6 votes
Accepted

Why Is This the Ground State of a Toric Code?

The crucial point it seems you are missing is to recognize that $A_s^2=1$ (for that matter, $B_p^2=1$ also) and therefore $(1+A_s)/2$ is a projector onto the $+1$ eigenspace of $A_s$. We would then ...
nervxxx's user avatar
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6 votes
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What is the Stabilizer of a Code?

A group action of a group $G$ on a set $X$ is a map $\phi:\, G\times X \rightarrow X$ such that $$ \phi(e,x) = x, \quad\text{and}\quad \phi(g, \phi(h,x) ) = \phi(gh,x), $$ for all $x\in X$ and $g,h\...
Markus Heinrich's user avatar
6 votes

How does the size of a toric code torus affect its ability to protect qubits?

The Toric code is an error correcting code. The distance of the code (I.e. the number of local operations required to convert one logical state into an orthogonal one) is equal to $N$, where the Toric ...
DaftWullie's user avatar
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5 votes
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is the minimum weight perfect matching decoder optimal

Is [minimum weight perfect matching] an optimal decoder? No, it's not optimal. For example, it uses the weight of the shortest path between two detection events as an approximation for the ...
Craig Gidney's user avatar
3 votes
Accepted

How to get the matched nodes using Matching.decode() in PyMatching?

Currently you are correct that PyMatching outputs the edges in the solution, but not the pairs of detection events. You can find a valid pairing of in Python by tracing the paths between detection ...
oscarhiggott's user avatar
2 votes

How does the size of a toric code torus affect its ability to protect qubits?

The length of the loops determines the quantity called the code distance. The key fact about its effect on the function of the code is the exponential suppression of errors. More precisely, if $d$ is ...
Adam Zalcman's user avatar
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2 votes
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Questions on Definitions and Concepts Regarding Toric Codes

Regarding your first question: These codes are especially well suited for fault-tolerant implementation, because the procedure for measuring the error syndrome is highly local. What is meant here is ...
Lior's user avatar
  • 1,210
2 votes

Is there a way to perform a defect-free logical CNOT on the toric code?

I cannot find any information on this [...] Does lattice surgery provide an option? If you go to google scholar and search "lattice surgery" it brings up the relevant papers. The very first ...
Craig Gidney's user avatar
2 votes
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How to compute Z logical operators of a toric code using Kunneth theorem?

I don't understand why and how to get $\mathcal{H}_0$ and $\mathcal{H}_1$ The repetition code can be written as a chain complex $$ C_1 \xrightarrow{\partial_1} C_0, $$ For example for the distance 4 ...
Peter-Jan's user avatar
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2 votes
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Simulating a small distance surface code with individual qubit addressing

Stim doesn't currently support recording errors when sampling from a circuit, but it does support recording errors when sampling from a detector error model via the ...
Craig Gidney's user avatar
2 votes

3D toric code (2D + measurement errors) has higher threshold error rate $p_{\text{th}}$ than 2D?

Hard to say where you're going wrong, but indeed, introducing measurement errors into the system should make $p_{\text{th}}$ lower. One helpful reference might be the following phase diagram computed ...
squiggles's user avatar
  • 900
1 vote

Construction of 3D Toric Code

Take a cube and tesselate it with smaller cubes. Put qubits on the edges, Z-stabilizers on the edges surrounding a face, and X-stabilizers on the edges incident to a vertex, and then identify the ...
squiggles's user avatar
  • 900
1 vote
Accepted

Pymatching: 'Matching' object has no attribute 'decode_batch'

Make sure you're on the latest version. pip install --upgrade pymatching.
Craig Gidney's user avatar
1 vote
Accepted

How to Switch Toric Code to Surface Code (no using STIM!)

The code sniplet in your question constructs the toric code as a hypergraph product (HGP) code using the non full-rank matrix of the repetition code : this is an $n \times n$ matrix with rank $n-1$. ...
unknown's user avatar
  • 2,146
1 vote

How to Switch Toric Code to Surface Code (no using STIM!)

It's almost always the case that it's easier to make the donut version of a code, rather than the planar version, because the planar version needs boundaries whereas the donut version can be all bulk. ...
Craig Gidney's user avatar
1 vote
Accepted

Energy and degeneracy of the ground state and excitations of the toric code

One of the major points of the Toric code Hamiltonian is that all the terms commute, each of which as $\pm 1$ eigenvalues. So, to find the ground state, you need something that is the $+1$ eigenstate ...
DaftWullie's user avatar
  • 58.1k
1 vote

is the minimum weight perfect matching decoder optimal

In light of the comments above, I'm going to try to answer this question, with the caveat that I am taking my first steps in this field so I might have errors here. TLDR - it seems to me that in an ...
Lior's user avatar
  • 1,210
1 vote
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how to simulate toric and surface codes with stim + PyMatching

An encoding circuit is needed [...] for toric and surface codes The notebook you linked does a surface code experiment at the end. It uses Stim's built-in surface code circuit generation (...
Craig Gidney's user avatar

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