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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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How to set the meas_level and optimization_level in Quantum Azure IonQ provider?

I am performing a benchmark process using the Azure Quantum IonQ provider. I use Qiskit in the Azure platform, and I was trying to see how to control the transpiling and compiling process. It seems ...
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Lattice surgery on small Surface Code (d=2, 7 qubits)

I'm currently trying to understand lattice surgeries. As an exercise I'm currently implementing the lattice surgery operations described in Entangling logical qubits with lattice surgery, by Alexander ...
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Scaling functions for the logical error rate of the surface code

I'm trying to understand where the common expression $P_L=A\left(\frac{p}{p_{th}}\right)^{\lceil \frac d 2 \rceil}$ comes from. When estimating the logical error rate of the surface code, people seem ...
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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 ...
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What is the actual probability of not losing information (in a depolarizing channel)

The probability that a depolarizing channel doesn't affect the information is usually assumed to be $1-3p$, while, for convenience, it is affected with same probability $p$ by any Pauli operator $X,Y,...
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Scaling of magic state distillation with single & two-qubit gate error

How does the resulting error in magic state distillation scale with the fidelity of single qubit gates and two qubit gates? In most experimental systems, two-qubit gate errors are much larger than ...
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6 votes
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Why does code switching not allow for universal fault-tolerant quantum computation?

In this paper, the authors briefly mention that one proposed method to bypass the Eastin-Knill theorem is to perform code-switching. That is, given codes $C_1$ and $C_2$ which permit a complementary ...
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Lattice surgery: how is merged code stabilized by 'merging stabilizers' if their measurement outcome can be -1?

I am trying to understand the lattice surgery technique performed in the paper Entangling Logical Qubits with Lattice Surgery and I'm running into some confusion with the merging procedure. The setup ...
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Prove that the Bures metric satisfies a contractive property and has unitary invariance

In this paper, the authors assert that the Bures metric satisfies a contractive property and has unitary invariance. These terms aren't defined or proved in the paper, nor is any reference given for a ...
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Idle time or reset, efficiency comparison on Qiskit

As usual in QEC there are ancillary qubits set to the ground state. In a QEC scheme, not necessarily such a qubit is used immediately. In Qiskit framework, i think, all the qubits are initialized to ...
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How many times does syndrome extraction need to be repeated?

As part of implementing a QEC code, we need to do some syndrome measurement. In order for this syndrome measurement, we need to make sure that any $k \leq t$ errors (with a distance $2t + 1$ code) ...
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The statevector of three-qubit bit-flip encoding circuit for entangled state

I was trying to implement a three-qubit bit-flip code for shared entangled state. I was curious to analyze this mathematically, but the problem is I can't calculate the statevector after encoding. ...
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2 votes
1 answer
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Is fault-tolerant computation easier than we thought?

In this paper, Bravyi and Haah introduce triorthogonal stabilizer codes, which arise from triorthogonal matrices defined as follows. An $m \times n$ binary matrix $G$ with rows $f_1, \dots, f_m$ is ...
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How can a finite set of gates be universal?

One of the critical arguments in the Eastin-Knill Theorem is that a finite set of gates cannot be universal since "a finite number of unitary operators cannot approximate infinitely many to ...
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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 ...
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4 votes
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Is the Eastin-Knill Theorem incorrect?

I am reading through this paper (the Eastin-Knill Theorem) and there is a step in the proof of the main theorem that I do not understand. Let $Q$ be a composite quantum system supporting a quantum ...
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2 answers
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Surface Code Eigenstates as Circles

I learned that logical $|0\rangle$ of surface code, is an eigenstate, where all stabilizers are +1 value, and since the z-stabilizer is enforcing an even amount of edged in each node, and the x-stab ...
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2 votes
1 answer
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How can I apply 2 qubit pauli operator error instead of CNOT operator in stim code?

