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Accepted

• 5,047
Accepted

### 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 ...
• 22.9k

### 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 ...
• 12.1k

### 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 ...
• 37.8k

### 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 $\pm 1$ ...
• 58.8k
Accepted

### 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 ...
• 6,719
Accepted

### 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 ...
• 58.8k

### 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 ...
• 37.8k

### 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 ...
• 58.8k
Accepted

### 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 ...
• 5,509
Accepted

### 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 ...
• 2,401
Accepted

### 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 ...
• 58.8k
Accepted

### 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 ...
• 315
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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 ...
• 37.8k
Accepted

### 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: ...
• 37.8k
Accepted

### 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: <...
• 37.8k
Accepted

• 5,509
Accepted

### 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 ...
• 2,066

### 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 ...
• 5,047
Accepted

### 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\...
• 22.9k

### Do stabilizer operations map stabilizer states to stabilizer states?

Yes - that's the definition of stabilizer circuits: They map stabilizer states to stabilizer states.
• 6,719

### 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 ...
• 58.8k
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, ...