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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 } .
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.
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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.
Current estimates put the threshold for the surface code on the order of 1%, [8] though estimates range widely and are difficult to calculate due to the exponential difficulty of simulating large quantum systems.
2.3 Quantum uncertainty. ... Download QR code; Print/export Download as PDF; Printable version; In other projects Wikidata item; Appearance. move to sidebar hide
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).