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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78 views

How to compute the error threshold for the $9$-qubit Shor code?

I was trying to calculate the threshold of $9$-qubit Shor's code. The error channel is $$E=(1-p)I+p/3X+p/3Y+p/3Z.$$ But I got the threshold is equal to 1. How can I get the right threshold (I believe ...
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Can Shor‘s code correct two- or three-qubit errors?

I have read some articles about Shor's code (e.g. this one). It is said that Shor's code can correct a single-qubit error. What about two qubit errors? Three qubit errors? It confused me a lot...
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CSS Code in disguise

Suppose I have a quantum error correcting code described by a set of stabilizers. Is there any easy way of checking whether or not it is equivalent to a code constructed using the CSS construction, ...
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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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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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76 views

Understanding transversal gates for the 7 qubit steane code

How can one derive the complete list of transversal operators for the 7-qubit Steane code? I can derive the Clifford operators that are transversal, but I do not understand an easy way to check for ...
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66 views

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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214 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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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 ...
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Arbitrary error correctable iff Pauli error are: misunderstanding from Preskill notes

I am following Preskill notes. What I want to understand is why it is in general enough to be able to correct n-qubit Pauli errors to say that an arbitrary error can be corrected. I call: $\mathcal{M}(...
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Phase flip error correction on state $|0\rangle$

When I read documentations about quantum error correction, it generally speak about bit flip error, and phase flip on $|1\rangle$ state, so for example, let's say my initial state is $|\psi\rangle = a|...
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In quantum computation, if the fidelity is high enough to promise over 50% success rate, is quantum error-correction still needed?

Let's say we have many qubits and gates. The errors happen randomly, for example with a probability of 0.1% at each place(at this place, no quantum error correction is implemented). If the probability ...
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Standard form Shor's code

I'm trying to solve exercise 10.57 in Nielsen-Chuang, where you have to obtain the standard form check matrix of Shor's code. I followed the procedure laid out in the earlier chapter but then realised ...
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How to define POVM measurement operators for a composite quantum state

I have an evolved quantum composite state $\hat{\rho}^{\otimes{N}}$ that I retrieved from a quantum channel $\mathcal{E}$, Now I do know how to define a POVM for the evolved states $\hat{\rho}$ that ...
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What does it mean "the N uses of classical-quantum channel"?

I was reading a paper Quantum Polar codes by Mark M. Wilde, where he discusses the N uses of the channel in the classical-quantum channel setting. What does he mean by "multiple channel uses"...
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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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CSS Code and Fault-tolerant Problem in Nielsen and Isaac Chuang‘s book

Could someone help me with $Problem-10.52$ in Nielsen and Isaac Chuang‘s book? The screenshot is shown below. I have no idea about that. Hope someone can give me some suggestions. By the way, if I ...
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Is there a good reason to use T-count minimization for circuits executed on current IBM open quantum systems (real hardware)?

As far as I understood from a series of papers, minimizing the T-count in Clifford+T circuits is essential for fault-tolerant ...
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Pauli Matrix tolerant and encoding

I'm stumped by these questions in Chuang's book. If I have a state $|ψ\rangle=1/2\sqrt2(1+M_0)(1+M_1)(1+M_2)|0\rangle_7$, where $$M_0=X_0X_4X_5X_6; M_1=X_1X_3X_5X_6; M_2=X_2X_3X_4X_6$$ I rewrite its ...
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Question about quantum error correction and density matrixs

I am now studying QEC and feel confused. If I have a density matrix before the correction: The circuit is: $\rho = p^0(1-p)^3|\varphi\rangle \langle\varphi|+p^1(1-p)^2\sum_{i=1}^3X_i|\varphi\rangle \...
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Intuition behind Stabilizer code conditions

My question is very related to Intuition about Knill-Laflamme QEC conditions but in the particular case of Stabilizer code. In Nielsen&Chuang, the theorem 10.8 on page 466 gives error-correction ...
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Is there a way to entangle to a dirty qubit?

Let's say I do something to a qubit, and I want to entangle it to a 2nd one, like this: ...
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Understanding "Restrictions on Transversal Encoded Quantum Gate Sets"

I am trying to understand a part from the article "Restrictions on Transversal Encoded Quantum Gate Sets". in the article they talk about the importance of transversal encoded gates for ...
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1answer
78 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 ...
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Why and how is quantum noise predictable?

I have been learning about quantum error correction using the zero-noise extrapolation method from this paper and have been pleased with the results. This method takes advantage of the fact that the ...
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44 views

Quantum error correction: block length and error rate definitions

I encountered notions of block length and error rate for quantum error correcting codes, which literature seems to just assume. Can someone please give precise definitions for these? Usually we ...
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Are there any implications of the automorphism group in QECC?

