Questions tagged [stabilizer-code]
A stabilizer quantum error-correcting code appends ancilla qubits to qubits that we want to protect. A unitary encoding circuit rotates the global state into a subspace of a larger Hilbert space. This highly entangled, encoded state corrects for local noisy errors.
119 questions from the last 365 days
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Why does the error is always logical NOT when MWPM fails?
Consider the decoding of noisy syndrome using pymatching (as shown here) for the repetition code. The logical operator in this case is simply identity matrix. One could write the python code as
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How to obtain the set of stabilizer generator of a code, from stabilizer of its circuit tableau simulation
What is the relationship between the stabilizer generator of a quantum stabilizer code, and the stabilizer generator of its circuit tableau? I notice some difference for cases.
I use Stim to simulate ...
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Does the distance of a stabilizer code depend on the logical basis?
Suppose we have a $[[n,2,d]]$ stabilizer code whose one choice of basis of logical operators is $\{\bar{X}_1,\bar{Z}_1\}$ and $\{\bar{X}_2,\bar{Z}_2\}$. We can also choose different basis $\{\bar{X}_1\...
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The weight distribution of uniformly random Clifford conjugation
For an $n$ qubit system, I fix a non-identity Pauli $P$ and perform the following experiment $N$ times:
Sample a Clifford gate $U_i$ uniformly at random from the Clifford group (iid).
Compute $w_i$, ...
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Can we increase code distance for stabilizer codes by fixing logical operators?
Assume we have a $[[n, k, d]]$ stabilizer code and each of its logical qubit has the same weight $d$. Can we generate a new code $[[n,k',d']]$ with $k'<k$ and $d'>d$ by fixing some logical ...
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Is there a way to calculate the distance of a stabilizer code with multiple logical qubits in Stim?
I am wondering if there is a way to calculate the distance of a stabilizer code given stabilizer generators. Especially, I want to calculate the distance of a code with multiple logical qubits, i.e., $...
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Question regarding the vanishing of syndromes(excitations) in the boundaries of the symmetric rotated surface code
I've begun looking at Logical blocks for fault-tolerant topological quantum computation, and I've come across something in their diagrams of the XZZX code that is confusing me, in Fig 4.
They state ...
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Is this a valid stabilizer state?
Suppose I have a code where the codewords are such that there are an equal number of $0$s and $1$s. For example, if $n=6$, codewords are $000111$, $101010$, $110001$ and so on as long as there are ...
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Fastest method to find any single error for a given stabilizer set and given syndrome
I am working on trellis structures of stabilizer codes, where I construct a trellis for a given set of stabilizer generators and a given syndrome. I want to find any arbitrary error out of all the ...
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Wrong encoding circuit for $[[5, 1, 3]]$ code in Gottesman's thesis
In Chapter 4 of Daniel Gottesman's thesis, he gave an encoding circuit for the famous $[[5, 1, 3]]$ code.
Note that in this figure, $R$ is the Hadamard gate. However, this circuit seems not working ...
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Stablizer expression(parity check matrix) to system Hamitonian for a quantum error correction code
Given a stablizer expression (parity check matrix) of a quantum error correction code, how to obtain its system Hamitonian? I am not sure whether this is correct: for toric code, its Hamitonian is ...
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PyMatching in presence of qubit error for the repetition code
I'm considering the repetition code in presence of measurement and qubit error as discussed in SEC IV B of this paper. I am trying to use pymatching for the syndrome decoding using mwvp in presence of ...
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Clarifications on stabilisers/generators for any state
An n-qubit state will have n generators. If I have a spin entangled state like $|0\rangle|11\rangle + |1\rangle|00\rangle$, I would say the generators are $XXX, IZZ, ZZI$.
First:
Is $ZZI$ correct for ...
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Why does this stabilizer flow hold?
I am reading https://arxiv.org/pdf/2302.02192 by Gidney and it shows how to construct stabilizer flow diagrams. See the following diagram (figure from Appendix A)
The claim here is that qubit s is ...
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MWPM decoding with PyMatching in presence of measurement error
I am trying to do decoding in presence of measurement error and bit-flip error for the repetition code. In the PyMatching version 0.2.2 https://pypi.org/project/PyMatching/0.2.3/ it mentions that
To ...
