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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Quantum Error Correction Difficulties: "Measurements destroy quantum information" Intuition
Context: In section 10.1.1, Nielsen and Chuang describe the difficulties QEC faces compared to classical error correction. Particularly that
Measurements destroy quantum information.
Shor states in ...
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Quantum error correction references
I would like to understand more about QEC, such as surface codes, error codes, and qLDPC codes.
For quantum algorithms beyond the Nielsen and Chuang book the lecture notes by Andrew Childs are a very ...
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Generating Equal Amplitude Superposition States from Another Equal Amplitude Superposition State
Can we prepare a state regarding a transformation in quantum computing that seems to generate another equal amplitude superposition state when applying a Hadamard gate?
Specifically, I observed that ...
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How to use stim to implement a specific circuit
I'm doing a QEC job, and I want to apply an error like $U_{err}(\theta) = \prod_j e^{-i\frac{\theta_j}{2}P_j} = \prod_j (I\cos(\frac{\theta_j}{2}) - iP_j\sin(\frac{\theta_j}{2}))$, where $\theta_j$ is ...
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How cat state is constructed?
Context:
On Wikipedia, a cat state (2-component) is defined as
$$
|\text{cat}_+\rangle = C_+(|\alpha\rangle + |-\alpha\rangle)
$$
$$
|\text{cat}_-\rangle = C_-(|\alpha\rangle - |-\alpha\rangle),
$$
...
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Doubt in Shor-Preskill's proof of security proof of BB84
I was going through the proof of security of the modified Lo-Chau protocol, which eventually leads to the proof of security of BB84. In the error correction step, Nielson-Chuang mention that any ...
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Are there standard forms for non-additive quantum codes?
Stabilizer codes have a well known standard form for the stabilizers, logicals and destabilizers.
Are there any counterparts for non-additive codes?
In particular I'd like to get a standard form ...
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Feedback control with feedback Hamiltonian vs quantum error correction
I stumbled upon a pretty old paper from 2003 by C. Ahn, H.M. Wiseman and G.J. Milburn. They use quantum control with feedback Hamiltonian to perform error correction. The paper proposes to encode $n-1$...
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Fault tolerance of turning off a stabilizer measurement for later rounds of error correction?
Suppose during a smooth lattice merge operation, I perform one round of error correction, treating the two code patches as one single lattice that stretches across the intermediate qubits. Then, one ...
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Can we somehow relate quantum volume with the total number of gates present in a circuit which the machine can succesfully run without large errors
Let's say I have a circuit which contains x number of gates and a machine with quantum volume y, Can I relate x and y somehow to determine if the circuit can be succesfully run on the machine without ...
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Surface code with measurement errors
I am trying to implement error correction to a surface code with circuit-based noise model. In this model, there is a probability $p$ for an error in: initialization, Hadamard, CZ (or CNOT) and ...
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Does the Quantum Fourier Transform require universality?
Background:
In most setups of fault-tolerant quantum computation, universality is achieved using Clifford gates such as $(S, H, \text{CNOT})$ and the $T$-gate.
The Eastin-Knill theorem can be ...
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What are the main motivations for pursuing CSS codes over general non-CSS codes?
It seems like most research regarding stabilizer codes focused on CSS codes, which can be understand as a "tensor product" of two classical linear codes. However, I feel like non-CSS codes ...
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Implement parity check gadget using stim
I want to implement the logical $\left|00\right\rangle$ state preparation for $C_4$([[4,2,2]]) code using Stim. And the circuit is as below:
I have written a piece of code that seems to be effective:
<...
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In context of stabilizer codes, are logical gates and Pauli operators the same?
I was reading this blog post and as I understood, a unitary operator is logical if and only if it's in $\mathcal{N}(\mathcal{S})$, and a Pauli operator is Logical if and only if it's in $\mathcal{C}(\...
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how to construct a ((7,2,3)) code and verify its properties
These two recent papers describe a process to construct (non-stabilizer) codes with "exotic" transversal gates :
paper1
paper2
Most of the codes have distance $d=2$ which makes them less ...
