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.

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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 ...
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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 ...
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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 $...
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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?...
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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}...
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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 ...
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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'...
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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
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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 ...
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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 ...
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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
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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
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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 ...
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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^{\...
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Stabilizer States - Calculating measurement probabilities with the rank of the stabilizer table's X-block

Consider a $n$-Qubit stabilizer state $\rho = \ket{\psi}\bra{\psi}$ and its $n \times 2n$ boolean stabilizer tableau. Any Stabilizer State can be expressed as an equally weighted superposition $$ \ket{...
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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 ...
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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 ...
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Destructive measurement surface code: parity interpretation

In the surface code the logical $Z_L$ operator is measured destructively at the end with the following procedure: Measure all data qubits in the $Z$ basis with outcome $D_i \in \{\pm 1\}$ Compute ...
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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 ...
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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 \...
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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 ...
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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 ...
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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
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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 \...
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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
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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 ...
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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 ...
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How to get the density matrix from a stabilizer table in qiskit

I am new to qiskit and quantum computing in general, so bear with me please. For my bachelor's thesis, I am programming qiskit to first generate a random Clifford circuit (qc) and to then measure the ...
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How are classical shadows stored efficiently?

The paper states that classical shadows can be stored efficiently by stabilizer formalism. But I'm confused what information is stored? I can't tell the efficiency come from. In the paper,the ...
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Non-magic non-stabilizer multi-qubit states

Does anyone know of any resources that show examples of simple multi-qubit states which are non-stabilizer states but that are still classically efficiently stimulable? Another way to phrase it is ...
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Printing Stabilizer State of a circuit using Cirq

How to print the stabilizers for a given circuit using Cirq (just like in Qiskit)? For example, if I have the following cluster state and make a circuit using Cirq. I give the circuit as input, and I ...
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How to use stabilizers to simulate the circuit?

Sorry for this easy questions. That is a simple textbook question but I don't know how to solve it. I have the following circuit: The question was "use stabilizers to simulate the circuit" ...
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Are there non-stabilizer multi-qubit states that are easy to simulate?

The Gottesman-Knill theorem states that the following process is efficiently simulatable on a classical computer: start of with a set of qubits in a computational basis apply any amount of $H, S$ and ...
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Exact Probabilities of Outcomes for Clifford Circuits with Mid-Circuit Measurements Using Stim

I am trying to find the exact probabilities of specific measurement outcomes for Clifford circuits with mid-circuit measurements. Essentially, I am looking for a function that takes an arbitrary ...
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How to write the state associated to a family of stabilizers

The answer is probably obvious but I am missing something. Let's say I have a quantum state $|\psi \rangle$ on $n$ qubits stabilized by $n$ Pauli operators $\{g_1,...,g_n\}$. My question is: How can I ...
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Gottesman Knill theorem - why $O(n)$ operations for **arbitrary** *unitary* gates

My question is closely related to this one but the answer focused mainly on measurements while my question is for unitary Clifford operations: why do we need $O(n)$ operations to update a quantum ...
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The construction of every element of the Clifford group using H,S and CNOT circuits

I am trying to understand the following theorem: Every element $U\in C_n$ of the Clifford group can be constructed using $H, S, CNOT$ gates. In Nielsen and Chuang's book this is left as an exercise (...
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Group of commuting Pauli matrices doesn't permit synthesis

I am working on learning grouped measurement and I began by reading this paper by a group out of UChicago showing a method for the synthesis of circuits for the grouped measurement of a set of ...
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Is there a way to use stabilizer formalism for non-computational basis input states?

In Nielsen and Chuang, exercise 10.42 is to use stabilizers to prove the teleportation circuit works as claimed. It has a footnote that it only works given a restricted class of inputs (it doesn't ...
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Quantum algorithm for hidden subgroup problems: question on cosets

We have a group $G$ and a function $f$ which hides a subgroup $H$, and we want to find $H$. The quantum algorithm for solving the problem involves the use of two registers, initially at $\left|0,0\...
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Surface Code - Convert Control Error to Clifford Error

I am simulating surface code, in order to find the logical error as a function of control error in my circuit. Each of my data qubits is multiplied in control error = a unitary matrix $U$ which is ...
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How to find the stabilizer generators for a post-measurement state?

My question is closely related to this one. A bit of vocabulary and a reminder of basic properties: I consider the total Hilbert space of the problem has dimension $2^n$. I call a "well defined ...
Marco Fellous-Asiani's user avatar
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Given a $|W_8\rangle$, perform a CCCZ using stabilizer operations

I know it's possible to perform a CCZ operation using only stabilizer operations (Cliffords + Pauli measurements + classical feedback) by consuming a $|CCZ\rangle$ state, and that a $|W_4\rangle$ ...
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What is the largest number of stabilizers a pure state can have?

What is the largest number of stabilizers a pure state can have? Elaborately put: Let $P(n)$ denote the Pauli group. Given an arbitrary pure state $|\psi\rangle$, what is the upper limit on how many ...
Quantum Guy 123's user avatar
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Can you measure sums of Paulis in the stabilizer formalism?

Suppose we wanted to measure the observable $Z_{1} + Z_{2} + \cdots + Z_{N}$ in a stabilizer state. Is it possible to do this using only Clifford operations, and possibly adding some auxiliary qubits? ...
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Computing expectation value of a Pauli string on stabilizer states

I need some help on stim, where I'm trying to compute expectation values of Pauli strings. Hopefully I did not overlook on the documentation an implementation of ...
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Quantum advantage with only Clifford gates (Gottesman Knill theorem)

Let's say I want to solve a computational task which input can be encoded in $n$ bits of information. The look for a quantum advantage is (usually) asking to find a quantum algorithm in which there ...
Marco Fellous-Asiani's user avatar
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Conditional lower bound on approximate stabilizer rank of magic states

I am currently reading about the approximate stabilizer rank and properties of the same. I will quote the definitions from this paper. The stabilizer rank of a quantum state $|\psi\rangle$ is the ...
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Getting exponential sequence of coefficients with not so many $T$-gates

Let $\Psi \in (\mathbb{C}^2)^{\otimes n}$ be a $n$-qubit quantum state. In the computational basis, we can write $\Psi$ as $$\Psi = \sum_{(i_1, \dots, i_n) \in \mathbb{F}_2^n} \Psi_{i_1, \dots, i_n} |...
Fritz Hefter's user avatar
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Are there any packages that can calculate stabilizer tableau of a QECC

I'm experimenting with some small quantum error correcting codes (QECC). For example $[[5,1,3]]$, $[[8,3,3]]$ or toric codes $[[2d^2,2,d]]$ ($d=2,3,\cdots$). The last one being defined by redundant ...
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