Currently I am building up quantum error correction code with stim code. According to this paper [https://journals.aps.org/pra/abstract/10.1103/PhysRevA.86.032324], I need to make an error channel ...
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Code Length of most Naive Repetition Code

What is the formula or answer to the following coding: I would like to get logical qubit with logical error rate of $P_L=1E-10$ or $P_L=1E-19$ for Shor's algorithm, but I would like to know how many ...
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How to set up a error detection and correction scheme on qiskit

Does, at the state of the art, Qiskit allows to do error detection and correction? If yes, how can I do it?
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2 votes
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Local Hermitian operators can be written as sums over local error operators?

In this paper, near the bottom of the left half of page 3, the authors claim that any local Hermitian operator (one which acts only on a single subsystem of a larger composite system) can be expressed ...
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Adding dephasing to stim's generated surface code circuits

I'm trying to find the threshold for dephasing error on a surface code. I've used the stim.Circuit.generated("surface_code_unrotated_memory_x") function to generate a surface code lattice. ...
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Surface code cycle time

I have been reading some papers related to surface codes, and I have a doubt regarding the cycle time of surface codes. Both of the papers I have been studying (How to factor 2048 bit RSA integers in ...
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2 votes
1 answer
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Laflamme's perfect quantum error correcting code (5 qubit) - Not getting the correct syndrome bits

I am trying to implement 5 qubit encoding and decoding circuit which results into generation of syndrome bits that pin points the location of error so that we can correct it afterwards. Please have a ...
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1 answer
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extracting the Pauli error on the data qubits at the end of circuit in stim

Let's say one implements the following circuit in stim from the stim tutorial: ...
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2 answers
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How do I specify the coordinates of dephasing errors in the surface code using Stim

I am trying to find the threshold of a surface code for dephasing error using pymatching and stim. I am unable to apply the Z_ERROR onto the qubits using the given format since a surface code will ...
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3 votes
1 answer
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Calculating bit flip and phase error using local operations for GHZ state

Suppose each qubit of $\text{GHZ}_3$ State is distribute to $3$ different parties at different locations through a noisy quantum channel. So each qubit can possibly go through: a bit flip error, a ...
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Is there any reasonably efficient way of performing qudit circuit simulations using Stim?

Is there any reasonably efficient way of performing qudit circuit simulations using Stim? If so, then how much worse would the computational complexity scale?
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2 votes
1 answer
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Diagrammatic language for fault-tolerance: I don't understand the difference between preparation correctness and propagation properties

I am learning the diagrammatic language for fault-tolerance which allows to show in an "easy" manner if circuits are fault-tolerant. I first put a reminder of the definitions and then I ask ...
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2 votes
1 answer
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Magic state distillation with the $15$ qubit code. How are the transversal $T$ performed? Also via a (lower level) state injection?

I have a code $C1$ that admits native fault-tolerant gates for any element in the Clifford group (for instance via transversal implementation), but that then doesn't have an easy way to perform the $T$...
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1 vote
1 answer
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Magic state distillation with the 15 qubit code. Why is the failure $O(p^2)$ if we include $|+_L\rangle$ preparation and decoding?

My goal is to understand why the magic state distillation (for instance based upon the $15$ qubit code) improves the fidelity of the magic state. I followed this lecture to try to understand it. I ...
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4 votes
3 answers
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Simulate Surface /Topological Code with Majorana - Huge Complexity Saving

This article "Correcting coherent errors with surface codes" is talking in the methods section, about simulating topological codes / surface codes, using Majorana equivalent. It is also ...
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2 answers
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What is (formally) a transversal operator?

This question concerns about a formal definition of transversal operator. I understood that transversal operator are a group of operators which are efficient in terms of circuit depth and can be used ...
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8 votes
2 answers
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If Majorana qubits are analogous to surface codes, why do the diagrams use lines instead of squares?

If you go to some random Majorana paper or talk, you will find a diagram like this one. Note that the diagram is using lines. Making the lines longer should exponentially increase the error ...
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1 answer
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What does "perfect measurement" mean in the context of fault tolerance?

