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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Four-qubit error correction code

$\newcommand{\ket}[1]{\left|#1\right>}$ Consider the following 4-qubit code which allows you to detect a bitflip and/or a phaseflip. The logical 0 and 1 are encoded as: \begin{align*} \ket{0}_{...
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Doubt in Simple proof of Security of the BB84 QKD

I am not able to understand an argument from Simple proof of Security of the BB84 QKD; I need your help. In page 2 it is mentioned that "Alice can measure her half of the encoded EPR pairs before ...
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66 views

How to calculate the threshold for gate fidelity?

I've been interested in gate fidelity lately. In the meantime, I came up with a specific example. Let's think about the encoding of situation $\alpha_0 |0\rangle + \alpha_1 |1\rangle \rightarrow \...
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351 views

What is the meaning of "shots" and "rounds" in Stim?

I'm running simulations using Stim to get logical error rate vs. physical error rates of some quantum error correcting codes (QECC). I looked into the documentation, but I'm confused about the meaning ...
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How to detect and correct swap errors in a quantum circuit?

Let's assume that I have a density matrix $\rho$ that consists of $N$ qubits. If this density matrix undergoes an error-channel that swaps any two qubits with an equal probability, i.e. $$ \mathcal{E}(...
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Quantum Operation - non trace increasing

I am reading Nielsen Chuang Chapter 8. They say that if a quantum operation is trace-preserving, then \begin{equation} Tr\left(\sum_k E_k^{\dagger}E_k \rho\right) = 1 \end{equation} which I understand....
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Nyquist–Shannon sampling theorem for Quantum Evolution

In classical digital signal processing one can try to identify the dynamics of a system by sampling its evolution from an initial time $t_0$ to a final time $t_1$. Sampling $N$ times results in a ...
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52 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 ...
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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 ...
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How to take 3-point or more correlators into edge probability computation in surface code decoding?

In Google's work of repetition code (Exponential suppression of bit or phase errors with cyclic error correction), they use the method of Correlation Matrix to ...
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Why we need real-time feedback of Surface Code Decoding results?

When we use surface code to detect and correct errors, we can track the errors in software and change the computation results accordingly. So why do we need real-time feedback on decoding results? Are ...
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Knill Laflamme conditon

In Preskill's notes on quantum error correcting codes in Section 7.2, there seems to be no condition on the environment part of the state, i.e. $|0\rangle_E$ in $|\psi\rangle \otimes |0\rangle_E$. ...
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How to perform encoding and syndrome measurement in stim

I can generate the encoding circuit of a stabilizer code and can read it into stim. For example for the $[[5,1,3]]$ code : ...
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255 views

What is the motivation for using dual lattice in the surface code?

I'm learning surface codes from a theoretical perspective. In all the literature I read, they just introduced the dual lattice for working with $Z$ strings without many words addressing why we need to ...
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56 views

Can error correction for a classical algorithm with bit flips be easier than for a general quantum circuit?

Assume one runs a purely classical algorithm on $n$ logical qubits on a physical device with some bit flip probability. Can implementing error correction in this case be any easier than in the case of ...
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How to import a generic stabilizer code in stim

I just started using stim to simulate the performance of quantum error correcting codes. I hope this is the right place to ask questions about the subject. The example code with the repetition code ...
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78 views

Conditional lower bound on approximate stabilizer rank of magic states

I am currently reading about the approximate stabilizer rank and properties of the same. I will quote the definitions from this paper. The stabilizer rank of a quantum state $|\psi\rangle$ is the ...
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Confusion about Mitiq folding for zero noise extrapolation

I'm reading the mitiq zero noise extrapolation documentation and I just can't make sense of how the scale_factor for folding work. "The minimum scale factor is one (which corresponds to folding ...
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Where does the "correction" in quantum error correction occur, specifically when using repetition codes?

