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I am looking for a small stabilizer code that protect one logical qubit from $X$ and $Z$ noise.
Maybe not necessarily all of the physical qubits.
Something that has a good tradeoff between number of detections and number of physical qubits.
I know about the Stean code, which use 7 qubits. I also know about the [[5,1,3]] code.
Is there a code with less physical qubit doing something similar?
The [[5,1,3]] code that you mention is the smallest possible distance 3 quantum error correcting code. It is known as the "perfect" quantum code because it exactly saturates one of the size bounds for non-degenerate codes.
The Quantum Hamming Bound says that for a distance 3 code on $n$ qubits, there are $2^n$ possible states. These must include the two logical states and space for the $3n$ possible single-qubit errors acting on each of those logical states, so that distinct errors map to distinct states, which are therefore detectable and correctable. So, you have $$ 2^n\geq 2(1+3n). $$ If you try a few values of $n$, you'll find that $n=5$ is the smallest values allowed.
