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6

If you are trying to implement a fault-tolerant quantum computation, you need to implement unitary gates that act on logical qubits. You typically have a finite set of these gates available, and what you really care about is making your operations in such a way as to keep the fault-tolerant threshold as small as possible. If you calculate a fault-tolerant ...


5

The matrix $$ M = \frac{1}{\sqrt{2}}\begin{bmatrix}-i & 1\\-1 & i\end{bmatrix} $$ resembles $$ X/2 = \frac{1}{\sqrt{2}}\begin{bmatrix}1 & -i\\-i & 1\end{bmatrix}\tag1 $$ where we follow the notation $\pm X/2$ for the $\pm\frac{\pi}{2}$ rotation around the $X$ axis as used in the table B.6 on page 101 in Julian Kelly's PhD thesis. We can make ...


2

Almost all error mitigation methods (including CDR) help reduce errors in expectation values and are not suitable to mitigate single-shot experiments. So, in the context of a quantum variational circuit associated to a MaxCut problem, error mitigation can be used only for better approximating the cost function improving: The variational optimization process....


3

You can decompose the T gates themselves to create a Toffoli Gate. Here is one way of doing this:- You can refer to this Qiskit chapter if you are interested and want to understand gate decomposition: https://qiskit.org/textbook/ch-gates/more-circuit-identities.html


5

What the author wrote is completely correct, they did not make a mistake. The subgroup of Cliffords fixing $X_n$ and $Z_n$ is indeed isomorphic to $C_{n-1}$ as a group, this is simply because this subgroup acts by assumption as $$ U (\sigma_1 \otimes \dots \otimes \sigma_n) U^\dagger = \tilde U (\sigma_1\otimes\dots\otimes\sigma_{n-1})\tilde U^\dagger \...


2

A Clifford $C_n$, defined by how it maps each of $X_i$ and $Z_i$ for $1 \leq i \leq n$, via the functions $g_i(\sigma_i)$ where $$\sigma_i = \{\pm I_i, \pm X_i, \pm Y_i, \pm Z_i\},$$ can be seen as the operation $g_1(\sigma_1) \cdot g_2(\sigma_2) \cdots \cdot g_n(\sigma_n)$ that acts on any arbitrary Pauli, $$P = \sigma_1 \cdot \sigma_2 \cdots \cdot \sigma_n....


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