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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 ...
Alfred Huang's user avatar
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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 $\...
池田隼's user avatar
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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 ...
user890890's user avatar
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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, ...
Jun_Gitef17's user avatar
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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'...
Daniel Mandragona's user avatar
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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 ...
Son100's user avatar
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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 ...
Hans Schmuber's user avatar
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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}(\...
Hamed's user avatar
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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 $O(n^3)$. ...
Jonah Sachs's user avatar
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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 ...
Zee's user avatar
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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 ...
Solarflare0's user avatar
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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 ...
Mohan's user avatar
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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-...
Onur Danaci's user avatar
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Understanding the error operator representation $E = i^{\lambda}X(a)Z(b)$

Question regarding exercise $27.3.2$ in "Concise Encyclopedia of Coding Theory". The exercise states: We write $E = X((0,1))Z((0,0))$ and $E' = iX((0,1))Z((1,1))$. We choose the ordering $(...
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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 ...
Jacob's user avatar
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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 ...
Feng Pan's user avatar
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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> = $...
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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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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 $...
Jonas Anderson's user avatar
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2 answers
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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 ...
Physics Penguin's user avatar
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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 ...
Giorgos Giapitzakis's user avatar
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1 answer
132 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 ...
Star21's user avatar
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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 ...
3nondatur's user avatar
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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', ...
Yossarian's user avatar
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3 answers
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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 ...
Yossarian's user avatar
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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 \...
Yossarian's user avatar
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1 answer
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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 (...
Radu M.'s user avatar
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Applying a single-qubit Pauli measurement to 3 or more pure non-orthogonal $n$-qubit stabilizer states

Given 3 (or more) pure non-orthogonal $n$-qubit stabilizer states where the number of qubits $n \ge 2$, say $|\psi_1\rangle,|\psi_2\rangle,|\psi_3\rangle$, define $|\nu\rangle\langle \nu |$ as a ...
Si Chen's user avatar
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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 -- ...
JMark's user avatar
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1 answer
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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 ...
Michael.Andy's user avatar
2 votes
1 answer
155 views

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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