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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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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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
If we have 3 or more pure non-orthogonal $n$-qubit stabilizer states, where $n \ge 2$, is it true that there always exists a single-qubit Pauli measurement that will map these states to a set of post-...
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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 ...
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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 ...
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Locally commuting operators and positivity on tensor product spaces
I am reading this paper https://arxiv.org/abs/quant-ph/0501020 and two questions have been arisen for me: 1. In page 3 (left column) has been written: "Hence it follows that if $\tilde{S}_{k}^{(...
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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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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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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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Encoded Fusion Based Ring Resource State
In the article on Fusion Based Quantum Computing, they presented that a fusion of 4qubits encoded states, is creating an encoded ring resource state.
My question is - when we make fusion (Type 2) to 2 ...
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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 ...
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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 ...
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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 ...
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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} |...
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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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What is a stabilizer state?
I am reading through the paper "Direct Fidelity Estimation from Few Pauli Measurements" (arXiv:1104.4695) and it mentions 'stabilizer state'.
"The number of repetitions depends on the ...
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What is the effect of measuring a node in a cluster state?
I'm still quite confused about the post-measurement state when a node of a cluster state is measured.
As far as I'm aware, for a cluster state, a given node $j$ will have a stabilizer
$$X_j\bigotimes_{...
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Degenerate vs non-degenerate errors
One of the key features of quantum error correction that makes it different from classical error correction. When code is non-degenerate, an error $E$ takes codewords to different words. codewords.
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How to calculate the generators of a list of stabilizer?
Let's say we have a list of stabilizers: {'YZY*', 'XYY*', 'YZYZ', 'Z*Z*', '*XZ*', 'XZX*', 'ZX*Z', 'XZXZ', 'YYX*', 'Z*ZZ'}. Is there any existed formula or function (eg. in qiskit) that can calculate ...
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Measuring entanglement entropy using a stabilizer circuit simulator
I'm trying to simulate stabilizer circuits using the Clifford tableau formalism that lets you scale up to hundreds of qubits. What I want to do is find the entanglement entropy on by splitting my ...
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Entanglement Witnesses close to GHZ states
Consider page 2 of Toth's paper 'Entanglement detection in the stabilizer formalism (2005)'. To detect entanglement close to GHZ states, they construct entanglement witnesses of the form $$\mathcal{W} ...
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Stabilizer code: error detection why does it matter?
State vectors take more space when we have to specify certain states. In stabilizer formalism, we can specify these states in a much more compact way.
But in error correction then we say that the ...
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Gottesman Knill theorem: why $O(n^2)$ classical operation to keep track of a Clifford gate
Starting from a state stabilized by Pauli matrices, and using only Clifford operations Gottesman Knill theorem ensures us that such algorithm can be classically simulated.
Indeed, if I call my initial ...
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How to generate all stabilizer states numerically?
I would like to obtain a list of all stabilizer states in the given dimension (not necessarily qubit systems). What is an efficient way of generating this list numerically?
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Inner Product of Stabilizer States
The end of Section III in the paper Improved Simulation of Stabilizer Circuits by Aaronson and Gottesman shows how to compute the inner product of stabilizer states $|\psi\rangle$ and $|\varphi\rangle$...
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What is a Qiskit stabilizer
I wonder what is a Qiskit stabilizer and why we need it. Also I want to know the difference between stabilizer and extended stabilizer.
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Is there a function in Qiskit to measure tensor products of Pauli operators?
Is there a function in Qiskit similar to this function in Q# which measures strings of Pauli operators applied to different qubits?
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Algorithm to find stabilizer states
I'm working on a project in which I have to find stabilizer states based on a few criteria, the main one being that it has to have a certain amount of coherence, I'm using the following equation to ...
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