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That these codes allow indeed for quantum computations of arbitrary length is the content of the quantum threshold theorem, found by Michael Ben-Or and Dorit Aharonov, which asserts that you can correct for all errors if you concatenate quantum codes such as the CSS codes—i.e. re-encode each logical qubit by the same code again, and so on, on ...
In this code, 5 physical qubits are used to encode the logical qubit. [2] With X {\displaystyle X} and Z {\displaystyle Z} being Pauli matrices and I {\displaystyle I} the Identity matrix , this code's generators are X Z Z X I , I X Z Z X , X I X Z Z , Z X I X Z {\displaystyle \langle XZZXI,IXZZX,XIXZZ,ZXIXZ\rangle } .
Quantum error-correcting codes restore a noisy, decohered quantum state to a pure quantum state. A stabilizer quantum error-correcting code appends ancilla qubits to qubits that we want to protect. A unitary encoding circuit rotates the global state into a subspace of a larger Hilbert space.
In a quantum error-correcting code, the codespace is the subspace of the overall Hilbert space where all logical states live. In an n {\displaystyle n} -qubit stabilizer code , we can describe this subspace by its Pauli stabilizing group, the set of all n {\displaystyle n} -qubit Pauli operators which stabilize every logical state.
Forney et al. provided an example of a rate-1/3 quantum convolutional code by importing a particular classical quaternary convolutional code (Forney and Guha 2005). Grassl and Roetteler determined a noncatastrophic encoding circuit for Forney et al.'s rate-1/3 quantum convolutional code (Grassl and Roetteler 2006).
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A qubit (/ ˈ k juː b ɪ t /) or quantum bit is a basic unit of quantum information—the quantum version of the classic binary bit physically realized with a two-state device. A qubit is a two-state (or two-level) quantum-mechanical system , one of the simplest quantum systems displaying the peculiarity of quantum mechanics.
In addition to investigating fault tolerant quantum computation, the Eastin–Knill theorem is also useful for studying quantum gravity via the AdS/CFT correspondence and in condensed matter physics via quantum reference frame [2] or many-body theory. [3] The theorem is named after Bryan Eastin and Emanuel Knill, who published it in 2009. [1]