We often see that classically automorphism group of an error-correcting code plays a crucial role in many computational problems. Are there any important implications that depend on this in quantum ...
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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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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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57 views

Is there any difference between a quantum- and classically-controlled gate if I know my basis?

Consider an unrealistic 2-qubit plus 1 ancilla bit-flip error correction code (images generated by quirk), where I know by some means or other that an error may have happen on qubit0 (represented by ...
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Example of Quantum Error Correction [closed]

Shor's 9 Qubit code. Imagine that we encode the state $| \psi \rangle = \alpha | 0 \rangle + \beta | 1 \rangle$ using Shor's 9 qubit code, then an X error occurs on the 8th qubit of the encoded state $...
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(Why) does every CSS code allow for transversal measurement?

I recently encountered the statement that any CSS code [that encodes single-qubit logical states] has a property that measurements can be performed qubit-wise, but the measurement outcome of every ...
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How to correct a phase error of the form $e^{ix}$

Is there a method to correct an error of the form of $e^{ix}$, acted on a two-qubit state without knowing the nature (the value of x) of the error? For example, $$\alpha|00\rangle +\beta|01\rangle+\...
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1answer
37 views

Where do errors occur during error syndrome and recovery?

I'm studying quantum error correction on Nielsen's Quantum Computation and Quantum Information, but I don't understand how it identifies the number of places where an error may occur. It uses ...
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1answer
59 views

Obtaining hermitian matrix using Knill and Laflamme condition?

Let $E$ be the set of all correctable errors and $E_a, E_b \in E$. Let $\lbrace \vert c_1\rangle, \vert c_2\rangle, \ldots \vert c_k\rangle\rbrace$ be the basis of codewords in the codespace. It is ...
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What are magic states?

I wonder what are magic states, and a magic state gadget. While I'm reading a paper, these terms frequently appear.
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Need mathematical Calculation of input(T1,T2,f) and output(Readout Error)

I would like to know about the mathematical relations among T1, T2, frequency, readout error, and single-qubit error? This screenshot is from the ibmq-16 Melbourne excel data file.
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1answer
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Finding the irreducible representation of Kraus operators of a dephasing channel

I would like to understand an example of finding a noiseless subsystem of a quantum channel from the irreducible representation of its Kraus operators. Assume we have $2$ dephasing channels acting on ...
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1answer
55 views

Circuit of quantum error correction code

I read an interesting paper of QEC with only two extra qubits. In the paper, there is a circuit. Note that . Here are some contexts of the paper: "The stablizer is $XZZXI$. If the second gate ...
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1answer
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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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How to calculate distance of $k=0$ stabilizer code?

This could be seen as a followup to the question "How to calculate the distance of stabilizer code?". Summarizing the accepted answer : distance is the minimum weight of the set $$E = \bigl\{...
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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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Is there a proof or example to show that a noiseless subsystem is not necessarily closed under addition?

In a text (section 3.6 page 92) about noiseless subsystems by D. Lidar, it is mentioned:'A subsystem is a tensor factor in a tensor product, and this does not have to be a subspace (e.g., in general ...
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How is it not a contradiction that it is possible to build fault tolerant circuits with strictly contractive (e.g.: depolarizing noise) channels?

This paper discusses strictly contractive channels, i.e. channels that strictly decrease the trace distance between any two input quantum states. It is shown that if a quantum circuit is composed of ...
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Dose Quantum threshold theorem apply to IBM Quantum Experience

According to Quantum threshold theorem, error rate can be arbitrarily low. But when I use IBM Quantum Experience to measure a simple $|+\rangle$, it gives result of $|1\rangle$ with probability of 52....
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1answer
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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
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What is the certain error rate in a quantum channel?

Quantum error correction is a fundamental aspect of quantum computation. I have read some material about "Quantum Channel" and "Quantum error correction". I have known the formula ...
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In the three-qubit bit flip code, why can the first bit flip without impacting the entanglement with the other qubits?

The principle of the three-qubit Bit Flip Code is straight forward at first sight. Using CNOT you basically encode $$a|0\rangle + b|1\rangle $$ to $$ a|000\rangle + b|111\rangle$$ using ...
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1answer
118 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 ...
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Is the restriction of a strictly contractive channel (SCC) to a subspace necessarily still SCC? (impossibility of perfect QEC for SCCs)

This paper shows the impossibility of perfect error correction for strictly contractive quantum channels, i.e., for channels such that $||\mathcal{E}(\rho)-\mathcal{E}(\sigma) ||\leq k ||\rho-\sigma||$...