# Questions tagged [fault-tolerance]

For questions about achieving (almost) perfect logical operations with imperfect gates. The tag also applies to questions about the amount of tolerable noise in a quantum circuit, as well as designing circuits to prevent errors from spreading.

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### Transversal logical gate for Stabilizer (or at least Steane code)

I know that for Steane code, we can implement transversally some gates like cNOT, Hadamard and Pauli. What I am looking for is a resource in which it is explained why implementing those gate give rise ...
3answers
127 views

### CSS Code in disguise

Suppose I have a quantum error correcting code described by a set of stabilizers. Is there any easy way of checking whether or not it is equivalent to a code constructed using the CSS construction, ...
1answer
27 views

### Can one easily compute the weight enumerator of a CSS code constructed from two binary codes $M_1,M_2$?

Consider a CSS code, which is constructed from two binary codes $M_1$ and $M_2$, so its symplectic matrix is of the form: $$\begin{pmatrix} M_1 & 0 \\ 0 & M_2 \end{pmatrix}.$$ Suppose one ...
2answers
239 views

### How does the size of a toric code torus affect its ability to protect qubits?

The Toric code Hamiltonian is: $\sum_{x,y}\left( \prod_{i\in p(x,y)} Z_{ixy} + \prod_{i\in v(x,y)} X_{ixy} \right),$ where the $v$ and $p$ are defined according to this picture (courtesy of James ...
0answers
31 views

### How to do the counting when computing the fault tolerant threshold of quantum codes?

Here, I want to ask a basic question about how to compute the fault tolerance threshold of quantum codes. As I know, maybe, the most usual way is to do some simulations. Howvever, I am more intersted ...
1answer
86 views

### (Why) does every CSS code allow for transversal measurement?

I recently encountered the statement that any CSS code [that encodes single-qubit logical states] has a property that measurements can be performed qubit-wise, but the measurement outcome of every ...
1answer
35 views

### What are the simplest examples of codes with transversal non-Clifford gates?

There are certain codes which have transversal non-Clifford gates. My question is, what are the simplest examples of such codes and how does one check for a non-Clifford gate? Is it a computationally ...
1answer
56 views

### Understanding transversal gates for the 7 qubit steane code

How can one derive the complete list of transversal operators for the 7-qubit Steane code? I can derive the Clifford operators that are transversal, but I do not understand an easy way to check for ...
0answers
32 views

### Fault-tolerant qudit computation: advantage? [closed]

What are advantages of a fault-tolerant qudit computation compared with fault-tolerant quantum computation for qubits?
3answers
1k views

### What are magic states?

I wonder what are magic states, and a magic state gadget. While I'm reading a paper, these terms frequently appear.
1answer
76 views

### CSS Code and Fault-tolerant Problem in Nielsen and Isaac Chuang‘s book

Could someone help me with $Problem-10.52$ in Nielsen and Isaac Chuang‘s book? The screenshot is shown below. I have no idea about that. Hope someone can give me some suggestions. By the way, if I ...
0answers
45 views

### Is there any published research on quantum fault tolerant circuits that are designed for a specific task or algorithm?

I have searched for works that focus on designing a fault-tolerant quantum circuit for a specialized application and only found 'Error-corrected quantum annealing with hundreds of qubits' by Pudenz, ...
2answers
144 views

### What is the pseudo threshold of a QECC using stabilizer formalism

Can someone explain what is the threshold and the pseudo threshold of a Quantum Error Correction Code , for instance the 9-qubit code, and how to calculate it using the stabilizer formalism simulation ...
1answer
35 views

### Where do errors occur during error syndrome and recovery?

I'm studying quantum error correction on Nielsen's Quantum Computation and Quantum Information, but I don't understand how it identifies the number of places where an error may occur. It uses ...
1answer
50 views

### Circuit of quantum error correction code

I read an interesting paper of QEC with only two extra qubits. In the paper, there is a circuit. Note that . Here are some contexts of the paper: "The stablizer is $XZZXI$. If the second gate ...
2answers
140 views

### How is it not a contradiction that it is possible to build fault tolerant circuits with strictly contractive (e.g.: depolarizing noise) channels?

This paper discusses strictly contractive channels, i.e. channels that strictly decrease the trace distance between any two input quantum states. It is shown that if a quantum circuit is composed of ...
1answer
65 views

### Quantum error correction - approximate vs exact

When one has a noisy quantum channel over which one wishes to send some quantum state, one usually comes up with an encoder and decoder. Then, the error between the input state and the output state is ...
2answers
128 views