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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.

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Can a stabilizer tableau with a single row be updated?

The context for this problem is that I wish to know how logical operators evolve over a circuit that is composed of Clifford gates and single qubit measurements. Note that the measurements may be mid-...
Arsenal_fan's user avatar
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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
2 votes
1 answer
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How are these T-distillation circuits derived?

I'm trying to wrap my head around T-distillation circuits, specifically for the 15-to-1 distillation code. My question is, how does one discover these circuits from the description of the code? There ...
quantum_squirrel's user avatar
1 vote
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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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What is the measurement algorithm that the PyClifford's authors used?

I am studying how to perform measurements on pure stabilizer states of the Pyclifford package. The author used the convention to represent stabilizer states the same as in the paper: Improved ...
Việt Nguyễn's user avatar
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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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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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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
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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
1 vote
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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 ...
iknownothing's user avatar
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Uniqueness, absolutely maximally entangled states, and the 3 qutrit code

There is a well known $ [[3,1,2]]_3 $ qutrit stabilizer code with stabilizer generators $ XXX $ and $ ZZZ $. This code is related to a $ [[4,0,3]]_3 $ qutrit stabilizer state with stabilizer ...
Ian Gershon Teixeira's user avatar
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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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
3 votes
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204 views

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 ...
Vincent's user avatar
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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 <...
compp's user avatar
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Difficulties Encountered in Simulating Quantum Purification Circuits for Magic State Distillation

Recently, I've been studying Kitaev's paper on magic state distillation. Referring to Nathan Babcock's presentation, I wrote a program in MATLAB to simulate the purification circuit. Instead of the ...
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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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If the encoding circuit for a quantum code is a Clifford operation then the logical pauli errors correspond to pauli physical errors?

I was reading Lecture $2$ from M2 ICEP- Quantum Information Theory (https://perso.ens-lyon.fr/omar.fawzi/teaching/qit-paris/lecture_notes/QEC-lec2.pdf). I am a little confused on the reasoning for the ...
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Are there standard forms for non-additive quantum codes?

Stabilizer codes have a well known standard form for the stabilizers, logicals and destabilizers. Are there any counterparts for non-additive codes? In particular I'd like to get a standard form ...
unknown's user avatar
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Can we find a product of single-qubit states that is orthogonal to a stabilizer codespace?

Let's consider a single-qubit pure state $$\rho=\rho(x, y,z)=\frac{1}{2}(I+xX+yY+zZ)$$ with $x^2+y^2+z^2=1$. Can we show the following statement is true: For any $n$-qubit tensor product states $\rho^{...
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How to find a set of independent logical operators for a stabilizer code with Stim

As a simple example, say I am studying the stabilizer code with $n=4$ physical qubits and stabilizers $Z_0Z_1Z_2Z_3$ and $X_0X_1X_2X_3$ (I'll be using 0-based indexing throughout). Using Stim, I could ...
Imperishable Night's user avatar
3 votes
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What are the main motivations for pursuing CSS codes over general non-CSS codes?

It seems like most research regarding stabilizer codes focused on CSS codes, which can be understand as a "tensor product" of two classical linear codes. However, I feel like non-CSS codes ...
Yunzhe's user avatar
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3 votes
1 answer
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In context of stabilizer codes, are logical gates and Pauli operators the same?

I was reading this blog post and as I understood, a unitary operator is logical if and only if it's in $\mathcal{N}(\mathcal{S})$, and a Pauli operator is Logical if and only if it's in $\mathcal{C}(\...
Hamed's user avatar
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How to Use Stim for Noisy Simulation with Adaptive Circuit and Noise Tracking?

I am trying to learn stim to simulate noisy quantum circuits. My goal is to construct an adaptive circuit in which I can add quantum gates one by one and observe the changes in the circuit's check ...
iknownothing's user avatar
2 votes
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What information about the logical state does one obtain from the stabilizers?

Consider the initialization of a surface code into $\vert 0\rangle_L$. This is done by initializing all the data qubits into $\vert 0\rangle$ respectively and then turning on the $XXXX$ and $ZZZZ$ ...
user1936752's user avatar
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How can I obtain the rotated surface code from the original unrotated one?

I am trying to understand the equivalence between the original planar surface code and the rotated version. Looking at Tomita and Svore's paper (2014), they say "The number of qubits [from ...
Ezequiel Rodriguez Chiacchio's user avatar
2 votes
3 answers
140 views

Rotated surface code vs unrotated surface code - Boundary stabilizers

I use these image for the questions below: When we look at the surface code, we can define it in two slightly different ways. The first figure shows a way where we keep all the data qubits and ...
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How to deal with logical gates which map Logical Paulis to non-Pauli products?

I am trying to figure out the action of logical gates on quantum codes. I am trying to check the logical operations that the logical gate performs on the code. I can do it in simple cases, for example ...
am567's user avatar
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Stabilizer codes with no "pure" stabilizers

For all stabilizer codes, we can represent its stabilizer generators using a standard form $$ H = \left(\begin{array}{ccc|ccc} I & A_1 & A_2 & B & 0 & C \\ 0 & 0 & 0 &...
Yunzhe's user avatar
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5 votes
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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
3 votes
1 answer
125 views

For stabilizer codes, is a certain logical operation unique?

