Efficiently Controllable Graphs

TL;DR

This paper demonstrates that certain sparse graphs enable efficient quantum control and computation, with the ability to perform universal quantum operations by controlling a small fraction of the network, and shows polynomial complexity for related classical problems.

Contribution

It introduces the concept of efficiently controllable graphs and proves their applicability to quantum networks, including lattices, percolation clusters, and random graphs, with implications for classical computational complexity.

Findings

Quantum networks on such graphs are efficiently controllable.

Universal quantum computation is achievable with minimal control.

Estimating ground states of related Hamiltonians is polynomial-time solvable.

Abstract

We investigate graphs that can be disconnected into small components by removing a vanishingly small fraction of their vertices. We show that when a quantum network is described by such a graph, the network is efficiently controllable, in the sense that universal quantum computation can be performed using a control sequence polynomial in the size of the network while controlling a vanishingly small fraction of subsystems. We show that networks corresponding to finite-dimensional lattices are efficently controllable, and explore generalizations to percolation clusters and random graphs. We show that the classical computational complexity of estimating the ground state of Hamiltonians described by controllable graphs is polynomial in the number of subsystems/qubits.