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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|...
Adam Zalcman's user avatar
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14 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). ...
Yack's user avatar
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13 votes
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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+...
Craig Gidney's user avatar
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10 votes

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 ...
Adam Zalcman's user avatar
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9 votes
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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 ...
Niel de Beaudrap's user avatar
8 votes
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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)...
Markus Heinrich's user avatar
8 votes
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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 ...
Adam Zalcman's user avatar
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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 ...
Niel de Beaudrap's user avatar
6 votes

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 ...
Craig Gidney's user avatar
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6 votes

Why do stabilizer cut the Hilbert space into two halves?

More carefully, if you have $n$ qubits and $n-k$ linearly independent stabilisers, +1 eigenspace has dimension $2^k$. The way that I like to think about this is that each stabiliser yields $\pm 1$ ...
DaftWullie's user avatar
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6 votes
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Why do stabilizer cut the Hilbert space into two halves?

Take independent stabilizers $S_1,S_2,\dots$. First, note that $\mathrm{tr}(S_i)=0$, as well as $\mathrm{tr}(S_i S_j)=0$, and the same for any other product of the $S_i$ (as those all are Pauli ...
Norbert Schuch's user avatar
6 votes
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Is it always possible to write the state corresponding to a set of stabilizer generators?

Given an $n$-qubit system and $n$ generators $g_i$ (which commute and square to identity), then the state that you are after satisfies $$ g_i|\psi\rangle=|\psi\rangle. $$ (This defines it uniquely, up ...
DaftWullie's user avatar
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5 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 ...
Craig Gidney's user avatar
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5 votes

How to verify whether a state is a stabilizer state?

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 ...
DaftWullie's user avatar
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5 votes
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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 ...
JSdJ's user avatar
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5 votes
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What information about the logical state does one obtain from the stabilizers?

TL;DR Yes, getting $+1$ outcomes for all stabilizers only tells you that it is a logical state but gives no information as to which state it is. You cannot conclude anything about the logical state ...
FDGod's user avatar
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4 votes
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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 ...
DaftWullie's user avatar
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4 votes
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Measuring entanglement entropy using a stabilizer circuit simulator

With the help of Craig Gidney and some others, I was able to pin down the procedure to calculate the entropy. Here are the steps. Create your circuit with a stabilizer simulator This can be done with ...
Germ's user avatar
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4 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 ...
Craig Gidney's user avatar
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4 votes
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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: ...
Craig Gidney's user avatar
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4 votes
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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: <...
Craig Gidney's user avatar
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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 $...
Craig Gidney's user avatar
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4 votes
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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 ...
Adam Zalcman's user avatar
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4 votes

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 $\...
JSdJ's user avatar
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4 votes
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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 ...
Sam Jaques's user avatar
  • 2,066
4 votes

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 ...
Markus Heinrich's user avatar
4 votes
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Does Gottesman-Knill theorem apply with any computational basis input?

The two constraints on preparations are equivalent in the presence of Clifford gates. More precisely, for any $x=x_1x_2\dots x_n\in\{0,1\}^n$, we can prepare $|x\rangle$ in $O(1)$ time using $$|x\...
Adam Zalcman's user avatar
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4 votes

Do stabilizer operations map stabilizer states to stabilizer states?

Yes - that's the definition of stabilizer circuits: They map stabilizer states to stabilizer states.
Norbert Schuch's user avatar
3 votes

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 ...
DaftWullie's user avatar
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3 votes
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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, ...
DaftWullie's user avatar
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