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Questions tagged [error-correction]

Quantum error correction (QEC) is a collection of techniques to protect quantum information from decoherence and other quantum noise, to realise fault-tolerant quantum computation. Quantum error correction is expected to be essential for practical quantum computation in the face of noise on stored quantum information, faulty quantum gates, faulty state preparations, and faulty measurements. (Wikipedia)

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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
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Parameters for which there is a unique stabilizer code

Two stabilizer codes are said to be equivalent if they can be related by non-entangling Cliffords, i.e. by local Cliffords and SWAP gates. There are unique stabilizer codes for the parameters $ [[2,0,...
Ian Gershon Teixeira's user avatar
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$\langle Z \rangle_L$ in the Distance Two Surface Code

In an experimental realization of the distance 2 surface code, the codewords are: $$|0\rangle_L = \frac{1}{\sqrt{2}} (|0000\rangle + |1111\rangle), |1\rangle_L = \frac{1}{\sqrt{2}} (|0101\rangle + |...
clunky monkey's user avatar
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Will logical clock cycle time be a limiting factor for quantum computations?

Fault-tolerant quantum computation promises to strongly suppress the errors by scaling up the size of the systems. Right now, different physical implementations of proto quantum computers have very ...
Frederik Ravn Klausen's user avatar
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Quantum error detection

I'm a bit confused regarding the definition of error detection. Let $H$ be a Hilbert space, $C$ a subspace, $P\colon H\to C$ the projection, and $E$ a linear operator on $H$. Consider these two ...
Tron's user avatar
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What is the domain of the dual map of a quantum channel?

Possibly a naive question...if the dual map of a quantum channel gives the evolution of the system in the Heisenberg picture by acting on observables, and observables are self-adjoint operators on the ...
Mara Jade's user avatar
2 votes
1 answer
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How to verify that a certain gate was applied to a quantum code

Suppose I have a quantum error correcting code $|\psi \rangle = \alpha | 0 \rangle + \beta | 1 \rangle$, say the $[[7,1,3]]$ Steane code for concreteness. Suppose there is a black box that either ...
Eric Kubischta's user avatar
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Is there any machine learning method for finding quantum error correction codes?

To define a quantum error correction code, first one needs to model noise, such as Pauli noise, dephasing noise, etc. Then according to the noise, look for the code space, stabilizer, and logical ...
mingo's user avatar
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Obtaining and Applying XX and ZZ Parity in Stim for Controlled Paulis

I am attempting to perform a CNOT between two surface code qubits in Stim, based on this paper by Daniel Litinski and Felix von Oppen. The CNOT they perform is shown in the figure below (Figure 3 from ...
Maxwell Poster's user avatar
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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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What is the definition of color codes?

Is there a generally accepted definition of what a color code is? I have found two definitions that I am not able to reconciliate with each other: The error correction zoo defines color codes via ...
qubitzer's user avatar
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explain_detector_error_model_errors complains "no single circuit error had these exact symptoms"

I am using the explain_detector_error_model_errors() method, unfortunately I am receiving this: ...
edp23's user avatar
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Does there exist a general way of finding the size of the stabilizer group $|S|$?

So I know that, for a stabilizer code, the stabilizer group $S$ has $n-k$ commuting generators. Is there a general way of knowing what the order of the full group of $S$ is, aside from writing out all ...
am567's user avatar
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Why do Surface codes generate so much measurement data?

I read in the Sparse Blossom paper: "A surface code superconducting quantum computer with a million physical qubits will generate measurement data at a rate of around 1 terabit per second". ...
Daniel Mandragona's user avatar
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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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Is there a simple condition under which X-and-Z-error correctability leads automatically to Y-error correctability?

I had the impression and guess that in a quantum error correction code, once it can correct any single-qubit X and Z errors, it automatically can also correct all single-qubit Y errors. Now after ...
aystack's user avatar
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For stabilizer codes, why does the error syndrome not depend on the codeword?

