New answers tagged error-correction
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Best implementation for logical CNOT on Shor's code?
I'm led to think that I may define a better implementation, involving just 3 CNOTs.
I doubt that it's possible.
The first obstacle that you will run into is that the support of the two logical ...
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A question on dimensions of the basis vectors for the $[[6,4]]$ code
$\underline{x} \in \mathbb{F}_{2}^{4}$ as you say, but $x_j \in \mathbb{F}_{2}$ is a scalar. $\underline{g_j}$ is a row of $G_{C/C^{\perp}}$, thus has length 6. Therefore, $\sum_{j=1}^{4}x_{j}\...
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The definition of Hastings and Haah honeycomb code
They are the same, up to single qubit basis rotations. If you compile them both into $M_{ZZ}$ + single qubit gates then you will get identical circuits.
It's the same distinction as the CSS surface ...
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What is the syndrome in Hastings and Haah honeycomb code?
It's actually pretty varied.
There are three edge types: X, Y, and Z. But for the purposes here it makes more sense to label them geometrically: the edge type cut along the top and bottom boundary (V),...
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`predictions` and `fault_ids` in Stim and PyMatching for surface code decoding
In an X memory experiment, the X observable is either flipped or not flipped. There's only one logical qubit, regardless of the code distance. So there's always one bit of output. If there were two ...
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Effect of rounds in surface-code simulation with Stim
More rounds is more time that the logical qubit has to survive the noise it is being subjected to. So, in a memory experiment, logical error rate goes up with rounds.
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How exactly does measuring a syndrome operator work for 'non-discrete' errors?
When measuring the syndrome operators, the system is projected into the code subspace. You can express the state of the repetition code you are describing is any linear combination of the logical ...
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How exactly does measuring a syndrome operator work for 'non-discrete' errors?
The inner product as you've written it calculates the expectation value of the measurement of the stabilizer outcome; it does not describe the actual projection associated with measurement. This is ...
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Can IBM Quantum hardware handle any CSWAP at all?
This is the transpiled circuit of a single $\text{CSWAP}$ gate onto one of IBM's QPUs:
It is a very deep circuit, so it is not surprising your results are just noise.
Looking at the paper Shallow ...
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Deriving |𝐶𝐶𝑍⟩ magic states from |𝐶𝐶𝐶𝑍⟩?
The simplest thing you could possibly do is to turn one $|C^nZ\rangle$ state into $O(1/2^n)$ $|CCZ\rangle$ states by measuring out $n-3$ of the qubits and discarding if any of the measurement results ...
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Obtaining the stabilizer outcome by small operators
This can actually get complicated to solve, because of lattice surgery. For example, measuring these Pauli strings will measure $X_1X_2X_3X_4X_5X_6$ and no other operators over qubits 1-6:
$Z_{98}$
$...
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Obtaining the stabilizer outcome by small operators
As long as don't measure operators which anti commute, you are fine. Once you did, the value of the previous operators is random (it is like measuring a single qubit in X and then in Z).
So, in i), ...
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Does one have multiple degrees of freedom in defining logical states and logical operators of a QEC?
Almost...
Yes, the stabilizers of a code define a space. You are nominally free to pick any pair of states within that to be logical 0 and 1, and hence define logical Z.
You then have freedom to ...
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The understandings of logical operator
Consider a code with one logical qubit (for the sake of argument). In the same way that any single-qubit state is the +1 eigenstate of some linear combination of Pauli operators, there's a logical ...
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What is the physical operation behind "moving edges" and "moving corners" in Litinski's game of surface codes paper?
Moving Edges
Each edge has a basis (X or Z, shown as dashed or not dashed in the diagrams). When you move an edge, you sweep an area with it. Thus edge motions correspond to adding or removing area ...
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What is the logical gate speed of a superconducting quantum computer?
Very much depends on the machine. I'm not a hardware engineer but my rules of thumb are:
Unitary Layer of two qubit gates followed by layer of single qubit gates: 10ns to 100ns.
Dissipative Layer of ...
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How do Union-Find Decoders deal with Measurement errors through multiple measurement rounds?
It doesn't do anything special at all to deal with the measurement errors. The circuit noise problem is conceptually identical to the code capacity noise problem. Errors cause 2 or 1 detection events (...
