Skip to main content

Questions tagged [stabilizer-state]

Stabilizer states are quantum states that can be efficiently represented by some set of Pauli operators of which the state is a +1 eigenstate. Stabilizer states are used commonly in many areas of quantum computation, such as error correction, teleportation and state verification.

Filter by
Sorted by
Tagged with
4 votes
0 answers
112 views

Can the W state, or any non-stabilizer state for that matter, be considered a magic state?

Some examples of magic states include: the $ |T\rangle $ state for implementing the $ T $ gate is $$ T | + \rangle = \frac{1}{\sqrt{2}}(|0\rangle + e^{i\pi/4}|1\rangle) $$ A $ |\text{CS}\rangle $ ...
Ian Gershon Teixeira's user avatar
7 votes
1 answer
137 views

What is the stabilizer rank of the W state?

The $ n $ qubit $ W $ state is defined here https://en.wikipedia.org/wiki/W_state The stabilizer rank of a quantum state $|\psi\rangle$ is the minimal $r$ such that \begin{equation} |{\psi}\rangle = \...
Ian Gershon Teixeira's user avatar
2 votes
1 answer
85 views

Common ways to fault tolerantly prepare a stabilizer state

It is my impression that it is much easier to fault tolerantly prepare a stabilizer state than it is to prepare a magic state. What are some common ways to fault tolerantly prepare a given stabilizer ...
Ian Gershon Teixeira's user avatar
0 votes
0 answers
12 views

What is the measurement algorithm that the PyClifford's authors used?

I am studying how to perform measurements on pure stabilizer states of the Pyclifford package. The author used the convention to represent stabilizer states the same as in the paper: Improved ...
Việt Nguyễn's user avatar
0 votes
0 answers
14 views

Extended Stabilizer Simulator:Get Statistics of the quantum circuit

I am trying to simulate a Clifford+T gate quantum circuit using extended stabilizer simulator. But as i see it only returns the measurement output. The measurement output(counts) depends on the shots ...
Asim Sharma's user avatar
0 votes
1 answer
43 views

How to find the $+1$ eigenvectors of the stabilizers for the Shor code

I am currently working through chapter $3$ of "Stabilizer Codes and Quantum Error Correction" (Daniel Gottesman's thesis). I would like to know the general method for finding the $+1$ ...
am567's user avatar
  • 597
3 votes
0 answers
40 views

Codes from orthogonal stabilizer states

This article discusses the equivalence between code word stabilized (CWS) and union stabilizer codes. This family with two names is also equivalent to the graph codes discussed in Nonadditive Quantum ...
Ian Gershon Teixeira's user avatar
3 votes
0 answers
38 views

Are these two codes the same?

There is an $ ((11,2,5)) $ CWS code given in table B2 at the end of https://kth.diva-portal.org/smash/get/diva2:894450/FULLTEXT01.pdf I know that some CWS codes are equivalent to stabilizer codes, but ...
Ian Gershon Teixeira's user avatar
2 votes
1 answer
58 views

Measuring stabilizers for qutrit stabilizer code

For stabilizer codes we do many things like extracting error syndromes and preparing states and stuff like that by measuring the stabilizer generators. For qubit stabilizer codes measuring the ...
Ian Gershon Teixeira's user avatar
1 vote
1 answer
40 views

Efficiently Generating Expectation Values of Operators Using Stim

I am trying to generate expectation values of operators based on several shots. using the stim library and noticed that it has the option to get the expectation of ...
iknownothing's user avatar
2 votes
0 answers
74 views

Uniqueness, absolutely maximally entangled states, and the 3 qutrit code

There is a well known $ [[3,1,2]]_3 $ qutrit stabilizer code with stabilizer generators $ XXX $ and $ ZZZ $. This code is related to a $ [[4,0,3]]_3 $ qutrit stabilizer state with stabilizer ...
Ian Gershon Teixeira's user avatar
2 votes
2 answers
83 views

Kraus decomposition of merging in lattice surgery

I am reading about lattice surgery from this paper. I am interested in the merge operation which takes 2 qubits to 1 qubit. I want to understand the logical-level Kraus operation that the merge does. ...
qubit's user avatar
  • 23
3 votes
2 answers
81 views

Commutation of $XX$ and $ZZ$ operators

It is known that the Pauli operators $XX$ and $ZZ$ commute. Consider the state $\vert{++}\rangle$ which is an eigenstate of $XX$. But we also know that $$ZZ\vert{++}\rangle = \vert{--}\rangle$$ so ...
Kieran's user avatar
  • 33
0 votes
0 answers
15 views

If the encoding circuit for a quantum code is a Clifford operation then the logical pauli errors correspond to pauli physical errors?

