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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 to do a Bell state measurement on encoded qubits fault tolerantly?

Experimentally how do you do a Bell state measurement? Is it just a CNOT and a Hadamard and then a measurement in the computational basis? Or is it even simpler than that because it is easy to measure ...
Ian Gershon Teixeira's user avatar
6 votes
0 answers
105 views
+50

Universal gate set for the $ [[15,1,3]] $ code

The $ [[15,1,3]] $ triorthogonal code implements transversal $ T $. Since it is a CSS code, two blocks will also have a transversal $ CNOT $ gate. To get a universal gate set all that is required is ...
Ian Gershon Teixeira's user avatar
1 vote
1 answer
48 views

Definition of fault tolerant gate gadgets

I am trying to understand fault tolerance for gate gadgets a bit better. Here Gottesman provides the following definition: I generally understand this definition but I have two questions. (1) Why is ...
Eric Kubischta's user avatar
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3 answers
62 views

Does the threshold estimation of the surface code not need a fault-tolerant setup?

The threshold estimation of the $[[9,1,3]]$ surface code in stim's getting started tutorial extracts the syndromes by having a single ancilla for each stabilizer generator. It is the same setup as in ...
qubitzer's user avatar
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1 answer
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Is there a proof in the literature of magic state injection for surface codes?

Proofs of correctness for fault-tolerance schemes rarely give tight constants for error correction performance. However, they do provide formal reasoning that a scheme is correct and all the pieces ...
Chris Pattison's user avatar
2 votes
1 answer
91 views

Common ways to fault tolerantly prepare a stabilizer state

It is my impression that it is much easier to fault tolerantly prepare a stabilizer state than it is to prepare a magic state. What are some common ways to fault tolerantly prepare a given stabilizer ...
Ian Gershon Teixeira's user avatar
1 vote
1 answer
27 views

Simulating an ideal circuit with surface code

In a seminal paper AGP, the authors proved the threshold theorem and obtained a threshold for concatenated Steane code $[[7^k,1,3^k]]$. One thing to note is that, to ensure fault-tolerance, error ...
AndyLiuin's user avatar
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3 votes
2 answers
473 views

Does QEC always eventually fail in a deep circuit?

My understanding is that any QEC code is able to detect and correct a certain number of physical errors on its qubits and this is the code distance $d$. If we have more than $d$ errors in a given ...
Josph's user avatar
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2 answers
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Definition of threshold/pseudothreshold/breakeven in QECC

Although there has been discussions on this1, I still have some questions. I will firstly summarize my understanding of these concepts, threshold, pseudo-threshold (please correct if I am wrong), ...
AndyLiuin's user avatar
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0 answers
23 views

Is there a term for fault-tolerance of logical gates that depends on the underlying hardware?

Is there a notion of fault-tolerance of logical gates that also accounts for the underlying hardware? If so, how is this notion / term called? An example: A possible choice of logical basis states in ...
qubitzer's user avatar
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What could be intermediate steps towards a fully fault-tolerant quantum algorithm with added value for each step?

Starting from an algorithm without any error correction or detection, what incremental improvements can be made towards fully fault-tolerant encoding and operation? It seems that fault-tolerant error ...
qubitzer's user avatar
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3 votes
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How to initilize logical qubits stabilizer states in a fault-tolerant way

I have what might be an elementary question for experts. Suppose I wish to carry out fault tolerant quantum computation, with an error correcting code such as the 7-qubit code or the 5-qubit code (of ...
user173611's user avatar
1 vote
1 answer
51 views

Why are the following gates implemented transversally/weakly transversally?

I have read that logical gates can be implemented transversally ($U_{L} = V^{\otimes n}$) or weakly transversally ($U_{L} = \otimes _{i=1} ^{n} V_{i}$). I have verified that the $[[8,3,2]]$ colour ...
am567's user avatar
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4 votes
2 answers
115 views

Fault-tolerant syndrome extraction

My question concerns fault-tolerant measurement of the syndrome of an quantum error-correcting code. Somewhat recently, Chao and Riechardt, in https://arxiv.org/abs/1705.02329, proposed a method ...
user173611's user avatar
2 votes
1 answer
49 views

Can we perform teleported-error-correction with noisy Bell pairs?

