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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How can fault tolerance be understood from the perspective of detecting regions in a circuit?

The language of detectors and detecting regions in the context of quantum error correction is introduced in https://arxiv.org/pdf/2302.02192.pdf. Detecting regions seem to be capable of specifying ...
Ezequiel Rodriguez Chiacchio's user avatar
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1 answer
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Circumventing the Eastin-Knill theorem by means of Shor's fault tolerant Toffoli gate

I have been reading about the Eastin-Knill theorem, which states that no quantum error correction code can can transversely implement a universal gate set. In this context, for example, the surface ...
Josu Etxezarreta Martinez's user avatar
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Is there a no-go theorem for or an upper bound on code threshold?

My understanding of the code threshold for an $ [[n,1,3]] $ code is something roughly like the setup given in How to compute the error threshold for the $9$-qubit Shor code? where the error channel is ...
Ian Gershon Teixeira's user avatar
4 votes
1 answer
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Can Wiesner's quantum money be realized (with logical qubits) today?

Consider Wiesner's quantum money scheme. With today's devices and today's error correction and mitigation schemes, how long can we hold $n$ logical qubits such that they are all (logically) in a ...
Mark Spinelli's user avatar
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Why is the first logical unitary gate in this example fault tolerant?

From Arthur Pesah's blog on "Computing with Quantum Codes using Transversal Gates", found here: https://arthurpesah.me/blog/2023-12-25-transversal-gates/ He gives the following examples of ...
am567's user avatar
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Recent hardware advances towards fault-tolerant quantum computing and quantum error correction

Please excuse the broad nature of the question. I am interested in understanding recent developments by various hardware companies and their approaches towards quantum error correction. My knowledge ...
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What are the differences between code deformation, code conversion, code switching and gauge fixing?

From my understanding so far: Code switching is essentially just a fault-tolerant mapping between two codes. It's called code conversion if it's gradational, i.e. changes to stabilizer generators are ...
tomek's user avatar
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Solution Nielsen and Chuang exercize 10.71

Exercise 10.71: Verify that when $M = e^{−iπ/4}SX$ the procedure we have described gives a fault-tolerant method for measuring $M$. The book describes a procedure to perform the measurement. Instead ...
Maria 's user avatar
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How to devise a flag qubit scheme for concatenated codes?

This flag qubit idea was introduced by Rui Chao and Ben Reichardt, last year when I read the paper [arXiv:1705.02329] I thought I understood fully. But today as I revisit I found myself still a bit ...
AndyLiuin's user avatar
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$X$ and $Z$ stabilizer generators are on identical sites. What does $\otimes_{i=1}^n H_i$ do?

I am learning about the CSS construction in Nielsen and Chuang, and I'm trying to understand the effect of $\otimes_{i=1}^n H_i$ on a code with $X$ and $Z$ stabilizer generators on the same sites. (...
user196574's user avatar
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Preparation of logical state with qLDPC code

I'm recently learning qLDPC code. Although I've seen literature talking about the code construction, parity check, possible logical operation and decoding. How exactly do I prepare e.g. a logical phi ...
AndyLiuin's user avatar
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Can we distill magic states with arbitrary angle $\theta$?

There seems to be numerous work about the distillation protocol of the $T$-magic state $$ \frac{1}{\sqrt{2}}(|0\rangle+e^{i\pi/4}|1\rangle). $$ Similarly, I am wondering if it is possible to distill a ...
Yunzhe's user avatar
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Leakage Detection Circuit

Leakage happens when the state of the system is no longer in the computational subspace that defines a qubit. In Preskill's notes on Fault Tolerant Quantum Computation (Pg 40), a circuit for leakage ...
user1752323's user avatar
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Prospects of topological/anyon approach to QC

Given the recent articles/preprints by Google and Quantinuum claiming implementation of braiding of non-Abelian anyons on a quantum processor, what is the likely impact of these findings, and ...
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Effect of too many syndrome measurement cycles on surface code threshold

Assume I have a surface code with distance $d$ and an i.i.d error model with both single qubit depolarization and measurement errors, both with probability $p$. In this case, one usually repeats the ...
Yaron Jarach's user avatar
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Shor FTEC scheme vs Steane FTEC scheme and other schemes

In the common stabilizer-based error correction procedure, one uses one syndrome qubit to measure the eigenvalue of a weight "w" syndrome by "w" times entangling the syndrome ...
Yaron Jarach's user avatar
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The Output of Transversal Bell Measurement in Knill's Method of Fault-Tolerant Error Correction (FTEC)