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Is this quantum circuit design more prone to noise?
I am designing a quantum circuit which uses one qubit as the control of many $\text{CSWAP}$ operations. And then, this qubit will be measured.
Will the result be more prone to noise since all the ...
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How to convert stim circuit to a detector check matrix and a logical observable matrix?
I would like to output a detector check matrix and a logical observable matrix defined in this paper.
I've also learned from the related question: How to convert stim encoder circuit to a parity ...
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How to Use Stim for Noisy Simulation with Adaptive Circuit and Noise Tracking?
I am trying to learn stim to simulate noisy quantum circuits. My goal is to construct an adaptive circuit in which I can add quantum gates one by one and observe the changes in the circuit's check ...
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How is Magic State Distillation (MSD) compatible with Stabiliser Measurement?
Suppose I have the rotated surface code in which I want to inject a $T$-state. What I don't get it is: suppose we do end up with the logical state $|0_L\rangle+e^{i\pi/4}|1_L\rangle$, then we must ...
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How is a surface code state measured?
What is the exact operation required to measure the logical state of a surface code? Does one need to measure every data qubit, or only some subset of them to perform the measurement?
For simplicity, ...
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What is the role of color in color code? ( theoritically and exprimentally)
We know that the color code has an extra element relative toric code . It is color . I want to know what is the role of color ?
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How can quantum error correction correct small rotations/continuous errors?
I'm having trouble understanding the so-called "digitization of errors" argument in QEC.
Suppose I have to encode my logical qubit into $n$ physical qubits to do error correction. I will use ...
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What is the definition of physical gate error rate?
The fidelity of two quantum states $\rho$ and $\sigma$ is a well-defined (up to discussions about a square):
$$
F(\rho, \sigma) = \text{Tr}\left( \sqrt{ \sqrt{\rho} \sigma \sqrt{\rho}}\right)^2.
$$
...
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Can you produce Non-Clifford gates via Lattice Surgery?
Is there any theorem stating that lattice merge/split operations (or perhaps more generally, so-called gauge-fixing) cannot produce a non-Clifford gate that can complete the universal gateset for the ...
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How can I obtain the rotated surface code from the original unrotated one?
I am trying to understand the equivalence between the original planar surface code and the rotated version.
Looking at Tomita and Svore's paper (2014), they say
"The number of qubits [from ...
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Rotated surface code vs unrotated surface code - Boundary stabilizers
I use these image for the questions below:
When we look at the surface code, we can define it in two slightly different ways.
The first figure shows a way where we keep all the data qubits and ...
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Can Interaction with the environment cause $|0\rangle$ to evolve into a superposition like $|+\rangle$?
We say if a qubit is in state $|0\rangle$, and it is possible if it interacts with the environment it can result in a bit flip to a state $|1\rangle$, so I was wondering why can't it go to a ...
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How to deal with logical gates which map Logical Paulis to non-Pauli products?
I am trying to figure out the action of logical gates on quantum codes. I am trying to check the logical operations that the logical gate performs on the code.
I can do it in simple cases, for example ...
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Initialization of surface code into a logical state
I read this nice answer about how to initialize a surface code into a logical state but have a basic question about it.
If I take a surface code patch with all physical qubits in the $\vert 0\rangle$ ...
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Question about fault-tolerant measurement in Nielsen&Chuang page 490
The fault-tolerant procedures for state preparation and logical operation consider the procedure successful if only a one-qubit error occurs in each encoded block of the output.
So, I thought that the ...
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Magic state distillation, eigenvectors of Clifford gates, and transversal gates
I have some (possibly mistaken) ideas about magic state distillation. Please disabuse me of them?
In section III "Universal Quantum Computation with Magic States" of the original paper on ...
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Fault Tolerance of 2-transversal gates
Suppose I have a single block $n$-qubit stabilizer code that can correct a weight 1 error (so the distance is $d=3$). If I apply a $1$-transversal gate of the form $U = U_1 \otimes U_2 \otimes \cdots \...
