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What is a stabilizer state?

Let $\mathcal{G}_n$ denote the Pauli group on $n$ qubits. An $n$-qubit state $|\psi\rangle$ is called a stabilizer state if there exists a subgroup $S \subset \mathcal{G}_n$ such that $|S|=2^n$ and $A|...
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Quantum advantage with only Clifford gates (Gottesman Knill theorem)

Are there examples of quantum algorithms only composed of Clifford operations that show [...] A reduction in the "same spirit" of the $n^{800}→n$ for instance. No. An $n$ qubit Clifford+...
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7 votes
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How many N-qubit stabilizer states are there?

There are $S(n) = 2^n \prod_{i=1}^n (2^i + 1)$ $n$-qubit stabilizer states, as per Corollary 21 of D. Gross, Hudson's Theorem for finite-dimensional quantum systems, J. Math. Phys. 47, 122107 (2006). ...
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Quantum advantage with only Clifford gates (Gottesman Knill theorem)

Quantum advantage using Clifford gates Gottesman-Knill theorem applies to stabilizer circuits only, not to all circuits consisting of Clifford gates. The former satisfy the stronger requirements of ...
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How to find the stabilizer generators for a post-measurement state?

Yes, there exists a relatively straightforward algorithm for finding the stabilizer generators of the post-measurement state. TL;DR: Instead of "forgetting" about the stabilizer generators ...
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Why is the $N$-qubit stabilizer group abelian?

It is not necessary to define the group as commuting —$\def\ket#1{\lvert#1\rangle}$ by virtue of every element in the group stabilising the state $\ket{\psi}$, this property follows. Because we ...
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6 votes

Is there a tool that shows me all $2^n$ stabilizers for a given graph state?

To obtain the stabilisers of a graph state, from its adjacency matrix: Change all 1s to Zs Change all 0s to identity operators Put X operators on the diagonal Each row then represents a stabiliser ...
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How to generate all stabilizer states numerically?

First of all, keep in mind that there is no efficient way of generating them, simply because the number of stabiliser states increases super-exponentially with the number of qudits $n$ (like $p^{O(n^2)...
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How to find the stabilizer generators for a post-measurement state?

Let $n$ be the observable you measured, and let $S$ be the set of stabilizer generators for the state before the measurement. Some of the old stabilizers generators anti-commute with the new ...
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Stabilizer state QFI lower limit query

The state $\psi$ (this is denoting the density matrix, even though it's a pure state) can be described as a sum of all the products of the stabilizers. We are promised that $X_i$ is not in the ...
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4 votes

Measuring entanglement entropy using a stabilizer circuit simulator

I think stim is the right tool for the job here, because it gives you access to the stabilizer generators and also it defines stim.PauliString which you can use to ...
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4 votes
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Can you measure sums of Paulis in the stabilizer formalism?

No, it's not possible. For example, being able to directly measure $X+Y$ would allow you to prepare T states and thereby perform T gates, which are not stabilizer operations. If the fact that $X$ and $...
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What is the largest number of stabilizers a pure state can have?

Every $n$-qubit pure quantum state has at most $2^n$ stabilizers. There are at least two approaches to proving this bound. One makes explicit use of the language of symplectic bilinear forms and the ...
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What is the largest number of stabilizers a pure state can have?

TL/DR The dimension of the stabilizer $\mathcal{S}$ is $2^{n}$ because there are exactly $n$ generators to make the dimension of the stabilizer's eigenspace exactly $1$. Then, each element in $\...
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Quantum algorithm for hidden subgroup problems: question on cosets

Consider that if we enumerate the cosets of $H$ as $g_0+H,g_1+H,\dots, g_n+H$, then every $g\in G$ can be written as $g_i+h$ for some $i$ and some $h\in H$, and this correspondence is 1-to-1. This ...
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Are there any packages that can calculate stabilizer tableau of a QECC

You can use stim for this, although you do have to write the stabilizer projection procedure for yourself. Write some methods to project a system into the +1 eigenstate of several stabilizers: ...
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3 votes
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Gottesman Knill theorem: why $O(n^2)$ classical operation to keep track of a Clifford gate

You can pretty easily prove by counting that specifying a stabilizer operation or a stabilizer state requires $\Omega(n^2)$ bits. If you're not tracking some of those bits, your simulator is ...
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How to get the stabilizer group for a given state?

You need to watch out when you say 'stabilizer' or 'stabilizers' because there is a little bit of ambiguity in that terminology$^{1}$. The stabilizer $\mathcal{S}$ of a state $|\psi \rangle$ is the ...
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Degenerate vs non-degenerate errors

As you say, non-degenerate codes have a lot of well-understood machinery that's brought in from the classical side. That helps us from a conceptual stance, and a mathematical one (making rigorous ...
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How to calculate the generators of a list of stabilizer?

The way that I'd do it is to write out the stabilizers in a $10\times 8$ matrix in this case (number of rows= number of stabilizers, number of columns is double the number of qubits). For each row, ...
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What is the stabilizer group of a $|W\rangle$ state?

To cite from my answer from over at physics.SE: The W state is not a stabilizer state - for a stabilizer state, the 1-site reduced density matrices must be maximally mixed or pure, which they aren't. ...
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Constructing an eigenbasis of graph states for a set of stabilizers

If you start with one graph state, which is an eigenstate of stabilizer (each of which comprises an X tensored with a bunch of Zs), then the other eigenstates of those stabilizers are the original ...
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Computing expectation value of a Pauli string on stabilizer states

You can use stim.TableauSimulator.peek_observable_expectation to compute the expected value of Pauli product observables without affecting the simulator's state: <...
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3 votes

Given a $|W_8\rangle$, perform a CCCZ using stabilizer operations

It's possible to perform a CCCZ by consuming a $|W_{8}\rangle$ state. The key idea is to use the state $\text{AND}_{1,2} \cdot \text{AND}_{1,3} \cdot \text{AND}_{2,3} \cdot \text{AND}_{1,2,3} \cdot |+\...
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3 votes
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Surface Code - Convert Control Error to Clifford Error

Answer from the literature Background The question of the validity of the incoherent approximation is explored in detail in this publication: Bravyi, S., Englbrecht, M., König, R. et al. Correcting ...
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Gottesman Knill theorem - why $O(n)$ operations for **arbitrary** *unitary* gates

You're correct. The $O(n)$ update cost is precisely for Cliffords with constant support, in particular the usual generators $S,H$, and $CX$. That's what the standard Aaranson-Gottesman simulator and ...
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2 votes

Deriving a state vector from a stabilizer state's generators

The post-selection idea can be made to work without involving an entangled state. All that matters is that the initial state, before post-selection, has some overlap with the correct final state. I'm ...
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2 votes

Constructing an eigenbasis of graph states for a set of stabilizers

I'm not sure I understand the question, since this seems quite straightforward. Graph states are Clifford states, so for a state on $n$ qubits, the set of stabilizers has $n$ generators looking like ...
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2 votes

Stabilizer for quantum error correction code

Here are a couple of observations which will hopefully clarify things. Only some states have Pauli stabilisers. You have correctly identified that not all states have Pauli stabilisers. An example ...
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2 votes

Stabilizer state verification and specification from state vector

Here's a necessary condition that might help recognise potential stabilizer states. I'll state it for qubits as that's what I'm used to thinking about, but I suspect it can be generalised: all the ...
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