I was reading Lecture $2$ from M2 ICEP- Quantum Information Theory (https://perso.ens-lyon.fr/omar.fawzi/teaching/qit-paris/lecture_notes/QEC-lec2.pdf). I am a little confused on the reasoning for the ...
am567's user avatar
  • 597
1 vote
1 answer
97 views

Efficient Clifford simulation and entropy of reduced density matrices

Suppose I have a Clifford circuit $C$ and I want to estimate the entanglement entropy of a subset of two qubits, say, $\{q_0, q_1\}$, i.e. the quantity $$S(\rho_{q_0 q_1}) = - \text{Tr}[\rho_{q_0 q_1} ...
jth's user avatar
  • 368
0 votes
1 answer
79 views

How to Use Stim for Noisy Simulation with Adaptive Circuit and Noise Tracking?

I am trying to learn stim to simulate noisy quantum circuits. My goal is to construct an adaptive circuit in which I can add quantum gates one by one and observe the changes in the circuit's check ...
iknownothing's user avatar
2 votes
1 answer
200 views

What information about the logical state does one obtain from the stabilizers?

Consider the initialization of a surface code into $\vert 0\rangle_L$. This is done by initializing all the data qubits into $\vert 0\rangle$ respectively and then turning on the $XXXX$ and $ZZZZ$ ...
user1936752's user avatar
  • 3,033
3 votes
1 answer
98 views

How do I sample from a stim Tableau's stabilizer state?

(Copied from https://github.com/quantumlib/Stim/issues/708) [In stim], is there any way to directly generate the samples starting from a tableau and not a circuit? I understand that the standard way ...
Craig Gidney's user avatar
  • 38.4k
3 votes
2 answers
252 views

Is it always possible to write the state corresponding to a set of stabilizer generators?

Given a set of stabilizer generators, is it always possible to write down the state corresponding to it? Is there a way to write down the quantum state corresponding to a stabilizer generator?
Dotman's user avatar
  • 133
1 vote
1 answer
75 views

Toffoli gate not included in the normalizer group

While reviewing the stabilizer formalism in Nielsen and Chuang, I could not understand why the Toffoli and the $\frac{\pi}{8}$ cannot be described as normalizers, even though they map Pauli group ...
Amazon Forrest's user avatar
1 vote
1 answer
46 views

Do stabilizer operations map stabilizer states to stabilizer states?

Stabilizer operations comprised of stabiliser state preparations, Clifford gates, Pauli measurements, classical randomness and conditioning. Does stabilizer operations map stabilizer states to ...
Michael.Andy's user avatar
1 vote
1 answer
107 views

On difference in the number of two-qubit stabilizer states that are separable (36) vs those that are maximally entangled (24) and partial entanglement

We have a set of two-qubit stabilizer states. There are 60 of them: 36 separable and 24 maximally entangled (MES). I was wondering whether we can somehow compare the size of the set of partially ...
Sutasu's user avatar
  • 153
1 vote
1 answer
59 views

Is there a proof that any pure two-qubit Partially entangled state lies somewhere in between a separable and maximally entangled state?

Intuitively this seems true, but is there a proof of the following: We have two-qubit pure state. Given a Partially entangled state (PES) $|P\rangle$ we can always find a separable state $|S\rangle$, ...
Sutasu's user avatar
  • 153
5 votes
0 answers
74 views

Finding all small (e.g. 7 qubit) stabilizer codes

Given some choice of parameters $ [[n,k,d]] $ with $ n $ small, is there any computationally easy way to find all of (or at least many of) the stabilizer codes with those parameters? For certain ...
Ian Gershon Teixeira's user avatar
1 vote
1 answer
81 views

Given a set of stabilizers, what is an efficient way to compute the logical states and logical operators?

