Description
In this work, we introduce a notion of duality between Hamiltonians, given as conjugation by a polynomial-depth quantum circuit. In this context, we firstly prove analytically that the 2D toric code is dual to two decoupled classical Ising chains, for any system size. Our argument builds upon the algorithm presented by [1] and studied in [2] to explicitly construct the conjugation circuit. Using the same tools we show---up to large system sizes---that many Hamiltonians composed of commuting Pauli operators are dual to classical models, most of which 1D Ising-like. The observations strongly suggest that the dualities hold for arbitrary system sizes. Should this be true, one could perform efficient quantum Gibbs sampling of all these models by resorting to efficiently sampling their classical counterparts. Additionally, we extend the above notion of duality to Lindbladians in order to show that mixing times and other quantities such as the spectral gap or the modified logarithmic Sobolev inequality are preserved under duality. As a result, by combining both notions of duality, given a Hamiltonian and an associated efficiently implementable and primitive Lindbladian for which there is rapid or fast mixing to the corresponding Gibbs state, we can obtain efficient Gibbs sampling of its dual Hamiltonians by sampling with their corresponding dual Lindbladians.
[1] Aaronson, Scott, and Daniel Gottesman. "Improved Simulation of Stabilizer Circuits." Phys. Rev. A 70 (2004), no. 5: 052328.
[2] Van Den Berg, Ewout, and Kristan Temme. "Circuit Optimization of Hamiltonian Simulation by Simultaneous Diagonalization of Pauli Clusters." Quantum 4 (2020): 322.
