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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Entanglement-assisted hashing bound for asymmetric depolarizing channels
I reading the paper EXIT-Chart Aided Quantum Code Design
Improves the Normalised Throughput
of Realistic Quantum Devices, which proposes the use of QTCs in order to do quantum error correction for ...
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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 ...
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Why is combined amplitude and phase damping considered sufficient for noise modeling?
In QECC literature, I often come across the "combined amplitude and phase damping channel" as being representative of a realistic noise model which makes sense (as amplitude damping and de-...
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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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In what situation are three rounds of syndrome measurement required for fault-tolerance in the surface code?
I have heard multiple times the claim that three rounds of syndrome measurement required for fault-tolerance in the surface code. I'm not sure what situation would require this, as I think less would ...
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Why can a point in anti-de Sitter space be modeled as a logical qutrit and how is its error correction done?
This isn't my area but the recent Quanta article How Space and Time Could Be a Quantum Error-Correcting Code struck me as interesting. They mention:
In their paper[1] conjecturing that holographic ...
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Smallest qudit error correcting/detecting codes
Consider encoding a qubit into $n$ qubits. It is well known that the smallest error detecting code is in $n=4$ qubits and the smallest error correcting code is in $n=5$ qubits.
Is there a similar ...
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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 ...
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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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Which improvements can we expect in Quantum Error Correction?
According to this article:
The surface code is currently the most widely pursued quantum error correction scheme
(Emphasis mine). It is also said:
However, there are drawbacks to the surface code, ...
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Quantum Belief Propagation decoding
I have been reading about a family of quantum error correction codes called Quantum Turbo Codes, which are the quantum analog of the well-known classical Turbo codes. This codes were introduced in ...
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When do we need to decode live the syndromes? Various examples to understand
I would like to precisely understand why we need to decode syndrome "live" when we want to perform $T$ gates. For this reason I consider different scenario (some are fictive but helpful to ...
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What is the smallest system to apply nonlocal quantum LDPC codes?
I just learned that there are very promising nonlocal low-density parity check codes (LDPC) codes for quantum error correction. What is the smallest number of qubits necessary to carry out these ...
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If you had a 100 qubit fully fault-tolerant quantum computer, what would you do?
As quantum computers improve, eventually we may have error-corrected devices that have very low error rate. However, many applications (Shor's algorithm, quantum chemistry) appear to require thousands ...
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Why is it necessary for the Fault-tolerant error correction condition to take into account two qubit errors?
The Fault-tolerant error correction (FTEC) condition states that for a code distance of 3, if there are no input errors, even if up to one gate fault (including a two qubit error occurring on a two ...
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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 ...
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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 ...
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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?
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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 $GF(4)$...
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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,...
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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 ...
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Eastin Knill Theorem and global phase
In quantum we don't care about global phases, but I want to ask a question about global phases anyway.
The original Eastin-Knill theorem paper https://arxiv.org/abs/0811.4262 says
$$
CP = \Pi_{i=1}^k ...
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Why the perfect 5-qubit code was used for magic state distillation?
I am currently trying to understand magic state distillation. So far, my understanding is that the general idea is to find a code where a non-Clifford gate is transversal (very well explained in https:...
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Why are all the error cosets $Q.S$ given the erasure chain $\mathcal E$ and a syndrome $\sigma$ equiprobable? (Delfosse-Zémor)
In arXiv 1703.01517 (published here), a maximum likelihood decoding for qubit loss is explained.
A quantum erasure channel erases each qubit independently with some probability $p$ and replaces it ...
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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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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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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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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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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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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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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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Calculating length of code words in quantum information(compression)
I was studying this article by Boestrom and Felbinger.
We define the significant length of the codewords in the preparation of the communication protocol : $$L_c(w_i) = \lceil log_k(i) \rceil$$
We ...
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How to do the counting when computing the fault tolerant threshold of quantum codes?
Here, I want to ask a basic question about how to compute the fault tolerance threshold of quantum codes. As I know, maybe, the most usual way is to do some simulations. Howvever, I am more intersted ...
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Proper definition of logical operation in Q.E.C
My question is related to this topic
I consider working with error correcting code on which I want to define logical operations.
Let's assume I want to define a logical operation on logical qubit. In ...
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Which codes can implement transversal non-Clifford gates
A paper Three-dimensional surface codes: Transversal gates and fault-tolerant architectures discusses 3D surface codes and shows that
CZ and CCZ gates are transversal in [these] codes.
They give a ...
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The importance of length-4 cycles in Quantum LDPC codes
It is a proven and well-known fact that length-4 cycles are detrimental to the performance of classical LDPC codes. This is due to the fact that such short cycles impair the decoding algorithm (Sum ...
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Is the quantum Singleton bound compatible with the Toric Code?
Note: Cross-posted on Physics SE.
The quantum Singleton bound states that for an error-correcting code with $n$ physical qubits and $k$ encoded qubits, and some subsystem $R$ of $m$ qubits that can '...
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How to address the 2 logical qubits on the toric code individually? In general, how to address $k$ logical qubits in a $[[n,k,d]]$ code independently?
Suppose I have a $[[n,k,d]]$-quantum error correction code. Let us take the toric code $(T^2=S^1 \times S^1)$ as an example. We have 2 logical qubits whose logical operators lie along the different ...
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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 ...
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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 ...
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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 are the advantages of building CSS codes out of dual-containing classical codes?
Is there any advantage to building CSS codes out of dual-containing classical codes, as opposed to general CSS codes? Other than satisfying the commutation conditions directly, are there any useful ...
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Simulating erasures with stim
We're trying to simulate erasure errors on the surface code using Stim. The threshold for erasure errors on the data qubits (after initialization) is 50%.
We followed the following post: How do I ...
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Show that ${\cal C}(\rho)={\rm Tr}[U(\rho\otimes|0\rangle\!\langle0|)U^\dagger]$ can be corrected iff $t=0$, where $U=I+(e^{it}-1)|\Psi^-〉〈\Psi^-|$
Let $ \mathcal{C} $ be the quantum channel defined by $$
\mathcal{C}(\rho)=\operatorname{Tr}_{2}\left[U(\rho
\otimes|0\rangle\langle 0|) U^{\dagger}\right], $$ where $ U $ is the
two-qubit unitary ...
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How does Teleportation based Error Correction (TEC) detect and correct loss/erasure errors?
I'm looking at papers like Demonstration of teleportation-based error correction in the IBM quantum computer (page) 10 and Role of syndrome information on a one-way quantum repeater using ...
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Obtaining low weight stabilizer generators
Suppose I know a set of stabilizer generators of a qubit quantum code. Is there a systematic (and possibly efficient) way to transform this set of generators to a different set (generating the same ...
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How is the theory of modular representation involved with error-correction?
Recently, I have heard that the theory of modular representation can be involved with error correction in quantum information theory. However, from a point of view of mathematician, I am still quite ...
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Density Evolution to Optimize QLDPC code design
Density Evolution is a simulative tool that models the behaviour of SPA (sum-product) decoders. It is useful because it enables the optimization of code designs so that extensive simulations can be ...
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There seems to be a problem with the implementation of identity gates on Qiskit as compared to the circuit composer
I have been experimenting with slowing down the decay of qubits, and as a control, I implemented a circuit with just identity gates for comparison. However, when I implement such circuits in Qiskit, ...
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Recent references establishing connections between stabilizer error correction and entanglement distillation procedures
I know some concrete entanglement distillation protocols (from this review).
I also know there are connections between error-correction codes and entanglement distillation (again this review but also ...
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