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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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: ...
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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 ...
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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".
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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 ...
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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 ...
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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'...
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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 ($...
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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 ...
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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 ...
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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), ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $+1$ ...
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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 ...
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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:
...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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.
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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 ...
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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 ...
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State injection in a surface code
I am reading this paper on lattice surgery by Dominic Horsman
et al, and am struggling to understand a simple method they put forward in their Figure 8(a)(b)(c) to inject an arbitrary state into a ...
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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 ...
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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 ...
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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.
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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&...
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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 ...
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How to implement fault tolerant cat state construction with qiskit?
I want to build a fault-tolerant error correction gadget in qiskit. As a part of that, I want to construct cat states (for the syndrome measurements) in a fault-tolerant way.
One way to do this, is to ...
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Error budget for a quantum memory experiment
Is there any standard way to make error budget for a quantum memory experiment simulation? To be more specific, suppose I ran the simulation for a distance d surface code under circuit-level noise, ...
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Transversal QLDPC Code Confusions
I have a few rudimentary questions about QLDPC codes.
Do any admit transversal Paulis? My intuition is that many such constructions are CSS, they should. But it's hard to see if this is obviously ...
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Why isn't quantum error correction used in quantum key distribution protocols like BB84?
In QKD protocols like BB84 there are some post-processing steps after obtaining the sifted key. One of them is error correction so that Alice and Bob end up with the same string of bits. This is to ...
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How to get the surviving stabilisers of a fusion network
In Fig. 5 of PsiQuantum's Fusion-based quantum computation paper, it's clear to me how to get R, F, and C.
How do I get S? Just by looking at (a), I could obviously tell you that we end up with <...
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What is the relation between Shor's code and the GHZ states?
The logical states for Shor's code $|0_L\rangle$ and $|1_L\rangle$ are
$$
\begin{equation}
\begin{aligned}
|0_L\rangle = \left(\frac{|000\rangle+|111\rangle}{\sqrt{2}}\right)^{\otimes{}3}\...
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In the QECC condition $\langle\psi|E_a^\dagger E_b|\phi\rangle=C_{ab}\langle\psi|\phi\rangle$, what is $C_{ab}$?
In this book, Theorem 2.7 has the QECC conditions. I attach a snippet here
Theorem 2.7 (QECC Conditions). $(Q, \mathcal{E})$ is a $Q E C C$ iff $\forall|\psi\rangle,|\phi\rangle \in Q, \forall E_a, ...
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How can I find out how compatible a quantum circuit is to a backend without executing the circuit
Say I have multiple backends, I need a score to determine how well a backend is suitable for that particular circuit, I have it's noise model too, I tried transpiling the circuit and calculating noise ...
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When does $G^{\otimes n}$ vs $G^{\dagger \otimes n}$ implement $G$ transversally
Given an $[[n,1]]$ or $((n,2))$ code and a $2 \times 2$ matrix $G$ in its transversal group.
There are cases when $G^{\otimes n}$ acts on $|\bar 0 \rangle,|\bar 1\rangle$ the same way as $G$ on $|0\...
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Fault-tolerant syndrome extraction
My question concerns fault-tolerant measurement of the syndrome of an quantum error-correcting code.
Somewhat recently, Rui Chao and Ben Reichardt, in https://arxiv.org/abs/1705.02329, proposed a ...
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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 ...
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