The Complexity of Translationally Invariant Spin Chains with Low Local Dimension

Abstract

Abstract. We prove that estimating the ground state energy of a translationally invariant, nearest-neighbour Hamiltonian on a 1D spin chain is QMA EXP -complete, even for systems of low local dimension (≈ 40). This is an improvement over the best previously known result by several orders of magnitude, and it shows that spin-glass-like frustration can occur in translationally invariant quantum systems with a local dimension comparable to the smallest-known non-translationally invariant systems with similar behaviour. While previous constructions of such systems rely on standard models of quantum computation, we construct a new model that is particularly well-suited for encoding quantum computation into the ground state of a translationally invariant system. This allows us to shift the proof burden from optimizing the Hamiltonian encoding a standard computational model, to proving universality of a simple model. Previous techniques for encoding quantum computation into the ground state of a local Hamiltonian allow only a linear sequence of gates, hence only a linear (or nearly linear) path in the graph of all computational states. We extend these techniques by allowing significantly more general paths, including branching and cycles, thus enabling a highly efficient encoding of our computational model. However, this requires more sophisticated techniques for analysing the spectrum of the resulting Hamiltonian. To address this, we introduce a framework of graphs with unitary edge labels. After relating our Hamiltonian to the Laplacian of such a unitary labelled graph, we analyse its spectrum by combining matrix analysis and spectral graph theory techniques.

Translationally Invariant Universal Classical Hamiltonians

“…An important open question is whether it is possible to reduce the local state dimension in these translationally invariant constructions, while preserving universality. One possible approach would be to apply the techniques from [BCO17], which were used to reduce the local dimension of qudits used in translationally invariant QMA-complete local Hamiltonian constructions.…”

The complexity of translationally invariant low-dimensional spin lattices in 3D

“…The construction we propose in this paper with at most 4-local interactions between spins of dimension 4 yields 4 8 degrees of freedom, a roughly two orders-of-magnitude improvement over a straightforward embedding of the best one-dimensional construction, and en par with the best non-translationally-invariant result. It also shows that there is only about three orders of magnitude left between this construction and spin systems that we encounter every day (e.g.…”

“…Enforcing translational invariance, we can regard e.g. [8]-nearest-neighbour interactions between spins of dimension ≈ 50-which would give roughly (50 2 ) 2 ≈ 6 × 10 6 parameters to choose from.…”

“…QMA EXP is similar to QMA, the quantum analogue of NP, but with an exponential-time verifier instead of polynomialtime-a necessary technicality for any translationally invariant result [8,13], since an n-qudit instance can only encode poly(log n) bits of information (in this case the side lengths of the lattice, which encode the input in unary). In essence, while a QMA EXP -hard problem can be verified in exponential time on a quantum computer, just as in the P vs. NP case it is not expected to be solved as efficiently (see appendix A for details).…”

“…Kitaev's seminal paper proving quantum-NP-hardness of the local Hamiltonian problem for the case that each interaction couples at most five spins [1] motivated significant progress towards understanding the computational complexity that arises in different variants of the local Hamiltonian problem [2][3][4][5][6][7][8][9][10]. These results are especially interesting from a computational perspective, answering which families of Hamiltonians are "complicated enough" to perform universal quantum computation [11,12].…”