No-go Theorem for One-way Quantum Computing on Naturally Occurring Two-level Systems

TL;DR

This paper proves that it is impossible for naturally occurring two-level (qubit) systems to serve as universal resource states for one-way quantum computing when constrained by realistic Hamiltonian properties.

Contribution

It establishes a no-go theorem showing that no genuinely entangled qubit ground state can be the unique ground state of a suitable two-body frustration-free Hamiltonian.

Findings

Genuinely entangled qubit states cannot be non-degenerate ground states of two-body frustration-free Hamiltonians.

Every spin-1/2 frustration-free Hamiltonian with two-body interactions has a ground state that is a product of single- or two-qubit states.

The result rules out natural qubit systems as universal resource states for one-way quantum computing under specified conditions.

Abstract

One-way quantum computing achieves the full power of quantum computation by performing single particle measurements on some many-body entangled state, known as the resource state. As single particle measurements are relatively easy to implement, the preparation of the resource state becomes a crucial task. An appealing approach is simply to cool a strongly correlated quantum many-body system to its ground state. In addition to requiring the ground state of the system to be universal for one-way quantum computing, we also want the Hamiltonian to have non-degenerate ground state protected by a fixed energy gap, to involve only two-body interactions, and to be frustration-free so that measurements in the course of the computation leave the remaining particles in the ground space. Recently, significant efforts have been made to the search of resource states that appear naturally as ground states in spin lattice systems. The approach is proved to be successful in spin-5/2 and spin-3/2 systems. Yet, it remains an open question whether there could be such a natural resource state in a spin-1/2, i.e., qubit system. Here, we give a negative answer to this question by proving that it is impossible for a genuinely entangled qubit states to be a non-degenerate ground state of any two-body frustration-free Hamiltonian. What is more, we prove that every spin-1/2 frustration-free Hamiltonian with two-body interaction always has a ground state that is a product of single- or two-qubit states, a stronger result that is interesting independent of the context of one-way quantum computing.