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Journal of the Optical Society of America B

Measurement-based quantum computation with an optical two-dimensional Affleck–Kennedy–Lieb–Tasaki state

Kai Liu, Wendong Li, Yongjian Gu

Published Oct 14, 2014• 4 citations
Publisher

Abstract

Measurement-based quantum computation can achieve universal quantum computation via simply performing single-qubit measurements alone on an entangled resource state. Instead of the canonical resource state–cluster state, we pay great attention to another resource state, the Affleck–Kennedy–Lieb–Tasaki (AKLT) state, which is a nondegenerate ground state of a nearest-neighbor two-body interaction Hamiltonian. We propose an optical scheme to generate a two-dimensional AKLT state defined on the honeycomb lattice. In our proposal, three photons from different singlet states are projected onto a spin-32 particle comprising the AKLT state with maximum success probability 38. We also show the controlled-NOT gate can be implemented on this photonic AKLT state via projective measurement using linear optics and photodetection.

DOI
10.1364/josab.31.002689

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Measurement-based quantum computation can achieve universal quantum computation via simply performing single-qubit measurements alone on an entangled resource state. Instead of the canonical resource state–cluster state, we pay great attention to another resource state, the Affleck–Kennedy–Lieb–Tasaki (AKLT) state, which is a nondegenerate ground state of a nearest-neighbor two-body interaction Hamiltonian. We propose an optical scheme to generate a two-dimensional AKLT state defined on the honeycomb lattice. In our proposal, three photons from different singlet states are projected onto a spin-32 particle comprising the AKLT state with maximum success probability 38. We also show the controlled-NOT gate can be implemented on this photonic AKLT state via projective measurement using linear optics and photodetection.
Universal quantum computation can be achieved by simply performing single-spin measurements on a highly entangled resource state, such as cluster states. The family of Affleck-Kennedy-Lieb-Tasaki (AKLT) states has recently been explored; for example, the spin-1 AKLT chain can be used to simulate single-qubit gate operations on a single qubit, and the spin-3/2 two-dimensional AKLT state on the honeycomb lattice can be used as a universal resource. However, it is unclear whether such universality is a coincidence for the specific state or a shared feature in all two-dimensional AKLT states. Here we consider the family of spin-3/2 AKLT states on various trivalent Archimedean lattices and show that in addition to the honeycomb lattice, the spin-3/2 AKLT states on the square octagon (4,82)(4,{8}^{2})(4,82) and the ``cross'' (4,6,12)(4,6,12)(4,6,12) lattices are also universal resource, whereas the AKLT state on the ``star'' (3,122)(3,{12}^{2})(3,122) lattice is likely not due to geometric frustration.
Universal quantum computation can be achieved by simply performing single-spin measurements on a highly entangled resource state, such as cluster states. The family of Affleck-Kennedy-Lieb-Tasaki (AKLT) states has recently been explored; for example, the spin-1 AKLT chain can be used to simulate single-qubit gate operations on a single qubit, and the spin-3/2 two-dimensional AKLT state on the honeycomb lattice can be used as a universal resource. However, it is unclear whether such universality is a coincidence for the specific state or a shared feature in all two-dimensional AKLT states. Here we consider the family of spin-3/2 AKLT states on various trivalent Archimedean lattices and show that in addition to the honeycomb lattice, the spin-3/2 AKLT states on the square octagon (4,82)(4,{8}^{2})(4,82) and the ``cross'' (4,6,12)(4,6,12)(4,6,12) lattices are also universal resource, whereas the AKLT state on the ``star'' (3,122)(3,{12}^{2})(3,122) lattice is likely not due to geometric frustration.
Universal quantum computation can be achieved by simply performing single-qubit measurements on a highly entangled resource state. Resource states can arise from ground states of carefully designed two-body interacting Hamiltonians. This opens up an appealing possibility of creating them by cooling. The family of Affleck-Kennedy-Lieb-Tasaki (AKLT) states are the ground states of particularly simple Hamiltonians with high symmetry, and their potential use in quantum computation gives rise to a new research direction. Expanding on our prior work [T.-C. Wei, I. Affleck, and R. Raussendorf, Phys. Rev. Lett. 106, 070501 (2011)], we give a detailed analysis to explain why the spin-3/2 AKLT state on a two-dimensional honeycomb lattice is a universal resource for measurement-based quantum computation. Along the way, we also provide an alternative proof that the 1D spin-1 AKLT state can be used to simulate arbitrary one-qubit unitary gates. Moreover, we connect the quantum computational universality of 2D random graph states to their percolation property and show that these states whose graphs are in the supercritical (i.e., percolated) phase are also universal resources for measurement-based quantum computation.
We demonstrate that the spin-2 Affleck-Kennedy-Lieb-Tasaki (AKLT) state on a square lattice is a universal resource for measurement-based quantum computation. Our proof is done by locally converting the AKLT to two-dimensional random planar graph states and by certifying that with a high probability the resulting random graphs are in the supercritical phase of percolation using Monte Carlo simulations. One key enabling point is the exact weight formula that we derive for arbitrary measurement outcomes according to a spin-2 positive operator-valued measure on all spins. We also argue that the spin-2 AKLT state on a three-dimensional diamond lattice is a universal resource, the advantage of which would be the possibility of implementing fault-tolerant quantum computation with topological protection. In addition, as we deform the AKLT Hamiltonian, there is a finite region in which the ground state can still support a universal resource before making a transition in its quantum computational power.
Measurement-based quantum computing, a powerful alternative to the standard circuit model, proceeds using only local adaptive measurements on a highly entangled resource state of many spins on a graph or lattice. Along with the canonical cluster state, the valence-bond solid ground state on a chain of spin-1 particles, studied by Affleck, Kennedy, Lieb, and Tasaki (AKLT), is such a resource state. We propose a simulation of this AKLT state using linear optics, wherein we can make use of the high-fidelity projective measurements that are commonplace in quantum-optical experiments, and describe how quantum logic gates can be performed on this chain. In our proposed implementation, the spin-1 particles comprising the AKLT state are encoded on polarization biphotons: three-level systems consisting of pairs of polarized photons in the same spatio-temporal mode. A logical qubit encoded on the photonic AKLT state can be initialized, read out, and have an arbitrary single-qubit unitary applied to it by performing projective measurements on the constituent biphotons. For MBQC, biphoton measurements are required which cannot be deterministically performed using only linear optics and photodetection.