scholarly journals Optimal verification of stabilizer states

2020 ◽  
Vol 2 (4) ◽  
Author(s):  
Ninnat Dangniam ◽  
Yun-Guang Han ◽  
Huangjun Zhu
Keyword(s):  
Stabilizer States

scholarly journals Transforming graph states using single-qubit operations

2018 ◽  
Vol 376 (2123) ◽  
pp. 20170325 ◽  
Author(s):  
Axel Dahlberg ◽  
Stephanie Wehner
Keyword(s):  
Quantum Information ◽  
Error Correcting Codes ◽  
Graph States ◽  
Target State ◽  
Single Qubit ◽  
Minor Problem ◽  
Fixed Parameter ◽  
Source State ◽  
Quantum Error ◽  
Stabilizer States

Stabilizer states form an important class of states in quantum information, and are of central importance in quantum error correction. Here, we provide an algorithm for deciding whether one stabilizer (target) state can be obtained from another stabilizer (source) state by single-qubit Clifford operations (LC), single-qubit Pauli measurements (LPM) and classical communication (CC) between sites holding the individual qubits. What is more, we provide a recipe to obtain the sequence of LC+LPM+CC operations which prepare the desired target state from the source state, and show how these operations can be applied in parallel to reach the target state in constant time. Our algorithm has applications in quantum networks, quantum computing, and can also serve as a design tool—for example, to find transformations between quantum error correcting codes. We provide a software implementation of our algorithm that makes this tool easier to apply. A key insight leading to our algorithm is to show that the problem is equivalent to one in graph theory, which is to decide whether some graph G ′ is a vertex-minor of another graph G . The vertex-minor problem is, in general, -Complete, but can be solved efficiently on graphs which are not too complex. A measure of the complexity of a graph is the rank-width which equals the Schmidt-rank width of a subclass of stabilizer states called graph states, and thus intuitively is a measure of entanglement. Here, we show that the vertex-minor problem can be solved in time O (| G | 3 ), where | G | is the size of the graph G , whenever the rank-width of G and the size of G ′ are bounded. Our algorithm is based on techniques by Courcelle for solving fixed parameter tractable problems, where here the relevant fixed parameter is the rank width. The second half of this paper serves as an accessible but far from exhausting introduction to these concepts, that could be useful for many other problems in quantum information. This article is part of a discussion meeting issue ‘Foundations of quantum mechanics and their impact on contemporary society’.

scholarly journals Validating and certifying stabilizer states

2019 ◽  
Vol 99 (4) ◽  
Author(s):  
Amir Kalev ◽  
Anastasios Kyrillidis ◽  
Norbert M. Linke
Keyword(s):  
Stabilizer States

scholarly journals Entanglement on mixed stabilizer states: normal forms and reduction procedures

2005 ◽  
Vol 7 ◽  
pp. 170-170 ◽  
Author(s):  
Koenraad M R Audenaert ◽  
Martin B Plenio
Keyword(s):  
Normal Forms ◽  
Stabilizer States

scholarly journals Perfect many-to-one teleportation with stabilizer states

2008 ◽  
Vol 77 (3) ◽  
Author(s):  
Guoming Wang ◽  
Mingsheng Ying
Keyword(s):  
Stabilizer States

scholarly journals Graphical description of the action of Clifford operators on stabilizer states

2008 ◽  
Vol 77 (4) ◽  
Author(s):  
Matthew B. Elliott ◽  
Bryan Eastin ◽  
Carlton M. Caves
Keyword(s):  
Stabilizer States

scholarly journals Determination of stabilizer states

2015 ◽  
Vol 92 (1) ◽  
Author(s):  
Xia Wu ◽  
Ying-hui Yang ◽  
Yu-kun Wang ◽  
Qiao-yan Wen ◽  
Su-juan Qin ◽  
...  
Keyword(s):  
Stabilizer States

scholarly journals Schur–Weyl Duality for the Clifford Group with Applications: Property Testing, a Robust Hudson Theorem, and de Finetti Representations

2021 ◽  
Author(s):  
David Gross ◽  
Sepehr Nezami ◽  
Michael Walter
Keyword(s):  
Quantum Information ◽  
Duality Theory ◽  
Fault Tolerant ◽  
Property Testing ◽  
List Type ◽  
Clifford Group ◽  
Weyl Duality ◽  
De Finetti ◽  
Tensor Powers ◽  
Stabilizer States

AbstractSchur–Weyl duality is a ubiquitous tool in quantum information. At its heart is the statement that the space of operators that commute with the t-fold tensor powers $$U^{\otimes t}$$ U ⊗ t of all unitaries $$U\in U(d)$$ U ∈ U ( d ) is spanned by the permutations of the t tensor factors. In this work, we describe a similar duality theory for tensor powers of Clifford unitaries. The Clifford group is a central object in many subfields of quantum information, most prominently in the theory of fault-tolerance. The duality theory has a simple and clean description in terms of finite geometries. We demonstrate its effectiveness in several applications: We resolve an open problem in quantum property testing by showing that “stabilizerness” is efficiently testable: There is a protocol that, given access to six copies of an unknown state, can determine whether it is a stabilizer state, or whether it is far away from the set of stabilizer states. We give a related membership test for the Clifford group. We find that tensor powers of stabilizer states have an increased symmetry group. Conversely, we provide corresponding de Finetti theorems, showing that the reductions of arbitrary states with this symmetry are well-approximated by mixtures of stabilizer tensor powers (in some cases, exponentially well). We show that the distance of a pure state to the set of stabilizers can be lower-bounded in terms of the sum-negativity of its Wigner function. This gives a new quantitative meaning to the sum-negativity (and the related mana) – a measure relevant to fault-tolerant quantum computation. The result constitutes a robust generalization of the discrete Hudson theorem. We show that complex projective designs of arbitrary order can be obtained from a finite number (independent of the number of qudits) of Clifford orbits. To prove this result, we give explicit formulas for arbitrary moments of random stabilizer states.

scholarly journals Stabilizer extent is not multiplicative

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 400
Author(s):  
Arne Heimendahl ◽  
Felipe Montealegre-Mora ◽  
Frank Vallentin ◽  
David Gross
Keyword(s):  
Open Problem ◽  
Efficient Algorithm ◽  
Tensor Products ◽  
Affirmative Answer ◽  
The State ◽  
Quantum Circuits ◽  
Classical Simulation ◽  
Important Open Problem ◽  
Classical Computer ◽  
Stabilizer States

The Gottesman-Knill theorem states that a Clifford circuit acting on stabilizer states can be simulated efficiently on a classical computer. Recently, this result has been generalized to cover inputs that are close to a coherent superposition of logarithmically many stabilizer states. The runtime of the classical simulation is governed by the stabilizer extent, which roughly measures how many stabilizer states are needed to approximate the state. An important open problem is to decide whether the extent is multiplicative under tensor products. An affirmative answer would yield an efficient algorithm for computing the extent of product inputs, while a negative result implies the existence of more efficient classical algorithms for simulating largescale quantum circuits. Here, we answer this question in the negative. Our result follows from very general properties of the set of stabilizer states, such as having a size that scales subexponentially in the dimension, and can thus be readily adapted to similar constructions for other resource theories.

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