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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13
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1answer
1k views

What is the Helstrom measurement?

I have been reading the paper Belief propagation decoding of quantum channels by passing quantum messages by Joseph Renes for decoding Classical-Quantum channels and I have crossed with the concept of ...
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Is the Pauli group for $n$-qubits a basis for $\mathbb{C}^{2^n\times 2^n}$?

The Pauli group for $n$-qubits is defined as $G_n=\{I,X,Y,Z \}^{\otimes n}$, that is as the group containing all the possible tensor products between $n$ Pauli matrices. It is clear that the Pauli ...
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Is Gil Kalai's argument against topological quantum computers sound?

In a lecture, recorded on Youtube, Gil Kalai presents a 'deduction' for why topological quantum computers will not work. The interesting part is that he claims this is a stronger argument than the ...
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Why is the action of controlled-Z unaltered by exchanging target control qubits?

In the book "Quantum Computer Science", when explaining the error correction code, it uses this picture and says "the action of controlled-z is unaltered by exchanging the target and ...
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296 views

Transversal logical gate for Stabilizer (or at least Steane code)

I know that for Steane code, we can implement transversally some gates like cNOT, Hadamard and Pauli. What I am looking for is a resource in which it is explained why implementing those gate give rise ...
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Intuition about Knill-Laflamme QEC conditions

The Knill Laflamme QEC conditions are stated this way: We consider a code space $C$ and its associated projector $P_C$. We consider a noise acting on our system: $E(\rho)=\sum_a E_a \rho E_a^{\...
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Quantum circuit for a three-qubit bit-flip code

I know a three-qubit bit-flip code has a common encoding circuit as follows, Further, as in page 35 in Gottesman's paper, the encoding circuit can also be constructed through stabilizer generators. ...
12
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454 views

What is the leading edge technology for creating a quantum computer with the fewest errors?

Which technological path seems most promising to produce a quantum processor with a greater quantum volume (preferring fewer errors per qubit over more qubits), than Majorana fermions? The preferred ...
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How to implement if statement based on measurement results in qiskit?

I tried to implement three qubit bit flip code in qiskit and need to get the result of measurements and then apply recovery quantum operations conditioned on the measurement results. The following is ...
3
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1answer
129 views

What is the standard noise channel that is applied in simulations?

I know there are various quantum noise channels, which include the depolarizing channel, the dephasing channel and the bit-flip channel; We can apply them in simulators easily. However, is there any ...
4
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1answer
108 views

Are applications with only polynomial speedup worth chasing after? (since error correction adds a heavy overhead)

A number of ML algorithms have demonstrated to have polynomial speed-up: But this (I'm assuming) is without error correcting qubits. How practical are algorithms that only exhibit polynomial speed-up ...
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2answers
344 views

Mitigating the noise in a quantum circuit

I'm using Qiskit and I have a Quantum Circuit (say, circuit) that gives reasonable results when using the simulator, namely ...
3
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77 views

Link between distance of a stabilizer code and number of errors it is able to correct

I am confused by a property. In the N&Chuang it is said that an $[n,k,2t+1]$ stabilizer code is able to correct up to $t$ errors. But for me if the code has distance $d$ it should be able to ...
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Necessary and sufficient condition to define logical operation (stabilizer code)

My question is highly related to this topic It is about defining logical operation on a Stabilizer code. I call $S$ the stabilizer group of a code space $C$, and I assumed it is generated by a family $...
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What level of "confidence" of the result from a quantum computer is possible?

At a very basic level, reading or measuring a qubit forces it to be in one state or the other, so the operation of a quantum computer to gain a result collapses the state into one of many ...
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Is error correction necessary?

Why do you need error correction? My understanding is that error correction removes errors from noise, but noise should average itself out. To make clear what I'm asking, why can't you, instead of ...
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4answers
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What is a Bacon-Shor code and what is its significance?

I'm at the AQC conference at NASA and everybody seems to suddenly be talking about the Bacon-Shor code but there is no Wikipedia page and the pdf that I gave a link to does not really explain what it ...
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5answers
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Why do error correction protocols only work when the error rates are already significantly low to begin with?

Quantum error correction is a fundamental aspect of quantum computation, without which large-scale quantum computations are practically unfeasible. One aspect of fault-tolerant quantum computing that ...
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310 views

Violation of the Quantum Hamming bound

The quantum Hamming bound for a non-degenerate $[[N,k,d]]$ quantum error correction code is defined as: \begin{equation} 2^{N-k}\geq\sum_{n=0}^{\lfloor d/2\rfloor}3^n\begin{pmatrix}N \\ n\end{...
8
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1answer
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How to calculate the distance of stabilizer code?

How to calculate the distance of the stabilizer code [[n,k,d]]? It's better if you can make a 3-qubit example. And what's the relationship between d and Pauli group?
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427 views

Nielsen&Chuang 5-qubit quantum error-correction encoding gate

$\newcommand{\bra}[1]{\left<#1\right|}\newcommand{\ket}[1]{\left|#1\right>}\newcommand{\bk}[2]{\left<#1\middle|#2\right>}\newcommand{\bke}[3]{\left<#1\middle|#2\middle|#3\right>}$ In ...
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1answer
114 views

How to measure syndromes in QEC

Shor's $9$ Qubit code. Imagine that we encode the state $|\psi \rangle =α|0\rangle+β|1\rangle$ using Shor's $9$ qubit code, then an $X$ error occurs on the 8th qubit of the encoded state $|E(\psi) \...
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Which quantum error correction code has the highest threshold (as proven at the time of writing this)?

Which quantum error correction code currently holds the record in terms of the highest threshold for fault-tolerance? I know that the surface code is pretty good ($\approx10^{-2}$?), but finding exact ...
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2answers
281 views

How does the size of a toric code torus affect its ability to protect qubits?

