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There are two possible outcomes for the measurement of a qubit—usually taken to have the value "0" and "1", like a bit. However, whereas the state of a bit can only be binary (either 0 or 1), the general state of a qubit according to quantum mechanics can arbitrarily be a coherent superposition of all computable states simultaneously. [2]
[1] [2] A logical qubit is a physical or abstract qubit that performs as specified in a quantum algorithm or quantum circuit [3] subject to unitary transformations, has a long enough coherence time to be usable by quantum logic gates (c.f. propagation delay for classical logic gates).
A 5-qubit code is the smallest possible code that protects a single logical qubit against single-qubit errors. A generalisation of the technique used by Steane , to develop the 7-qubit code from the classical [7, 4] Hamming code , led to the construction of an important class of codes called the CSS codes , named for their inventors: Robert ...
A qubit is a two-level system, and when we measure one qubit, we can have either 1 or 0 as a result. One corresponds to odd parity, and zero corresponds to even parity. This is what a parity check is. This idea can be generalized beyond single qubits. This can be generalized beyond a single qubit and it is useful in QEC.
Just as the bit is the basic concept of classical information theory, the qubit is the fundamental unit of quantum information.The same term qubit is used to refer to an abstract mathematical model and to any physical system that is represented by that model.
The purpose of quantum computing focuses on building an information theory with the features of quantum mechanics: instead of encoding a binary unit of information (), which can be switched to 1 or 0, a quantum binary unit of information (qubit) can simultaneously turn to be 0 and 1 at the same time, thanks to the phenomenon called superposition.
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 } .
Now the (real) dimension of U(n) is n 2. This is easy to see since the exponential map is a local homeomorphism from the space of self-adjoint complex matrices to U(n). The space of self-adjoint complex matrices has real dimension n 2. Corollary. The real dimension of the pure state space of H n is 2n − 2. In fact,