It is well-known that in https://arxiv.org/abs/quant-ph/0312190 that error correction can be performed by teleportation on the logical level, so prepare one logical Bell pair and (logical?) Bell ...
AndyLiuin's user avatar
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1 answer
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Trying to prove $U$ implements a logical $CCZ$ gate on the $[[8,3,2]]$ code?

I know from several different papers that the gate $U=TT^{\dagger}TT^{\dagger}TT^{\dagger}TT^{\dagger}$ actually implements a logical $CCZ$ gate on the $[[8,3,2]]$ quantum code. However, I am having ...
am567's user avatar
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Question about fault-tolerant measurement in Nielsen&Chuang page 490

The fault-tolerant procedures for state preparation and logical operation consider the procedure successful if only a one-qubit error occurs in each encoded block of the output. So, I thought that the ...
이호준's user avatar
5 votes
0 answers
90 views

Magic state distillation, eigenvectors of Clifford gates, and transversal gates

I have some (possibly mistaken) ideas about magic state distillation. Please disabuse me of them? In section III "Universal Quantum Computation with Magic States" of the original paper on ...
Ian Gershon Teixeira's user avatar
2 votes
2 answers
91 views

Fault Tolerance of 2-transversal gates

Suppose I have a single block $n$-qubit stabilizer code that can correct a weight 1 error (so the distance is $d=3$). If I apply a $1$-transversal gate of the form $U = U_1 \otimes U_2 \otimes \cdots \...
Eric Kubischta's user avatar
2 votes
0 answers
31 views

Procedures for fault-tolerant nondestructive logical measurements

Definition 2 on page 20 in Daniel Gottesman's An Introduction to Quantum Error Correction and Fault-Tolerant Quantum Computation states that a nondestructive logical measurement procedure is fault ...
Peter-Jan's user avatar
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3 votes
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How to perform logical operations between logical qubits in a $[\![8,3,2]\!]$ quantum code?

I have a small question on codes like $[\![8,3,2]\!]$, which encodes physical qubits to multiple logical qubits. How do we perform logical operations in between these logical qubits? Of course, if I ...
AndyLiuin's user avatar
  • 376
3 votes
1 answer
73 views

Fault-tolerant correlated measurement on logical qubits

So I have an encoded Bell pair $(|\overline{0}\overline{0}\rangle+|\overline{1}\overline{1}\rangle)$ in CSS code(e.g. Steane code) and I hope to perform stabilizer measurement of $\overline{XX}$ non-...
AndyLiuin's user avatar
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6 votes
1 answer
447 views

What exactly is a subsystem code?

I'm sure I'm just tripping up on terminology here. I'm new to the wonderful field of QEC, and so far am familiar with the stabiliser formalism and basic codes such as the generic surface code. I keep ...
compp's user avatar
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1 answer
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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
3 votes
1 answer
104 views

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
7 votes
2 answers
1k views

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
339 views

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
3 votes
1 answer
80 views

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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3 votes
1 answer
447 views

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 ...
user1936752's user avatar
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0 votes
0 answers
45 views

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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2 votes
0 answers
137 views

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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3 votes
1 answer
174 views

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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1 vote
2 answers
114 views

$X$ 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
2 votes
2 answers
580 views

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
  • 376
7 votes
0 answers
100 views

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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2 votes
0 answers
45 views

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
4 votes
0 answers
62 views

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 ...
user avatar
1 vote
1 answer
83 views

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
2 votes
1 answer
85 views

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
0 votes
1 answer
76 views

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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1 vote
1 answer
104 views

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
263 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-...
FDGod's user avatar
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4 votes
0 answers
73 views

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
  • 51
2 votes
1 answer
138 views

Steane's method for fault-tolerant syndrome measurement

In Steane's method for fault-tolerant syndrome measurement, as explained in https://arxiv.org/abs/quant-ph/0504218, ancilla is prepared in the logical 0 state for the circuit performing the X-type ...
david's user avatar
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0 votes
1 answer
141 views

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 ...
david's user avatar
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2 votes
1 answer
175 views

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
4 votes
0 answers
45 views

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
72 views

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
8 votes
2 answers
910 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
2 votes
1 answer
303 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