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Leibniz international proceedings in informatics

The Spin-2 AKLT State on the Square Lattice is Universal for Measurement-based Quantum Computation

Tzu-Chieh Wei, Robert Raussendorf

Published Jan 1, 2015• 0 citations
Publisher

Abstract

One-way quantum computation was first invented using the cluster state. Since then graph states, the generalization of the cluster state, were investigated and understood when they would enable such a measurement-based approach for quantum computation. Are there any other family of states, i.e., states with different entanglement structures, that can also serve as the universal resource for quantum computation? Recent study shows that the Affleck-Kennedy-Lieb-Tasaki (AKLT) states also provide a useful source. Here, we show that the spin-2 state on the square lattice is a universal resource for measurement-based quantum computation. We employ a POVM on all sites that convert the local 5-level system to 2-level, and the post-POVM state is a graph state, whose graph is in general non-planar. We then follow with another round of measurement to recover the planarity of the graphs by thinning. The resultant typical graphs are shown to reside in the supercritical phase of percolation via Monte Carlo simulations and the associated graph states are universal, implying the AKLT state is also universal.

DOI
10.4230/lipics.tqc.2015.48

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One-way quantum computation was first invented using the cluster state. Since then graph states, the generalization of the cluster state, were investigated and understood when they would enable such a measurement-based approach for quantum computation. Are there any other family of states, i.e., states with different entanglement structures, that can also serve as the universal resource for quantum computation? Recent study shows that the Affleck-Kennedy-Lieb-Tasaki (AKLT) states also provide a useful source. Here, we show that the spin-2 state on the square lattice is a universal resource for measurement-based quantum computation. We employ a POVM on all sites that convert the local 5-level system to 2-level, and the post-POVM state is a graph state, whose graph is in general non-planar. We then follow with another round of measurement to recover the planarity of the graphs by thinning. The resultant typical graphs are shown to reside in the supercritical phase of percolation via Monte Carlo simulations and the associated graph states are universal, implying the AKLT state is also universal.
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.
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.
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.
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.
The spin-3/2 Affleck-Kennedy-Lieb-Tasaki (AKLT) valence-bond state on a hexagonal lattice was shown to be a universal resource state for measurement-based quantum computation (MBQC). Can AKLT states of higher spin magnitude support universal MBQC? We demonstrate that several hybrid two-dimensional AKLT states involving a mixture of spin-2 and other lower-spin entities, such as spin-3/2 and spin-1, are also universal for MBQC. This significantly expands universal resource states in the AKLT family. Even though frustration may be a hindrance to quantum computational universality, lattices can be modified to yield AKLT states that are universal. The family of AKLT states thus provides a versatile playground for quantum computation.