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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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 ...
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0answers
32 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 ...
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3answers
135 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, ...
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1answer
38 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 ...
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1answer
60 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 ...
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1answer
148 views

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 ...
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0answers
33 views

Fault-tolerant qudit computation: advantage? [closed]

What are advantages of a fault-tolerant qudit computation compared with fault-tolerant quantum computation for qubits?
2
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1answer
82 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 ...
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1answer
90 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 ...
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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 ...
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3answers
2k 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.
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1answer
52 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 ...
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2answers
141 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 ...
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2answers
149 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 ...
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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, ...
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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 ...
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2answers
129 views

Why does quantum error correction work?

I read about some QEC codes such as Shor code. It encodes a logical qubit to 9 physical qubits, to correct the bit-flip and phase-flip error. To do this, it needs multiple $\mathrm{CNOT}$ and $\mathrm{...
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72 views

How's quantum noise and fault-tolerance related to symplectic geometry and geometric quantization?

Gil Kalai often speaks of the apparent connection between symplectic geometry, geometric quantization, and quantum noise. He is known to describe one of his focus areas as: (...) properties and ...
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2answers
89 views

Where does error correction go in a quantum algorithm?

Where does QEC fit in the larger picture of the execution of a fault tolerant quantum algorithm? For QEC, there is an initial preparation of the qubit state and the ancillary qubits; then a "noisy ...
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109 views

In what situation are three rounds of syndrome measurement required for fault-tolerance in the surface code?

I have heard multiple times the claim that three rounds of syndrome measurement required for fault-tolerance in the surface code. I'm not sure what situation would require this, as I think less would ...
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2answers
251 views

A question on Eastin-Knill theorem

I am reading the paper Restrictions on Transversal Encoded Quantum Gate Sets, Bryan Eastin, Emanuel Knill. I am unable to understand the following lines in the proof. As the set of all unitary ...
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1answer
197 views

Magic State Distillation Understanding Check

I'm currently trying to understand the T magic state distillation algorithm described in "Universal Quantum Computation with Ideal Clifford Gates and Noisy Ancillas" [1] (Section V starting on Page 6)....
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2answers
848 views

Where do we put error correction code in quantum circuit?

First of all : I am a beginner in quantum computing. I would like to have a resource (or an answer if it is not complicated) explaining where we put the error correction codes in a quantum circuit. ...
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2answers
130 views

Fault tolerant quantum measurement: how is implemented the “majority vote”

As a fundamental component for a quantum computation, the measurement needs to be implemented in a fault tolerant way. As indicated in Chuang and Nielsen Quantum Computation and Quantum Information, ...
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1answer
854 views

How are magic states defined in the context of quantum computation?

Quoting from this blog post by Earl T. Campbell: Magic states are a special ingredient, or resource, that allows quantum computers to run faster than traditional computers. One interesting example ...
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401 views

Reference on MITxPRO Applications of Quantum Computing Professional Certificate Program

Recently I found out the Applications of Quantum Computing Professional Certificate Program that MITxPRO is offering for people interested in quantum computing. I saw that it is consisted of four ...
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2answers
243 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 ...
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188 views

Why or how is 'cat' state preparation via a C-not-not operation not fault tolerant?

On p490 of Nielsen and Chuang, 2010 the authors say that the preparation of the 'cat' state ($|000\ldots 0\rangle+|111\ldots 1\rangle$) is not Fault Tolerant. Below is my mock up of the diagram they ...
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1answer
204 views

What set of quantum logical operations can one use to benchmark spin Hamiltonians?

Chemistry background: In magnetic molecules, it is sometimes the case that one can adjust the time-independent Hamiltonian by chemical design. This means there is freedom to adjust parameters in the ...
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5answers
514 views

Why do error correction protocols only work when the error rates are already significantly low to begin with?

Quantum error correction is a fundamental aspect of quantum computation, without which large-scale quantum computations are practically unfeasible. One aspect of fault-tolerant quantum computing that ...
12
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1answer
434 views

What is the leading edge technology for creating a quantum computer with the fewest errors?

Which technological path seems most promising to produce a quantum processor with a greater quantum volume (preferring fewer errors per qubit over more qubits), than Majorana fermions? The preferred ...
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1answer
245 views

How does approximating gates via universal gates scale with the length of the computation?

I understand that there is a constructive proof that arbitrary gates can be approximated by a finite universal gate set, which is the Solovay–Kitaev Theorem. However, the approximation introduces an ...
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3answers
601 views

Which quantum error correction code has the highest threshold (as proven at the time of writing this)?

Which quantum error correction code currently holds the record in terms of the highest threshold for fault-tolerance? I know that the surface code is pretty good ($\approx10^{-2}$?), but finding exact ...
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3answers
1k views

What level of “confidence” of the result from a quantum computer is possible?

At a very basic level, reading or measuring a qubit forces it to be in one state or the other, so the operation of a quantum computer to gain a result collapses the state into one of many ...