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1

Each of the itemized steps specifies the part of a quantum circuit where a single error occurs. So, for that part of the circuit, essentially what you want to do is write out the circuit diagram. Then, you "slice up" the circuit, i.e. put in vertical lines that just subdivide the circuit into distinct slices where all the gates in the slice can be ...

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I cannot figure out how equations $(1)$ and $(2)$ are equivalent. My humble explanation suggests that for $(1)$, they are equivalent since $E^{\dagger}_a E_b$ would cancel out to $I$ and we are left with $\langle c_i\vert c_j \rangle = 0$, since they are orthogonal. For $(2)$ we have $E^{\dagger}_a E_b$ would cancel out to $I$ and we are left with $\langle ... 3 Consider a quantum computer that can: Prepare qubits in state$|0\rangle$Apply unitary gates from the Clifford group Measure qubits in the$X$,$Y$, and$Z$bases This seems ideal because: We know how to implement all three functionalities quite easily (compared to more complicated gates or measurements) We can design algorithms for such a quantum ... 3 In the classic paper https://arxiv.org/pdf/quant-ph/9608006.pdf, on page 10, the distance of an$[n,0]$code is defined as the smallest non-zero weight of any stabilizer in the code. The physical interpretation for this definition given is, "An$[[n, 0, d]]$code is a quantum state such that, when subjected to a decoherence of$[(d − 1)/2]$coordinates, ... 8 Magic states are certain states that have very nice properties with respect to fault-tolerant quantum computation. In the vast landscape of quantum gates, there is a crude but useful distinction to be made between Clifford gates and all other gates (also referred to as the inspired non-Clifford gates). The set of Clifford gates is in technical terms the ... 2 I do not understand how if some quantum states$\rho, \sigma \in \mathbb{C}^n \otimes \mathbb{C}^m$then how can their addition not be in$\mathbb{C}^n \otimes \mathbb{C}^m$? The author is not claiming that this is false. This is certainly true, you are right about that! In a [text] (section 3.6 page 92) about noiseless subsystems by D. Lidar, it is ... 1 To clear up my own confusion, these DFSs exist for the channel that maps$|0\rangle$to$|0\rangle$and$|1\rangle$to$e^{j\phi}|1\rangle\$. This is not the channel discussed in the question.

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As you can see in th picture, the (I,Z) error occurs on 2 qubits going into a CNOT. I happens on the control and Z error happens on the target ancilla qubit, and then propagates upwards through the next CNOT, then continues from the ancilla to the following CNOT and becomes an X error on the 4th data qubit because of the Hadamard gate, and finally the 5 ...

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