Abstract
We prove that estimating the ground state energy of a translationally invariant, nearest-neighbour Hamiltonian on a 1D spin chain is textsf {QMA}_{{textsf {EXP}}}-complete, even for systems of low local dimension (approx 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.
Highlights
Turing’s Wheelbarrow is our constructive proof of a quantum Thue system (QTS) with properties as mentioned in Lemma 59
A more interesting question is whether there exists a universal QTS which can run any computation of a certain class of promise problems C, i.e. is complete for C
This work was motivated by the idea of finding a simple, translationally invariant and physically interesting system, for which the ground state energy problem is QMAEXP-hard
Summary
Turing’s Wheelbarrow is our constructive proof of a QTS with properties as mentioned in Lemma 59. This witness section, problem instance and the leftover ancillas are fed into Cl, and the output wire contains |out = cos((pa +pout)/3) |0 +sin((pa +pout)/3) |1 for the amplitudes pa—all ancillas being 0—and pout—the circuit output of Cl on the ancillas and problem instance Hastings proved that if the Hamiltonian describing the system is gapped, the ground state entanglement has to follow an area law [1]. It was believed that even for non-gapped Hamiltonians, area-law violations would contribute at most log corrections in the system size. Such long-range correlations in a spin chain’s ground state which scale with the system’s size are a common indicator of criticality, i.e. they show that the system is close to a quantum phase transition. The entanglement entropy is expected to scale logarithmically with the number of spins, since critical spin chains can often be related to a conformal field theory
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