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)

Filter by
Sorted by
Tagged with
0 votes
1 answer
54 views

Knill-Laflamme error correction conditions for impure(mixed) code states

Knill-Laflamme error correction conditions for correctable error set $\{\hat{E}_j\}$ are $ \hat{P}_{\mathcal{C}} \hat{E}_k^\dagger\hat{E}_l\hat{P}_{\mathcal{C}}\propto\hat{P}_{\mathcal{C}} $ where $\...
Saurabh Shringarpure's user avatar
2 votes
3 answers
471 views

Is the $[\![7,1,3]\!]$ Steane code a surface code?

All surface codes are CSS codes. The smallest error correcting ($ d=3 $) CSS code is for 7 qubits. So we can be sure that the $ [\![5,1,3]\!] $ code is not equivalent to a surface code since no ...
Ian Gershon Teixeira's user avatar
1 vote
1 answer
237 views

Is there a standard form for CSS codes?

For a generic stabilizer code, there's a standard form for $H,X,Z$; see for example previuos post. For CSS codes, there are at least these simplifications : $B=0$; $C_2=0$; I think $A_1$ and $D$ can ...
unknown's user avatar
  • 2,122
2 votes
0 answers
38 views

Exploiting virtual-$Z_{\theta}$ corrections in QEC

The work done by IBM Researchers to perform virtual-$Z_{\theta}$ gates allowed them to: achieving zero-time relative phase shifts; correcting $Z_{\phi}$ errors occurring when applying some gates -- e....
Daniele Cuomo's user avatar
0 votes
1 answer
103 views

Definition of fault-tolerant measurement in QCQI

In the book "Quantum Computation and Quantum Information," one of the definitions for fault-tolerant measurement is mentioned as "any single component in the procedure results in an ...
david's user avatar
  • 61
0 votes
1 answer
85 views

Commutation relationship and measurement results

There are things I do not understand about the following circuit, and I would appreciate it if you could explain. ...
david's user avatar
  • 61
1 vote
1 answer
49 views

What is the structure of quantum graph code [[4,0,3]]? Or does it exist? [closed]

What is the structure of quantum graph code [[4,0,3]]? Or does it exist? How the edges are connected?
user22477's user avatar
0 votes
1 answer
176 views

How to implement Shor's 9-Qubit-Code in Qiskit?

I am new to qiskit and currently trying to implement Shor's 9-Qubit_Code. I am trying to orient my code on the example given in the official qiskit-tutorial and so far I have the following: ...
3nondatur's user avatar
  • 173
1 vote
0 answers
133 views

Correctability of X, Y, and Z Errors in Quantum Surface Codes and Color Codes

In surface codes and color codes, when the code distance is $d$, you can correct up to $[(d-1)/2]$ Pauli errors. I would like to know what this $[(d-1)/2]$ Pauli errors means for $X$, $Y$, and $Z$. ...
david's user avatar
  • 61
2 votes
1 answer
52 views

Recognizing a CWS code in a specific case

A CWS code can be defined in terms of stabilizer codes/stabilizer states and graph states. See What is an example of a non-additive code that is not a CWS code? Is the $ ((11,2,3)) $ nonadditive ...
Ian Gershon Teixeira's user avatar
2 votes
1 answer
156 views

Is there non-fault-tolerant quantum error correction?

See this sectino on Wikipedia for Quantum error correction Peter Shor's 9-qubit-code, a.k.a. the Shor code, encodes 1 logical qubit in 9 physical qubits and can correct for arbitrary errors in a ...
Jagerber48's user avatar
8 votes
2 answers
1k views

CSS codes are the only stabilizer codes with transversal CNOT?

