Nonlocality as a benchmark for universal quantum computation in Ising anyon topological quantum computers

Abstract Adiabatic quantum computation (AQC) is a promising counterpart of universal quantum computation, based on the key concept of quantum annealing (QA). QA is claimed to be at the basis of commercial quantum computers and benefits from the fact that the detrimental role of decoherence and dephasing seems to have poor impact on the annealing towards the ground state. While many papers show interesting optimization results with a sizable number of qubits, a clear evidence of a full quantum coherent behavior during the whole annealing procedure is still lacking. In this paper we show that quantum non-demolition (weak) measurements of Leggett Garg inequalities can be used to efficiently assess the quantumness of the QA procedure. Numerical simulations based on a weak coupling Lindblad approach are compared with classical Langevin simulations to support our statements.

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GEOMETRIC PHASES AND TOPOLOGICAL QUANTUM COMPUTATION

International Journal of Quantum Information ◽

10.1142/s0219749903000024 ◽

2003 ◽

Vol 01 (01) ◽

pp. 1-23 ◽

Cited By ~ 30

Author(s):

VLATKO VEDRAL

Keyword(s):

Quantum Computation ◽

Large Scale ◽

Quantum Computer ◽

Fault Tolerant ◽

Geometric Phases ◽

Quantum Phases ◽

Topological Quantum ◽

Universal Quantum ◽

The Subject ◽

Topological Quantum Computation

In the first part of this review we introduce the basics theory behind geometric phases and emphasize their importance in quantum theory. The subject is presented in a general way so as to illustrate its wide applicability, but we also introduce a number of examples that will help the reader understand the basic issues involved. In the second part we show how to perform a universal quantum computation using only geometric effects appearing in quantum phases. It is then finally discussed how this geometric way of performing quantum gates can lead to a stable, large scale, intrinsically fault-tolerant quantum computer.

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A Short Introduction to Topological Quantum Computation

SciPost Physics ◽

10.21468/scipostphys.3.3.021 ◽

2017 ◽

Vol 3 (3) ◽

Cited By ~ 45

Author(s):

Ville Lahtinen ◽

Jiannis Pachos

Keyword(s):

Quantum Computation ◽

Condensed Matter ◽

Quantum Computers ◽

Short Introduction ◽

Entry Level ◽

Zero Modes ◽

Topological Quantum ◽

Topological Quantum Computation ◽

Superconducting Nanowire ◽

Energy Physics

This review presents an entry-level introduction to topological quantum computation -- quantum computing with anyons. We introduce anyons at the system-independent level of anyon models and discuss the key concepts of protected fusion spaces and statistical quantum evolutions for encoding and processing quantum information. Both the encoding and the processing are inherently resilient against errors due to their topological nature, thus promising to overcome one of the main obstacles for the realisation of quantum computers. We outline the general steps of topological quantum computation, as well as discuss various challenges faced by it. We also review the literature on condensed matter systems where anyons can emerge. Finally, the appearance of anyons and employing them for quantum computation is demonstrated in the context of a simple microscopic model -- the topological superconducting nanowire -- that describes the low-energy physics of several experimentally relevant settings. This model supports localised Majorana zero modes that are the simplest and the experimentally most tractable types of anyons that are needed to perform topological quantum computation.

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Quantum Computation and Measurements from an Exotic Space-Time R4

Symmetry ◽

10.3390/sym12050736 ◽

2020 ◽

Vol 12 (5) ◽

pp. 736

Author(s):

Michel Planat ◽

Raymond Aschheim ◽

Marcelo M. Amaral ◽

Klee Irwin

Keyword(s):

Hilbert Space ◽

Quantum Computing ◽

Fundamental Group ◽

Quantum Computation ◽

Quantum Measurement ◽

Space Time ◽

Fano Plane ◽

Topological Quantum ◽

Universal Quantum ◽

Complete Quantum

The authors previously found a model of universal quantum computation by making use of the coset structure of subgroups of a free group G with relations. A valid subgroup H of index d in G leads to a ‘magic’ state ψ in d-dimensional Hilbert space that encodes a minimal informationally complete quantum measurement (or MIC), possibly carrying a finite ‘contextual’ geometry. In the present work, we choose G as the fundamental group π 1 ( V ) of an exotic 4-manifold V, more precisely a ‘small exotic’ (space-time) R 4 (that is homeomorphic and isometric, but not diffeomorphic to the Euclidean R 4 ). Our selected example, due to S. Akbulut and R. E. Gompf, has two remarkable properties: (a) it shows the occurrence of standard contextual geometries such as the Fano plane (at index 7), Mermin’s pentagram (at index 10), the two-qubit commutation picture G Q ( 2 , 2 ) (at index 15), and the combinatorial Grassmannian Gr ( 2 , 8 ) (at index 28); and (b) it allows the interpretation of MICs measurements as arising from such exotic (space-time) R 4 s. Our new picture relating a topological quantum computing and exotic space-time is also intended to become an approach of ‘quantum gravity’.

