6 votes
Accepted

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 ...
Craig Gidney's user avatar
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6 votes

Are these gate sets proven to be not universal?

I think the authors haven't tried to prove it, hence the formulation. In fact, it is simple to see that $CX$ and $R_z(\theta)$ is not universal as both gates map computational basis states to ...
Markus Heinrich's user avatar
4 votes

Does the Quantum Fourier Transform require universality?

Yes, the QFT requires universality. No, there isn't a non-universal gate set that implements the QFT. Just having the QFT as an operation is already computationally universal, because it can generate ...
Craig Gidney's user avatar
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3 votes

How to translate between continuous variable model and discrete model?

Here is a partial answer. In general, the connection between continuous and discrete variables gets pretty complicated... However, if you restrict yourself to Gaussian unitaries on the CV side, and ...
Cole Comfort's user avatar
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 ...
DaftWullie's user avatar
3 votes

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 ...
DaftWullie's user avatar
3 votes
Accepted

How to transpile a quantum circuit using the $\{H,T,CNOT\}$ universal set of gates?

Solovay-Kitaev algorithm is implemented in SolovayKitaev class. You can use it by transpiling your circuit using ...
Egretta.Thula's user avatar
3 votes
Accepted

Calculating the classical OR gate in a 3 qubit (+1 ancillary qubit) circuit

There are two errors in your code: You only initialize the first two bits, the last bit of the input is unused. Thus, you should replace: ...
Tristan Nemoz's user avatar
  • 6,162
3 votes
Accepted

Decomposition of rotational matrix using {$H, T$} only

There are (asymptotically) far more efficient ways of performing the calculation than using the Solovay-Kitaev algorithm. I like the package provided here, assuming your rotation can easily be related ...
DaftWullie's user avatar
3 votes
Accepted

Error correction simulation after state injection

There are a few approaches you can take, which have various degrees of difficulty. Most of these would also work as experiments on hardware. Inject stabilizer states. Instead of trying to inject $|T\...
Craig Gidney's user avatar
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2 votes
Accepted

How is quantum universality related to efficient classical simulation?

We assume that there are tasks that a universal quantum computer can do efficiently, and a classical computer cannot. Therefore, if one can simulate efficiently the action of the Clifford group with a ...
Yaron Jarach's user avatar
2 votes
Accepted

Which gate should one add to CNOT in order to have a universal set for Quantum Computing?

I think there's a typo here. I think it's supposed to say that without loss of generality $S$ is a rotation by angle $\theta$ (and hence your proposed parametrisation of $S$ is correct). The most ...
DaftWullie's user avatar
2 votes

Decomposition of rotational matrix using {$H, T$} only

For single-qubit gates this can be done with the (indeed rather complex) Solovay-Kitaev algorithm. A solid pedagogical review is here. I put a working implementation here (it took me forever to get ...
rhundt's user avatar
  • 998
1 vote

Does the Quantum Fourier Transform require universality?

First, let have a look how QFT circuit is built. The main component is controlled version of gate defined as $$ R_k = \begin{pmatrix} 1 & 0 \\ 0 & \mathrm{e}^{i2\pi/2^k} \end{pmatrix}. $$ ...
Martin Vesely's user avatar
1 vote
Accepted

Decomposing a $4 \times 4$ unitary matrix into 2-level unitary matrices

Yes, you do not need to use all types of 2-level unitaries. Controlled NOT gates can be used to exchange rows/columns 1 and 2, or 3 and 4. For example, you can exclude the type (2,3) by using $(2,3)=\...
Vladimir Lysikov's user avatar
1 vote

Are these gate sets proven to be not universal?

There is another way to see that CNOT+ $R_Y(\theta)$ is universal while CNOT+ $R_X(\theta)$ or CNOT+ $R_Z(\theta)$ is not. The general mathematical result says that one has to satisfy Theorem 3.1 of ...
R.G.J's user avatar
  • 241
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 ...
Yaron Jarach's user avatar
1 vote

Calculating the classical OR gate in a 3 qubit (+1 ancillary qubit) circuit

Maybe I'm not answering the root of your question. I would like to point out that Qiskit has a boolean expression synthesiser that automatically does that: ...
luciano's user avatar
  • 5,763

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