Suppose I have $n$ qubits and I specify $n - k$ independent stabilizer generators. I have defined a Hilbert space with $k$ logical qubits. Moreover, there exist $2k$ operators that obey the Pauli ...
user1936752's user avatar
  • 3,033
3 votes
2 answers
57 views

Equivalent definition of distance for nondegenerate code

Let $ \mathcal{C} $ be a nondegenerate quantum code. Is it true that $ \mathcal{C} $ has distance $ d $ if and only if $ d $ is the minimum nonzero weight of an error that preserves the codespace? For ...
Ian Gershon Teixeira's user avatar
3 votes
3 answers
960 views

Why do stabilizer cut the Hilbert space into two halves?

I am trying to follow the logic of Slide 8 in this deck. The result is that if you have $n-k$ stabilizers in the set of stabilizers, then the dimension of the +1 eigenspace of all the stabilizers is $...
user1936752's user avatar
  • 3,033
0 votes
0 answers
44 views

Why does the Stinespring dilation of stabilizer operations have the form $\mathcal{E}(\rho) = tr_E(U \rho \otimes \rho_E U^\dagger)$?

Why does the Stinespring dilation of a stabilizer operation have the form $\mathcal{E}(\rho) = tr_E(U \rho \otimes \rho_E U^\dagger)$ where $U$ is a Clifford unitary and $\rho_E$ is a stabilizer state?...
Si Chen's user avatar
  • 129
1 vote
0 answers
96 views

Unitary equivalent stabilizer codes

The trivial stabilizer code is defined by $$T=\{|0\rangle^{\otimes(n-k)}\otimes|\Psi\rangle:|\Psi\rangle\in(\mathbb{C}^{2})^{k}\}\tag{1}$$ which is stabilized by the Pauli operators $Z_1, ...., Z_{n-k}...
Star21's user avatar
  • 55
4 votes
2 answers
345 views

how to go from a stabilizer state to a graph

A comment (by Marcus Heinrich) in a previous post says : "any stabiliser state is locally Clifford equivalent to a graph state and vice versa". I can go from a graph (defined by its ...
unknown's user avatar
  • 2,217
2 votes
0 answers
49 views

lattice surgery in state picture

I was following Surface code quantum computing by lattice surgery. A few questions about this paper have been asked in this forum, but I believe my question is new. The main text took a 'state picture'...
quTANum's user avatar
  • 103
1 vote
0 answers
151 views

The standard form of a CSS code is CSS?

I suspect that if the standard form of a code is $$H = \begin{pmatrix} H_X & 0 \\ 0 & H_Z \end{pmatrix}~, \quad(1)$$ then I can claim that the code is CSS. They way I'm thinking about ...
Jan Olle's user avatar
1 vote
2 answers
87 views

tricks to finding possible stabilisers for $|GHZ_{3} \rangle$

The famous 3 - qubit Greenberger, Horne and Zeilinger state: $|GHZ_{3} \rangle = \frac{1}{\sqrt{2}}[|000\rangle + |111\rangle]$. A stabiliser for $|GHZ_{3} \rangle$ is the 3 - tensor product X Pauli ...
Physkid's user avatar
  • 518
3 votes
0 answers
24 views

Are there non trivial two-party stabilizers in bipartite entanglement for product states?

In this recent paper where the authors discuss finite classification of entanglement types, on pg. 29 in appendix A, it is claimed that in bipartite entanglement for product state $|00\rangle$ there ...
Sanjana's user avatar
  • 131
1 vote
1 answer
87 views

Why are global phases neglected in the check matrix representation of stabilizers?

