Questions tagged [pauli-group]
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18
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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 ...
- 595
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103
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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 ...
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45
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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> = $...
- 295
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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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How to find the order of the error group $E$
The following is taken from "Quantum Error Correction Via Codes Over GF(4)" Calderbank, Rains, Shor, Sloane.
We are told that the group $E$ of tensor products $\pm w_{1} \otimes \dots \...
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1
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79
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When is a block diagonal matrix a tensor product of Pauli matrices?
$U = |0\rangle\langle 0|\otimes U_1 + |1\rangle\langle 1|\otimes U_2$ is a block-diagonal unitary matrix. For this question we will assume $U$ acts on qubits. Then for some integer $N\ge 1$, $U$ is a $...
- 457
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151
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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 ...
- 103
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180
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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
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1
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114
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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 ...
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60
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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 ...
- 173
4
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133
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Minimum-weight presentation for stabilizer group $S$ and logical Pauli group $N(S)/S$
Given some stabilizer group $S$ with presentation $\langle s_1, \dots, s_r \rangle$, what is known about finding a minimal-weight presentation for it? By this, I mean a new presentation $\langle s_1', ...
- 113
4
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3
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262
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Simulating stabilizer groups
Can any existing software be used (either directly or with a bit of persuading) to work with general stabilizer groups? From what I can see, tableau-based options like Stim and Qiskit can be used to ...
- 113
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2
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60
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Efficiently finding an explicit presentation for $N(S)/S$, for any stabilizer group $S$
Let $P_n$ denote the $n$-qubit Pauli group. This has presentation $P_n = \langle iI, X_1, \ldots, X_n, Z_1, \ldots, Z_n \rangle$. Suppose we have a stabilizer group $S = \langle s_1, \ldots, s_k \...
- 113
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316
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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 (...
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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-...
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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 -- ...
- 65
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1
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86
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Tensor product of Pauli strings?
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 ...
- 505
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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 ...
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