# Questions tagged [pauli-group]

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### Shortcutting Clifford circuit computations using relations between stabilizers

Overview I am analyzing the output state of a Clifford circuit for various stabilizer state inputs. My circuit has midcircuit computational-basis measurements. I am curious if it is possible to save ...
103 views

### Moments of Pauli coefficients of Haar-random states

I want to evaluate the quantity $\sum_{P\in \rm{P}^n}\text{Tr}^{\alpha}(\rho P)$, where $P$ is an element of n-qubit Pauli group $\rm{P}^n$ and $\rho$ is a density matrix of a Haar random state. It is ...
1 vote
45 views

### What do the cosets of the group $E/Z(E)$ look like? (E is the quantum error group and Z(E) is the centre of E)

If I have the quantum error group $E$ which contains elements of the form: {$\pm w_{1} \otimes \dots \otimes w_{n}, \pm i w_{1} \otimes \dots \otimes w_{n}$} The centre of $E$: $Z(E) = <iI> =$...
37 views

### Does the fact that the elements of the normalizer group commute with elements of the stabilizer group imply that the normalizer is abelian?

The following question is from a paper I am reading called "Quantum Error Correction Via Codes Over GF(4)" It says: Let $E$ be the quantum error group. Let $S' \leqslant E$ which specifies ...
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### Efficient way to calculate trace of product of Pauli string and matrix?

Basically the title, but more formally: is there a way to efficiently calculate the trace of the product of a Pauli string $P$ and a $2^n \times 2^n$ matrix $M$? That is, is there a way to calculate ...
180 views

### Proof that for an $[\![n,k,d]\!]$ code we have $N(S)/S\simeq{\cal G}_k$ with $S$ stabilizer

I am reading about Quantum Error Correction and more specifically about the stabilizer formalism. Nielsen's textbook introduces the selection of logical Pauli as a kind of "ad hoc" process ...
1 vote
114 views

### A question on the structure of the Clifford group

Let $P_n$ be the Pauli group on $n$ qubits defined by $P_n=\{i^k \sigma_1...\sigma_n: k=0, 1, 2 \; \mathrm{or}\;3, \sigma_j\in \{I, X, Y, Z\}\}$, where $I, X, Y, Z$ are Pauli matrices. The Clifford ...
1 vote
60 views

### What does Pauli's $Y$ matrix represent?

It is easy to see that Pauli's $X$ matrix represents the bit flip operation, i.e. $X \lvert 0 \rangle = \lvert 1 \rangle$ and $X \lvert 1 \rangle = \lvert 0 \rangle$. Similarly, Pauli's $Z$ matrix ...
133 views

1 vote
316 views

### How grouping of Pauli strings is handled in Qiskit when running VQE?

When performing a VQE algorithm, the electronic problem Hamiltonian of the physical system under study needs to be mapped to a qubit Hamiltonian written as a sum of tensor products of Pauli operators (...
1 vote
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### Applying a single-qubit Pauli measurement to 3 or more pure non-orthogonal $n$-qubit stabilizer states

If we have 3 or more pure non-orthogonal $n$-qubit stabilizer states, where $n \ge 2$, is it true that there always exists a single-qubit Pauli measurement that will map these states to a set of post-...
94 views

### Clifford gates constructed from CNOT, H and S gates

Trying to prove that all Clifford gates can be constructed with CNOT, H and S gates, I'm following the classical path by induction (Nielsen and Chuang, Quantum Computation and Quantum Information -- ...
We define \begin{equation} \sum_{i=1}^{4^l} P_i \otimes P_i, \sum_{i=1}^{4^m} Q_i \otimes Q_i, \end{equation} where $P_l$ is the $n$ qubit Pauli string and $Q_m$ is the $m$ qubit Pauli string. Does ...
### What are the elements of quotienting the Pauli group $\mathcal{P}_n := \widetilde{\mathcal{P}}_n / N$, and how to do calculations with it?
Let $\widetilde{\mathcal{P}}_n = \langle X_1,X_2,\dots,X_n,Z_1,\dots,Z_n\rangle$ together with all the phases $\{\pm 1, \pm i\}$ the regular Pauli group, and $N = \langle \pm i I\rangle$. I would ...