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31 votes
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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 ...
JSdJ's user avatar
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15 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 ...
forky40's user avatar
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14 votes
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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 ...
Mithrandir24601's user avatar
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14 votes
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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}...
Adam Zalcman's user avatar
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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 ...
James Wootton's user avatar
10 votes
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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 ...
Mithrandir24601's user avatar
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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%. ...
James Wootton's user avatar
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 ...
Josu Etxezarreta Martinez's user avatar
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 ...
Adam Zalcman's user avatar
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9 votes
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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 ...
DaftWullie's user avatar
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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 ...
Niel de Beaudrap's user avatar
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 ...
user1271772 No more free time's user avatar
9 votes
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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 ...
Craig Gidney's user avatar
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9 votes
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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 ...
Adam Zalcman's user avatar
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8 votes
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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 ...
Andrew O's user avatar
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8 votes
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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 ...
Craig Gidney's user avatar
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8 votes
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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 ...
squiggles's user avatar
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7 votes
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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 ...
Craig Gidney's user avatar
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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 ...
James Wootton's user avatar
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 ...
whurley's user avatar
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7 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 ...
DaftWullie's user avatar
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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 ...
user1271772 No more free time's user avatar
7 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 ...
Adam Zalcman's user avatar
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7 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 ...
DaftWullie's user avatar
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7 votes
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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 ...
Craig Gidney's user avatar
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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 ...
DaftWullie's user avatar
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6 votes

CSS Code in disguise

If it were me, I'd write out the $k$ $N$-qubit stabilizer generators in a $k\times 2N$ binary matrix, where each row corresponds to a stabilizer generator, with the first $N$ bits being the positions ...
DaftWullie's user avatar
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6 votes

Why does fault-tolerant transversal phase gate $P$ only work with doubly-even codes?

It might help to think about what the logical states actually look like. Up to normalisation, the logical 0 is $$ |0\rangle_L=\left(\prod_x(I+g_x)\right)|00\ldots 0\rangle. $$ Now, if you want a ...
DaftWullie's user avatar
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6 votes

What is the smallest code for a single logical qubit

The [[5,1,3]] code that you mention is the smallest possible distance 3 quantum error correcting code. It is known as the "perfect" quantum code because it exactly saturates one of the size ...
DaftWullie's user avatar
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6 votes
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What are the transversal gates of Shor's $[\![9,1,3]\!]$ code?

Superpositions are too entangled The logical computational basis of Shor's $9$-qubit code is $$ \begin{align} |0\rangle_L&=\frac{(|000\rangle+|111\rangle)(|000\rangle+|111\rangle)(|000\rangle+|111\...
Adam Zalcman's user avatar
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