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Procedures for fault-tolerant nondestructive logical measurements
Definition 2 on page 20 in Daniel Gottesman's An Introduction to Quantum Error Correction and Fault-Tolerant Quantum Computation states that a nondestructive logical measurement procedure is fault ...
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How to perform logical operations between logical qubits in a $[\![8,3,2]\!]$ quantum code?
I have a small question on codes like $[\![8,3,2]\!]$, which encodes physical qubits to multiple logical qubits. How do we perform logical operations in between these logical qubits?
Of course, if I ...
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Fault-tolerant correlated measurement on logical qubits
So I have an encoded Bell pair $(|\overline{0}\overline{0}\rangle+|\overline{1}\overline{1}\rangle)$ in CSS code(e.g. Steane code) and I hope to perform stabilizer measurement of $\overline{XX}$ non-...
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How to get smaller stabilizer codes with Gottesman's construction?
I'm following the construction from Section 3.5 of Daniel Gottesman's thesis (third paragraph), which is supposed to allow us to construct an $\left[ \hspace{-1pt} \left[ n-1, \, k + 1, \, d - 1 \...
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Why I am continuously getting "failed to decompose hyperedges into edges" in Stim?
I have written a Hypergraph product quantum error correction circuit on Stim. The code is $[\![65,9,4]\!]$. I have put circuit level noise and I am using the sinter ...
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Relating different surface code diagrams
How do I relate the surface code plaquette and vertex stabilisers of this depiction of the surface code (where the qubits are the edges):
to this depiction (measurement qubits are the black dots, ...
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What happens after the ancilla is measured in a parity measurement?
Measurement of qubits results in collapsing the wavefunction/state in $|0\rangle$ or $|1\rangle$. I have a doubt about Parity Measurement (Source).
We entangle physical qubits with ancilla qubits by $\...
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Error while plotting the resulting distribution
I am using this code, as a test of my qiskit, but I am having errors on using the simulator
Building the circuit
We begin by importing Qiskit.
from qiskit import *
...
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How to model the IBM quantum errors?
I'm conducting a study of the quantum channel errors, and a question has arisen: Is it possible to model the errors of an IBM quantum computer with Kraus operators?
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What exactly is a subsystem code?
I'm sure I'm just tripping up on terminology here. I'm new to the wonderful field of QEC, and so far am familiar with the stabiliser formalism and basic codes such as the generic surface code. I keep ...
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Compute the exact minimum distance of a QECC with integer linear programming method
Recently I'm reading the paper of Bivariate Bicycle(BB) QLDPC code proposed by IBM. The paper mentioned that:
The actual distance of each candidate code was computed using the integer linear ...
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Teleportation of Transversal Hadamard Gate from the $[[8,3,2]]$ to $[[4,2,2]]$ codes
I'm trying to understand the circuit from Appendix A of the paper Fault-Tolerant One-Bit Addition with the Smallest Interesting Colour Code.
Here the top 3 qubits represent the 3 logical qubits of ...
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Toffoli gate not included in the normalizer group
While reviewing the stabilizer formalism in Nielsen and Chuang, I could not understand why the Toffoli and the $\frac{\pi}{8}$ cannot be described as normalizers, even though they map Pauli group ...
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How do I perform an two-qubit erasure error in stim?
I try to simulate two-qubit erasure error by using CORRELATED_ERROR, ELSE_CORRELATED_ERROR, and an ancilla which is mentioned in How do I perform an erasure error in stim?. Then I simulate two-qubit ...
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Understanding Polynomial Quantum Error Correcting Codes
I am currently studying the paper Secure Multiparty Quantum Computation with (Only) a Strict Honest Majority and had a really hard time understanding the Quantum Authentication Scheme provided by the ...
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question about z-stabalizer
I'm learning quantum error correction. I also had a hard time understanding the following stabilizer form. I read this paper, which gives two circuits for Z-Stabilizer, I can understand a but I don't ...
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GKP code or Cat code Any rotation gate
In a bosonic code such as GKP code or cat code,
is there any way to perform the rotation operation in a single operation without using the expansion T-H-gates by Kitaev's theorem?
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