Graph states as ground states of two-body frustration-free Hamiltonians

Abstract

The framework of measurement-based quantum computation (MBQC) allows us to view the ground states of local Hamiltonians as potential resources for universal quantum computation. A central goal in this field is to find models with ground states that are universal for MBQC and that are also natural in the sense that they involve only two-body interactions and have a small local Hilbert space dimension. Graph states are the original resource states for MBQC, and while it is not possible to obtain graph states as exact ground states of two-body Hamiltonians, here we construct two-body frustration-free Hamiltonians that have arbitrarily good approximations of graph states as unique ground states. The construction involves taking a two-body frustration-free model that has a ground state convertible to a graph state with stochastic local operations, then deforming the model such that its ground state is close to a graph state. Each graph state qubit resides in a subspace of a higher dimensional particle. This deformation can be applied to two-body frustration-free Affleck-Kennedy -Lieb-Tasaki (AKLT) models, yielding Hamiltonians that are exactly solvable with exact tensor network expressions for ground states. For the star-lattice AKLT model, the ground state of which is not expected to be a universal resource for MBQC, applying such a deformation appears to enhance the computational power of the ground state, promoting it to a universal resource for MBQC. Transitions in computational power, similar to percolation phase transitions, can be observed when Hamiltonians are deformed in this way. Improving the fidelity of the ground state comes at the cost of a shrinking gap. While analytically proving gap properties for these types of models is difficult in general, we provide a detailed analysis of the deformation of a spin-1 AKLT state to a linear graph state.

Algebraic approach to the study of zero modes of Haldane pseudopotentials

“…There has been much interest in general properties of such frustration free lattice models recently, in both 1D and in higher dimensions, [42][43][44][45][46][47][48][49][50][51][52] especially, in connection with matrix-product like ground states such models may have. In particular, for the cylinder and torus geometries, the operators Q m R are related by lattice translations.…”

Hybrid valence-bond states for universal quantum computation

“…Graph states do not arise as unique ground states of two-body interacting Hamiltonians [17], which might be a disadvantage from the viewpoint of creating universal resource states by cooling. A finite gap in the Hamiltonian separating a unique resource state as a ground state from excited states is thus a desirable feature [24]. With suitably chosen boundary conditions, AKLT states are unique ground states of certain two-body interacting Hamiltonians [23], some of which are believed to possess a finite spectral gap [25,26].…”

Universal measurement-based quantum computation in two-dimensional symmetry-protected topological phases

“…From the viewpoint of SPT order and higher dimensionality, the only exception so far is the family of AKLT states in two and higher dimensions [47][48][49][50][51][52] and many of them have been shown to provide universal resource for quantum computation even beyond the AKLT points [53][54][55]. But their SPT order requires translation invariant symmetry to be respected.…”