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)

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Does the Quantum Fourier Transform require universality?

Background: In most setups of fault-tolerant quantum computation, universality is achieved using Clifford gates such as $(S, H, \text{CNOT})$ and the $T$-gate. The Eastin-Knill theorem can be ...
2 votes
1 answer
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how to construct a ((7,2,3)) code and verify its properties

These two recent papers describe a process to construct (non-stabilizer) codes with "exotic" transversal gates : paper1 paper2 Most of the codes have distance $d=2$ which makes them less ...
2 votes
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demotivation for 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 ...
1 vote
1 answer
23 views

Implement parity check gadget using stim

I want to implement the logical $\left|00\right\rangle$ state preparation for $C_4$([[4,2,2]]) code using Stim. And the circuit is as below: I have written a piece of code that seems to be effective: <...
2 votes
1 answer
49 views

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}(\...
1 vote
1 answer
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How is Magic State Distillation (MSD) compatible with Stabiliser Measurement?

Suppose I have the rotated surface code in which I want to inject a $T$-state. What I don't get it is: suppose we do end up with the logical state $|0_L\rangle+e^{i\pi/4}|1_L\rangle$, then we must ...
7 votes
2 answers
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What is the definition of physical gate error rate?

The fidelity of two quantum states $\rho$ and $\sigma$ is a well-defined (up to discussions about a square): $$ F(\rho, \sigma) = \text{Tr}\left( \sqrt{ \sqrt{\rho} \sigma \sqrt{\rho}}\right)^2. $$ ...
2 votes
1 answer
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How to convert stim circuit to a detector check matrix and a logical observable matrix?

I would like to output a detector check matrix and a logical observable matrix defined in this paper. I've also learned from the related question: How to convert stim encoder circuit to a parity ...
1 vote
0 answers
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Is this quantum circuit design more prone to noise?

I am designing a quantum circuit which uses one qubit as the control of many $\text{CSWAP}$ operations. And then, this qubit will be measured. Will the result be more prone to noise since all the ...
1 vote
1 answer
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How to fix two flip-bit errors in a 3 qubit input

Aware of how a 1-bit flip can be fixed for 3-qubit input by having 2 Ancilla bits, encoding, and using the Toffoli gate to fix the error and decode. How can this idea be extended for a 2-bit flip (...
5 votes
1 answer
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Regarding the inductive proof that any Clifford gate can be made of Hadamard, phase and c-not

In Exercise 10.40 of Nielsen and Chunang's textbook, the reader is supposed to construct an inductive proof of Theorem 10.6 that any Clifford gate can be made of Hadamard, phase and c-not. There it is ...
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1 answer
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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 ...
2 votes
1 answer
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How is a surface code state measured?

What is the exact operation required to measure the logical state of a surface code? Does one need to measure every data qubit, or only some subset of them to perform the measurement? For simplicity, ...
1 vote
1 answer
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How to know exactly error threshold of surface code in terms of number of physical qubit

I asked some question about surface code properties and I recommended following research paper from what I asked before. Research paper : https://journals.aps.org/pra/pdf/10.1103/PhysRevA.86.032324 In ...
3 votes
1 answer
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Single-shot error correction for the surface code with measurement errors

I am trying to implement a single-shot error correction for the surface code with data + measurement errors (both with prob. p), using the (build-in) BP+OSD decoder. I am mostly following these papers:...
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What is the role of color in color code? ( theoritically and exprimentally)

We know that the color code has an extra element relative toric code . It is color . I want to know what is the role of color ?
7 votes
2 answers
296 views

How can quantum error correction correct small rotations/continuous errors?

I'm having trouble understanding the so-called "digitization of errors" argument in QEC. Suppose I have to encode my logical qubit into $n$ physical qubits to do error correction. I will use ...
4 votes
1 answer
603 views

Why is it not a good idea to run Shor's algorithm on a big noisy quantum computer?

Suppose I try to run Shor's algorithm on a big noisy quantum computer. For a given input I repeat the circuit many times and collect statistics for the measurements of each qubit. Now based on the ...
3 votes
1 answer
60 views

Can you produce Non-Clifford gates via Lattice Surgery?

Is there any theorem stating that lattice merge/split operations (or perhaps more generally, so-called gauge-fixing) cannot produce a non-Clifford gate that can complete the universal gateset for the ...
2 votes
3 answers
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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 ...
6 votes
2 answers
573 views

Example non-stabilizer code?

A code is a non-stabilizer code if it is not equivalent by local unitaries to a stabilizer code. What is an example of a non-stabilizer code with distance $ d \geq 2 $? Are there any non-stabilizer ...
1 vote
1 answer
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Get parameters of a CU gate for implementing an erroneous CX gate with fidelity = 0.81

Let's say the definition of an erroneous CNOT gate is given as - CNOT is a 2-qubit quantum gate when the control qubit is at state 0, applies identity operation on the target qubit, when the control ...
1 vote
2 answers
392 views

How to encode an unknown state into stabilizer code?

For $[n,k]$ stabilizer code, we can prepare an encoded state by simply measuring an initial state $|0\rangle^{\otimes n}$ with the stabilizers $g_1, \dots ,g_{n-k}$ and fixing it according to the ...
2 votes
0 answers
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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 ...
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 ...
3 votes
1 answer
102 views

What are overlooked noise sources in circuit-level noise?

In the research of quantum error correction, noise is often classified into code capacity noise, phenomenological noise, and circuit-level noise. Among these, circuit-level noise is considered the ...
4 votes
1 answer
138 views

How to correct error during the syndrome measurement or recovery process?

