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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What informs which choice of stabilizer generators to measure?

A stabilizer code is a code with codespace defined as the joint $+1$ eigenspace of some set of stabilizers $\mathcal S = \{s_1, s_2,\ldots,s_j\}$ with eigenvalues $\pm 1$. Error correction involved ...
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4 votes
2 answers
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What are the transversal gates of Shor's [[9,1,3]] code?

What are the transversal gates of Shor's $ [[9,1,3]] $ code? Since this is a stabilizer code we have transversal Pauli gates. Indeed, transversal $X^{\otimes 9}$ implements logical $\overline Z$ and ...
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1 answer
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What are the transversal gates of the [[5,1,3]] code?

What are the transversal gates of the perfect 5-qubit code ( [[5,1,3]] code) ? Since this is a stabilizer code we have transversal Pauli gates. Are there any others? I am primarily interested in ...
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3 votes
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How many times does syndrome extraction need to be repeated?

As part of implementing a QEC code, we need to do some syndrome measurement. In order for this syndrome measurement, we need to make sure that any $k \leq t$ errors (with a distance $2t + 1$ code) ...
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1 vote
1 answer
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What is the time overhead for fault-tolerant gate teleportation?

The gate teleportation protocol (https://arxiv.org/abs/quant-ph/9908010) has a step where it utilizes cat state measurement of transversal operators. Since cat states can not detect phase errors it is ...
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1 answer
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What is the smallest code for a single logical qubit

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 ...
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1 vote
1 answer
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Why the boundary data qubits have to be put in the ground state for lattice surgery rough merging

I am trying to understand why we need to put the boundary data qubits in the ground state $|0\rangle$ for a rough merging procedure. From my current understanding, if no errors occurred (perfect ...
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2 votes
1 answer
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Questions on Definitions and Concepts Regarding Toric Codes

Currently, I reading the toric code section in the beginning of the paper Topological Quantum Memory. Here are a couple of sections that are somewhat confusing to me. (1) This quote below is under a ...
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A Particular Way of Erasing Ancilla Qubits

I am currently reading the paper Topological Quantum Memory by Dennis, Kitaev et. al. My main interest is to conceptually understand the idealized assumptions / requirements of a quantum computer ...
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4 votes
2 answers
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Why does fault-tolerant transversal phase gate $P$ only work with doubly-even codes?

I read in this topic that to get a fault-tolerant transversal relative phase gate of $\pi/2$ -- $P$ gate in the reference --, one has to select a CSS code which satisfies doubly-even distances. I don'...
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1 answer
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Diagrammatic language for fault-tolerance: I don't understand the difference between preparation correctness and propagation properties

I am learning the diagrammatic language for fault-tolerance which allows to show in an "easy" manner if circuits are fault-tolerant. I first put a reminder of the definitions and then I ask ...
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3 votes
1 answer
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Magic state distillation with the $15$ qubit code. How are the transversal $T$ performed? Also via a (lower level) state injection?

I have a code $C1$ that admits native fault-tolerant gates for any element in the Clifford group (for instance via transversal implementation), but that then doesn't have an easy way to perform the $T$...
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2 votes
1 answer
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Magic state distillation with the 15 qubit code. Why is the failure $O(p^2)$ if we include $|+_L\rangle$ preparation and decoding?

My goal is to understand why the magic state distillation (for instance based upon the $15$ qubit code) improves the fidelity of the magic state. I followed this lecture to try to understand it. I ...
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2 votes
0 answers
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How to encode one stabilizer code for $H, CX, \sqrt{Z}$ operators

I understood that to get fault-tolerant computing, one can start from any stabilizer group and then unitarily modifying it to get a new stabilizer group which relates to some logical operators. For ...
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2 votes
1 answer
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Eastin Knill Theorem and groups of transversal gates

The Eastin-Knill Theorem shows that the transversal gates always form a group and that moreover this group is a finite subgroup of the group of all unitaries. For many codes, for example all self dual ...
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What's the map from any linear classical codes to entanglement-assisted stabilizers codes

When a linear code is self-dual, its parity check matrix can be used to easily define a stabilizer code. From my understanding, thanks to entanglement-assisted stabilizer codes, it is possible to ...
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1 vote
1 answer
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What does "perfect measurement" mean in the context of fault tolerance?

