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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What is the physical operation behind "moving edges" and "moving corners" in Litinski's game of surface codes paper?
I was reading Litinski's A game of surface codes (https://quantum-journal.org/papers/q-2019-03-05-128/pdf/). In the introduction (page 2), the paper talks about operations like "moving edges"...
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What is the logical gate speed of a superconduting quantum computer?
What is the logical gate speed of a photonic quantum computer? says
In a simple world the speed of a photonic quantum computer would just be the speed at which it’s possible to make small (fixed ...
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AGP Fault-tolerance of the flag qubit QEC for 7-qubit Steane code
I was trying to apply the flag qubit QEC (https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.121.050502) for 7-qubit Steane code.
From the AGP method (https://arxiv.org/pdf/quant-ph/0504218),
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How do Union-Find Decoders deal with Measurement errors through multiple measurement rounds?
I've read a few papers regarding to Surface Code and its decoding algorithms. I've learned that a Union-Find decoder need up to $d$ measurement rounds to deal with measurement errors.
These ...
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How to find the undetected errors for general stabilizer codes in Stim?
In Stim, we use the detectors to track syndrome flips and infer the error pattern. However, the syndrome stays the same if the actual error pattern is a logical operator of the code by coincidence. It'...
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Are transversal entangling gates possible for stabilizer codes other than CSS?
It is well known that CSS codes can have lots of transversal entangling gates. For example, $ CNOT $ is exactly transversal on 2 blocks of any $ [[n,1,d]] $ CSS code. And https://arxiv.org/abs/1304....
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Is working with the |+> , |-> basis any harder than the |0>, |1> basis?
Say I have a code, for example the $ [[5,1,3]] $ code, and I want to (fault tolerantly) prepare the logical $ |+ \rangle $ state. Is that any harder than preparing the logical $ | 0 \rangle $ state? ...
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Highest theoretical threshold to fight single-qubit depolarizing noise for noiseless error-correction
Let's consider that each qubit in the lab faces a single-qubit depolarizing channel $\mathcal{N}(\rho)=(1-p) \rho + p \mathbb{I}/2$.
Is there a theoretical result indicating the largest value of $p$ ...
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Is it sufficient to assume a constant coherent error?
I've recently started working with quantum errors and noise and came across an intriguing but simple question. When we consider coherent errors in quantum gate operations, it's common to model them as ...
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Boundary conditions for surface code
I have a question about boundary conditions for surface codes. Do any surface codes have torus-like boundary conditions? Are there any surface codes that don't actually have boundary conditions, i.e. ...
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How is $(\langle \psi| E_{a}^\dagger E_{b} | \psi \rangle)^\dagger = C_{ba}^*\langle \psi| \psi \rangle $
I am reading through Daniel Gottesmans surviving as a quantum computer in a classical world. On page 36, he presents the following theorem:
Theorem 2.7 (QECC Conditions). $(Q, \mathcal{E})$ is a $Q E ...
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Is every code with a universal set of transversal gates trivial?
The quantum repetition code is an $ [[n,1,1]] $ stabilizer code with stabilizer generators $ Z_iZ_{i+1} $ for $ i=1, \dots, n-1 $.
The Eastin-Knill theorem states that a $ d >1 $ code cannot have a ...
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What does DETECTORs mean in the example circuit for rotated surface code in Stim?
In Stim, an example circuit for rotated surface code is provided:
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Advantages and disadvantages of rotated surface code
I think one of the advantages of rotated surface code is that it can express surface code with fewer physical bits. Are there any other advantages?
Also, are there any disadvantages compared to ...
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What is the easiest way to get path graph from Stim?
In Stim, we can get a detector graph with the probability of each error mechanism occurring. Now I want to construct a path graph from the detector graph, which is usually done by Dijkstra's algorithm....
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How does measurement based quantum computing (MBQC) behave under error propagation?
In the quantum circuit model, we know how to handle error propagation if we implement a unitary $U'$, which is $\varepsilon$-close to the ideal unitary $U$ and a state $|\psi'\rangle$, which is also $\...
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Are close states still close after measurement (regarding trace distance)?
We are given two states $|\psi_1\rangle, |\psi_2\rangle \in \mathbb{C}^2 \otimes \mathbb{C}^2$ with trace distance $\leq \varepsilon$, so they are very close to each other.
Now, assume we measure the ...
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Necessary condition for transversal Hadamard by family of stabilizer codes
A necessary and sufficient condition for a stabilizer code having transversal $CNOT$ is that the code is a CSS code (see Theorem 11.5 here or the question here).
I know that a sufficient condition for ...
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Define the $k$-local transversal logical operation
For a $[[n, 1]]$ QEC code $\mathcal{Q}$, we say single logical gate $R$ is transversal if the logical $\bar{R}$ can be implemented with $R^{\otimes n}$.
I am wondering if we could expand the ...
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Why can Pauli errors $E$ be decomposed as $E=T(S)LG$ with $T(S)$ "pure errors"?
I have a question about the decomposition of Pauli errors.
Pauli error $E \in \{I,X,Y,Z\}^{{\bigotimes}n}$ that satisfies the syndrome $S$ can be decomposed into a product of pure error $T(S)$, ...
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Why focus on T gates and not some other single qubit rotation R making Clifford + R universal?
Background: In many error correction codes in particular the surface code, the Clifford operations generated by the S,H and CNOT are transversal for quantum computation (meaning that these logical ...
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Does two quantum error correcting codes having the same CSS Tanner Graph imply that they are locally equivalent?
I am studying CSS quantum LDPC codes and I am curious as to whether the Tanner Graph structure necessarily must have long-range connections in order to be a non-local qLDPC code. This is because the ...
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See error samples in Stim+Pymatching [duplicate]
I have a surface code circuit written in Stim. Following Stim's intro I can use sinter to get logical error rates. I'd like to see what error patterns could lead to ...
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What sinter.plot_error_rate is actually doing with the data?
I don't quite understand what the **sinter.plot_error_rate** function is actually doing. From looking at the code, it seems to perform some kind of binomial fit. I'...
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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 ...
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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,...
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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 + |...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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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