Quantum error correction encodes logical states into a larger Hilbert space so that a specified family of physical errors can be detected and reversed.
The distance is the minimum weight of a Pauli operator that preserves the code space but acts nontrivially on its logical information.
An error syndrome records which stabilizer generators commute or anticommute with an error.
The Steane code is a seven-qubit CSS stabilizer code encoding one logical qubit with distance three.
A color code places qubits on a three-colorable lattice and assigns X-type and Z-type stabilizers to each colored plaquette.
A string operator creates excitations at its endpoints. Multiplying by a successive segment moves an endpoint because the shared intermediate excitation is toggled twice.
The surface code is a topological stabilizer code whose electric and magnetic excitations are bosons with mutual-semion braiding.
The toric code has commuting star and plaquette stabilizers. Its electric and magnetic excitations are bosons with mutual-semion statistics, and their fusion is a fermion.
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Quantum error correction (QEC) is a crucial aspect of quantum computing that aims to protect quantum information from errors due to decoherence, noise, and operational imperfections. Quantum bits, or qubits, are the fundamental units of quantum information. Unlike classical bits, which can be either 0 or 1, qubits can exist in superpositions of both states. This property makes quantum systems particularly susceptible to errors, as even small interactions with the environment can lead to significant loss of information.
Technique that uses multiple non-ideal qubits (physical qubits) to simulate/produce one perfect qubit (logical).
One is philosophically reminded of classical error correction codes, where we also have multiple input bits per actual information bit.
TODO understand in detail. This appears to be a fundamental technique since all physical systems we can manufacture are imperfect.
Part of the fundamental interest of this technique is due to the quantum threshold theorem.
For example, when PsiQuantum raised 215M in 2020, they announced that they intended to reach 1 million physical qubits, which would achieve between 100 and 300 logical qubits.
Video "Jeremy O'Brien: "Quantum Technologies" by GoogleTechTalks (2014)" youtu.be/7wCBkAQYBZA?t=2778 describes an error correction approach for a photonic quantum computer.
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