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In this work, we introduce a new family of [[6k, 2k, 2]] codes designed specifically to be compatible with adiabatic quantum computation. These codes support computationally universal sets of weighttwo logical operators and are particularly well-suited for implementing dynamical decoupling error suppression. For Hamiltonians embeddable on a planar graph of fixed degree, our encoding maintains a planar connectivity graph and increases the graph degree by only two. These codes are the first known to possess these features.

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The time or cost of simulating a quantum circuit by adiabatic evolution is determined by the spectral gap of the Hamiltonians involved in the simulation. In "standard" constructions based on Feynman's Hamiltonian, such a gap decreases polynomially with the number of gates in the circuit, L. Because a larger gap implies a smaller cost, we study the limits of spectral gap amplification in this context. We show that, under some assumptions on the ground states and the cost of evolving with the Hamiltonians (which apply to the standard constructions), an upper bound on the gap of order 1/L follows. In addition, if the Hamiltonians satisfy a frustration-free property, the upper bound is of order 1/L 2 . Our proofs use recent results on adiabatic state transformations, spectral gap amplification, and the simulation of continuous-time quantum query algorithms. They also consider a reduction from the unstructured search problem, whose lower bound in the oracle cost translates into the upper bounds in the gaps. The impact of our results is that improving the gap beyond that of standard constructions (i.e., 1/L 2 ), if possible, is challenging.

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