Given a stabilizer code $\mathcal{C}$ then $$ \mathcal{C} \text{ is CSS} \iff \text{CNOT} \text{ is transversal}. $$ The forward implication is well known, see for example Transversal logical gate ...
Eric Kubischta's user avatar
2 votes
1 answer
143 views

Are there qiskit or stim examples of how to decode a subsystem code

I can manage decoding a stabilizer code in qiskit and stim. I'd like to try a subsystem code next. For a stabilizer code, all stabilizers commute so simultaneous measurement of these produces ...
unknown's user avatar
  • 2,122
1 vote
1 answer
43 views

How to best represent 5-Qbit codeword operator in Qiskit [duplicate]

We learned that in the 5-Qibit error correcting code, the logical zero state $ | \bar{0} \rangle $ is encoded in the form of $ \frac{1}{4} (\mathbb{1}+M_0)(\mathbb{1}+M_1)(\mathbb{1}+M_2)(\mathbb{1}+...
Blackwidow's user avatar
1 vote
2 answers
137 views

Can all stabilizer and logical generators of a stabilizer code be written without any Pauli $Y$s?

For an $[[n,k,d\ge2]]$ error-detecting qubit stabilizer code there are $n-k$ stabilizer generators and $2k$ logical generators ($k$ logical $X$ generators and $k$ logical $Z$ generators). Allowing for ...
Jonas Anderson's user avatar
4 votes
0 answers
42 views

Can we have a constant-overhead threshold theorem?

The threshold theorem states that any abstract circuit in BQP can be computed by another polynomial-depth circuit that succeeds in the presence of noise. The original construction from 1996 requires ...
Dudu Ponar's user avatar
1 vote
1 answer
62 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 ...
LittleBlue's user avatar
1 vote
1 answer
116 views

Proofs that lattice surgery does what it claims to do

I'm interested in making a little list of papers that include a proof that lattice surgery really does what it claims to do; namely, performs a pairwise logical measurement. The original paper (http://...
Yossarian's user avatar
  • 123
1 vote
2 answers
181 views

How to show that the three-qubit repetition code only corrects up to 1-bit flip errors?

From Nielsen and Chuang, the error correction criteria is $$P E_i^{\dagger} E_j P=\alpha_{i j} P$$ $P$ is the projector onto the correct codespace, $E_{j}$ are error operations and $\alpha_{i j} $ is ...
Aubrey Sharansky's user avatar
5 votes
1 answer
168 views

Error correction conditions for phase flip errors in repetition code

I am very confused about the error correction condition in the 3-qubit repetition code when considering phase flip ($Z_i$) errors. It is well known that this code fails to protect from phase flip ...
user25295's user avatar
0 votes
0 answers
35 views

Uncorrectable error due to error on ancilla qubit

Consider a controlled-NOT (CX) gate between the two qubits, implemented with an interaction of the form $ \widehat{H}_{\mathrm{CX}}=V\left[\left(\frac{\hat{I}_1+\widehat{Z}_1}{2}\right) \otimes \hat{I}...
Aubrey Sharansky's user avatar
0 votes
1 answer
130 views

Using Deterministic Errors in Stim Detector Circuit and decoding with Pymatching: X_ERROR(1) failing in PyMatching

the Question Is there a standard way to use deterministic errors (so, for instance, X_ERROR(1)) in Stim with PyMatching? Currently, sometimes if I use ...
bumble13's user avatar
0 votes
1 answer
58 views

Different Decodings for Union Find

I read a paper about the implementation of a quasilinear UnionFind Decoder for quantum error correction and it says that the last step of it is building a sparse forest and peeling it. My question is ...
feder34's user avatar
5 votes
4 answers
362 views

A 32x32 matrix with the best rank * span_weight

Answers must contain a 32x32 matrix (or smaller). An answer is better if its matrix has a higher product of rank and span weight. In "Subsystem codes with spatially local generators". Sergey ...
Craig Gidney's user avatar
  • 36.7k
4 votes
0 answers
72 views

Resource Estimation for Ion Trap Architecture

Most resource estimation papers (computing physical qubit count, algorithm run time, power consumption, etc.) target the superconducting qubit architecture. Some examples include How to factor 2048 ...
quantum123's user avatar
7 votes
2 answers
742 views

Can you give an intuitive idea behind how the Minimum Weight Perfect Matching (MWPM) decoder work?