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Permutational quantum computing

Quantum Information and Computation ◽

10.26421/qic10.5-6-7 ◽

2010 ◽

Vol 10 (5&6) ◽

pp. 470-497

Author(s):

S.P. Jordan

Keyword(s):

Quantum Computation ◽

Polynomial Time ◽

Particle Trajectory ◽

Quantum Computers ◽

Matrix Elements ◽

Spin Network ◽

Spin Foam Model ◽

Topological Quantum ◽

One Step ◽

Topological Quantum Computation

In topological quantum computation the geometric details of a particle trajectory are irrelevant; only the topology matters. Taking this one step further, we consider a model of computation that disregards even the topology of the particle trajectory, and computes by permuting particles. Whereas topological quantum computation requires anyons, permutational quantum computation can be performed with ordinary spin-1/2 particles, using a variant of the spin-network scheme of Marzuoli and Rasetti. We do not know whether permutational computation is universal. It may represent a new complexity class within BQP. Nevertheless, permutational quantum computers can in polynomial time approximate matrix elements of certain irreducible representations of the symmetric group and approximate certain transition amplitudes from the Ponzano-Regge spin foam model of quantum gravity. No polynomial time classical algorithms for these problems are known.

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Systematic distillation of composite Fibonacci anyons using one mobile quasiparticle

Quantum Information and Computation ◽

10.26421/qic12.9-10-10 ◽

2012 ◽

Vol 12 (9&10) ◽

pp. 876-892

Author(s):

Ben W. Reichardt

Keyword(s):

Quantum Computation ◽

Quantum Computer ◽

Fault Tolerant ◽

Success Probability ◽

Quantum Algorithm ◽

Systematic Construction ◽

Topological Quantum ◽

Universal Quantum ◽

Measurement Devices ◽

Topological Quantum Computation

A topological quantum computer should allow intrinsically fault-tolerant quantum computation, but there remains uncertainty about how such a computer can be implemented. It is known that topological quantum computation can be implemented with limited quasiparticle braiding capabilities, in fact using only a single mobile quasiparticle, if the system can be properly initialized by measurements. It is also known that measurements alone suffice without any braiding, provided that the measurement devices can be dynamically created and modified. We study a model in which both measurement and braiding capabilities are limited. Given the ability to pull nontrivial Fibonacci anyon pairs from the vacuum with a certain success probability, we show how to simulate universal quantum computation by braiding one quasiparticle and with only one measurement, to read out the result. The difficulty lies in initializing the system. We give a systematic construction of a family of braid sequences that initialize to arbitrary accuracy nontrivial composite anyons. Instead of using the Solovay-Kitaev theorem, the sequences are based on a quantum algorithm for convergent search.

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Majorana Zero Modes in Networks of Cooper-Pair Boxes: Topologically Ordered States and Topological Quantum Computation

Annual Review of Condensed Matter Physics ◽

10.1146/annurev-conmatphys-031218-013618 ◽

2020 ◽

Vol 11 (1) ◽

pp. 397-420 ◽

Cited By ~ 4

Author(s):

Yuval Oreg ◽

Felix von Oppen

Keyword(s):

Quantum Computation ◽

Coulomb Blockade ◽

Cooper Pair ◽

Quantum Computer ◽

Zero Modes ◽

Topological Superconductors ◽

Topological Quantum ◽

Recent Developments ◽

Universal Quantum ◽

Coulomb Effects

Recent experimental progress introduced devices that can combine topological superconductivity with Coulomb-blockade effects. Experiments with these devices have already provided additional evidence for Majorana zero modes in proximity-coupled semiconductor wires. They also stimulated numerous ideas for how to exploit interactions between Majorana zero modes generated by Coulomb charging effects in networks of Majorana wires. Coulomb effects promise to become a powerful tool in the quest for a topological quantum computer as well as for driving topological superconductors into topologically ordered insulating states. Here, we present a focused review of these recent developments, including discussions of recent experiments, designs of topological qubits, Majorana-based implementations of universal quantum computation, and topological quantum error correction. Motivated by the analogy between a qubit and a spin-1/2 degree of freedom, we also review how coupling between Cooper-pair boxes leads to emergent topologically ordered insulating phases.

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Universal Qudit Hamiltonians

Communications in Mathematical Physics ◽

10.1007/s00220-021-03940-3 ◽

2021 ◽

Author(s):

Stephen Piddock ◽

Ashley Montanaro

Keyword(s):

State Energy ◽

Entangled State ◽

Quantum Computers ◽

List Type ◽

Universal Family ◽

Finite Dimensional ◽

Universal Quantum ◽

Aklt Model ◽

Almost All ◽

Natural Way

AbstractA family of quantum Hamiltonians is said to be universal if any other finite-dimensional Hamiltonian can be approximately encoded within the low-energy space of a Hamiltonian from that family. If the encoding is efficient, universal families of Hamiltonians can be used as universal analogue quantum simulators and universal quantum computers, and the problem of approximately determining the ground-state energy of a Hamiltonian from a universal family is QMA-complete. One natural way to categorise Hamiltonians into families is in terms of the interactions they are built from. Here we prove universality of some important classes of interactions on qudits (d-level systems): We completely characterise the k-qudit interactions which are universal, if augmented with arbitrary Hermitian 1-local terms. We find that, for all $$k \geqslant 2$$ k ⩾ 2 and all local dimensions $$d \geqslant 2$$ d ⩾ 2 , almost all such interactions are universal aside from a simple stoquastic class. We prove universality of generalisations of the Heisenberg model that are ubiquitous in condensed-matter physics, even if free 1-local terms are not provided. We show that the SU(d) and SU(2) Heisenberg interactions are universal for all local dimensions $$d \geqslant 2$$ d ⩾ 2 (spin $$\geqslant 1/2$$ ⩾ 1 / 2 ), implying that a quantum variant of the Max-d-Cut problem is QMA-complete. We also show that for $$d=3$$ d = 3 all bilinear-biquadratic Heisenberg interactions are universal. One example is the general AKLT model. We prove universality of any interaction proportional to the projector onto a pure entangled state.

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