In the check matrix representation of stabilizers, one does not care about the global phase. Now why is that? As far as I understand if I have a quantum computation, it can be computationally more ...
Lagrange's user avatar
0 votes
1 answer
36 views

Recovering phases in $2n$-bit binary representation of n-qubit Paulis

I am currently going through a paper discussing Pauli sampling strategies for VQE: https://arxiv.org/abs/1908.06942 I want to code and test their strategy. They explain how to create a circuit ...
Saturnin's user avatar
1 vote
1 answer
160 views

Gottesman-Knill theorem -- last measurement step

In the Gottesman-Knill theorem, the stabilizer set is updated after each Clifford gate. These steps are quite simple. At the end, the measurement is simulated. In some on-line explanations, I have ...
JMark's user avatar
  • 173
2 votes
1 answer
227 views

How to write a Stabilizer state in terms of sum of Pauli strings?

I'm reading the paper which introduces a method to characterize the Pauli noise channel. In eq(5) the authors state that the stabilizer state can be written as the following form $$ \left|\phi_e^{\...
Sherlock's user avatar
  • 695
0 votes
1 answer
109 views

Stabilizer States - Calculating measurement probabilities with the rank of the stabilizer table's X-block

Consider a $n$-Qubit stabilizer state $\newcommand{\ket}[1]{\vert#1\rangle}\newcommand{\bra}[1]{\langle#1\vert}\rho = \ket{\psi}\bra{\psi}$ and its $n \times 2n$ boolean stabilizer tableau. Any ...
Coryn7's user avatar
  • 37
0 votes
2 answers
105 views

Why does the qubit give random results in the circuit with rearranged CNOTs for Steane's seven qubit code in Stim?

The following is a part of the syndrome measurement circuit for Steane's seven qubit code in Stim(For ease of viewing, the TICK is omitted.). Since we are considering the detection of X errors, we use ...
lassel's user avatar
  • 51
1 vote
1 answer
71 views

Stabilizer Matrices for Mutually Unbiased Bases - what goes wrong here?

In section VIII D of this paper, the authors describe a circuit synthesis procedure to find the unitary transformation (as a quantum circuit) which diagonalizes a set of mutually commuting pauli ...
Juri V's user avatar
  • 105
4 votes
3 answers
420 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
  • 123
1 vote
2 answers
64 views

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
  • 123
5 votes
1 answer
208 views

Does Gottesman-Knill theorem apply with any computational basis input?

On Wikipedia, the Gottesman-Knill theorem is said to state the following: A quantum circuit using only the following elements can be simulated efficiently on a classical computer. Preparation of ...
trillianhaze's user avatar
1 vote
0 answers
43 views

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
  • 129
4 votes
1 answer
94 views

Example of code where codewords cannot be stabilizer states

Are there any known examples of $ ((n,K,d)) $ codes with $ d \geq 2 $ for which it is not possible to find a basis of codewords that are stabilizer states? A code word stabilized (CWS) code is defined ...
Ian Gershon Teixeira's user avatar
7 votes
1 answer
215 views

Fidelity concentration bound for random stabilizer states

Let $|\Phi\rangle$ be a normalized vector in $\mathbb{C}^d$ and let $|\psi\rangle$ be a random stabilizer state. I am trying to compute the quantity $$\mathsf{Pr}\big[|\langle \Phi|\psi \rangle|^2 \...
BlackHat18's user avatar
  • 1,363
2 votes
1 answer
70 views

Codes with codewords that aren't uniform modulus superposition

All stabilizer codes and also all non stabilizer codes that I am aware of, for example the ones here, Example non-stabilizer code? have a basis of codewords which are all uniform modulus ...
Ian Gershon Teixeira's user avatar
1 vote
1 answer
268 views

Stabilizer State - efficient calculation of measurement probabilities - Qiskit

I would like to calculate the probability of measuring some state $U\rho U^\dagger$ in the basis state $b \in (0,1)^{\otimes n}$, i.e. $<b|U\rho U^\dagger|b>$. Now, according to Gottesmann and ...
Coryn7's user avatar
  • 37
2 votes
1 answer
78 views

Stabilizer witness for entanglement detection

I am studying on entanglement detection applying stabilizer operators. In page 4 of this paper https://arxiv.org/abs/quant-ph/0501020 ,"for the detection of $N$-qubit entanglement we have to make ...
Star21's user avatar
  • 55