The Minimum Weight Perfect Matching (MWPM) decoder seems to be the most popular choice for decoding error syndromes in Surface Code quantum error correction. Can anyone give an intuitive idea of how ...
Aubrey Sharansky's user avatar
0 votes
2 answers
95 views

Why does the qubit give random results in the circuit with rearranged CNOTs for Steane's seven qubit code in Stim?

The following is a part of the syndrome measurement circuit for Steane's seven qubit code in Stim(For ease of viewing, the TICK is omitted.). Since we are considering the detection of X errors, we use ...
lassel's user avatar
  • 51
1 vote
2 answers
168 views

A Simple way of Encoding qubit in Stabilizer Codes with Stim

In Stim, is there a easy way to encode qubits into a code state $| 0_L \rangle$ in stabilizer codes, other than constructing an encoding quantum circuit? For example, in Steane's seven-qubit code, can ...
lassel's user avatar
  • 51
4 votes
0 answers
65 views

QEC code with transversal Hadamard on one logical qubit

Is there a known QEC code that encodes more than one logical qubit, and which has a transversal logical Hadamard gate on only one of the logical qubits?
Peter-Jan's user avatar
  • 1,529
4 votes
0 answers
55 views

Which kinds of codes have their automorphism group in the Clifford Group

Two quantum codes are said to be equivalent if they are related by a non-entangling gate. The automorphism group of a quantum code is the group of all code equivalences. Every automorphism of a $GF(4)$...
Eric Kubischta's user avatar
2 votes
1 answer
120 views

Generators for $[[9,1,3]]$ linear quantum code

In Theorem 7 of the paper Quantum Error Correction Via Codes Over GF(4) the authors come up with a sort of reduction algorithm. Right after they claim the $[[85,77,3]]$ Hamming code implies the ...
Eric Kubischta's user avatar
4 votes
1 answer
85 views

Example $ GF(4) $ linear $ [[9,1,3]] $ code

Many well known stabilizer codes are $ GF(4) $ linear. For example, the perfect $ [[5,1,3]] $ code and the $ [[7,1,3]] $ Steane code are both $ GF(4) $ linear. The $ [[9,1,3]] $ Shor code is not $ GF(...
Ian Gershon Teixeira's user avatar
2 votes
1 answer
213 views

What are the transversal gates of the $ [[4,2,2]] $ code?

Consider the $ [[4,2,2]] $ stabilizer code with stabilizer generators $ XXXX,ZZZZ $. Logical $ X $ on the first qubit is $$ XIXI $$ Logical $ Z $ on the first qubit $$ ZZII $$ Logical $ X $ on the ...
Ian Gershon Teixeira's user avatar
5 votes
0 answers
90 views

Famous qutrit codes with no qubit analogue

No $ [[3,1,2]] $ qubit code exists (more generally no $ ((3,2,2)) $ qubit code exists). See Why can't there be an error detecting code with fewer than 4 qubits? However there does a exist a $ [[3,...
Ian Gershon Teixeira's user avatar
1 vote
0 answers
38 views

anyone know how to solve the error plz? [closed]

...
juhaina's user avatar
  • 11
0 votes
0 answers
29 views

Logical error rate of topological cluster state under erasure channel

Assume I use the topological cluster state (https://arxiv.org/abs/quant-ph/0703143) to protect against erasure. Assume also that the only noise or erasure that I have in my system is loss in the ...
Yaron Jarach's user avatar
0 votes
1 answer
169 views

Comparing the Erasure and Depolarizing Noise Channels in QEC

Are there any quantum error-correction codes which have been found to exhibit high thresholds for either Pauli-based depolarizing noise or Erasure noise channels, but not for both? Are there reasons ...
MatthewS1990's user avatar
6 votes
0 answers
140 views

Is every $ ((11,2,5)) $ code equivalent to the $ [[11,1,5]] $ stabilizer code?

