Questions tagged [quantum-error-correction]

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Sufficient Existence of a CPTP map based on piecewise state mappings

Suppose I want to determine whether there exists a CPTP map, $\mathcal{E}$, such that $\mathcal{E}(\rho_i) \rightarrow \rho_i^{'}$ for $i \in S$. I'm specifically interested in the case where $|S| = 2$...
corduroy0898's user avatar
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How to check that an element belongs to a stabilizer group?

My direct Question: For a given stabilizer $S=\langle g_1,...,g_n\rangle$ (only represented by the generator) and an operator $g$, how can we check that $g\in S$? A wrong answer is to check that "...
hengyue li's user avatar
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Is it enough to recovery density matrix for error correction in quantum computing?

When we discuss error correction, we always talk about the recovery of the density matrix of a single qubit. But somehow I feel that this is not enough. Consider a circuit that contains more than 1 ...
hengyue li's user avatar
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Well-definedness of commutation relation in commuting local Hamiltonians

I'm reading the famous paper by Haah: Local stabilizer codes in three dimensions without string logical operators. In the last sentence of the introduction, he wrote: A logical operator is a Pauli ...
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Photon loss errors versus photon subtraction errors

Cat codes, $|0_L\rangle\approx|\alpha\rangle + |-\alpha\rangle $ and $|1_L\rangle\approx|i\alpha\rangle + |-i\alpha\rangle$, are said correct single-photon loss errors in the literature. Similarly, ...
Saurabh Shringarpure's user avatar
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How is the measurement of the $Z_0Z_2$ error syndrome realised by these two circuits?

I am told that the following quantum circuit diagram implements the $Z_0Z_2$ error syndrome, but I don't follow. I understand error syndromes when expressed in matrix form and how performing $Z_0Z_2|...
redpanda2236's user avatar
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What's spin liquid and how it connects quantum error correction?

I understand how a bit flip code in a 2D lattice is like a 2D Ising model in Kitaev's 2001 paper Topological Quantum Memory. But generally, can someone explain how quantum error correction connects to ...
Jiakai Wang's user avatar
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Important examples of degenerate quantum error correcting codes

A degenerate quantum error correcting code is one in which distinct error operators (e.g. $E_1$ and $E_2$) apply the same transformation to the code's logical states (i.e. $E_1|\psi_L\rangle = E_2|\...
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Error happened in quantum error correction

I have a question about the correction step in the quantum error correction. Consider the three-qubit encoding in the quantum error correction, The following operations are performed to $|\phi\rangle=...
SUSY's user avatar
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What are the logical 0 and 1 states in the 9 qubit 'surface 17' code?

I am trying to implement the 9 qubit 'surface 17' code, however it appeared to me that I couldn't find in the literature what the encoding states for such a physical system are. I have found in the ...
Julien Bréhier's user avatar
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Physical meaning of projection operator acting on a quantum state

Let's say I have a quantum state such as $$ | \psi \rangle = \alpha |00\rangle + \beta |11\rangle $$ for some pair of qubits. I am wondering how to interpret an operator like $$ P_1 = | 0 0 \rangle \...
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Degenerate quantum error correction code

I have been reading this to understand quantum error correction a bit. On page 13, there are paragraphs about degenerate QEC. It says, if $E_{1}E_{2} \in H $, $E_1 $ and $E_2$ will have the same ...
Blackwidow's user avatar
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Tensor product decomposition from Schur's lemma

I was reading a paper on subsystem codes for quantum error correction: https://arxiv.org/abs/quant-ph/0506023 In section IV.B, equation (16), they have arrived at a tensor decomposition of Hilbert ...
Dhiraj Madan's user avatar
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Code distance and other questions about Quantum double model as an error correcting code

Kitaev's quantum double model is an error correcting code, see: https://arxiv.org/abs/1908.02829 I am in a class on quantum error correction and the professor commented that a quantum double model for ...
Ian Gershon Teixeira's user avatar
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$n$-fold degeneracy, ground state, toric code [closed]

I'm having a hard time understanding the degeneracy of the Kitaev toric code. Can anyone explain to me with details the following: How do we find the ground states of the toric code and its gape? ...
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Deducing fusion rules of non-abelian fluxons

I have been reading about non-abelian fluxons in John Preskill's lectures notes on topological quantum computing and I do not understand how he deduced the fusion rules for fluxons in the example he ...
Hermitian_hermit's user avatar
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How to check if a qubit in a circuit is equal to another qubit?

