Measurement-based quantum computation in a two-dimensional phase of matter

Abstract

Abstract. Recently, it was shown that the non-local correlations needed for measurement-based quantum computation (MBQC) can be revealed in the ground state of the Affleck-Kennedy-Lieb-Tasaki (AKLT) model involving nearest-neighbour spin-3/2 interactions on a honeycomb lattice. This state is not singular but resides in the disordered phase of the ground states of a large family of Hamiltonians characterized by short-range-correlated valence bond solid states. By applying local filtering and adaptive single-particle measurements, we show that most states in the disordered phase can be reduced to a graph of correlated qubits that is a scalable resource for MBQC. At the transition between the disordered and Néel ordered phases, we find a transition from universal to non-universal states as witnessed by the scaling of percolation in the reduced graph state.

Universal measurement-based quantum computation with spin-2 Affleck-Kennedy-Lieb-Tasaki states

“…The operator D(a) is then used to deform the original spin-2 AKLT Hamiltonian into (11) where h AKLT i,j 's are the two-body terms between pairs of neighboring sites i, j in the original AKLT Hamiltonian. Similar deformation was discussed in the spin-1 and spin-3/2 cases [35,46,47]. It is easy to see that the following one-parameter deformed AKLT state…”

“…They can be deformed so as to maintain universality in a range of the deformation parameter [35]. We shall also argue that there is a finite region around the spin-2 AKLT point such that the ground state of a deformed AKLT Hamiltonian still supports a universal resource; see below in Sec.…”

“…Moreover, when a 1, |ψ D (a) is essentially the Néel ordered state, or more precisely, a superposition of two different Néel patterns, |S z = 3/2 ⊗ |S z = −3/2 ⊗ · · · and |S z = −3/2 ⊗ |S z = 3/2 ⊗ · · · , which does not have sufficient entanglement for MBQC, similar to the spin-3/2 case [35]. As the spin-2 AKLT state on the square lattice (and the diamond lattice) is not Néel ordered, the wavefunction in (10) must possess a transition point in terms of phases of matter at certain value of a.…”

“…V. Furthermore, frustrated AKLT states that are not likely universal [31] can be deformed in a such way that they are connected continuously to a cluster state [36]. These can be used to study the connection of transitions in phases of matter and in quantum computational power [12,34,35,37]. With suitably chosen boundary conditions, AKLT states are unique ground states of certain two-body interacting Hamiltonians [21], some of which are believed to possess finite spectral gap [38,39], including the spin-3/2 and the spin-2 AKLT states on the honeycomb and square lattices, respectively.…”

Symmetry-protected topologically ordered states for universal quantum computation

“…A local measurement in MBQC corresponds to a joint measurement on partons of the VBS (which are to be understood as virtual qubits) within one physical site [6] and the resources in the two equivalent models are related by a mathematical projection. Since the discovery of the one-way computer [2] on the square-lattice cluster state [9], there have been quite a few other universal resources states uncovered, including, for example, graph states [10], the tricluster state [11], weighted graph states [12], modified toric code states [12], and AffleckKennedy-Lieb-Tasaki (AKLT) states [13] on various lattices [14][15][16][17][18] and their deformations [7,19]. However, as of present, there still does not exist a complete characterization of all universal resources.…”