While reading through some lecture notes on quantum error correction, I read the statement: "In particular, the syndrome doesn’t depend on the specific codeword, only on the Pauli error." I'...
Daniel Mandragona's user avatar
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Is there a relation between number of non-equivalent logical operators and number of logical qubits?

I've been reading this blog post about surface code and it says There are four non-equivalent types of loops: the trivial ones (stabilizers), the horizontal ones ($X_1$ operator), the vertical ones ($...
Hamed's user avatar
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When does a channel between two error correcting codes preserve logical information?

I am trying to understand when a quantum channel preserves (part of) the information stored in an error correction code. Take some $[[n, k, d]]$ quantum code with stabilizer set $\mathcal{S}$. The ...
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Are $\delta$-close logical states of a code also close in terms of physical states?

If I have two logical states of a quantum code that are close in trace distance i.e. $\vert 0\rangle_L$ and $U_L\vert 0\rangle_L$ where $\|U_L - I\|_{\diamond} \leq \delta$, what is an upper bound (if ...
John Doe's user avatar
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2 answers
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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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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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What problems in chemistry or materials science could be solved with 100 fault-tolerant qubits?

Background IBM, Infleqtion, QuEra, and other quantum hardware companies have announced roadmaps where they expect to have 100 or more fault-tolerant qubits by the end of the decade. It seems ...
taciteloquence's user avatar
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What effect does phase flip error have in computation?

Phase flip error changes $\alpha|0\rangle + \beta|1\rangle$ into $\alpha|0\rangle - \beta|1\rangle$, but outcome probabilities are still $|\alpha|^2$ and $|-\beta|^2=|\beta|^2$. So, what is the effect ...
Hamed's user avatar
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1 vote
2 answers
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Motivation for simulating multiround syndrome extraction circuits for quantum error correction code

I am a new learner in quantum error correction and I am curious about the motivation for simulating multiround syndrome extraction circuits of quantum error correction code. The purpose for single ...
Chris H's user avatar
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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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How to find the $+1$ eigenvectors of the stabilizers for the Shor code

I am currently working through chapter $3$ of "Stabilizer Codes and Quantum Error Correction" (Daniel Gottesman's thesis). I would like to know the general method for finding the $+1$ ...
am567's user avatar
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error correction after lattice surgery for merging

Imagine you have merged two surface codes with lattice surgery by measuring all the stabilizers and treating both surface codes as one surface code, then are we doing error correction before or after ...
Shahrooz Pooryousef's user avatar
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Parallelizing decode_to_edges_array (PyMatching)

I am trying to parallelise the function: matching.decode_to_edges_array() to somewhat batch decode, but I am consistently running into the same problem: ...
edp23's user avatar
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Implementing CNOT logical gate for the Steane code

I am looking at the transversal logical gates for the Steane code. I am having trouble trying to implement the $CNOT$ logical gate for this code. I understand that we can transversally implement a ...
am567's user avatar
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"Session has been closed" Error in solving Maxcut problem by QAOA

I am trying to run this code- https://learning.quantum.ibm.com/tutorial/quantum-approximate-optimization-algorithm same way as they mentioned. After running this code, my qiskit Dashboard show 10-12 ...
Maths_hawk'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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Codes from orthogonal stabilizer states

This article discusses the equivalence between code word stabilized (CWS) and union stabilizer codes. This family with two names is also equivalent to the graph codes discussed in Nonadditive Quantum ...
Ian Gershon Teixeira's user avatar
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Are these two codes the same?

There is an $ ((11,2,5)) $ CWS code given in table B2 at the end of https://kth.diva-portal.org/smash/get/diva2:894450/FULLTEXT01.pdf I know that some CWS codes are equivalent to stabilizer codes, but ...
Ian Gershon Teixeira's user avatar
1 vote
1 answer
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Moving pauli product rotations past measurements

I'm trying to understand how the clifford + T compiler works in "A Game of Surface codes". How do I move a pauli product rotation block past a pauli product measurement block? More ...
Palash Goiporia's user avatar
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Does the Steane code have characteristics which make it especially useful for trapped-ion devices?