On page 26 of arXiv:quant-ph/0504218, it is written that in Knill's method of fault-tolerant error correction (FTEC), the output of the transversal bell measurement becomes $(P_m \otimes I) | \Phi_0 \...
kong's user avatar
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Logical |+> State Preparation

From this paper (page 2), they said that Surface code qubits can be easily initialized in the logical +1-eigenstates Logical |0> and Logical |+> of Logical Z and Logical X by initializing all ...
Mr. Morgan's user avatar
4 votes
2 answers
219 views

List of codes which can perform non-Clifford gates fault-tolerantly

I am looking to make a comprehensive list of all known quantum error correction codes which can perform one non-Clifford gate fault-tolerantly. Currently, I have found two codes. 3D Color code Tri-...
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Why is it necessary for the Fault-tolerant error correction condition to take into account two qubit errors?

The Fault-tolerant error correction (FTEC) condition states that for a code distance of 3, if there are no input errors, even if up to one gate fault (including a two qubit error occurring on a two ...
lan's user avatar
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Steane's method for fault-tolerant syndrome measurement

In Steane's method for fault-tolerant syndrome measurement, as explained in arXiv:quant-ph/0504218, ancilla is prepared in the logical 0 state for the circuit performing the X-type stabilizer ...
david's user avatar
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Definition of fault-tolerant measurement in QCQI

In the book "Quantum Computation and Quantum Information," one of the definitions for fault-tolerant measurement is mentioned as "any single component in the procedure results in an ...
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Is there non-fault-tolerant quantum error correction?

See this sectino on Wikipedia for Quantum error correction Peter Shor's 9-qubit-code, a.k.a. the Shor code, encodes 1 logical qubit in 9 physical qubits and can correct for arbitrary errors in a ...
Jagerber48's user avatar
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Can we have a constant-overhead threshold theorem?

The threshold theorem states that any abstract circuit in BQP can be computed by another polynomial-depth circuit that succeeds in the presence of noise. The original construction from 1996 requires ...
Dudu Ponar's user avatar
4 votes
0 answers
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Resource Estimation for Ion Trap Architecture

Most resource estimation papers (computing physical qubit count, algorithm run time, power consumption, etc.) target the superconducting qubit architecture. Some examples include How to factor 2048 ...
quantum123's user avatar
7 votes
2 answers
609 views

Can you give an intuitive idea behind how the Minimum Weight Perfect Matching (MWPM) decoder work?

The Minimum Weight Perfect Matching (MWPM) decoder seems to be the most popular choice for decoding error syndromes in Surface Code quantum error correction. Can anyone give an intuitive idea of how ...
Aubrey Sharansky's user avatar
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1 answer
176 views

What are the transversal gates of the $ [[4,2,2]] $ code?

Consider the $ [[4,2,2]] $ stabilizer code with stabilizer generators $ XXXX,ZZZZ $. Logical $ X $ on the first qubit is $$ XIXI $$ Logical $ Z $ on the first qubit $$ ZZII $$ Logical $ X $ on the ...
Ian Gershon Teixeira's user avatar
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Logical error rate of topological cluster state under erasure channel

Assume I use the topological cluster state (https://arxiv.org/abs/quant-ph/0703143) to protect against erasure. Assume also that the only noise or erasure that I have in my system is loss in the ...
Yaron Jarach's user avatar
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1 answer
160 views

Weakly transversal gates for the $ [[5,1,3]] $ code

For the $ [[5,1,3]] $ code https://en.wikipedia.org/wiki/Five-qubit_error_correcting_code $ X^{\otimes 5} $ implements logical $ X $ and $ Z^{\otimes 5} $ implements logical $ Z $. A less common gate ...
Ian Gershon Teixeira's user avatar
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1 answer
227 views

"Rigorous" understanding that one should correct Pauli drift before non-Clifford gates

It is a well-known fact that in fault-tolerant quantum computing, it is sufficient to actually correct the Pauli errors only right before the next non-Clifford gate. My purpose is to check all the ...
Marco Fellous-Asiani's user avatar
4 votes
2 answers
164 views

Threshold estimate for 7-qubit code with flag-qubit syndrome extraction

I am looking for a paper where the threshold for 7-qubit code having a syndrome extraction based on flag qubits has been done. In the original paper about flag qubits, there is an estimation of pseudo-...
Marco Fellous-Asiani's user avatar
4 votes
1 answer
120 views

How to correct error during the syndrome measurement or recovery process?