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. It is ...
Ian Gershon Teixeira's user avatar
0 votes
0 answers
175 views

How can I perform syndrome measurements for both X-type and Z-type stabilizers in Stim?

I would like to simulate the logical error rate of the surface code under depolarizing noise using Stim. The decoder I am using decodes from both X-type and Z-type stabilizer syndrome values. ...
lan's user avatar
  • 51
0 votes
0 answers
121 views

Phenomenological threshold estimates for rotated surface code

I am trying to find threshold estimates for the rotated surface code for depolarizing phenomenological/circuit-level noise with and without measurement errors (I am aware that those might be ...
jerol's user avatar
  • 13
2 votes
1 answer
86 views

Are there countably many quantum codes up to equivalence?

Two quantum codes are said to be equivalent if they are related by a non-entangling gate. Over the range of all parameters $((n,K,d))$ with $d> 1$ we know there are at least countably many non-...
Eric Kubischta's user avatar
0 votes
1 answer
110 views

What is the Stim color_code:memory_xyz circuit executing?

In Stim's color_code:memory_xyz, the all data qubits are first initialized to the |0> state, then the Z-type stabilizer measurement circuit is executed, and ...
lan's user avatar
  • 51
1 vote
1 answer
125 views

Implementing circuits with post-selection in Stim

I want to use Stim to calculate logical error rates with circuits that involve post-selection on measurement outcomes. Is it possible to do so with Stim and if yes, how do I go about it? To provide ...
transistor's user avatar
1 vote
1 answer
96 views

Is there a way to perform two simulations with the same noise in Stim?

I would like to calculate the logical error probabilities of surface codes under depolarizing noise using Stim. In doing so, if all data qubits are initialized to |0>, it is not possible to account ...
lan's user avatar
  • 51
1 vote
1 answer
62 views

parameters for which all codes are degenerate

Throughout, assume that $ n,K,d $ are integers such that an $ ((n,K,d)) $ code exists Recall the quantum Hamming bound: For a non-degenerate ((n,K,d)) qubit code we must have $$ K(\sum_{j=0}^{\lfloor ...
Ian Gershon Teixeira's user avatar
5 votes
2 answers
144 views

Decoding of quantum non CSS code using Minimum Weight Perfect Matching (MWPM)

I am trying to implement a decoder for a quantum non-CSS code, where the syndrome defects are always created in pairs. First question is more theoretical: Why and how does MWPM work for non-CSS codes? ...
Zohar's user avatar
  • 53
2 votes
2 answers
167 views

How are quantum error-correction conditions in Nielsen and Chuang implemented in practice?

Quantum error-correction conditions in Nielsen and Chuang, 10th-anniversary edition (Theorem 10.1) state that the error operation $\mathcal{E}$ with operation elements $\{E_i\}$ is correctable if and ...
EugeneB's user avatar
  • 167
0 votes
0 answers
47 views

Initialize data qubits in the first step of syndrome measurement

I saw many papers that say that in syndrome measurement of the stabilizer codes, such as surface codes and color codes, firstly, all data qubits are initialized to $|0\rangle$. What is the reason? I ...
lan's user avatar
  • 51
0 votes
1 answer
124 views

How can I determine if the error correction by estimated error results in a logical error in circuit-level noise?

I am trying to simulate the logical error probability of color code under circuit-level noise. In the code capacity noise and the phenomenological noise, the Pauli errors to the data qubits that occur ...
lan's user avatar
  • 51
0 votes
1 answer
289 views

How can I extract actual Pauli error in Stim?

I am using a quantum circuit simulator "Stim" to simulate the logical error probability of color code under circuit-level noise. In the code capacity noise and the phenomenological noise, ...
lan's user avatar
  • 51

1
3 4
5
6 7
16