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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147 views

Is Toffoli a fault tolerant transversal gate of the repetition code on bit-flip-error-only qubits?

Suppose you have physical qubits that only have bit flip noise (no phase flip noise, no unwanted measurements, no leakage, no amplitude decay, etc; just unwanted X rotations). Such qubits can be kept ...
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How can I add readout error in constructing a quantum noise model?

I'm trying to construct a quantum noise model with readout error only: qiskit.providers.aer.noise.NoiseModel.add_readout_error. However, when I try to specify the ...
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Is there any software package tracking Pauli errors in stabilizing circuits with GUI?

I want to track the Pauli errors propagating in stabilizer circuits like surface code. I wonder whether there is any developed package that can do this elegantly. It will be better if it has GUI that ...
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What are the typical gate times for single-qubit and 2-qubit gates for ion trap, superconducting, neutral atom, photonic, spin QC?

What are the typical gate times for single-qubit and 2-qubit gates for -- ion trap, -- superconducting, -- neutral atom, -- photonic, -- spin quantum computers based on today's technologies?
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How many physical qubits are needed to encode a logical qubit on ion trap, superconducting, neutral atom, photonic QC?

How many physical qubits are needed to encode a logical qubit on an -- ion trap, -- superconducting, -- neutral atom, -- photonic, -- spin quantum computer based on today's technologies?
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When discussing error correction, what are the objects in the expression $PE_i^\dagger E_j P=\alpha_{ij} P$?

I've started reading the book "Quantum Computation and Quantum Information" by Michael A. Nielsen and Issac L. Chuang, specifically chapter 10 (about quantum error correction), and I'm ...
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How to determine the threshold value of the quantum error correction code

How to determine the threshold value of the quantum error correction code, what is the specific method, such as surface code, how to determine the threshold value of the color code with a decoder, I ...
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Stabilizer codes to Classical codes over GF(4)

In the work [1] it was shown that stabilizer codes could be mapped to binary codes (albeit with a symplectic norm) which further can be mapped to additive codes over GF(4). Can someone explain why it ...
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Is it interesting to look at surface codes where the boundary conditions give the shape of a Mobius strip, Klein bottle, or the projective plane?

I was reading about the Toric code and how the boundary codes give it a shape of of a torus. Is it interesting to look at other ways to orient the edges of a square lattice? I.e. a mobius strip, Klein ...
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Are there differences in how quantum error correcting codes are applied on superconducting vs. trapped ion systems?

Are there any differences in how quantum error correcting codes are currently applied on superconducting vs. trapped ion systems?
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Magic state distillation: why is it harder to prepare the encoded $|A_{\pi/4}\rangle$ than $|0 \rangle$

My question is the following Let's assume I am using Steane concatenated code to do error correction. This code admits transversal Clifford operations but not the $T$-gate. In order to implement this ...
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Why aren’t repetition codes used to encode qubits in superposition states?

I just finished reading the section of the qiskit textbook on quantum error correction using repetition codes(https://qiskit.org/textbook/ch-quantum-hardware/error-correction-repetition-code.html) and ...
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Doubt in CSS error correction step

At the end of page 3 in Simple proof of security of the BB84 QKD, the following equation (equation 4) is given : $$ \begin{array}{r} \frac{1}{2^{n}\left|C_{2}\right|} \sum_{z}\left[\sum_{w_{1}, w_{2} ...
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Analytical estimation of physical qubits per logical qubits for various code

I saw many papers providing the number of physical per logical qubit required for different kind of codes. However the estimations are usually numerical. I was wondering if there exist in the ...
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1answer
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Measurement bitstrings to detection event bitstrings in stim

Does Stim have exposed functionality for converting a measurement outcome string into a detection event string? I.e. I feed stim a string of 0,1's corresponding to measurements in a circuit along with ...
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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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1answer
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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 \...
3
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1answer
369 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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Why does a quantum operation being trace-preserving imply that $\sum_k E_k^\dagger E_k=I$?

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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1answer
67 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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42 views

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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2answers
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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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1answer
260 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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2answers
61 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 ...
2
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1answer
75 views

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 ...
3
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1answer
82 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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1answer
31 views

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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1answer
47 views

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 ...
4
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1answer
75 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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1answer
35 views

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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53 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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1answer
35 views

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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1answer
37 views

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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195 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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1answer
50 views

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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1answer
43 views

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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1answer
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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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1answer
39 views

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 ...
4
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1answer
67 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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1answer
36 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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1answer
121 views

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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1answer
85 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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