8
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 ...
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8
votes
Why are there eight $T$ magic state and twelve $H$ magic states?
It becomes very simple when you look at it in the Bloch representation
(Figure by me. Ref: Robustness of Magic and Symmetries of the Stabiliser Polytope)
The depicted octahedron is the polytope ...
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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 ...
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7
votes
Accepted
Why are there eight $T$ magic state and twelve $H$ magic states?
Your reasoning is in the correct direction, but realize that $C_{1}$ permutes the $3$ Paulis, and adds a $\pm 1$ on any of the Paulis. See for instance Table 1 on page 20 of Entanglement in ...
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6
votes
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Universal Gate Set, Magic States, and costliness of the T gate
The T state $Z^{1/4}|+\rangle$ has four core advantages over most other states:
You can physically inject T states at pretty high fidelity.
It has a reasonably cheap distillation circuit, as far as ...
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5
votes
Accepted
Is the plus state a magic state for the Hadamard gate?
Start with a state $|\psi\rangle|+\rangle$. Measure the operator $X_1Z_2$ using either lattice surgery or an ancilla qubit and $CZ,CX$ gates. Call this result $m_{xz}$ Then measure the first qubit in ...
- 647
5
votes
Accepted
Good references to learn magic state distillation for fault tolerance
Disclaimer: these links do not all fit the criterion recent.
First and foremost, it is important to realize that there is no 'one' magic state, and no 'one' magic state distillation.
The term 'magic ...
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4
votes
Why do magic state consumption circuits work?
First, note that you can phase a qubit by preparing an ancilla equal to it and phasing the ancilla:
You can pick $\theta=1/4$ to get a T gate. And because the bottom qubit is being discarded, you can ...
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4
votes
Magic state distillation: why is it harder to prepare the encoded $|A_{\pi/4}\rangle$ than $|0 \rangle$
To prepare $|0_L\rangle$ in a CSS code, all you have to do is separately initialize all the data qubits into $|0\rangle$ and then start measuring the stabilizers of the code. The reason this works is ...
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3
votes
Universal Gate Set, Magic States, and costliness of the T gate
Imagine you're interested in implementing a gate
$$
P_k=\left(\begin{array}{cc} 1 & 0 \\ 0 & e^{i\pi/2^k} \end{array}\right),
$$
so $Z=P_0$, $S=P_1$ and $T=P_2$. Now imagine that you're going ...
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3
votes
Accepted
Magic state distillation with the $15$ qubit code. How are the transversal $T$ performed? Also via a (lower level) state injection?
You start with physically injected magic states, expanded into $C_1$. In the case of the surface code you could use e.g. Li injection.
The error rate of this part is extremely important, as it limits ...
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3
votes
Accepted
Magic state distillation with the 15 qubit code. Why is the failure $O(p^2)$ if we include $|+_L\rangle$ preparation and decoding?
Here, I assumed that steps 1. and 3. are perfect. In practice, they will not be.
Encoding and decoding are protected by C1 (the code that doesn't support a T gate). So encoding and decoding are as ...
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3
votes
Accepted
Given a $|W_8\rangle$, perform a CCCZ using stabilizer operations
It's possible to perform a CCCZ by consuming a $|W_{8}\rangle$ state.
The key idea is to use the state $\text{AND}_{1,2} \cdot \text{AND}_{1,3} \cdot \text{AND}_{2,3} \cdot \text{AND}_{1,2,3} \cdot |+\...
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3
votes
Getting intuition on the state-injection relations for the generalized $\exp(-iP \pi/8)$ $T$-gates (ideally using ZX calculus)
TL;DR: The key idea that helps to translate these circuits into ZX diagrams is the interpretation of two strongly complementary spiders as the copy and parity operations$^1$. As expected, the ZX ...
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2
votes
Is the plus state a magic state for the Hadamard gate?
The above circuit shows how to inject a Hadamard gate by means of a magic state $|+\rangle$ and an Ising gate.
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2
votes
Pros/cons of the different schemes to have complete fault-tolerant gatesets
Magic state distillation is the most popular because it is (as of yet) the most efficient. Also, magic state distillation fuels gate teleportation, so I wouldn't say it's used instead of gate ...
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2
votes
Accepted
How to you initialize a qubit into a magic state in IBM Quantum composer
First, use a Hadamard gate to create the superposition
$$\frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$$
If you write down what you want next, you will find that you need to act with the unitary gate with ...
1
vote
Universal Gate Set, Magic States, and costliness of the T gate
To shorten Craig's answer, T is a single qubit gate, diagonal in the Z basis, and still a relatively simple, easy-to-understand rotation. H for example is not a rotation.
For your last question - the ...
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1
vote
Accepted
Intuition behind observable rotations during magic state injection
One way to rotate a multi-term observable is to temporarily xor each of the terms onto an ancilla qubit, and phase the ancilla qubit, like this:
If that ZZ term is one of the observables of a logical ...
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1
vote
Accepted
Which phase gates can be catalyzed by Clifford+Toffoli circuits with post-selection?
This has a lot to do with work that my colleagues and I presented at QIP in 2022, with the first of this series of papers now available on arXiv. In particular, in the linked paper we show that a ...
1
vote
Accepted
Scaling of magic state distillation with single & two-qubit gate error
The physical gate error rates determine the initial magic state fidelity you are starting from, and how big of a code distance you need for the factories as you boost the fidelity.
The initial ...
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1
vote
Accepted
Universal quantum computation by Clifford gates plus magic state
From Sec. V on in the Bravyi-Kitaev paper, it is all about noise - in particular magic state distillation, which is the process of distilling the magic state from many copies of some noisy state $\rho$...
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