Questions tagged [pauli-group]
Questions about or related to the Pauli group.
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Can an $n$-qubit observable be constructed from single-qubit Pauli X and Z observables? [duplicate]
Any $2\times2$ matrix can be expressed as the sum and product of the identity, Pauli $X$ and Pauli $Z$ matrices. This is often used to express a singe-qubit state using just Pauli matrices. One can ...
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Higher spin Clifford gates
We're familiar with Clifford gates for qubits, which have attracted a lot of attention and research effort. Clifford circuits normalize Pauli strings, i.e., under conjugation, Clifford circuits map ...
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Determine if an operator is in the stabilizer group
I was working on error correction algebraically but I now want to use some computational method.
I have a set of generators of a stabilizer group which I represent as a rectangular matrix over $\...
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Why is the group membership problem hard for general matrices but not for the stabilizer group?
In this answer, it is claimed that the problem of answering whether an invertible matrix $A$ is an element of the group $\langle B_1, B_2,..., B_n\rangle$ for invertible matrices $B_i$ is in NP.
For ...
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Minimum number of qubits to express given commutation relations (and linear dependences) of Pauli terms
I'm interested in the question written in the title. To explain what I mean, let's take the following set of 9 Pauli terms for 3 qubits:
\begin{equation}
X_1X_2, X_2X_3, X_3X_1,~ Y_1Y_2, Y_2Y_3, ...
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Finding a succinct representation of a CPTP map
Consider a single qubit CPTP map $\mathcal{N}$ such that
$$\mathcal{N}(I) = I + pZ,~~~~~~\mathcal{N}(Z) = (1-p)Z,$$
where $I$ and $Z$ are Pauli operators. For an $n$ qubit Pauli operator $P$, made ...
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For stabilizer codes, why does the error syndrome not depend on the codeword?
While reading through some lecture notes on quantum error correction, I read the statement:
"In particular, the syndrome doesn’t depend on the specific codeword, only on the Pauli
error."
I'...
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Algorithm for computationally generating the single qudit clifford group
For a d dimensional single qudit, knowing the generators of the group being the Hadamard gate and the Phase gate, how would I generate the entire group computationally in python? Or for any finite ...
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Better optimization of bounds on sums of Pauli strings?
I'm trying to bound a quantity $||\sum_i \alpha_i P_i ||$ where the $P_i$ are arbitrary Pauli strings, $||.||$ is the operator norm (max eigenvalue) and $\alpha_i$ are arbitrary real coefficients.
If ...
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In context of stabilizer codes, are logical gates and Pauli operators the same?
I was reading this blog post and as I understood, a unitary operator is logical if and only if it's in $\mathcal{N}(\mathcal{S})$, and a Pauli operator is Logical if and only if it's in $\mathcal{C}(\...
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Complexity of Variational Quantum Eigensolvers
I am doing research surrounding VQE and am a bit confused about the complexity and its comparison to classical systems. My brief research has yielded me that classical eigenvalue solving is $O(n^3)$. ...
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Calculating Nested Commutator using a program
The $\tilde\alpha_{\text{comm}}$ mentioned in Theory of Trotter Error paper is calculated via the nested commutators. For a Hamiltonian $H = \sum_\gamma H_\gamma$, the formula for pth order is as ...
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Under what conditions are two sets of Pauli operators Clifford-equivalent?
Suppose I have two set of $N$-qubit Pauli operators $\mathcal{S} = \{P_1,\ldots,P_K\}$ and $\mathcal{T} = \{Q_1,\ldots,Q_K\}$. In this context, a Pauli operator is a Hermitian element of the Pauli ...
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What unitary commutes with all local Pauli operators?
I was thinking about this problem of identifying a set of unitary operations (other than the identity operation) that commute with local pauli $\sigma_X$ and $\sigma_Z$ matrices, i.e. find $U$ such ...
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Dimension reduction to subsystem in Pauli-Liouville basis: How to implement partial trace of Pauli-Transfer Matrices?
I have a complementary question to my previous (thankfully answered, but I can't verify for larger systems) question:
Multi-qubit quantum channels in Pauli-Liouville basis: Tensor product of Pauli-...
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