scholarly journals Counting single-qubit Clifford equivalent graph states is #P-complete

2020 ◽  
Vol 61 (2) ◽  
pp. 022202 ◽  
Author(s):  
Axel Dahlberg ◽  
Jonas Helsen ◽  
Stephanie Wehner
Keyword(s):  
Graph States ◽  
Single Qubit ◽  
Equivalent Graph

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 Mapping graph state orbits under local complementation

Quantum ◽  
2020 ◽  
Vol 4 ◽  
pp. 305
Author(s):  
Jeremy C. Adcock ◽  
Sam Morley-Short ◽  
Axel Dahlberg ◽  
Joshua W. Silverstone
Keyword(s):  
Quantum Computing ◽  
Graph Theory ◽  
Quantum Entanglement ◽  
Graph State ◽  
Graph States ◽  
Potential Applications ◽  
Graph Operation ◽  
Local Complementation ◽  
Equivalent Graph ◽  
Component Graph

Graph states, and the entanglement they posses, are central to modern quantum computing and communications architectures. Local complementation – the graph operation that links all local-Clifford equivalent graph states – allows us to classify all stabiliser states by their entanglement. Here, we study the structure of the orbits generated by local complementation, mapping them up to 9 qubits and revealing a rich hidden structure. We provide programs to compute these orbits, along with our data for each of the 587 orbits up to 9 qubits and a means to visualise them. We find direct links between the connectivity of certain orbits with the entanglement properties of their component graph states. Furthermore, we observe the correlations between graph-theoretical orbit properties, such as diameter and colourability, with Schmidt measure and preparation complexity and suggest potential applications. It is well known that graph theory and quantum entanglement have strong interplay – our exploration deepens this relationship, providing new tools with which to probe the nature of entanglement.

scholarly journals Transforming graph states to Bell-pairs is NP-Complete

Quantum ◽  
2020 ◽  
Vol 4 ◽  
pp. 348
Author(s):  
Axel Dahlberg ◽  
Jonas Helsen ◽  
Stephanie Wehner
Keyword(s):  
Large Scale ◽  
Entangled States ◽  
Graph State ◽  
Quantum Networks ◽  
Classical Communication ◽  
Graph States ◽  
Single Qubit ◽  
Fundamental Building Block ◽  
Np Complete ◽  
Quantum Internet

Critical to the construction of large scale quantum networks, i.e. a quantum internet, is the development of fast algorithms for managing entanglement present in the network. One fundamental building block for a quantum internet is the distribution of Bell pairs between distant nodes in the network. Here we focus on the problem of transforming multipartite entangled states into the tensor product of bipartite Bell pairs between specific nodes using only a certain class of local operations and classical communication. In particular we study the problem of deciding whether a given graph state, and in general a stabilizer state, can be transformed into a set of Bell pairs on specific vertices using only single-qubit Clifford operations, single-qubit Pauli measurements and classical communication. We prove that this problem is NP-Complete.

scholarly journals DECOMPOSITION OF UNITARY MATRICES AND QUANTUM GATES

2013 ◽  
Vol 11 (01) ◽  
pp. 1350015 ◽  
Author(s):  
CHI-KWONG LI ◽  
REBECCA ROBERTS ◽  
XIAOYAN YIN
Keyword(s):  
General Scheme ◽  
Quantum Gate ◽  
Quantum Gates ◽  
Unitary Matrix ◽  
Additional Structure ◽  
Unitary Matrices ◽  
Decomposition Scheme ◽  
Single Qubit ◽  
Matlab Program

A general scheme is presented to decompose a d-by-d unitary matrix as the product of two-level unitary matrices with additional structure and prescribed determinants. In particular, the decomposition can be done by using two-level matrices in d - 1 classes, where each class is isomorphic to the group of 2 × 2 unitary matrices. The proposed scheme is easy to apply, and useful in treating problems with the additional structural restrictions. A Matlab program is written to implement the scheme, and the result is used to deduce the fact that every quantum gate acting on n-qubit registers can be expressed as no more than 2n-1(2n-1) fully controlled single-qubit gates chosen from 2n-1 classes, where the quantum gates in each class share the same n - 1 control qubits. Moreover, it is shown that one can easily adjust the proposed decomposition scheme to take advantage of additional structure evolving in the process.

scholarly journals Blind quantum computation where a user only performs single-qubit gates

2021 ◽  
Vol 142 ◽  
pp. 107190
Author(s):  
Qin Li ◽  
Chengdong Liu ◽  
Yu Peng ◽  
Fang Yu ◽  
Cai Zhang
Keyword(s):  
Quantum Computation ◽  
Single Qubit ◽  
Blind Quantum Computation

Arbitrary single-qubit transformations on a quantum frequency processor

2020 ◽  
Author(s):  
Hsuan-Hao Lu ◽  
Emma M. Simmerman ◽  
Pavel Lougovski ◽  
Andrew M. Weiner ◽  
Joseph M. Lukens
Keyword(s):  
Quantum Frequency ◽  
Single Qubit

scholarly journals Experimental test of the Greenberger–Horne–Zeilinger-type paradoxes in and beyond graph states

2021 ◽  
Vol 7 (1) ◽  
Author(s):  
Zheng-Hao Liu ◽  
Jie Zhou ◽  
Hui-Xian Meng ◽  
Mu Yang ◽  
Qiang Li ◽  
...  
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Success Probability ◽  
Quantum Foundations ◽  
Mixed States ◽  
Graph States ◽  
Hardy Type ◽  
Realistic Description ◽  
Entanglement Detection ◽  
Quantum Paradoxes

AbstractThe Greenberger–Horne–Zeilinger (GHZ) paradox is an exquisite no-go theorem that shows the sharp contradiction between classical theory and quantum mechanics by ruling out any local realistic description of quantum theory. The investigation of GHZ-type paradoxes has been carried out in a variety of systems and led to fruitful discoveries. However, its range of applicability still remains unknown and a unified construction is yet to be discovered. In this work, we present a unified construction of GHZ-type paradoxes for graph states, and show that the existence of GHZ-type paradox is not limited to graph states. The results have important applications in quantum state verification for graph states, entanglement detection, and construction of GHZ-type steering paradox for mixed states. We perform a photonic experiment to test the GHZ-type paradoxes via measuring the success probability of their corresponding perfect Hardy-type paradoxes, and demonstrate the proposed applications. Our work deepens the comprehension of quantum paradoxes in quantum foundations, and may have applications in a broad spectrum of quantum information tasks.

scholarly journals ESTIMATING PURITY AND ENTROPY IN STABILIZER STATE EXPERIMENTS

2010 ◽  
Vol 08 (01n02) ◽  
pp. 325-335 ◽  
Author(s):  
HARALD WUNDERLICH ◽  
MARTIN B. PLENIO
Keyword(s):  
Quantum Information ◽  
Measurement Data ◽  
Simulated Data ◽  
The State ◽  
Graph State ◽  
Graph States ◽  
Underlying Graph ◽  
Moderate Number ◽  
Full State ◽  
State Tomography

Many experiments in quantum information aim at creating graph states. Quantifying the purity of an experimentally achieved graph state could in principle be accomplished using full-state tomography. This method requires a number of measurement settings growing exponentially with the number of constituents involved. Thus, full-state tomography becomes experimentally infeasible even for a moderate number of qubits. In this paper, we present a method to estimate the purity of experimentally achieved graph states with simple measurements. The observables we consider are the stabilizers of the underlying graph. Then, we formulate the problem as: "What is the state with the least purity that is compatible with the measurement data?" We solve this problem analytically and compare the obtained bounds with results from full-state tomography for simulated data.

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