Generalized flow and determinism in measurement-based quantum computation

Abstract

We extend the notion of quantum information flow defined by Danos and Kashefi [1] for the one-way model [2] and present a necessary and sufficient condition for the deterministic computation in this model. The generalized flow also applied in the extended model with measurements in the (X, Y ), (X, Z) and (Y, Z) planes. We apply both measurement calculus and the stabiliser formalism to derive our main theorem which for the first time gives a full characterization of the deterministic computation in the one-way model. We present several examples to show how our result improves over the traditional notion of flow, such as geometries (entanglement graph with input and output) with no flow but having generalized flow and we discuss how they lead to an optimal implementation of the unitaries. More importantly one can also obtain a better quantum computation depth with the generalized flow rather than with flow. We believe our characterization result is particularly essential for the study of the algorithms and complexity in the one-way model.

Supercurrents in Unidirectional Channels Originate from Information Transfer in the Opposite Direction: A Theoretical Prediction

“…This suggests that the supercurrent can probe the potential for information transfer without actually transferring the information. This may be useful in the context of defining quantum information flows [15,16]. On the practical side, this property of the supercurrent makes it feasible to check if in more complex QH states, all of the edge channels actually flow in the same direction [17].…”

Computational Power of Correlations

“…Two distinct groupings can be found in the literature. Graph states [5,13], which employ solely the algebra of Pauli operators ⊕L → BQP P → BQP ⊕L → P cluster states lattice graph states [5] certain CTN states [4] ! ?…”

Garbled Quantum Computation

“…Those dependencies are within the two previous "layers" (past neighbours or past neighbours of past neighbours). This is evident from the flow construction [41,42], which guarantees that corrections can be done, and the explicit form of dependencies involves only neighbours and next-to neighbours in the graph G. A flow is defined by a function ( f : O c → I c ) from measured qubits to non-input qubits and a partial order ( ) over the vertices of the graph such that ∀i : i f (i) and ∀j ∈ N G (i) : f (i) j, where N G (i) denotes the neighbours of i in G. Each qubit i is X-dependent on f −1 (i) and Z-dependent on all qubits j, such that i ∈ N G (j).…”