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Erasure Error Correction Protocol in Kah Jen Wo's Thesis
Referring to this thesis (download at bottom of page): https://repository.tudelft.nl/record/uuid:649f5fe9-0266-4be2-a3a5-4b5ffd13073e
In section 4.2, I understood things quite well up to Table 4.1. ...
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non-CSS codes with transversal logical gates
The five-qubit $[[5, 1, 3]]$ perfect code is a non-CSS code and has transversal $HS$ gate.
Do we also have examples of some other non-CSS codes that has transversal $HS$ gate? Or do we in general know ...
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Measurement of a surface code patch - how many detectors in Stim after the final round?
I am looking at the Stim code for a surface code patch after the final measurement. This usually has a detector that looks at multiple elements. I am struggling to understand which elements should be ...
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Stabilizer of Color code
I have a question about stabilizers in color codes. In color codes, neighboring faces share two qubits, and the weight of the stabilizers is even.
My questions are:
1.Can the product of stabilizers ...
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Commuting measurements performed in the different order?
The question is in the context of stabilizer measurements where we have commuting stabilizers and we can choose to measure them in a different order, as is commonly done in the surface code. I choose ...
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Necessary and sufficient conditions for a stabilizer code to have certain transversal gates
We have many sufficient conditions to judge if a stabilizer code has certain transversal gates. For example:
CSS codes will have block-wise transversal CNOTs.
self-dual codes will have transversal ...
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If $g \in S$, does this necessitate that $g^{\dagger} \in S$?
In Lecture 5 of Aram Harrow's lecture series "Quantum Information Science II", he sets about proving that $\Pi_{S} = \frac{1}{|S|} \sum_{g \in S} g$ can be written as $\Pi_{S}= \prod_{i=1}^{...
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Proof for how logical operators generated systematically will satisfy Pauli commutation
Let me assume a CSS code.
Suppose I have $n$ qubits and have $(n-k)$ stabilizers which we label by the set $S = \{S_1, S_2, .. S_{n-k}\}$. Let me then find $2k$ Pauli operators $L_i$ that are
Not ...
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Showing that $V_{S} \subseteq \Pi_{S}$?
From Aaaron Harrow's Lecture series: Quantum Information Science II, Lecture on stabilizers, I am a little confused with one of the details of the following proof.
Claim 1. $$\Pi_{S} = \frac{1}{|S|} \...
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Why does measuring the $X$ stabilisers at the boundary for rough merging give $X_{L}X_{L}$
In the paper Surface code quantum computing by lattice surgery, for rough merging, they state that measuring the $X$ stabilisers at the boundary is equivalent to $X_{L}X_{L}$.
What I don't understand ...
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Smallest codes with transversal cliffords
Is there a classification or database of codes which allow transversal implementation of the Clifford group? The only ref I know is this non-exhaustive list on the error correction zoo for any ...
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Are rough and smooth boundaries (or primal and dual) always Z stabilisers and X stabilisers?
I've been going through a lot of literature on surface codes and lattice surgery, and I keep running into the issue of different papers, at least to me, changing their definition of rough and smooth (...
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Repetition code applied to decoherence-free subspaces
The repetition code (figure below) can be used to detect bit-flip errors. Such a code is useful for studying, but it can't be used in more general scenarios where also phase errors occur.
I was ...
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Why parity-check matrices not reduced when working with Bivariate Bicycle codes?
Bivariate Bicycle (BB) codes were introduced in [1]. These families of QEC codes are defined via parity check matrices containing a two block structure of the form $H_{X} = (A|B)$ and $H_{Z} = (B^{T}|...
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How are stabilizers defined with monomial labellings for Bivariate Bicycle codes?
I have been recently looking at getting familiarized with Bivariate Bicycle (BB) codes [1], and in general, Multivariate Bicycle codes [2]. I need some help figuring out the usage of the monomial ...
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Intuition behind constructions of good qLDPC codes
A good $[[n, k, d]]$ qLDPC code is one where $\lim_{n \rightarrow \infty} \frac{k}{n} > 0$ and $k = O(n)$ and $\lim_{n \rightarrow \infty} \frac{d}{n} > 0$ and $d = O(n)$. A CSS code is one ...
