STEINSPRING QUANTUM ERROR CORRECTION CODE
Every useful code has n > k this physical redundancy is necessary to detect and correct errors without disturbing the logical state. A stabilizer code is characterized by three parameters ], where n is the number of physical qubits, k is the number of encoded logical qubits, and d is the minimum distance of the code (the smallest number of simultaneous qubit errors that can transform one valid codeword into another). The code space is the joint +1 eigenspace of a set of commuting Pauli operators on n qubits, called stabilizer generators the error syndrome is determined by measuring these operators, which allows errors to be diagnosed and corrected. The most widely used codes are the stabilizer codes, which are closely related to classical linear codes. These can be physically realized by the states of a spin-1/2 particle, the polarization of a single photon, two distinguished levels of a trapped atom or ion, the current states of a microscopic superconducting loop, or many other physical systems. Most work on quantum error correction has focused on systems of quantum bits, or qubits, which are two-level quantum systems. An error set is represented by a set of operators that can multiply the codeword state. No code that stores information can protect against all possible errors instead, codes are designed to correct a specific error set, which should be chosen to match the most likely types of noise. In general, codewords of a quantum code are entangled states. It is possible to determine whether an error has occurred by a suitable measurement and to apply a unitary correction that returns the state to the code space without measuring (and hence disturbing) the protected state itself. This code is designed so that the most common errors move the state into an error space orthogonal to the original code space while preserving the information in the state. The information is stored in a quantum error-correcting code, which is a subspace in a larger Hilbert space. Quantum error correction is a set of methods to protect quantum information-that is, quantum states-from unwanted environmental interactions ( decoherence) and other forms of noise. Experimental Implementation of Quantum Error Correction
3.2 Completely Positive, Trace-Preserving Maps.A Brief History of Quantum Error Correction
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