Theory of quantum error-correcting codes
Abstract
Quantum error correction will be necessary for preserving coherent states against noise and other unwanted interactions in quantum computation and communication. We develop a general theory of quantum error correction based on encoding states into larger Hilbert spaces subject to known interactions. We obtain necessary and sufficient conditions for the perfect recovery of an encoded state after its degradation by an interaction. The conditions depend only on the behavior of the logical states. We use them to give a recovery-operator-independent definition of error-correcting codes. We relate this definition to four others: the existence of a left inverse of the interaction, an explicit representation of the error syndrome using tensor products, perfect recovery of the completely entangled state, and an information theoretic identity. Two notions of fidelity and error for imperfect recovery are introduced, one for pure and the other for entangled states. The latter is more appropriate when using codes in a quantum memory or in applications of quantum teleportation to communication. We show that the error for entangled states is bounded linearly by the error for pure states. A formal definition of independent interactions for qubits is given. This leads to lower bounds on the number of qubits required to correct e errors and a formal proof that the classical bounds on the probability of error of e-error-correcting codes applies to e-error-correcting quantum codes, provided that the interaction is dominated by an identity component.
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Quantum information is spread over multiple physical qubits to generate a logical qubit.
“The quantum information is spread over multiple physical qubits to generate a logical qubit (Calderbank and Shor, 1996; Gottesman, 1997; Knill and Laflamme, 1997; Shor, 1995).”
From Noisy intermediate-scale quantum algorithms (2022) · cites this paper as Knill and Laflamme, 1997
Kitaev and Solovay showed that universality of a set G guarantees that any quantum computation can be approximated by a circuit using G with only polylogarithmic cost.
“Kitaev[39] and Solovay[66] showed, by a beautiful proof which uses Lie algebra and Lie groups, that universality of a set G guarantees that any quantum computation can be approximated by a circuit which uses G with only polylogarithmic cost.”
From Fault-Tolerant Quantum Computation with Constant Error Rate (2008) · cites this paper as [66]
Theorem 6 (Kitaev, Solovay): For a universal set of gates G over U(m), there exists a polynomial p_m such that any element of U(m) can be approximated up to ε by a word from G not longer than p_m(log(1/ε)), and this word can be found efficiently by a classical Turing machine.
“Theorem 6. Kitaev, Solovay,: Consider a universal set of gates G over U ( m ) , for some integer m . Then there exists a polynomial p m such that any element of U ( m ) can be approximated up to ǫ by a word from G which is not longer than p m ( log (1 /ǫ )) , and this word can be found efficiently (in time which is p m ( log (1 /ǫ )) ) by a classical Turing machine.”
From Fault-Tolerant Quantum Computation with Constant Error Rate (2008) · cites this paper as [66]
Recent results on fault tolerance in non-Markovian settings make a crucial assumption about the smallness of the norm of the system-bath interaction Hamiltonian, which severely restricts the class of physical reservoirs.
“recent results on fault tolerance in non-Markovian settings [19, 20, 21, 22], while representing a significant step forward, make a crucial assumption about the smallness of the norm of the system-bath interaction Hamiltonian, which severely restricts the class of physical reservoirs.”
From Internal consistency of fault-tolerant quantum error correction in light of rigorous derivations of the quantum Markovian limit (2006) · cites this paper as [22]
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References 18 · Cited by 1,593
- Authors
- Knill, E., & Laflamme, R.
- Affiliations
- Los Alamos National Laboratory (US)
- Venue
- Physical Review A
- Publisher
- American Physical Society (APS)
- Published
- 1997-02-01
- Volume
- 55
- Issue
- 2
- Pages
- 900-911
- ISSN
- 0556-2791 (ISSN-L)
- 1050-2947
- 1094-1622
- OpenAlex
- W2763302490
- Cited by
- 1,593
- References
- 18
- Citations by year
- 2026: 86 · 2025: 125 · 2024: 87 · 2023: 67 · 2022: 69 · 2021: 94 · 2020: 73 · 2019: 59 · 2018: 51 · 2017: 47 · 2016: 39 · 2015: 43 · 2014: 44 · 2013: 49 · 2012: 40
- FWCI
- 75.70
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Physical Review A (publishedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Information and Cryptography
- Physical Sciences › Computer Science › Artificial Intelligence › Coding theory and cryptography
- Keywords
- quantum error correction, quantum error-correcting codes, error syndromes, entangled states, fidelity, quantum memory, quantum teleportation, qubit interactions
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