An Introduction to Quantum Error Correction and Fault-Tolerant Quantum Computation

By Daniel Gottesman. 2009

705 citations recorded by the source corpus.

Quantum states are very delicate, so it is likely some sort of quantum error correction will be necessary to build reliable quantum computers. The theory of quantum error-correcting codes has some close ties to and some striking differences from the theory of classical error-correcting codes. Many quantum codes can be described in terms of the stabilizer of the codewords. The stabilizer is a finite Abelian group, and allows a straightforward characterization of the error-correcting properties of the code. The stabilizer formalism for quantum codes also illustrates the relationships to classical coding theory, particularly classical codes over GF(4), the finite field with four elements. To build a quantum computer which behaves correctly in the presence of errors, we also need a theory of fault-tolerant quantum computation, instructing us how to perform quantum gates on qubits which are encoded in a quantum error-correcting code. The threshold theorem states that it is possible to create a quantum computer to perform an arbitrary quantum computation provided the error rate per physical gate or time step is below some constant threshold value.

Subfields: Artificial Intelligence, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/0b57d234d4a4ab5c88b5d063a2d625aa06f9787e

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Citations in the corpus

Quantum error mitigation (2022, background): “One useful approximation, which is widely employed in the field of quantum error correction, is to model noise as discrete, probabilistic events named faults that can occur at the various locations in the circuit including gates, idling steps and measurements (Gottesman, 2009; Terhal, 2015).”
New class of quantum error-correcting codes for a bosonic mode (2016, background): “Hence, typical quantum error-correction schemes based on models of independent single qubit errors cannot be easily transferred to this problem [63,64].”
Quantum error correction: an introductory guide (2019, methodology): “rator yields a (+1) eigenvalue when applied to the logical state Z1Z2 |ψiL= Z1Z2(α|00i +β|11i) = (+1)|ψiL. (18) The Z1Z2 operator is said to stabilize the logical qubit |ψiL as it leaves it unchanged [28]. Conversely, the Z1Z2 operator projects the errored states, X1 |ψiL and X2 |ψiL, onto the (−1) ”
Diagnostics of Mixed-State Topological Order and Breakdown of Quantum Memory (2023, background): “The major roadblock to realizing quantum computers is the presence of errors and decoherence from the environment which can only be overcome by adopting quantum error correction (QEC) and fault tolerance [1].”
Dynamically protected cat-qubits: a new paradigm for universal quantum computation (2013, neutral): “Here, by ﬁ rst order fault-tolerance, we mean that we prevent the errors due to single-photon loss or photon dephasing to propagate through various quantum operations (see e.g. de ﬁ nition 4 of [27]).”
A quantum processor based on coherent transport of entangled atom arrays (2021, neutral): “The Steane code, surface code, and toric code all have transversal single-qubit Clifford operations on the logical qubit [8, 34] (including in-software rotations of the lattice), which is a high-fidelity operation in our system since the transveral rotations are implemented in parallel with our glob”
Quantum error correction for quantum memories (2013, neutral): “(4), can be slightly reformulated; see Theorem 3 in Gottesman (2009): A code C can correct a set of errors E ∈ E if and only if for all states j ψ i in the code space C we have where the constant c ð E Þ is independent of j ψ i .”
Fault-tolerant measurement-based quantum computing with continuous-variable cluster states. (2013, methodology): “[33] for a review) is the ability to reduce logical errors in a quantum computation to arbitrarily low levels if the physical error rate of the individual gates comprising the computation is below a ﬁxed, positive value called the fault-tolerance threshold .”
Holographic quantum error-correcting codes: toy models for the bulk/boundary correspondence (2015, background): “her holographic codes. 5.7 Holographic stabilizer codes Stabilizer codes have been extensively studied in quantum coding theory, and are often used in applications to fault-tolerant quantum computing [43]. Here we describe how to construct a family of holographic codes which are also stabilizer code”
Quantum Computing in the NISQ era and beyond (2018, background): “materials and control and fabrication to get where we are now. Eventually we expect to be able to protect quantum systems and scale up quantum computers using the principle of quantum errorcorrection [10]. The essential idea of quantum error correction is that if we want to protect a quantum system ”
The resource theory of stabilizer quantum computation (2013, background): “For an overview of the stabilizer formalism in the context of fault tolerance see [16, 18].”