On the web page of the Error Correction Zoo for the [[7,1,3]] Steane Code, there are quite a few realizations on trapped-ion devices. In fact, there are a lot of implementations generally compared to ...
vollautomatthi's user avatar
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Confusion about lattice-surgery operation

According to https://arxiv.org/abs/1111.4022, to perform a logical CNOT using lattice surgery, three patches of surface codes are needed and nearest neighbour operations are performed. However, I ...
AndyLiuin's user avatar
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2 votes
1 answer
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Measuring stabilizers for qutrit stabilizer code

For stabilizer codes we do many things like extracting error syndromes and preparing states and stuff like that by measuring the stabilizer generators. For qubit stabilizer codes measuring the ...
Ian Gershon Teixeira's user avatar
3 votes
0 answers
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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
42 views

Efficiently Generating Expectation Values of Operators Using Stim

I am trying to generate expectation values of operators based on several shots. using the stim library and noticed that it has the option to get the expectation of ...
iknownothing's user avatar
3 votes
1 answer
76 views

How does the surface code satisfy constraints of relativity?

Consider a surface code in the shape of a rectangle that has very very long horizontal edges and very short vertical edges. I initialize the surface code into the logical state $\vert 0\rangle$. Let's ...
user1936752's user avatar
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2 votes
2 answers
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Kraus decomposition of merging in lattice surgery

I am reading about lattice surgery from this paper. I am interested in the merge operation which takes 2 qubits to 1 qubit. I want to understand the logical-level Kraus operation that the merge does. ...
qubit's user avatar
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2 votes
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Does a 3 qudit code always exist? (except for qubits)

There is a famous $ [[3,1,2]]_3 $ qutrit stabilizer code with stabilizer generators $$ XXX $$ $$ ZZZ $$ where it should be clear from context that $ X $ and $ Z $ here denote the appropriate qudit ...
Ian Gershon Teixeira's user avatar
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2 answers
76 views

Definition of good quantum LDPC code

I am a bit confused by the exact definition of a "good" quantum LDPC code. I call $k$ the number of logical qubits, $n$ the number of physical qubits and $d$ the distance of the code. I ...
Marco Fellous-Asiani's user avatar
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State injection in a surface code

I am reading this paper on lattice surgery by Dominic Horsman et al 2012 New J. Phys. 14 123011, and am struggling to understand a simple method they put forward in their Figure 8(a)(b)(c) to inject ...
quantumctrl's user avatar
1 vote
1 answer
65 views

Deformation rules for ZX "pipe" diagrams

If I consider the 3-dimensional structures from https://arxiv.org/abs/1905.08916, these would form a nice 3-dimensional structure for surface code layouts, except I'm not sure what the equivalence ...
Sam Jaques's user avatar
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1 answer
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Remove edges from PyMatching matching graph / Copy Matching object

I am simulating a surface code where the edges in the syndrome graph change slightly from shot to shot based on erasures that occur in the circuit (i.e. I have a syndrome graph corresponding to the ...
J Watty's user avatar
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4 votes
2 answers
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Transversal CNOTs on CSS codes with multiple logical qubits

I am interested in the theory of implementing logical gates on quantum error correcting codes. From a practical view, transversal gates are very attractive. I have a question about transversal gates. ...
Vincent's user avatar
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3 votes
2 answers
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Stabilizers of the [[12, 2, 4]] "Carbon" code

In a recent experimental paper, Microsoft with IonQ claimed that they demonstrated the "repeated error correction" with both the $[[7, 1, 3]]$ Steane code and the $[[12, 2, 4]]$ "Carbon&...
Yunzhe's user avatar
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Codewords of Subsystem Codes

I have a question about subsystem codes that came up when reading the article Universal transversal gates with color codes - a simplified approach. Subsystem codes are introduced in Section IV. A The ...
Vincent's user avatar
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