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Measurement of a logical operator in a Foliated Surface Code
I am reading Universal Fault Tolerant Measurement Based Quantum Computing. On page 6, under the section Measurement-based qubit transmission, it states that measurement of the logical operators is ...
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What are the logical X and Z operators of the two qubits of the toric code?
I am familiar with the surface code, which encodes one logical qubit and the logical X and Z operators are horizontal and vertical lines of X and Z physical gates respectively.
However, the toric code ...
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Bell measurement for the $ [[5,1,3]] $ code
The $ [[5,1,3]] $ code is the smallest quantum error-correcting code (and it is the unique 5 qubit error-correcting code see Corollary 10 of Quantum codes of minimum distance two ), so sometimes it is ...
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How to find the matrices which generate the physical realizations of logical Pauli operators for the $[[6,4]]$ stabilizer code?
Question surrounds example given on page 8 of this paper.
The rows of the following matrix determine the physical states of the $[[6,4]]$ code:
$$G= \begin{bmatrix}
1 & 1 & 0 & 0 & 0 &...
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Given a stabilizer state, check if all k qubit subsets are maximally mixed
Problem
I’m checking if all k-qubit subsets of a n-qubit stabilizer state are maximally mixed.
Approach that works, but is slow
The following approach works. But it's numerically painful, so I'm ...
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Fault-tolerant CNOT gates
The controlled-not gate
$$
CNOT = \begin{bmatrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\0 & 0 & 0 & 1 \\0 & 0 & 1 & 0 \end{bmatrix}
$$
is important in many ...
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Variant of Quantum Hamming bound for CSS codes
The arguments leading to the Quantum Hamming Bound for non-degenerate $[[n,k,d = 2t+1]]$ codes
$$\sum_{j=0}^t 3^j \binom{n}{j} ≤ 2^{n-k}$$
can be manipulated to find a stronger related bound for non-...
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Dephasing noise and error correction
It is my impression that $ Z $ errors are the dominant source of noise in many platforms used for quantum computing. For example, Ultrahigh Error Threshold for Surface Codes with Biased Noise says ...
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Finding stabilizers from a T distillation code
Consider $T$ state distillation as per Figure 14 in this paper.
I am trying to figure out what the stabilizers are. I thought of trying to evolve them starting from e.g. $S = XI\cdots I$ acting on the ...
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$[[10, 2, 2]]$ codes with transversal T gates
There is a 10-to-2 magic distillation protocols for the $T$ state based on the four-qubit codes (https://arxiv.org/pdf/1204.4221). As the protocol has order-2 distillation efficiency, we can use a $[[...
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A question on dimensions of the basis vectors for the $[[6,4]]$ code
I am working from page $8$ of this paper.
The generator for the coset representatives of $C^{\perp}$ in $C$ for the $[6,5,2]$ classical code is given by $$G_{C/C^{\perp}} = \begin{bmatrix} 1 & 1 &...
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Determine if an operator is in the stabilizer group
I was working on error correction algebraically but I now want to use some computational method.
I have a set of generators of a stabilizer group which I represent as a rectangular matrix over $\...
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Why is the group membership problem hard for general matrices but not for the stabilizer group?
In this answer, it is claimed that the problem of answering whether an invertible matrix $A$ is an element of the group $\langle B_1, B_2,..., B_n\rangle$ for invertible matrices $B_i$ is in NP.
For ...
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How exactly does measuring a syndrome operator work for 'non-discrete' errors?
I am reading a summary of the stabilizer formalism in this paper, which considers the following encoding of a single qubit in three qubits:
$$
|\overline{0}\rangle=\frac{1}{\sqrt{2}}(|000\rangle+|111\...
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Does one have multiple degrees of freedom in defining logical states and logical operators of a QEC?
Consider a rotated surface code. Let the surface code have $Z$ stabilizers along the top and bottom boundary and $X$ stabilizers along the left and right boundary.
If I initialize all the physical ...
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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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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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Realization of Surface code, can an eigenvalue of -1 be taken?
I have a question about the implementation of Surface Code. I understand that Surface Code is a stabilizer state defined by plaquette operators and vertex operators, meaning it is a state where the ...
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