A simple nearest-neighbor two-body Hamiltonian system for which the ground state is a universal resource for quantum computation

Abstract

We present a simple quantum many-body system - a two-dimensional lattice of qubits with a Hamiltonian composed of nearest-neighbor two-body interactions - such that the ground state is a universal resource for quantum computation using single-qubit measurements. This ground state approximates a cluster state that is encoded into a larger number of physical qubits. The Hamiltonian we use is motivated by the projected entangled pair states, which provide a transparent mechanism to produce such approximate encoded cluster states on square or other lattice structures (as well as a variety of other quantum states) as the ground state. We show that the error in this approximation takes the form of independent errors on bonds occurring with a fixed probability. The energy gap of such a system, which in part determines its usefulness for quantum computation, is shown to be independent of the size of the lattice. In addition, we show that the scaling of this energy gap in terms of the coupling constants of the Hamiltonian is directly determined by the lattice geometry. As a result, the approximate encoded cluster state obtained on a hexagonal lattice (a resource that is also universal for quantum computation) can be shown to have a larger energy gap than one on a square lattice with an equivalent Hamiltonian.

5 pages, 1 figure; v2 has a simplified lattice, an extended analysis of errors, and some additional references; v3 published version

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• Resource-efficient linear optical quantum computation
• Valence Bond Solids for Quantum Computation
• Absence of magnetic order for the spin-half Heisenberg antiferromagnet on the star lattice
• Simple proof of fault tolerance in the graph-state model

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• Phase transitions and localizable entanglement in cluster-state spin chains with Ising couplings and local fields
• Graph states as ground states of many-body spin-1/2 Hamiltonians
• Quantum computation in correlation space and extremal entanglement
• Adiabatic preparation without Quantum Phase Transitions
• Spin lattices with two-body Hamiltonians for which the ground state encodes a cluster state
• Quantum computation via measurements on the low-temperature state of a many-body system