34
votes
Accepted
What are magic states?
Magic states are certain states that have very nice properties with respect to fault-tolerant quantum computation.
In the vast landscape of quantum gates, there is a crude but useful distinction to be ...
- 5,639
18
votes
What are magic states?
In addition to the accepted answer and @user1271772's examples, here is a circuit primitive referred to explicitly as a "T-gate gadget" in [1] (originally appearing in [2]):
where ...
- 7,448
17
votes
Accepted
Transversal logical gate for Stabilizer (or at least Steane code)
Let $\mathcal{H}$ be the Hilbert space of a set of physical qubits and let $S$ be the stabilizer group of a stabilizer code $\mathcal{G} \subset \mathcal{H}$.
A transversal operator $U$ on $\mathcal{H}...
- 24.2k
14
votes
Accepted
What level of "confidence" of the result from a quantum computer is possible?
The majority of useful/relatively efficient algorithms1 for quantum computers belong to the 'bounded-error quantum polynomial time' (BQP) complexity class. By this definition, you want the 'failure ...
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12
votes
What is the leading edge technology for creating a quantum computer with the fewest errors?
That is indeed the most important question at the moment!
Superconducting qubits currently have the biggest devices. But will they continue to scale? Will short coherence times make it too hard for ...
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11
votes
Accepted
What exactly is a subsystem code?
I too find the various descriptions of subsystem codes a little more opaque than necessary.
At it's heart, a subsystem code is literally just a regular old "subspace" code with some ...
- 995
10
votes
Accepted
How does approximating gates via universal gates scale with the length of the computation?
Throughout this answer, the norm of a matrix $A$, $\left\lVert A\right\rVert$ will be taken to be the spectral norm of $A$ (that is, the largest singular value of $A$). The solovay-Kitaev theorem ...
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10
votes
Which quantum error correction code has the highest threshold (as proven at the time of writing this)?
As far as I’m aware, the surface code is still regarded as the best. With an assumption of all elements failing with equal probability (and doing so in a certain way) it has a threshold of around 1%.
...
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10
votes
Where do we put error correction code in quantum circuit?
Based on your question I think that you were not looking for the correct term. Error correction codes are methods in order to detect and correct possible errors that arise in qubits due to the effect ...
10
votes
Accepted
Definition of magic $T$ and $H$ states: are there different definitions for them?
Magic state names are not consistent across papers.
For example, the state used to perform T gates via gate teleportation, equal to $\frac{1}{\sqrt{2}}\left(|0\rangle + e^{i \pi/4}|1\rangle\right)$ up ...
- 40.7k
10
votes
Accepted
How to intuitively understand why $T$ gate can't be implemented transversally?
TL;DR: It is not true that $T$ gate can't be implemented transversally. However, universal gatesets, such as $\text{Clifford}{+}T$, cannot be implemented transversally by Eastin-Knill theorem. The ...
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10
votes
Accepted
Can you give an intuitive idea behind how the Minimum Weight Perfect Matching (MWPM) decoder work?
I recommend reading the sparse blossom paper.
Strings
In the surface code, errors produce pairs of detection events. Chains of errors can cancel out the detection events along the chain, leaving only ...
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10
votes
Is there a no-go theorem for or an upper bound on code threshold?
TL;DR: The threshold is not just a property of a quantum error correcting code. The existence and value of the threshold depends on the noise model (and also on the chosen decoding procedure). To ...
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9
votes
Accepted
How are magic states defined in the context of quantum computation?
It is any state that, if you have an unlimited supply of them, can be used to give you universal quantum computation when used in conjunction with perfect Clifford operations.
The standard example is ...
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9
votes
Accepted
Magic State Distillation Understanding Check
1) Magic state distillation is performed within the surface code
If you mean the distillation circuit is implemented with encoded logical qubits instead of raw physical qubits, then yes.
2) The ...
- 40.7k
9
votes
What level of "confidence" of the result from a quantum computer is possible?