If we have a set of qubits q1 ,q2 and q3 for example, $$ \vert{q_1}\rangle = a \vert{0}\rangle +b \vert{1}\rangle\\ \vert{q_2}\rangle=\vert{0}\rangle\\ \vert{q_3}\rangle=\vert{0}\rangle $$ the state ...
El-Mo's user avatar
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Smallest Distance-5 Quantum Error Correction Code? [closed]

Is it known/proven what the smallest quantum error correction code is that can correct arbitrary two-qubit Pauli errors? I can think of the nested/concatenated 5-qubit code or a 25-qubit version of ...
Iris Cong's user avatar
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Understanding surface code diagrams

I'm trying to understand the diagrams used in the following paper. I'm new to the area so I don't have experience with the stabilizer formalism, but I'm trying to figure out what's going on in the ...
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Can a Hadamard gate be realized with perfect precision?

This is my first question on the Physics Stack Exchange; I hope it is not be a duplicate… :-| I am not really a physicist (I am actually a professional mathematician), but I have already read quite a ...
Rémi Peyre's user avatar
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Is there a theoretical limit on the smallest angle that can be used in a quantum rotation gate?

Assuming ideal conditions and no error, what is the smallest $\theta>0$ that can be used in $R_x(\theta)$ or $R_y(\theta)$.
Mert's user avatar
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Why doesn't using quantum error-correction increase the production of entropy and decohrence?

Error-correction in quantum computing is designed to get around the decoherence "washing out" the answer to a computation. But wouldn't the introduction of error-correction procedures or apparatus ...
Michael C Price's user avatar
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Is HaPPY code a certain type of MERA?

Pastawski, Yoshida, Harlow, and Preskill introduced the HaPPY code in their (now famous) paper, arXiv:1503.06237, as a way to model the AdS/CFT correspondence as a quantum error-correcting code. ...
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Can we restore a state of a whole system from its subsystem?

I am thinking about the deletion error correcting codes for quantum information. In classical information theory, there exist some deletion error correcting codes. An easy example is the following ...
Jack's user avatar
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What happens to physical qubits when they decohere during quantum error correction?

Say you have a scaled up system with 98% fidelity so 2% decoheres. These are physical qubits doing error correction on information encoded on logical qubits. 98% of these entangled qubits will be ...
Genetic Avatar's user avatar
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2 answers
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Showing that a set of errors is correctable (Knill-Laflamme conditions)?

I am confused about how to apply the Knill-Laflamme Quantum Error-correction conditions, which are the following: A code $C \leq H$ is correctable for $\mathcal{E} = \sum_{i=1}^{n}E_i \rho E_i^*$ ...
TuringTester69's user avatar
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1 answer
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Why are only unitary operations allowed in quantum information theory?

In the quantum information theory, any operations for a quantum state have to be unitary operations. Why is this restriction needed? Can't we make a non-unitary operation to a state? I know that a ...
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Is the Quantum Singleton Bound Compatible with the Toric Code?

The Quantum Singleton Bound states that for an error-correcting code with $n$ physical qubits and $k$ encoded qudits, and some subsystem $R$ of $m$ qudits that can 'access the entire quantum code', it ...
Joe's user avatar
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How can I prove that the partial trace is well-defined?

When I define the partial trace as below, how can I prove it well-defined? I understand that I have to indicate $Tr_k(\rho)$ does not depend on how to take the ONB of $\mathbb{C}^2$ $$n\in \mathbb{Z}_{...
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Does the measuring instrument which counts the number of quantum particles exist?

I have recently been thinking about the quantum deletion error correcting codes, but the primary problem is whether the receiver can detect the loss of quantum particles or not. So my question is that ...
user avatar
2 votes
2 answers
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Can quantum deletion error-correcting codes be constructed?

I'm wondering whether or not we can construct quantum deletion error-correcting codes. The quantum deletion error is defined by the partial trace. If we can, could anyone give an example?
user avatar
2 votes
1 answer
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Uniquness of stabilizer generators of a state $\left| \psi \right>$?

Let $G_n$ be the $n$-qubit Pauli group and $S$ an abelian subgroup which does not contain the element ($-I$). I know that if $S$ has $n$ generators then it specifies a single state $\left| \psi \right&...
Quantum spaghettification's user avatar
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On the optimality of Stabilizer codes

I have been studying about quantum error correction codes, specifically aobut the so-called stabilzer codes, and I wonder if are this class of quantum codes the best codes known (in terms of capacity ...
Josu Etxezarreta Martinez's user avatar
10 votes
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Topological quantum error correcting codes which are not CSS codes

The most promising-seeming quantum error correction codes for the medium-to-long term are the topological codes, of which the toric code (and variants such as planar surface codes) and colour codes ...
Niel de Beaudrap's user avatar
5 votes
1 answer
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What are the quiescent states of the surface code?

Notation $$X \equiv \sigma_x \qquad Z \equiv \sigma_z$$ Introduction The surface code is a quantum error correction code. In the surface code, we have a two-dimensional grid of qubits each coupled ...
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Quantum Error Correction: Surface code vs. color code

Recently, two groups working on quantum computers published results on quantum error correction. The first was Rainer Blatt's group, who used trapped ions to perform a topologically encoded qubit ...
Mario Krenn's user avatar