I am reading the paper Quantum error correction with only two extra qubits. They say there that (in the section on the [[5, 1, 3]] code): If a flag is raised or a syndrome is nontrivial, then the ...
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1 answer
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What are the consequences of Shor's code being degenerate?

I read that Shor's code is degenerate and, as consequence, an error $\sigma_{1}^z - \sigma_{2}^z$ annihilates codewords. However, I also read that one can assume such kind of error to never occur, as ...
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2 votes
1 answer
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nonlinear behavior of $1/\Lambda$ in surface code simulations around threshold

I am simulating the dependence of $\Lambda$, as defined in google's Exponential suppression of bit or phase errors with cyclic error correction paper as a function of physical error rate. The $\Lambda$...
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when calculating code thresholds, why do we need to take a number of rounds that scales with the code distance?

In stim's error corretion example here, and in pymatching's toric code example here, the threshold calculation involves taking a number of rounds that scales as the code distance. Why is that the ...
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3 answers
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Surface Code - Convert Control Error to Clifford Error

I am simulating surface code, in order to find the logical error as a function of control error in my circuit. Each of my data qubits is multiplied in control error = a unitary matrix $U$ which is ...
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Replace condition in Knill-Laflamme quantum error conditions with probability

The Knill-Laflamme conditions say that in order to have a quantum error correcting code for some error set $\mathcal{E}$, we must satisfy both of the following conditions for all $E_a,E_b \in \mathcal{...
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4 votes
1 answer
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The merging operation is not symmetric on its inputs in surface code as it should naturally be

My misunderstanding is probably due to a trivial thing I am not seeing. In lattice surgery for surface code, we can perform a merging operation. This operation should be symmetric as indicated on Eq.5 ...
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6 votes
1 answer
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Why do we need to keep of history of $d$ cycle for surface code in order to resist against data *and* measurement errors

I am trying to understand why we need to keep in memory $d$ clock cycles in order to correct efficiently for the surface code where the syndrome qubits are also faulty ($d$ is the code distance). [...
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4 votes
1 answer
63 views

compatibility of stim's error model with conventional decoding schemes

I am working with stim's excellent code generator together with pymatching, using stim's generated error model and the glue code in the tutorial here. I am trying to understand better what the output ...
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2 votes
1 answer
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Analytical Approximation of Pseudothreshold for Steane Code

Is there an analytical way by which one can approximately find the pseudothreshold of the Steane code? My supervisor told me that there is a way, using the physical error rate and the number of gates ...
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2 votes
1 answer
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Kind of errors on entangled states

I see on this chapter that, when noise applies to an even number of entangled qubits, such noise can be considered as operating only on half of those qubits. This is interesting, but I'm not sure if I'...
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1 vote
1 answer
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Information about the output of %qiskit_backend_overview

Does anyone know where can I find more information about the meaning of some parameters outputted by %qiskit_backend_overview ? I mostly refer to "Avg. T1/T2&...
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5 votes
1 answer
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Algorithm resource estimation: why "usually" only the $T$ gate count is being provided. When the Clifford count is provided, does that include Id gate

[edit] I am not necessarily focused on topological error correction (as the answer already provided reasons in this context). Even in the case of topological error correction I am not sure to ...
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4 votes
2 answers
275 views

How to find the stabilizer generators for a post-measurement state?

My question is closely related to this one. A bit of vocabulary and a reminder of basic properties: I consider the total Hilbert space of the problem has dimension $2^n$. I call a "well defined ...
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1 vote
2 answers
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Are ancillary qubits necessary to represent a syndrome measurement in the circuit model?

From my understanding, in QEC with stabiliser codes, one can use the operator formalism and state to perform a measurement over a space which is orthogonal to the code space. This allows to not ...
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2 votes
1 answer
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Lattice splitting: understanding mathematically why it works based on the stabilizer formalism

I am struggling to understand why the lattice splitting procedure works precisely. I am following the appendix in this paper. [edit]: While the answer in this topic answers the question, the concept ...
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