Suppose we have a $[[n, 1]]$ stabilizer code $Q$ and a single-qubit unitary $U$. We define the logical counterpart of $U$ as $\bar{U}$. My question is: Is there just one $\bar{U}$ up to stabilizers of ...
Yunzhe's user avatar
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6 votes
1 answer
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How to get smaller stabilizer codes with Gottesman's construction?

I'm following the construction from Section 3.5 of Daniel Gottesman's thesis (third paragraph), which is supposed to allow us to construct an $\left[ \hspace{-1pt} \left[ n-1, \, k + 1, \, d - 1 \...
Ben Criger's user avatar
2 votes
2 answers
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Why I am continuously getting "failed to decompose hyperedges into edges" in Stim?

I have written a Hypergraph product quantum error correction circuit on Stim. The code is $[\![65,9,4]\!]$. I have put circuit level noise and I am using the sinter ...
Omprakash Chandra's user avatar
1 vote
1 answer
46 views

Relating different surface code diagrams

How do I relate the surface code plaquette and vertex stabilisers of this depiction of the surface code (where the qubits are the edges): to this depiction (measurement qubits are the black dots, ...
compp's user avatar
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3 votes
2 answers
255 views

Is it always possible to write the state corresponding to a set of stabilizer generators?

Given a set of stabilizer generators, is it always possible to write down the state corresponding to it? Is there a way to write down the quantum state corresponding to a stabilizer generator?
Dotman's user avatar
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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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2 votes
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Are the logical states of the corresponding unencoded state unique?

Given the generators of a stabilizer code and this code's orthogonal projector is $\Pi$. Take five qubit code as am example, the logical state of $\vert 0 \rangle$ is calculated as $\vert 0^L \rangle =...
Michael.Andy's user avatar
3 votes
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How many $ [[9,1,3]] $ surface codes are there?

Two codes are said to be equivalent if their code spaces are related by a non-entangling gate, i.e., a gate from $U(2)^{\otimes n} \rtimes S_n$, the local unitaries together with permutations. The ...
Ian Gershon Teixeira's user avatar
5 votes
0 answers
77 views

Finding all small (e.g. 7 qubit) stabilizer codes

Given some choice of parameters $ [[n,k,d]] $ with $ n $ small, is there any computationally easy way to find all of (or at least many of) the stabilizer codes with those parameters? For certain ...
Ian Gershon Teixeira's user avatar
6 votes
1 answer
148 views

Is the Steane code the only $ [\![7,1,3]\!] $ CSS code?

Is the Steane code the only $ [\![7,1,3]\!] $ CSS code? This paper claims there are 10 non-equivalent $ [\![7,1,3]\!] $ stabilizer codes. How many of these are CSS codes? Is it just the Steane code?
Ian Gershon Teixeira's user avatar
0 votes
1 answer
70 views

Ancilla qubit error spreading to multiple data qubits?

In this paper, the author considers a four qubit stabilizer circuit shown below. Note that the first four qubits from the top are data qubits and can be in any state (here we just put them in $\vert ...
user1936752's user avatar
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5 votes
1 answer
300 views

Decoding the Steane Code

The $[[5,1,3]]$ code is a perfect code basically meaning that the weight-0 and weight-1 error spaces completely fill out the $32$-dimensional Hilbert space. On the other hand, the $[[7,1,3]]$ Steane ...
Eric Kubischta's user avatar
3 votes
2 answers
65 views

How to calculate pseudo-threshold of the [[5,1,3]] code analytically to match simulation?

In this answer there is a simple method given to compute the "worst case pseudo-threshold" of the Shor code. I want to apply this to the $[[5,1,3]]$ perfect code. Namely, given a ...
Eric Kubischta's user avatar
1 vote
1 answer
82 views

Given a set of stabilizers, what is an efficient way to compute the logical states and logical operators?

Suppose I have $n$ qubits and I specify $n - k$ independent stabilizer generators. I have defined a Hilbert space with $k$ logical qubits. Moreover, there exist $2k$ operators that obey the Pauli ...
user1936752's user avatar
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6 votes
2 answers
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Codes saturating the bound d=(n+1)/2

An $ n $ qudit code always has distance bounded above by $ d \leq \frac{n+1}{2} $ (Edit: the correct bound appears to be $ d \leq \frac{n}{2}+1 $ but the $ [[6,0,4]] $ hexacode and the $ [[2,0,2]] $ ...
Ian Gershon Teixeira's user avatar
5 votes
1 answer
65 views

What are the properties of stabilizer codes that ensure $C(S) = N(S)$?

In Daniel Gottesman's thesis "Stabilizer Codes and Quantum Error Correction", he makes the following claim: The set of elements in $G$ that commute with all of $S$ is defined as the ...
am567's user avatar
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3 votes
2 answers
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Why can a quantum code correct $t$ errors only when $d \geq 2t + 1$?

I am working from chapter 7 from notes for ph229 by J. Preskill. The notes define the distance of a quantum code as: The distance $d$ is the the minimum weight of a Pauli operator $E$ such that: $$\...
am567's user avatar
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