I am reading the paper Quantum error correction with only two extra qubits. They say there that (in the section on the [[5, 1, 3]] code): If a flag is raised or a syndrome is nontrivial, then the ...
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Analytical Approximation of Pseudothreshold for Steane Code

Is there an analytical way by which one can approximately find the pseudothreshold of the Steane code? My supervisor told me that there is a way, using the physical error rate and the number of gates ...
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How to understand Clifford+T from a quantum information theory perspective [closed]

I'd like to study the important operator set Clifford+T from a quantum information theory perspective. The Clifford+T set is universal and it is important because allows for efficient error correction ...
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4 votes
1 answer
146 views

Threshold for quantum Repetition Code

I'm learning about the Threshold theorem but I struggle with the computations of the threshold which are usually presented (even the one in the book by Nielsen & Chuang). To clear my head, I would ...
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1 vote
0 answers
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What will be the most useful quantum algorithms in the fault-tolerant quantum computers era?

When we'll have fault tolerant quantum computers with a lot of qubits, what will be the most useful algorithms (studied so far)? I know about Shor, Grover and quantum phase estimation but I'm pretty ...
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2 answers
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Magic state distillation: why is it harder to prepare the encoded $|A_{\pi/4}\rangle$ than $|0 \rangle$

My question is the following Let's assume I am using Steane concatenated code to do error correction. I consider that the stabilizers are extracted fault-tolerantly through the Steane method. The ...
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1 vote
1 answer
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Pros/cons of the different schemes to have complete fault-tolerant gatesets

I am interested to have references and comment about pro/cons of the various methods that are used to implement complete gateset in a fault tolerant manner. Usually the Clifford operations have a ...
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5 votes
1 answer
131 views

Definition of magic $T$ and $H$ states: are there different definitions for them?

I am a bit confused by the definition of magic $T$ and $H$ states and I would like to check if their name is actually not uniformously spread in the litterature (or if I am not understanding something)...
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6 votes
1 answer
130 views

Good references to learn magic state distillation for fault tolerance

I need to learn magic state distillation procedure and their application to fault-tolerance. One of the original paper on this subject is the following: https://arxiv.org/pdf/quant-ph/0403025.pdf I am ...
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11 votes
1 answer
443 views

Why exactly are variational algorithms considered promising?

There is obviously a great deal of work happening at the moment on variational quantum algorithms. However, I'm struggling to understand why exactly are they considered promising? Looking through some ...
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4 votes
2 answers
174 views

Accuracy threshold theorem: what is the physical probability of failure exactly

In this famous paper is derived the quantum accuracy threshold theorem for concatenated codes. I have questions for the case errors are assumed to be local and Markovian. In the so called level-1 ...
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1 vote
1 answer
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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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2 votes
0 answers
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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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7 votes
3 answers
510 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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2 votes
1 answer
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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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3 votes
1 answer
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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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8 votes
1 answer
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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 ...
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2 votes
0 answers
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Fault-tolerant qudit computation: advantage? [closed]

What are advantages of a fault-tolerant qudit computation compared with fault-tolerant quantum computation for qubits?
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2 votes
1 answer
122 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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5 votes
1 answer
222 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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0 votes
1 answer
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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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22 votes
3 answers
4k 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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2 votes
1 answer
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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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2 votes
2 answers
152 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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4 votes
2 answers
401 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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1 vote
1 answer
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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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5 votes
2 answers
185 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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5 votes
0 answers
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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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4 votes
2 answers
105 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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6 votes
0 answers
133 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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3 votes
2 answers
280 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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5 votes
1 answer
291 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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12 votes
2 answers
902 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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5 votes
2 answers
212 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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