The Toric code Hamiltonian is: $\sum_{x,y}\left( \prod_{i\in p(x,y)} Z_{ixy} + \prod_{i\in v(x,y)} X_{ixy} \right),$ where the $v$ and $p$ are defined according to this picture (courtesy of James ...
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Practical Implementations of QECCs in IBM Q Experience

I am learning how to program the IBM Q Experience quantum computers in order to learn more about how does it work and in order to perform some experiments in it. By doing so I was wondering what are ...
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Lower bound for Degenerate Codes?

According to (Macchiavello, Palma, Zeilinger, 2001; pg82) a lower bound of the encoding Hilbert space of a non degenerate code is given by the quantum version of the Hamming bound: $$2^k \sum_{i=0}^t ...
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1answer
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Get result measurement into a circuit with Qiskit

I'm trying to implement the three qubit bit flip code in Qiskit. Therefore, I need to get the result of a measure of two ancilla qubits, to deduce which gate I need to use do recover my logical qubit....
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1answer
113 views

Is it possible to implement c_if statement based on a measurement in a single specified classical bit in qiskit?

I'm trying to write a 1-bit teleportation error correction code and there is one part in it where I need to add a gate that's dependent on a measurement from earlier in the circuit. The measurement ...
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1answer
161 views

Different QFT results when using Simulator or Quantum Machine

I'm performing QFT using the following code: ...
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2answers
278 views

How to get the stabilizer group for a given state?

Let's say we have the GHZ state with 3 qubits: $$ |\mathrm{GHZ}\rangle = \dfrac{1}{\sqrt{2}}\Big(|000\rangle + |111\rangle \Big)$$ I want to find the stabilizer group of this state, that is, the $...
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1answer
56 views

What's the relation between the sign of error correction code and commute of operators?

For example, the 5-qubit QECC. If $X_i, Y_i, Z_i$ commutes with $M_i$, the eigenvalue will be +1. Otherwise, the eigenvalue will be -1. What's the relation between the commute and the sign of the ...
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1answer
156 views

IBM Q calibration parameters

If I download calibration of a quantum processor on IBM Q website I see these parameters: T1 T2 frequency (GHz) readout error single qubit U2 error rate CNOT error rate T1 and T2 are relaxation and ...
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1answer
100 views

What does it mean to perform a measurement in correspondence with different projections?

In error correction, like the bit flip, you perform a measurement which corresponds to different projections so that the outcomes can teach you about the error. What does it mean? How do you actually ...
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2answers
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Allowed CNOT gates for IBM Q 5 quantum computer

I trying to do some tests in the IBM Q5 computer of IBM quantm experience for some simple error correction protocols, but as I can see, some operations between the qubits are not allowed. For example,...
8
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1answer
353 views

Intuition for Shor code failure probability

Consider the 9 qubit Shor code. This can detect and correct arbitrary single qubit errors, but if there are 2 or more single qubit errors before a correction round, the correction will fail. (In the ...
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1answer
542 views

What quantum channels are considered in quantum communication, and how does this choice affect the construction of error correction codes?

The so-called depolarizing channel is the channel model that is mostly used when constructing quantum error correction codes. The action of such channel over a quantum state $\rho$ is $$\rho\...
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1answer
159 views

Can we conclude that errors on Sycamore are Poisson-distributed Pauli errors?

In Martinis' recent Caltech lecture on the Sycamore paper, he appears to make much of the fact that FIG. 4 of the paper show straight-line fidelity - that is, the fidelity decreases log-linearly with ...
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0answers
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Degenerated vs non degenerated code: for both there always exist Kraus bringing to different orthogonal subspaces?

Context of my question I call: $\mathcal{M}(\rho)=\sum_a M_a \rho M_a^{\dagger}$ an error map, $C$ the code space. A CPTP recovery operation exists if and only if, the Kraus operator of the error map ...
3
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2answers
222 views

What is the pseudo threshold of a QECC using stabilizer formalism

Can someone explain what is the threshold and the pseudo threshold of a Quantum Error Correction Code , for instance the 9-qubit code, and how to calculate it using the stabilizer formalism simulation ...
2
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2answers
91 views

How does Surface-17 tell apart Z errors on Db and Dc?

I'm looking into this paper from DiCarlo's group Scalable quantum circuit and control for a superconducting surface code. I don't understand how it's supposed to identify specific single-qubit errors, ...
2
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1answer
81 views

Specifying qubits to achieve measurement error mitigation on Qiskit

I'm learning how to do error mitigation on Qiskit as my experiment result differs from the simulated result. I read the tutorial here, but I have some questions about it. If I have understood it ...
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1answer
153 views

Circuit for implementing Steane's code for Quantum Error Correction

I was reading about the Steane's code for QEC. However I could not find any implementation or circuit describing its creation. Can anyone share/explain how the Steane's code could be implemented. The ...
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1answer
163 views

In Shor’s 9-qubit code, is the error syndrome for qubit 5 phase-flip the same as that for qubit 6 phase-flip?

The Shor’s 9-qubit code has the following stabilizers $\hat{S}_1= \hat{Z}_1\hat{Z}_2$ , $\hat{S}_2= \hat{Z}_2\hat{Z}_3$, $\hat{S}_3= \hat{Z}_4\hat{Z}_5$ $\hat{S}_4= \hat{Z}_5\hat{Z}_6$, $\hat{S}_5= \...
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1answer
80 views

Considering quantum codes as codes over $F_2$

It is very common to look at stabilizer codes as codes over GF(2) or codes over GF(4). Mostly I have seen this for computations for distance of codes and syndromes. How do other notions like say ...