate physical limits to computation
Lloyd, S. (2000) · Nature · doi:10.1038/35023282 · Cited by 1,120 · see citing works
Self-citation
- Authors
- Lloyd, S.
- Affiliations
- Los Alamos National Laboratory (US)
- Corresponding author
- Lloyd, Seth
- ORCID
- Lloyd, Seth
- Venue
- Physical Review Letters
- Publisher
- American Physical Society (APS)
- Published
- 1993-08-09
- Volume
- 71
- Issue
- 6
- Pages
- 943-946
- ISSN
- 0031-9007 (ISSN-L)
- OpenAlex
- W1979105960
- Cited by
- 31
- References
- 1
- Citations by year
- 2025: 5 · 2023: 1 · 2022: 3 · 2021: 2 · 2020: 3 · 2018: 1 · 2017: 2 · 2015: 1 · 2013: 1
- FWCI
- 2.00
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Physical Review Letters (publishedVersion)
- PubMed (publishedVersion)
- Topics
- Physical Sciences › Computer Science › Computational Theory and Mathematics › Computability, Logic, AI Algorithms
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
- Keywords
- quantum computing hardware, uncomputability, diagonalization, evolution operator, universal computation, program state, spectral decomposition, theory of everything, incompleteness, quantum foundations
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