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Robustness of an AKLT State on a Honeycomb Lattice.

Philip Allen Mar

Published Jan 1, 2011• 0 citations

Abstract

Cluster states used in measurement based quantum computation do not exist as nondegenerate ground states of two-body Hamiltonians, making them difficult to realize physically [8]. Using Affleck-Kennedy-Lieb-Tasaki (AKLT) states on a honeycomb lattice, a universal resource can be obtained for performing MBQC, by transforming these states into cluster states via two transformations. The first transformation turns the AKLT states into a graph state by using a three-element positive operator value measures and analyzing the effects of these operators in yielded encoded qubits on groups of vertices (domains). The graph state is then transformed to a cluster state as described in [8]. The transformation to a cluster state requires that the graph state contain a connected path from the left side of the original lattice to the right side. Since POVMs are measurements, they may sometimes, in a physical implementation of MBQC, contain errors. Instead of a three- element POVM, we might have a nine-element POVM, with various effects on the groups of vertices they are applied to. This can cause entire groups of vertices to be rendered useless for quantum computation, in particular, they may prevent the existence of a connected path, so that the graph state cannot be transformed to a cluster state. We must make sure the AKLT state is robust, that is, even with certain errors in the POVM, it still can be transformed to a cluster state. We ran simulations to con rm that for certain errors in the POVM (dependent on a parameter pdelete), there always exists a connected path. The main result in this study was the formulation of an algorithm that helped drastically reduce the time to run these simulations. Since the AKLT states are large entangled systems, it is time consuming to produce `typical lattices, but the introduction of this algorithm should help in running further simulations for other POVM outcomes in the future.

DOI: 10.14288/1.0085966

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Cluster states used in measurement based quantum computation do not exist as nondegenerate ground states of two-body Hamiltonians, making them difficult to realize physically [8]. Using Affleck-Kennedy-Lieb-Tasaki (AKLT) states on a honeycomb lattice, a universal resource can be obtained for performing MBQC, by transforming these states into cluster states via two transformations. The first transformation turns the AKLT states into a graph state by using a three-element positive operator value measures and analyzing the effects of these operators in yielded encoded qubits on groups of vertices (domains). The graph state is then transformed to a cluster state as described in [8]. The transformation to a cluster state requires that the graph state contain a connected path from the left side of the original lattice to the right side. Since POVMs are measurements, they may sometimes, in a physical implementation of MBQC, contain errors. Instead of a three- element POVM, we might have a nine-element POVM, with various effects on the groups of vertices they are applied to. This can cause entire groups of vertices to be rendered useless for quantum computation, in particular, they may prevent the existence of a connected path, so that the graph state cannot be transformed to a cluster state. We must make sure the AKLT state is robust, that is, even with certain errors in the POVM, it still can be transformed to a cluster state. We ran simulations to con rm that for certain errors in the POVM (dependent on a parameter pdelete), there always exists a connected path. The main result in this study was the formulation of an algorithm that helped drastically reduce the time to run these simulations. Since the AKLT states are large entangled systems, it is time consuming to produce `typical lattices, but the introduction of this algorithm should help in running further simulations for other POVM outcomes in the future.

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,{8}^{2})

and the ``cross''

(4,6,12)

lattices are also universal resource, whereas the AKLT state on the ``star''

(3,{12}^{2})

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.

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,{8}^{2})

and the ``cross''

(4,6,12)

lattices are also universal resource, whereas the AKLT state on the ``star''

(3,{12}^{2})

lattice is likely not due to geometric frustration.

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.