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Fault-tolerant quantum computation with constant error
Open accessAbstract
In the past year many developments have taken place in the area of quantum error corrections. Recently Shor showed how to perform fault tolerant quantum computation when, , the probability for a fault in one time step per qubit or per gate, is polylogarithmically small. This paper closes the gap and shows how to perform fault tolerant quantum computation when the error probability, , is smaller than some constant threshold, 0 . The cost is polylogarithmic in time and space, and no measurements are used during the quantum computation. The same result is shown also for quantum circuits which operate on nearest neighbors only.
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In the citing paper
“First, the choice to work with circuits which allow parallelism is necessary, because sequential quantum computation, such as quantum Turing machines, cannot be noise resistant[3], since error corrections cannot be applied fast enough to prevent accumulation of errors.”
From Fault-Tolerant Quantum Computation with Constant Error Rate (2008) · cites this paper as [3]
In the citing paper
“In a previous paper, we have shown that quantum systems can be simulated efficiently by a classical Turing machine if the noise is large[3].”
From Fault-Tolerant Quantum Computation with Constant Error Rate (2008) · cites this paper as [3]
In the citing paper
“In [3] it was shown that for a very high noise rate, the quantum circuit behaves in a classical way.”
From Fault-Tolerant Quantum Computation with Constant Error Rate (2008) · cites this paper as [3]
In the citing paper
“Second, by considering more favorable schemes for Hamiltonian decomposition 256,257 , most modern-day simulation algorithms scale in terms of τ max = t ‖ H ‖ max instead of τ , where ‖ H ‖ max is the absolute value of the largest entry of H as a matrix in the computational basis.”
From Quantum Chemistry in the Age of Quantum Computing (2019) · cites this paper as [256]
In the citing paper
“In fact, arbitrarily large quantum computations can be performed with arbitrary accuracy, provided the error level of the elementary components of the quantum computer is below a certain threshold. This important result is called the threshold theorem of quantum computation 59-62”
From Measurement-based quantum computation (2009) · cites this paper as [60]
Citations
Cites
Algorithms for quantum computation: discrete logarithms and factoring
Shor, P. W. (2002) · Proceedings 35th Annual Symposium on Foundations of Computer Science · doi:10.1109/sfcs.1994.365700 · Cited by 8,713
Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer
Shor, P. W. (1997) · SIAM Journal on Computing · doi:10.1137/s0097539795293172 · Cited by 8,472
Simulating physics with computers
Feynman, R. P. (1982) · International Journal of Theoretical Physics · doi:10.1007/bf02650179 · Cited by 7,691
Quantum computation with quantum dots
Loss, D., & DiVincenzo, D. P. (1998) · Physical Review A · doi:10.1103/physreva.57.120 · Cited by 6,840
A single quantum cannot be cloned
Wootters, W. K., & Zurek, W. H. (1982) · Nature · doi:10.1038/299802a0 · Cited by 5,406 · cited 1×
“It was argued by scientists that due to the continuity of quantum states, and to the fact that quantum states cannot be cloned[ 74 ], it will be impossible to protect quantum information from noise[72, 47].”
BackgroundMixed-state entanglement and quantum error correction
Bennett, C. H., DiVincenzo, D. P., Smolin, J. A., & Wootters, W. K. (1996) · Physical Review A · doi:10.1103/physreva.54.3824 · Cited by 5,272
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
Scheme for reducing decoherence in quantum computer memory
Shor, P. W. (1995) · Physical Review A · doi:10.1103/physreva.52.r2493 · Cited by 4,479
Elementary gates for quantum computation
Barenco, A., Bennett, C. H., Cleve, R., DiVincenzo, D. P., Margolus, N., Shor, P., Sleator, T., Smolin, J. A., & Weinfurter, H. (1995) · Physical Review A · doi:10.1103/physreva.52.3457 · Cited by 4,311
Quantum Mechanics Helps in Searching for a Needle in a Haystack
Grover, L. K. (1997) · Physical Review Letters · doi:10.1103/physrevlett.79.325 · Cited by 4,306 · cited 1×
“10, 11] and quantum algorithms[25, 65, 36 ], in particular Shor’s factorization algorithm [62].”
