AI 中文总结
本文研究无冗余哈密顿量的复杂性，发现其配分函数近似在低温和高温下分别易于经典计算和NP困难，并给出制备热场态的量子算法，证明基态能量估计为QMA完全。
AI 中文摘要
我们考虑一类无冗余哈密顿量，其中对于任何哈密顿项乘积，若至少有一项出现奇数次，则该乘积的迹为零。这包括由泡利算符乘积之和构成且各项之间无关系的哈密顿量。这类哈密顿量自然出现在哈密顿量解码量子干涉测量（Hamiltonian Decoded Quantum Interferometry）中，该方法将制备量子态的难度分为两步：经典解码步骤，以及制备此类无冗余哈密顿量的热场双态（thermofield double state）的步骤。我们关注这些哈密顿量的难度，既出于内在兴趣，也为了理解哈密顿量DQI中态制备步骤的复杂性。我们给出了若干结果，表明在温度比一般哈密顿量低二次方时问题变得容易，在给定逆温度$\beta$下的复杂性取决于$\beta^2 d$，其中$d$是反对易图的度数。对于较小的$\beta^2 d$，我们给出一个有效的经典算法来近似配分函数；而对于较大的$\beta^2 d$，我们证明近似配分函数是NP困难的。为此，我们引入了所谓的反对易玻璃（anticommutation glass），其中阻挫纯粹源于反对易关系。我们给出了一个次指数时间的量子算法，用于在较小的$\beta^2 d$下制备热场态，并展示了基于Feiguin-Klich哈密顿量的多项式时间算法的一些结果。作为一个可能具有独立意义的结果，我们证明该哈密顿量导致了一个多项式时间算法，用于在度数$d$的相互作用图上制备一般哈密顿量的热场双态，当$\beta\lesssim 1/d$时。最后，我们证明估计无冗余哈密顿量的基态能量是QMA完全的。
英文摘要
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.
Comments39 pages, 3 figures
