On The Complexity of Redundancy-Free Quantum Hamiltonians

Abstract

We consider a class of redudancy-free Hamiltonians, which are those where the trace vanishes for any product of Hamiltonian terms in which at least one term appears an odd number of times. This includes Hamiltonians which are sums of products of Paulis with no relations between them. These Hamiltonians arise naturally in Hamiltonian Decoded Quantum Interferometry, which separates the hardness of preparing quantum states into two steps: a classical decoding step, and a step preparing the thermofield double state of such a redundancy-free Hamiltonian. We focus on the hardness of these Hamiltonians, both for intrinsic interest and to understand the complexity of the state preparation step in Hamiltonian DQI. We present several results showing that the problem becomes easy at quadratically lower temperature than for a general Hamiltonian, with the complexity at given inverse temperature $β$ depending on $β^2 d$, where $d$ is the degree of the anticommutation graph. We give an efficient classical algorithm for small $β^2 d$ to approximate the partition function, while for large $β^2 d$ we show that approximating the partition function is NP-hard. To this end, we introduce what we call an anticommutation glass, where frustration arises purely from anticommutation relations. We give a subexponential time quantum algorithm to prepare the thermofield state for small $β^2 d$, and present some results toward a polynomial time algorithm based on the Feiguin-Klich Hamiltonian. As a result of possible independent interest, we show that this Hamiltonian leads to a polynomial time algorithm for preparing thermofield double states of general Hamiltonians on an interaction graph of degree $d$ for $β\lesssim 1/d$. Finally, we show that estimating the ground state energy of redundancy-free Hamiltonians is QMA-complete.