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Questions tagged [pauli-group]

Questions about or related to the Pauli group.

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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
4 votes
2 answers
409 views

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 $(...
am567's user avatar
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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> = $...
am567's user avatar
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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 ...
am567's user avatar
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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 \...
am567's user avatar
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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
4 votes
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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
4 votes
1 answer
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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
1 vote
1 answer
125 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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4 votes
3 answers
380 views

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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2 votes
1 answer
520 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 (...
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
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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 ...
R.W's user avatar
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