6
votes
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 ...
- 867
4
votes
Accepted
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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3
votes
Accepted
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 ...
- 151
3
votes
Accepted
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 $\...
- 168
3
votes
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 ...
- 39.6k
3
votes
Generating Equal Amplitude Superposition States from Another Equal Amplitude Superposition State
TL;DR
Yes, it is actually very simple to prepare a state that has the wanted property. Namely, $$|i\rangle^{\otimes n} = S^{\otimes n} H^{\otimes n} |0\rangle^{\otimes n}$$ with $|i\rangle = \frac{1}{\...
- 745
2
votes
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 ...
- 39.6k
2
votes
Accepted
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}|...
- 2,738
2
votes
A better name for "weakly self-dual CSS codes"
In this paper the term "homogeneous" is used to describe a situation where $H_X=H_Z$
- 21
2
votes
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 ...
- 189
1
vote
Accepted
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}\...
- 138
1
vote
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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1
vote
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),...
- 39.6k
1
vote
Accepted
`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 ...
- 39.6k
1
vote
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 ...
- 39.6k
1
vote
Accepted
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 ...
- 59.9k
1
vote
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 ...
- 59.9k
1
vote
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 ...
- 39.6k
1
vote
Accepted
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 (...
- 39.6k
1
vote
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 ...
- 39.6k
1
vote
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.
- 39.6k
1
vote
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 ...
- 39.6k
1
vote
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 ...
- 59.9k
1
vote
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 ...
- 138
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