I'm reading the part of the qiskit textbook that deals with this (https://qiskit.org/textbook/ch-quantum-hardware/error-correction-repetition-code.html) and so far it seems as though they're just ...
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69 views

What is an algorithm to generate random error in depolarizing channel

The Qunatum Depolarizing Channel is parametrized by a single real variable $\lambda, 0 \leq \lambda \leq 1$. I have a system of $n$ qubits. I'd like to generate random errors from that channel. These ...
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Qiskit noise model question (from textbook)

I'm reading the chapter Introduction to Quantum Error Correction using Repetition Codes and a code example demonstrates how to add depolarizing and pauli error. I have several questions. Is it not ...
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52 views

What is the algorithm for the optimal decoder in a quantum erasure channel?

I'm reading this paper : Holographic Quantum Error Correcting Codes and on page 3 they describe an optimal decoder for erasure channel. The description is for CSS codes but they claim that "it is ...
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What is the process of error correction if I want to apply one $X$ gate to my logical qubit?

I am trying to understand the general process of error correction. E.g. I have a few physical qubits encoded as a logical qubit $\vert0\rangle_L = \vert 0 0 0...0\rangle$. What is the process of error ...
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Can someone please explain how the syndrome bit still ends up being 0 in this quantum error correction circuit using repetition code?

I'm not too great at dealing with superpositions and applying the CNOT gate when superpositions are involved. Can you go through it in detail each gate using math/matrices etc. It's based on the ...
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187 views

General conditions to define a logical operator: is there some general characterization of those elements (from group theory for instance)

I assume working with stabilizer codes where the stabilizer group is denoted $S$ and the code space is $C(S)$. The minimal requirement for a logical operation $M$ is to have: $$\forall |\psi\rangle \...
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Does the formula $\sum_k f_k{\rm Tr}(O_k U\rho_k U^\dagger)$ have any physical meaning, in the context of variational quantum algorithms?

I am reading a review about the variational quantum algorithm. And there is a definition of the cost function: $C(\theta)=\sum_k f_k (Tr[O_k U(\theta)\rho_kU^\dagger(\theta)])$ Where $U(\theta)$ is a ...
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Pros/cons of the different schemes to have complete fault-tolerant gatesets

I am interested to have references and comment about pro/cons of the various methods that are used to implement complete gateset in a fault tolerant manner. Usually the Clifford operations have a ...
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Definition of distance d of the QECC $T(S)$

Definition of distance $d$: $d$ of $T(S)$ is the weight of the smallest Pauli operator $N$ in $N(S)$ \ $S$. $S$ is the stabilizer, $T(S)$ is the corresponding QECC, and $N(S)$ are all errors commute ...
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Do stabilizers of Shor's 9-qubit code all have eigenvalue 1?

I am watching a recorded seminar on Youtube given by Prof.Daniel Gottesman (here). At 38:39 he claims that $M_{1}=Z_1\otimes Z_2$,$M_{2}=Z_2\otimes Z_3$,$M_{3}=Z_4\otimes Z_5$, $M_{4}=Z_5\otimes Z_6$,$...
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If there is quantum noise, can a time interval be short enough that only one-qubit errors (eg. as in the Shor code) occur?

The Shor code can be used to protect against an arbitrary one-qubit error. If the time interval over which quantum noise acts is short enough, can it be assumed only one-qubit errors occur, so one ...
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60 views

Good references to learn magic state distillation for fault tolerance

I need to learn magic state distillation procedure and their application to fault-tolerance. One of the original paper on this subject is the following: https://arxiv.org/pdf/quant-ph/0403025.pdf I am ...
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How to translate performance of two classical codes to a quantum CSS code?

I have a database of classical codes with simulation results in binary symmetric channel (BSC). The codes are defined by their parity check matrices $H$. I can pick pairs of codes and call them $H_x$ ...
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34 views

How can surface codes perform splitting and merging at the same time when they implement logical CNOT operation

Using lattice surgery technique, CNOT operation can be implemented as follows. step 1. smooth merging between the control qubit and intermediate qubit step 2. smooth splitting the merged qubit above ...
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Keep receiving Error - AttributeError: 'QuantumCircuit' object has no attribute 'save_statevector'

I am attempting to code a feed-forward simulation of a QECC in Qiskit. Qiskit (to my knowledge) does not directly support feed-forward simulation, so I attempted a work-around. I call the simulator ...
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82 views

Why is the quantum volume vastly greater than the number of qubits squared?

