# Questions tagged [toric-code]

The toric code is a topological quantum error correcting code, and an example of a stabilizer code defined on a two-dimensional spin-lattice. It is the simplest and most well studied of the quantum double models (first studied in the context of Z2 spin liquid in 1991). It is also the simplest example of topological order—Z2 topological order. The toric code can also be considered to be a Z2 lattice gauge theory in a particular limit. (Wikipedia)

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### How to address the 2 logical qubits on the toric code individually? In general, how to address $k$ logical qubits in a $[[n,k,d]]$ code independently?

Suppose I have a $[[n,k,d]]$-quantum error correction code. Let us take the toric code $(T^2=S^1 \times S^1)$ as an example. We have 2 logical qubits whose logical operators lie along the different ...
54 views

### Why does Stim/PyMatching evaluate the threshold error rate differently for even and odd distances?

For the toric code, the threshold error rate value for no measurement errors should be around $p_{th}\approx 0.109$. If simulated with even distances, one finds that this is true. However, when I ...
1 vote
63 views

### How to compute Z logical operators of a toric code using Kunneth theorem?

I'm going through pymatching tutorial on constructing a toric code using hypergraph product of two repetition codes. The hypergraph product code construction $H G P\left(H_1, H_2\right)$ takes as ...
1 vote
130 views

### Simulating a small distance surface code with individual qubit addressing

I want to simulate a small-distance surface code with either STIM or Qiskit. I want individual control over all the qubits (including the ancillas) to induce controlled errors in my circuit. I want to ...
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### Pymatching: 'Matching' object has no attribute 'decode_batch'

I am trying to run the example from documentation: pymatching docs Here is my code: ...
1 vote
41 views

### How horizontal loops on the toric code are all undetectable?

I understand that if we have, for example, the blue path doesn't make any check operator detect any error, but what about the red path? We have the check operator detecting since there is just one ...
112 views

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

I was curious to whether the two logical qubits on the toric code can be entangled through, for instance, a logical CNOT operation. However, I cannot find any information on this, only how you can do ...
1 vote
92 views

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

Here is the toric code example which I found from :https://pymatching.readthedocs.io/en/latest/toric-code-example.html ...
1 vote
167 views

### Pymatching Toric Code vs Surface Code

I am looking the example of toric code in Pymatching. Here is the code: ...
118 views

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

In the module matching.py of PyMatching, it is possible to construct Matching objects to decode matching graphs using the minimum-weight perfect matching decoder. The ...
59 views

### Why are all the error cosets $Q.S$ given the erasure chain $\mathcal E$ and a syndrome $\sigma$ equiprobable? (Delfosse-Zémor)

In arXiv 1703.01517 (published here), a maximum likelihood decoding for qubit loss is explained. A quantum erasure channel erases each qubit independently with some probability $p$ and replaces it ...
198 views

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

Recall the hamiltonian of the toric code: (information mainly extracted from https://arxiv.org/pdf/1610.09260.pdf) Consider Je=Jm=1. I've been trying to get the exact energies and degeneracies of the ...
264 views

### Why Is This the Ground State of a Toric Code?

I am currently trying to study the ground state of the Toric Code. I am currently reading this paper. The Hamiltonian is given by the following, where $A_s$'s are the star operators made out a tensor ...
402 views

### What is the Stabilizer of a Code?

I am currently reading the beginning of the paper Topological Quantum Memory and I am confused about one part: The check operators generate an Abelian group, the code’s stabilizer. When I learned ...
50 views

### Questions on Definitions and Concepts Regarding Toric Codes

Currently, I reading the toric code section in the beginning of the paper Topological Quantum Memory. Here are a couple of sections that are somewhat confusing to me. (1) This quote below is under a ...
667 views

### is the minimum weight perfect matching decoder optimal

The toric code and other popular codes can be decoded using minimum weight perfect matching. Is this an optimal decoder? Here by optimal, I mean it gives the best logical error rate vs physical error ...
626 views

### how to simulate toric and surface codes with stim + PyMatching

According to PyMatching's github page the package can be decode toric and surface codes. Stim's example uses stim + PyMatching combination to get logical error rate vs physical error rate curves for ...
Note: Cross-posted on Physics SE. The quantum Singleton bound states that for an error-correcting code with $n$ physical qubits and $k$ encoded qubits, and some subsystem $R$ of $m$ qubits that can '...
The Toric code Hamiltonian is: $\sum_{x,y}\left( \prod_{i\in p(x,y)} Z_{ixy} + \prod_{i\in v(x,y)} X_{ixy} \right),$ where the $v$ and $p$ are defined according to this picture (courtesy of James ...