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?

For the toric code, it's relatively easy to see how many erasure errors can be corrected. You can understand this by looking at how one can correct erasure errors. In the following figure a) (from https://arxiv.org/abs/0904.3556) erasure errors happen to data qubits indicated by the 2 red crosses.

As shown in b) one erasure error can be overcome by 'merging' two weight 4 stabilizers to form a weight 6 stabilizer. In the example shown in the figure, four weight 4 stabilizers are merged to form two weight 6 stabilizers.

A logical operator on the toric code is any cycle going around the torus (donut). Therefore the logical operator can be measured if there is at least one cycle that avoids all deleted qubits.

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 $d$ erasure errors in the toric code.

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?

For the toric code, it's relatively easy to see how many erasure errors can be corrected. You can understand this by looking at how one can correct erasure errors. In the following figure a) (from https://arxiv.org/abs/0904.3556) erasure errors happen to data qubits indicated by the 2 red crosses.

As shown in b) one erasure error can be overcome by 'merging' two weight 4 stabilizers to form a weight 6 stabilizer. In the example shown in the figure, four weight 4 stabilizers are merged to form two weight 6 stabilizers.

A logical operator on the toric code is any cycle going around the torus (donut). Therefore the logical operator can be measured if there is at least one cycle that avoids all deleted qubits.

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.

Also if I just need to open Nielsen and Chuang to a particular section, feel free to let me know!

This is a great question and as far as I'm aware it's not known for many codes how many delation/erasure errors they can tolerate.

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?

For the toric code, it's relatively easy to see how many erasure errors can be corrected. You can understand this by looking at how one can correct erasure errors. In the following figure a) (from https://arxiv.org/abs/0904.3556) erasure errors happen to data qubits indicated by the 2 red crosses.

As shown in b) one erasure error can be overcome by 'merging' two weight 4 stabilizers to form a weight 6 stabilizer. In the example shown in the figure, four weight 4 stabilizers are merged to form two weight 6 stabilizers.

A logical operator on the toric code is any cycle going around the torus (donut). Therefore the logical operator can be measured if there is at least one cycle that avoids all deleted qubits.

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 $d$ erasure errors in the toric code.

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?

For the toric code, it's relatively easy to see how many erasure errors can be corrected. You can understand this by looking at how one can correct erasure errors. In the following figure a) (from https://arxiv.org/abs/0904.3556) erasure errors happen to data qubits indicated by the 2 red crosses.

As shown in b) one erasure error can be overcome by 'merging' two weight 4 stabilizers to form a weight 6 stabilizer. In the example shown in the figure, four weight 4 stabilizers are merged to form two weight 6 stabilizers.

A logical operator on the toric code is any cycle going around the torus (donut). Therefore the logical operator can be measured if there is at least one cycle that avoids all deleted qubits.

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 $d$ erasure errors in the toric code.