According to Wikipedia, the Honeywell computer has quantum volume ("QV") equal to 1024 and has 10 qubits. QV should not be more than $n^2$ squared, i.e. 100. Is this any math error? Are the ...
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Simplifying equation for two qubit syndrome extraction code

In the paper Quantum Error Correction: An Introductory Guide, the author gives the following formula for a simple two qubit code (eq. 19 on the paper). $$ E|\psi\rangle_L|0\rangle_A \xrightarrow{\text{...
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Error correction on $n$ qubits all in same state except for a few

This might be a straightforward problem for you guys; it would be helpful if you can explain it in simple language. I have $n$-qubits given as $$\frac{1}{\sqrt{2}} \left(|0\rangle+ e^{\iota\theta_{1}}|...
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49 views

Circuit for fault tolerant syndrome measurements Steane Code

I was exploring this paper: https://arxiv.org/pdf/1505.07390.pdf The paper describes the syndrome measurement for the [7,1,3] code. I was looking a fault-tolerant syndrome measurement and found this ...
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45 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 ...
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56 views

Are there any packages that can calculate stabilizer tableau of a QECC

I'm experimenting with some small quantum error correcting codes (QECC). For example $[[5,1,3]]$, $[[8,3,3]]$ or toric codes $[[2d^2,2,d]]$ ($d=2,3,\cdots$). The last one being defined by redundant ...
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What do Z logical errors look like in 3d color codes?

I am trying to better understand (standard, not gauge) 3d color codes. In particular, I am working with the lattice proposed in 1. I understand how X error works, forming strings of the kind Vertex -&...
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What is the effect of measuring a node in a cluster state?

I'm still quite confused about the post-measurement state when a node of a cluster state is measured. As far as I'm aware, for a cluster state, a given node $j$ will have a stabilizer $$X_j\bigotimes_{...
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Construction of arbitrary Normalizer Gates using H, S and CNOT Gates

This question is in reference to Exercise 10.40 of Nielsen and Chuang's textbook, which is an attempt to prove the theorem that any $n$ qubit Normalizer gate can be built out of $H$, $S$, and $CNOT$ ...
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Stabilizer codes: why is $N(S)/S$ homomorphic to the Pauli group on $k$ qubits

Given an abelian subgroup $S$ of the Pauli group on $n$ qubits with $r$ independent generators we obtain an $2^{n-r}$-dimensional subspace stabilized by $S$. By picking a basis for this space, it is ...
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55 views

Degenerate vs non-degenerate errors

One of the key features of quantum error correction that makes it different from classical error correction. When code is non-degenerate, an error $E$ takes codewords to different words. codewords. ...
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QPE Circuit test on Quantum Computer ('ibmq_16_melbourne')

After several atempts, I cannot mitigate the error when running the code on a NISQ, via the qiskit library (more specifically on the 'ibmq_16_melbourne'). I've already mapped the connected qubits and ...
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55 views

translating between measurement based and circuit based quantum computation

I think I understand circuit based QC (CBQC) well enough; I know very little about MBQC. From what I read it seems that they are somehow "equivalent". I'd like to check this with a concrete ...
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Calculating symplectic dual of a code

Stabilizer codes can be treated as symplectic codes over $\mathbb{F}_2$ (or over $\mathbb{F}_p$ when taking about q-dits). While treating error class, symplectic dual of the code plays a crucial part (...
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81 views

Accuracy threshold theorem: what is the physical probability of failure exactly

In this famous paper is derived the quantum accuracy threshold theorem for concatenated codes. I have questions for the case errors are assumed to be local and Markovian. In the so called level-1 ...
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How do the extra energy levels of a transmon qubit affect computation/fidelity?

I was reading about transmon qubits, and I know that they are not true two-level systems. Are there any math/papers which talk about how those extra energy levels affect the computation? I'm assuming ...

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