New answers tagged error-correction
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What does "lift" mean in the Lifted Product (LP) Code?
An $\ell$-lift is just a shorter term for an $\ell$-fold covering graph, a very general idea from graph theory and topology to obtain $\ell$ times larger graph $X^\alpha$ out of a small base graph $X$ ...
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Accepted
Why $\sqrt{\rho} = P \sqrt{\rho}$ in the proof of quantum error correction conditions in Nielsen & Chuang?
Recall from the sentence immediately preceding $(10.20)$ that $\rho$ is a state in the code subspace $C$ and from the statement of theorem $10.1$ that $P$ is a projector onto $C$. We will show that $\...
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Why $\sqrt{\rho} = P \sqrt{\rho}$ in the proof of quantum error correction conditions in Nielsen & Chuang?
$\rho$ is a state in the code. $P$ is the projector onto the codespace. That means (by definition) that for any state $|\psi\rangle$ in the code, $P|\psi\rangle=|\psi\rangle$.
$\rho$ is one such state ...
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Can Shor‘s code correct two- or three-qubit errors?
The distance 3 Shor code can't correct 2 errors. There are pairs of errors that have symptoms that look identical to a single error, but require different corrections.
However, the distance 5 Shor ...
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Can Shor‘s code correct two- or three-qubit errors?
Let me provide a sketch proof for why it cannot detect 2 qbit errors:
Let's define two different 2 qubits errors one will be applied on the state 0 and the other on the state 1.
If we show that
$$
...
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Can we get which error mechanisms acutually happens in Stim?
As of 1.12 there's no way to be told which errors were sampled when simulating a circuit.
There is a way to do it when sampling detector error models, because their errors have much simpler structure. ...
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Preparation of logical state with qLDPC code
General states is impossible; your gate set must be discrete to have fault tolerance. Even preparing simple states is hard. It's still an open research question. For example, Tomas Jochym O'Connor's ...
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Preparation of logical state with qLDPC code
Under noiseless conditions, you can create a encoding circuit, using Gottesman's algorithm for any $[[n,k,d]]$ stabilizer code. You can input any arbitrary $k$-qubit physical state, known or unknown, ...
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How to convert stim encoder circuit to a parity check matrix?
I was also thinking of using stim.Tableau.to_stabilizers but was getting 'stim._stim_polyfill.Tableau' object has no attribute 'to_stabilizers' error.
This is probably because you have stim ...
- 33.3k
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How to properly generate circuit measurement results from detector error model
For our purposes, we simply aggregated the detection events from the DEM sampler across the time axis (mod 2). This yields 100% redundant information, of course, but it seemed to help training ...
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Current situation of quantum computing with respect to physical vs logical qubits
When a quantum computing company or university group speak of qubits, generally they speak of physical qubits. The number of logical qubits depends on the quantum error correction code that you employ,...
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If a quantum error correcting code can correct every single-qubit $X$ and $Z$ error, can it also correct every single-qubit $Y$ error?
TL;DR: No. The ability to correct single-qubit $X$ and $Z$ errors does not imply the ability to correct single-qubit $Y$ errors.
Stabilizer generators
Consider the $[\![7,3]\!]$ code with stabilizer ...
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How to fix two flip-bit errors in a 3 qubit input
I was able to figure it out now via taking 3 ancilla qubits instead of 2 and figuring out the invariant in case 2 bits are flipped.
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Regarding the inductive proof that any Clifford gate can be made of Hadamard, phase and c-not
This question is about the second part of the cited exercise. The first part is the single qubit case. In the second part, one is basically supposed to prove by induction that with the given ...
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do local clifford gates preserve code distance?
Since local Cliffords (LC) do not change the weight of a Pauli operator, the distance of two LC-equivalent stabilizer codes is the same.
As a remark: Any stabilizer code is also LC-equivalent to a ...
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For any given parameters, does there always exist a quantum code which saturates the Quantum Hamming Bound?
Why not take the special case of $t=1,k=2$ (because we know $t=1,k=1$ has a solution). You are looking for an integer value of $n$ such that
$$
1+3n=2^{n-2}.
$$
I claim there is not such solution. One ...
- 55.8k
3
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Accepted
For any given parameters, does there always exist a quantum code which saturates the Quantum Hamming Bound?
TL;DR: No. There exist $t$ and $k$ for which the Quantum Hamming Bound (QHB) cannot be saturated by any block size $n$.
Constraint on block size $n$
Consider the simplest case of $t=1$ in which the ...
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Get parameters of a CU gate for implementing an erroneous CX gate with fidelity = 0.81
tl;dr: The operation you're describing is not unitary and cannot be implemented with a CU gate.
A standard CX gate implements the operation
$$
CX = |0\rangle\langle 0| \otimes \mathbb{1} + |1\rangle\...
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