Towards a complete cohomology invariant for non-locality and contextuality

Carù, Giovanni

doi:10.48550/arxiv.1807.04203

Cited by 7 publications

(10 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Contextuality has recently been related to cohomology [7,8,30,31,33]. It would be a worthwhile goal to pursue such classifications further and bring them in contact with our result on computational complexity.…”

Section: Discussionmentioning

confidence: 81%

Contextuality as a resource for measurement-based quantum computation beyond qubits

Frembs¹,

Roberts²,

Bartlett³

2018

New J. Phys.

View full text Add to dashboard Cite

Contextuality-the obstruction to describing quantum mechanics in a classical statistical way-has been proposed as a resource that powers quantum computing. The measurement-based model provides a concrete manifestation of contextuality as a computational resource, as follows. If local measurements on a multi-qubit state can be used to evaluate nonlinear boolean functions with only linear control processing, then this computation constitutes a proof of strong contextuality-the possible local measurement outcomes cannot all be pre-assigned. However, this connection is restricted to the special case when the local measured systems are qubits, which have unusual properties from the perspective of contextuality. A single qubit cannot allow for a proof of contextuality, unlike higher-dimensional systems, and multiple qubits can allow for stateindependent contextuality with only Pauli observables, again unlike higher-dimensional generalisations. Here we identify precisely that strong non-locality is necessary in a qudit measurement-based computation (MBC) that evaluates high-degree polynomial functions with only linear control. We introduce the concept of local universality, which places a bound on the space of output functions accessible under the constraint of single-qudit measurements. Thus, the partition of a physical system into subsystems plays a crucial role for the increase in computational power. A prominent feature of our setting is that the enabling resources for qubit and qudit MBC are of the same underlying nature, avoiding the pathologies associated with qubit contextuality.

show abstract

Section: Discussionmentioning

confidence: 81%

Contextuality as a resource for measurement-based quantum computation beyond qubits

Frembs¹,

Roberts²,

Bartlett³

2018

New J. Phys.

View full text Add to dashboard Cite

show abstract

“…While conceptually different, MBQC is equivalent to the circuit model in computational power. Now turning to the lines of Diagram 1, proofs of quantum contextuality in terms of cohomology were first given in [7] (also see [22] - [25]), and symmetry transformations were first used for contextuality proofs in [26] and [27]. Refs.…”

Section: Approachmentioning

confidence: 99%

Cohomological framework for contextual quantum computations

Raussendorf¹

2019

QIC

View full text Add to dashboard Cite

We describe a cohomological framework for measurement-based quantum computation in which symmetry plays a central role. Therein, the essential information about the computation is contained in either of two topological invariants, namely two cohomology groups. One of them applies only to deterministic quantum computations, and the other to general probabilistic ones. Those invariants characterize the computational output, and at the same time witness quantumness in the form of contextuality. In result, they give rise to fundamental algebraic structures underlying quantum computation.

show abstract

“…In refs. [13][14][15][16], the authors point out the influence of cohomology in contextuality and nonlocality. We hope this bundle approach will reforce this relation.…”

Section: (B) Making Topology (More) Explicitmentioning

confidence: 99%

“…: a nontrivial homology class implies the nontriviality of some bundle and forbids such extension. In this sense, a nontrivial first homology group says that a scenario allows for contextuality, while a nontrivial cohomology class is a way of witnessing contextuality [13][14][15][16].…”

Section: (B) Making Topology (More) Explicitmentioning

confidence: 99%