Here's the figure 10.21 from Nielsen's Quantum Computation and Quantum Information to explain the fault-tolerant quantum computing, where the circuit in the figure corrects the error that happens in ...
SUSY's user avatar
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Difference between Repeat-until-success circuit and Distillation procedure

On the fault-tolerance quantum computer (FTQC), the errors will be corrected by the error correction code (like surface code). In this case, the small angle rotation gate is difficult to apply on the ...
劉承瀚's user avatar
3 votes
1 answer
233 views

Transversal Hadamard gates and multiple logical qubits

This question is linked to this post. I wanted to work through the details in the main answer to check I understood the conditions for the existence of logical Hadamard gates. But I am slightly ...
user289's user avatar
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Conditions for Fault-tolerant QC to work better than not doing any EC (except the ideal decoder)

I've been exploring fault-tolerant QC with the simplest 7-qubit Steane code, yes I'm able to obtain all the predicted thresholds for various transversal gates and the EC gadgets. I then tried to ...
AndyLiuin's user avatar
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4 votes
1 answer
129 views

Fault tolerance of MBQC

Is there a proof that measurement-based quantum computation can be fault tolerant under any noise model? Or at least, under any noise model with small enough correlation length?
Yaron Jarach's user avatar
1 vote
0 answers
43 views

Computation for threshold of CNOT gate

I'm trying to reproduce some threshold results using 7-qubit Steane code in those legacy papers. I managed to obtain the graph for circuit noise and single qubit gate e.g. the pseudocode for computing ...
AndyLiuin's user avatar
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1 vote
1 answer
163 views

Logical error rate for QECC using qiskit

I've been coding QECC with qiskit recently. As I'm not very skilled in programming I encountered a lot of questions. Now I'm trying to generate the logical error rate against physical error rate graph ...
AndyLiuin's user avatar
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0 answers
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Qiskit, possible to manipulate mid-circuit measurement outcomes?

I'm currently coding up an error correction gadget using qiskit. The one I'm doing now is 7-qubit Steane code with Steane's error correction gadget, in which the ancilla qubits are measured and then ...
AndyLiuin's user avatar
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2 votes
1 answer
327 views

Estimate of threshold values for 7-qubit Steane code with qiskit (or stim)

I have trouble recreating the graphs shown in this paper https://arxiv.org/pdf/quant-ph/0508176.pdf i.e. the quadratic trend of the logical error against physical single qubit error for [7,1,3] code I ...
AndyLiuin's user avatar
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0 answers
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Why the perfect 5-qubit code was used for magic state distillation?

I am currently trying to understand magic state distillation. So far, my understanding is that the general idea is to find a code where a non-Clifford gate is transversal (very well explained in https:...
Diego's user avatar
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How is the size of the circuit derived, in proving the threshold theorem?

In chapter 10.6.1 in Nielsen and Chuang, the section on concatenated codes and the threshold theorem (pages 480-481) states: The size of the simulating circuit goes as $d^k$ times the size of the ...
banercat's user avatar
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6 votes
1 answer
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Why can't we implement fault-tolerant single qubit rotations continuously?

My understanding is that in order to implement single-qubit rotations on logical fault-tolerant qubits one has to do smth like discretizing rotations and using $T$ gates. So, is it the case that given ...
mavzolej's user avatar
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4 votes
2 answers
311 views

Necessity of decoding in fault-tolerant quantum computation

On page 476 of Nielsen/Chuang's book it is stated: The basic idea of fault-tolerant quantum computation is to compute directly on encoded quantum states in such a manner that decoding is never ...
WuWu's user avatar
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7 votes
2 answers
657 views

How to intuitively understand why $T$ gate can't be implemented transversally?

Suppose we have all to all connectivity and can implement fault-tolerant logical gates in the Clifford group. Why can't we just apply a physical $T$ gate to every physical qubit in the code block? (...
Jiakai Wang's user avatar
2 votes
1 answer
151 views

Distance of one dimensional quantum error correcting code

Consider one dimensional quantum codes $[[ n,k=0]] $. One way to describe them is using stabilizer framework with $n$ independent Pauli matrices. Usually, one considers them in the graph state model ...
Root's user avatar
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4 votes
1 answer
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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 ...
itsabijection's user avatar
5 votes
2 answers
939 views

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 $\overline Z$ ...
Ian Gershon Teixeira's user avatar
2 votes
1 answer
274 views

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 ...
Ian Gershon Teixeira's user avatar
4 votes
0 answers
90 views

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) ...
itsabijection's user avatar