BackgroundQuantum Computations with Cold Trapped Ions
Cirac, J. I., & Zoller, P. (1995) · Physical Review Letters · doi:10.1103/physrevlett.74.4091 · Cited by 3,733
Rapid solution of problems by quantum computation
Deutsch, D., & Jozsa, R. (1992) · Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences · doi:10.1098/rspa.1992.0167 · Cited by 2,700
Decoherence and the Transition from Quantum to Classical
Zurek, W. H. (1991) · Physics Today · doi:10.1063/1.881293 · Cited by 2,632
Quantum Theory: Concepts and Methods
Peres, A., & Ballentine, L. E. (1995) · American Journal of Physics · doi:10.1119/1.17946 · Cited by 2,561
Completeness theorems for non-cryptographic fault-tolerant distributed computation
Ben-Or, M., & Wigderson, A. (1988) · ACM Symposium on Theory of Computing (STOC) · doi:10.1145/62212.62213 · Cited by 2,529
Self-citationGood quantum error-correcting codes exist
Calderbank, A. R., & Shor, P. W. (1996) · Physical Review A · doi:10.1103/physreva.54.1098 · Cited by 2,516
Cohen-Tannoudji, C. N., & Laloë, F. (1977) · Cited by 1,886
Probabilistic Logic and the Synthesis of Reliable Organisms from Unreliable Components
von Neumann, J. (1956) · Cited by 1,652
Theory of quantum error-correcting codes
Knill, E., & Laflamme, R. (1997) · Physical Review A · doi:10.1103/physreva.55.900 · Cited by 1,593
Bulk Spin-Resonance Quantum Computation
Gershenfeld, N. A., & Chuang, I. L. (1997) · Science · doi:10.1126/science.275.5298.350 · Cited by 1,528
Quantum computations: algorithms and error correction
Kitaev, A. Y. (1997) · Russian Mathematical Surveys · doi:10.1070/rm1997v052n06abeh002155 · Cited by 1,424
On the Power of Quantum Computation
Simon, D. R. (1997) · SIAM Journal on Computing · doi:10.1137/s0097539796298637 · Cited by 1,295
Bernstein, E., & Vazirani, U. (1993) · Proceedings of the twenty-fifth annual ACM symposium on Theory of computing - STOC '93 · doi:10.1145/167088.167097 · Cited by 1,284
Quantum computational networks
Deutsch, D. E. (1989) · Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences · doi:10.1098/rspa.1989.0099 · Cited by 1,270
Class of quantum error-correcting codes saturating the quantum Hamming bound
Gottesman, D. (1996) · Physical Review A · doi:10.1103/physreva.54.1862 · Cited by 1,227 · cited 1×
“error correcting codes was developed[43], and a group theoretical framework for almost all codes was found [14, 15, 31 ].”
Fault-tolerant quantum computation
Shor, P. W. (2002) · Annual Symposium on Foundations of Computer Science · doi:10.1109/sfcs.1996.548464 · Cited by 1,195
Two-bit gates are universal for quantum computation
DiVincenzo, D. P. (1995) · Physical Review A · doi:10.1103/physreva.51.1015 · Cited by 1,069
Perfect Quantum Error Correcting Code
Laflamme, R., Miquel, C., Paz, J. P., & Zurek, W. H. (1996) · Physical Review Letters · doi:10.1103/physrevlett.77.198 · Cited by 966
Theory of fault-tolerant quantum computation
Gottesman, D. (1998) · Physical Review A · doi:10.1103/physreva.57.127 · Cited by 945
Quantum computers and dissipation
(1996) · Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences · doi:10.1098/rspa.1996.0029 · Cited by 899
Multiple particle interference and quantum error correction
Steane, A. M. (1996) · Cited by 871
Quantum Error Correction and Orthogonal Geometry
Calderbank, A. R., Rains, E. M., Shor, P. W., & Sloane, N. J. A. (1997) · Physical Review Letters · doi:10.1103/physrevlett.78.405 · Cited by 811
Decoherence, Continuous Observation, and Quantum Computing: A Cavity QED Model
Pellizzari, T., Gardiner, S. A., Cirac, J. I., & Zoller, P. (1995) · Physical Review Letters · doi:10.1103/physrevlett.75.3788 · Cited by 786
Modern Quantum Mechanics, Revised Edition
Sakurai, J. J., & Commins, E. D. (1995) · American Journal of Physics · doi:10.1119/1.17781 · Cited by 710
Quantum data processing and error correction
Schumacher, B. W., & Nielsen, M. A. (1996) · Physical Review A · doi:10.1103/physreva.54.2629 · Cited by 689
Yao, A. C.-C. (2002) · Annual Symposium on Foundations of Computer Science · doi:10.1109/sfcs.1993.366852 · Cited by 680
Knill, E., Laflamme, R., & Zurek, W. H. (1998) · Science · doi:10.1126/science.279.5349.342 · Cited by 663
Maintaining coherence in quantum computers
Unruh, W. G. (1995) · Physical Review A · doi:10.1103/physreva.51.992 · Cited by 594
Implementation of a quantum search algorithm on a quantum computer
Jones, J. A., Mosca, M., & Hansen, R. H. (1998) · Nature · doi:10.1038/30687 · Cited by 558
Quantum circuits with mixed states
Aharonov, D., Kitaev, A., & Nisan, N. (1998) · ACM Symposium on Theory of Computing (STOC) · doi:10.1145/276698.276708 · Cited by 487