Elaborating somewhat on Mithrandir24601's response —
The feature you're worried about, that a quantum computer might produce a different answer on the next ...
- 12.3k
9
votes
Why exactly are variational algorithms considered promising?
"As far as I understand there aren't many rigorous results on performance of these algorithms, similar to many classic machine learning approaches."
You are correct in that, unlike Grover's ...
8
votes
Accepted
Reference on MITxPRO Applications of Quantum Computing Professional Certificate Program
I signed up for this series because I was interested in the 2nd and 3rd courses.
There are a lot of students from different backgrounds so I think that limits the depth of what the instructors can ...
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8
votes
Why do error correction protocols only work when the error rates are already significantly low to begin with?
Classical Version
Think about a simple strategy of classical error correction. You've got a single bit that you want to encode,
$$
0\mapsto 00000\qquad 1\mapsto 11111
$$
I've chosen to encode it into ...
- 60.7k
8
votes
CSS Code in disguise
Yes. We will define a procedure for checking whether a given stabilizer code $\mathcal{G}$ is a Calderbank-Shor-Steane (CSS) code using the following
Theorem. Stabilizer code $\mathcal{G}$ is a CSS ...
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8
votes
Magic state distillation: why is it harder to prepare the encoded $|A_{\pi/4}\rangle$ than $|0 \rangle$
The point is the logical $|0\rangle$ and $|+\rangle$ are (relatively) easy to prepare. You start with any bunch of qubits, it doesn't matter what state. You simply measure the stabilizers of the code ...
- 60.7k
7
votes
Where do we put error correction code in quantum circuit?
In fault-tolerant quantum computing, we make a distinction between physical qubits and logical qubits.
The logical qubits are the ones we use in our algorithm. So if our input is a number stored in ...
- 11.4k
7
votes
Which quantum error correction code has the highest threshold (as proven at the time of writing this)?
I believe that the Centre for Engineered Quantum Systems, School of Physics, The University of Sydney and the Center for Theoretical Physics, Massachusetts Institute of Technology use of a tensor ...
- 529
7
votes
What is the pseudo threshold of a QECC using stabilizer formalism
The first answer discusses what the pseudothreshold is and how to find it, but I will try to give a few details on the difference between thresholds and pseudothresholds, since your first question ...
- 1,416
7
votes
What are magic states?
Consider a quantum computer that can:
Prepare qubits in state $|0\rangle$
Apply unitary gates from the Clifford group
Measure qubits in the $X$, $Y$, and $Z$ bases
This seems ideal because:
We know ...
7
votes
Accepted
Qutrit Steane Code
$$
XXXXIII \\
XXIIXXI \\
XIXIXIX \\
ZZ^\dagger Z^\dagger Z III \\
ZZ^\dagger II Z^\dagger Z I \\
Z I Z^\dagger I Z^\dagger I Z
$$
After I asked the question I played around with the generators by ...
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7
votes
Accepted
Recent hardware advances towards fault-tolerant quantum computing and quantum error correction
This kind of thing is why:
(That's just the easiest image I could find with a google search. It's from https://twitter.com/jjgarciaripoll/status/1295752348124553217 . Don't know what paper, but ...
- 40.7k
6
votes
How does the size of a toric code torus affect its ability to protect qubits?
The Toric code is an error correcting code. The distance of the code (I.e. the number of local operations required to convert one logical state into an orthogonal one) is equal to $N$, where the Toric ...
- 60.7k
6
votes
Accepted
Why or how is 'cat' state preparation via a C-not-not operation not fault tolerant?
First of all, the two conditions for fault tolerant measurements are:
A single error gives no more than one error per block of qubits
The measurement result needs to be correct with probability $1-\...
- 3,756
6
votes
Why or how is 'cat' state preparation via a C-not-not operation not fault tolerant?
First a matter of terminology. I don't have my copy of Nielsen & Chuang to hand, but I would have thought that the bottom, extra qubit, is the one that is the ancilla. I am also not entirely ...
- 60.7k
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