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A better name for "weakly self-dual CSS codes"
In this paper the term "homogeneous" is used to describe a situation where $H_X=H_Z$
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How to find the undetected errors for general stabilizer codes in Stim?
It sounds like you're confused about how error correcting codes work, but first I'll answer the title question.
How to find the undetected errors for general stabilizer codes in Stim?
There are ...
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Is working with the |+> , |-> basis any harder than the |0>, |1> basis?
$[[5, 1, 3]]$ is tri-symmetric for $X,Y,Z$ basis so I believe it won't showcase any bias for different basis choices.
For CSS codes (without transversal $H$), the answer seems quite apparent when your ...
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Is working with the |+> , |-> basis any harder than the |0>, |1> basis?
It entirely depends on the code you are using. For CSS codes, Z-basis and X-basis states are typically equally easy. They both have transversal initialization, the CNOT treats them symmetrically, etc.
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Highest theoretical threshold to fight single-qubit depolarizing noise for noiseless error-correction
One way of defining the quantum capacity of a quantum channel $\mathcal{E}$ is to ask, asymptotically, as $n\to \infty$, how much quantum information can I send with error rate going to $0$ over $\...
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Highest theoretical threshold to fight single-qubit depolarizing noise for noiseless error-correction
Realistically, probably the zero-rate Hashing bound, so like 18-19%. Technically this could be violated? But I doubt by much.
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Highest theoretical threshold to fight single-qubit depolarizing noise for noiseless error-correction
If you assume the error correction to be noiseless and only consider a depolarizing channel on all qubits (say $p$ is the same on all qubits), then there will always be a probability of stabilizing a ...
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How is $(\langle \psi| E_{a}^\dagger E_{b} | \psi \rangle)^\dagger = C_{ba}^*\langle \psi| \psi \rangle $
Your equation states $C_{ab}^*=C_{ba}$. This is precisely what you want: $C$ is self-adjoint.
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What does DETECTORs mean in the example circuit for rotated surface code in Stim?
we intend to decode only X errors, so we need detectors for only Z-type stabilizers
First, because to do otherwise would be cheating at the benchmark.
Second, because no actually they are useful.
In ...
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Advantages and disadvantages of rotated surface code
One disadvantage of the rotated surface code is that hook errors can reduce the effective code distance if oriented the wrong way.
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What is the easiest way to get path graph from Stim?
The easiest way is to write glue code that iterates over the detector error model, converting error instructions into the edges of a networkx graph. Then use networkx for graph transformations like ...
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Are close states still close after measurement (regarding trace distance)?
If you could find a way to make close states become consistently far apart after measurement, you could use the deferred measurement principle to transform that into a process that did it with only ...
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How does measurement based quantum computing (MBQC) behave under error propagation?
Because of the deferred measurement principle, any circuit where measurement amplified errors could be translated into a circuit where unitary operations amplified errors. Therefore measurements can't ...
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Are close states still close after measurement (regarding trace distance)?
You can't say anything. Consider for example
$$\begin{align}
|\psi_1\rangle = \sqrt{\epsilon}|00\rangle + \sqrt{1-\epsilon}|11\rangle \\
|\psi_2\rangle = \sqrt{\epsilon}|01\rangle + \sqrt{1-\epsilon}|...
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Define the $k$-local transversal logical operation
Not exactly the same as what you're asking, but related, is the concept used here:
https://arxiv.org/abs/1206.1609
Once you go beyond transversal application of single-qubit operations, what becomes ...
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Why can Pauli errors $E$ be decomposed as $E=T(S)LG$ with $T(S)$ "pure errors"?
For stabilizer codes, a pure error is basically a correctable error that is not a stabilizer. This means that a pure error takes the code to a space orthogonal to the code (i.e. a space corresponding ...
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State injection in a surface code
I believe your reasoning is correct.
To put it in other words: you define the logical operators $Z_L$ and $X_L$ as a string of Pauli $Z$ and $X$ in the usual way for surface code (i.e., a row from ...
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Is there a simple condition under which X-and-Z-error correctability leads automatically to Y-error correctability?
As user1936752 said in his comment, "the code being CSS gives you a sufficient condition" (though, not a necessary condition, of course).
This is because CSS codes sort of consist of two ...
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