Self-citationNuclear magnetic resonance spectroscopy: An experimentally accessible paradigm for quantum computing
Cory, D. G., Price, M. D., & Havel, T. F. (1998) · Physica D: Nonlinear Phenomena · doi:10.1016/s0167-2789(98)00046-3 · Cited by 419
Phase uncertainty and loss of interference: A general picture
Stern, A., Aharonov, Y., & Imry, Y. (1990) · Physical Review A · doi:10.1103/physreva.41.3436 · Cited by 412
The ion trap quantum information processor
Steane, A. M. (1997) · Applied Physics B · doi:10.1007/s003400050225 · Cited by 331
Quantum Computers, Factoring, and Decoherence
Chuang, I. L., Laflamme, R., Shor, P. W., & Zurek, W. H. (1995) · Science · doi:10.1126/science.270.5242.1633 · Cited by 289
Lint, J. H. (1971) · Cited by 246
Bosonic quantum codes for amplitude damping
Chuang, I. L., Leung, D., & Yamamoto, Y. (1997) · Physical Review A · doi:10.1103/physreva.56.1114 · Cited by 217
Quantum Error Correction with Imperfect Gates
Kitaev, A. Y. (1997) · Quantum Communication, Computing, and Measurement · doi:10.1007/978-1-4615-5923-8_19 · Cited by 214
Quantum computation in quantum-Hall systems
Vladimir Privman, A., Vagner, I. D., & Kventsel, G. F. (1998) · Physics Letters A · doi:10.1016/s0375-9601(97)00974-2 · Cited by 184
Approximate quantum Fourier transform and decoherence
Barenco, A., EKERT, A. K., Suominen, K., & Törmä, P. (1996) · Physical Review A · doi:10.1103/physreva.54.139 · Cited by 175
Fault-Tolerant Quantum Computation with Higher-Dimensional Systems
Gottesman, D. (1999) · Lecture notes in computer science · doi:10.1007/3-540-49208-9_27 · Cited by 149
Non-binary unitary error bases and quantum codes
Lab., N. (. S. L. A. N., Knill, E., & USDOE, W. (1996) · doi:10.2172/373768 · Cited by 138
Aharonov, D. (1999) · WORLD SCIENTIFIC eBooks · doi:10.1142/9789812815569_0007 · Cited by 131
Self-citationThreshold Accuracy for Quantum Computation
Knill, E., Laflamme, R., & Zurek, W. (1996) · arXiv · doi:10.48550/arxiv.quant-ph/9610011 · Cited by 99
Landauer, R. W. (1995) · Springer eBooks · doi:10.1007/978-94-011-0041-0_33 · Cited by 74
Factoring in a dissipative quantum computer
Miquel, C., Paz, J. P., & Perazzo, R. P. J. (1996) · Physical Review A · doi:10.1103/physreva.54.2605 · Cited by 67
Quantum/classical correspondence in the light of Bell's inequalities
Khalfin, L. A., & Tsirelson, B. S. (1992) · Foundations of Physics · doi:10.1007/bf01889686 · Cited by 55
Polynomial simulations of decohered quantum computers
Aharonov, D., & Ben-Or, M. (2002) · Proceedings of 37th Conference on Foundations of Computer Science · doi:10.1109/sfcs.1996.548463 · Cited by 41
Self-citationBerthiaume, A., & Brassard, G. (2005) · Workshop on Physics and Computation (PhysComp) · doi:10.1109/phycmp.1992.615538 · Cited by 21
Self-Correcting Two-Dimensional Arrays.
Gács, P. (1989) · Cited by 17
Reliable storage of information in a system of unreliable components with local interactions
Cirel’son, B. S. (1978) · Lecture notes in mathematics · doi:10.1007/bfb0070081 · Cited by 6
Non-Clifford Gates Are Required for Long-Term Memory
Nelson, J., Rajakumar, J., & Gullans, M. J. (2026) · DROPS (Schloss Dagstuhl – Leibniz Center for Informatics) · doi:10.4230/lipics.ccc.2026.38
Self-citation
Cited by
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Fault-tolerant quantum computation by anyons
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Shor, P. W. (1999) · SIAM Review · doi:10.1137/s0036144598347011 · Cited by 3,904
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Unconditional Security of Quantum Key Distribution over Arbitrarily Long Distances
Lo, H., & Chau, H. F. (1999) · Science · doi:10.1126/science.283.5410.2050 · Cited by 1,867
Metadata
- Authors
- Aharonov, D., & Ben-Or, M.
- Affiliations
- Hebrew University of Jerusalem (IL)
- Venue
- Proceedings of the twenty-ninth annual ACM symposium on Theory of computing - STOC '97
- Publisher
- ACM Press
- Published
- 1997
- Pages
- 176-188
- OpenAlex
- W2052169397
- Cited by
- 631
- References
- 61
- Citations by year
- 2026: 25 · 2025: 44 · 2024: 29 · 2023: 18 · 2022: 17 · 2021: 26 · 2020: 29 · 2019: 21 · 2018: 18 · 2017: 19 · 2016: 19 · 2015: 23 · 2014: 19 · 2013: 16 · 2012: 20
- FWCI
- 16.52
- Type
- conference paper
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- ACM Symposium on Theory of Computing (STOC) (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Information and Cryptography
- Physical Sciences › Mathematics › Discrete Mathematics and Combinatorics › Coding theory and cryptography
- Keywords
- fault-tolerant quantum computing, quantum error correction, concatenated quantum codes, secret sharing
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