# Tag Info

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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### 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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### 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 ...
• 57.9k
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### 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 ...
• 5,449
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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 ...
• 36.7k
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### How do magic states circumvent the Eastin-Knill theorem?

The important distinction here is not between Clifford and non-Clifford. It's between transversal and non-transversal. The Eastin-Knill theorem simply says that you cannot create a universal set of ...
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### Clifford+T synthesis with imperfect T gates

If your T gates aren't perfect, then you will need to do state distillation to make them good enough that the error you accumulate from using $N$ of them is less than the error you incur from ...
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### 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 ...
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### 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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### 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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### 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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### Is it possible to implement the controlled-S gate, such that the inner gate between the CNOTs belongs to the Clifford?

TL;DR: After sending $D$ across the equals sign, the right hand side is similar to a controlled Pauli operator and the left hand side is a diagonal operator. Their spectra turn out to be incompatible, ...
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### Magic state distillation of physical vs logical qubit

You would distill at the physical level whenever the fidelity of the output state is lower than the fidelity of your physical operations and storage. For magic states, my guess is you would never do ...
• 36.7k
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### Does the preparation of magic states need magic gates?

An example of universal quantum computation scheme using magic states can be found in this paper. What is shown there is that if you have access to several copies of noisy magic states, you can purify ...
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### Which single-qubit mixed states work for magic state distillation?

I've not kept sufficiently up to date with the most recent literature, however, here are some partial results: Along certain axes of the Bloch sphere, the divide between the octahedron and the ...
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### 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 ...
• 22.3k
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### How can sub-logarithmic magic state distillation be possible?

I forgot to consider the fact that $\gamma$ can be reduced by increasing the number of outputs. The number of inputs does have to be at least $\Omega(\log \frac{1}{\epsilon})$, but the number of ...
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### 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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### 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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### 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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