# Are there other ways to understand/define code distance in stabilizer QEC codes?

I guess code distance is the same as "Hamming distance" which is related to the overlap between different "codewords" in the codespace?

More technically, my understanding is that a stabilizer quantum-error-correcting code (QECC) having code distance $$d$$ implies that any operator $$\mathcal{E}$$ capable of producing a logical error must have support (i.e., act nontrivially) on at least $$d$$ qubits? This implies that every logical operator acts nontrivially on at least $$d$$ qubits.

I'm curious if there are other (equivalent) definitions. Specifically, I'm wondering if the minimum number of errors applies equally to deletion / erasure errors (on at most $$d-1$$ qubits). However, I guess deletion errors aren't the same as logical errors (i.e., they don't change between codewords), so you can correct at most $$t = (d-1)/2$$ of them, so that's the maximum number of deletion errors that can be tolerated?

If so, then I would expect the following statement to hold: Given a QECC with code distance $$d$$ and nominal logical operators $$\{ \overline{X}, \overline{Y}, \overline{Z}\}$$, then for any region $$C$$ that contains $$\left| C \right| = d-1$$ qubits (or more likely $$\left| C \right| = t = (d-1)/2$$ qubits), there exist elements $$\mathcal{S}_x$$, $$\mathcal{S}_y$$, $$\mathcal{S}_z$$ of the stabilizer group such that $$\{ \mathcal{S}_x \overline{X} , ~\mathcal{S}_y \overline{Y} , ~ \mathcal{S}_z \overline{Z} \}$$ are also valid logical operators, each of which acts trivially on every qubit in $$C$$?

As a bonus question: If this definition is correct, are we guaranteed that the stabilizer elements $$\mathcal{S}_x$$, $$\mathcal{S}_y$$, and $$\mathcal{S}_z$$ are unique (assuming that $$\left| C\right|$$ is maximal for code distance $$d$$)? Do the numbers $$N$$ and $$k$$ of physical and logical qubits matter for this?

Also if I just need to open Nielsen and Chuang to a particular section, feel free to let me know! I'll have a copy soon. Also, if you have a random fun fact relating code distance to something else, feel free to comment it, since I'm generally interested.

• Can I just clarify what you mean by deletion/erasure errors? Do you mean what I'm imagining - a qubit gets removed from the system, and we can spot that it has been removed, and we know which qubit was removed? Commented Feb 3, 2023 at 7:16
• If so, there's some mention of a relevant result here: quantumcomputing.stackexchange.com/a/4802/1837 (I'm sure I went through more of the details in an answer at some point, but cannot immediately lay my hands on it) Commented Feb 3, 2023 at 7:21
• @DaftWullie yes, I mean to allow for both of these cases (we know which qubit was traced out / removed, OR we just know that some qubit was traced out / removed). I'll take a look at the "≥5" proof and see if any aspects of that answer my questions! Commented Feb 3, 2023 at 23:06

If you delete all qubits that form a cycle, you can't measure the logical operator anymore. The smallest cycle one can form consists of $$d$$ data qubits, and therefore we can correct upto $$d-1$$ erasure errors in the toric code.