Here's the figure 10.21 from Nielsen's Quantum Computation and Quantum Information to explain the fault-tolerant quantum computing, where the circuit in the figure corrects the error that happens in ...
6 votes
1 answer
373 views

State of the art decoding algorithms for surface code

I am interested to know what are the "most promising" decoder algorithms that are used in surface code today. Of course, different criteria can tell what we expect from a good decoder. In ...
1 vote
1 answer
48 views

Error correction conditions for bit flip code

I am reading Nielsen and Chuang's chapter on error correction. I have a very basic question about this theorem: Theorem 10.8 (Error-correction conditions for stabilizer codes): Let $S$ be the ...
2 votes
2 answers
55 views

Can Interaction with the environment cause $|0\rangle$ to evolve into a superposition like $|+\rangle$?

We say if a qubit is in state $|0\rangle$, and it is possible if it interacts with the environment it can result in a bit flip to a state $|1\rangle$, so I was wondering why can't it go to a ...
3 votes
1 answer
120 views

Quantum error correction circuit from stabilizer codes

I am trying to familiarize myself with QEC codes and their use in quantum communication. Right now, I am trying to implement some more widely known codes in Qiskit. My main problem is I cannot wrap my ...
0 votes
0 answers
22 views

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 ...
0 votes
1 answer
54 views

Initialization of surface code into a logical state

I read this nice answer about how to initialize a surface code into a logical state but have a basic question about it. If I take a surface code patch with all physical qubits in the $\vert 0\rangle$ ...
0 votes
1 answer
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The Output of Transversal Bell Measurement in Knill's Method of Fault-Tolerant Error Correction (FTEC)

On page 26 of arXiv:quant-ph/0504218, it is written that in Knill's method of fault-tolerant error correction (FTEC), the output of the transversal bell measurement becomes $(P_m \otimes I) | \Phi_0 \...
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1 answer
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Fault Tolerance of 2-transversal gates

Suppose I have a single block $n$-qubit stabilizer code that can correct a weight 1 error (so the distance is $d=3$). If I apply a $1$-transversal gate of the form $U = U_1 \otimes U_2 \otimes \cdots \...
2 votes
0 answers
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Question about fault-tolerant measurement in Nielsen&Chuang page 490

The fault-tolerant procedures for state preparation and logical operation consider the procedure successful if only a one-qubit error occurs in each encoded block of the output. So, I thought that the ...
1 vote
1 answer
59 views

How can fault tolerance be understood from the perspective of detecting regions in a circuit?

The language of detectors and detecting regions in the context of quantum error correction is introduced in https://arxiv.org/pdf/2302.02192.pdf. Detecting regions seem to be capable of specifying ...
4 votes
2 answers
117 views

Compute the exact minimum distance of a QECC with integer linear programming method

Recently I'm reading the paper of Bivariate Bicycle(BB) QLDPC code proposed by IBM. The paper mentioned that: The actual distance of each candidate code was computed using the integer linear ...
4 votes
0 answers
63 views

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 ...
1 vote
1 answer
36 views

I am getting an error on STIM: An error case in a composite error exceeded that max supported number of symptoms (<=15)

I tried putting some ELSE_CORRELATED_ERROR on my custom made surface code circuit and I am getting this error while counting logical errors. What does it mean? And how can I fix it?
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 \...
0 votes
1 answer
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How to perform below operation in Qiskit?

I want to implement the below equation in Qiskit. $(A \otimes B).\rho.(B^\dagger \otimes A^\dagger)$ where $\rho$ is a density matrix and $A$ and $B$ are CNOT gates. $$ A=\begin{bmatrix} 1 & 0 &...
3 votes
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60 views

How to perform logical operations between logical qubits in a $[\![8,3,2]\!]$ quantum code?

I have a small question on codes like $[\![8,3,2]\!]$, which encodes physical qubits to multiple logical qubits. How do we perform logical operations in between these logical qubits? Of course, if I ...
1 vote
1 answer
157 views

How to do error correction after encoded Bell measurement?

Need some help with the concepts of encoded/logical bell measurement. Please visualize the picture in your mind. Suppose I have a node with 7+7 qubits side by side, left 7 is $|0_{L}\rangle$ and right ...
3 votes
1 answer
65 views

Fault-tolerant correlated measurement on logical qubits

So I have an encoded Bell pair $(|\overline{0}\overline{0}\rangle+|\overline{1}\overline{1}\rangle)$ in CSS code(e.g. Steane code) and I hope to perform stabilizer measurement of $\overline{XX}$ non-...
2 votes
0 answers
29 views

Procedures for fault-tolerant nondestructive logical measurements

Definition 2 on page 20 in Daniel Gottesman's An Introduction to Quantum Error Correction and Fault-Tolerant Quantum Computation states that a nondestructive logical measurement procedure is fault ...
5 votes
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145 views

Does a $ [[23,1,9]] $ code exist?

The $ [[23,1,7]] $ Golay code is an excellent code. It is a doubly even CSS code with $ H_X=H_Z $ and allows transversal implementation of all Clifford gates. One way in which it is not optimal is ...
1 vote
1 answer
469 views

Best implementation for logical CNOT on Shor's code?

As the Shor's code is a CSS code, it admits a transversal implementation of logical CNOT. An immediate implementation may perform 9 (reversed) CNOT, by respecting the order of the qubits. However. ...
2 votes
2 answers
54 views

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 ...
1 vote
1 answer
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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, ...

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