A QMA-Complete Translationally Invariant Hamiltonian Problem and the Complexity of Finding Ground State Energies in Physical Systems

Oracle complexity classes and local measurements on physical Hamiltonians

The canonical problem for the class Quantum Merlin-Arthur (QMA) is that of estimating ground state energies of local Hamiltonians. Perhaps surprisingly, [Ambainis, CCC 2014] showed that the related, but arguably more natural, problem of simulating local measurements on ground states of local Hamiltonians (APX-SIM) is likely harder than QMA. Indeed, [Ambainis, CCC 2014] showed that APX-SIM is P^QMA[log]-complete, for P^QMA[log] the class of languages decidable by a P machine making a logarithmic number of adaptive queries to a QMA oracle. In this work, we show that APX-SIM is P^QMA[log]-complet…

深度平移对称神经量子态用于阻挫磁体的设计原理

Design principles of deep translationally-symmetric neural quantum states for frustrated magnets

本文通过将ConvNext架构应用于阻挫磁体基态研究，揭示了其与分解视觉Transformer的相似性，并通过数值实验优化设计，在Shastry-Sutherland和J1-J2模型上取得了与最先进方法相当的变分能量，为平移对称神经网络设计提供了指导。

The Complexity of the Separable Hamiltonian Problem

In this paper, we study variants of the canonical Local-Hamiltonian problem where, in addition, the witness is promised to be separable. We define two variants of the Local-Hamiltonian problem. The input for the Separable-Local-Hamiltonian problem is the same as the Local-Hamiltonian problem, i.e. a local Hamiltonian and two energies a and b, but the question is somewhat different: the answer is YES if there is a separable quantum state with energy at most a, and the answer is NO if all separable quantum states have energy at least b. The Separable-Sparse-Hamiltonian problem is defined similar…

The Complexity of the Consistency and N-representability Problems for Quantum States

QMA (Quantum Merlin-Arthur) is the quantum analogue of the class NP. There are a few QMA-complete problems, most notably the ``Local Hamiltonian'' problem introduced by Kitaev. In this dissertation we show some new QMA-complete problems. The first one is ``Consistency of Local Density Matrices'': given several density matrices describing different (constant-size) subsets of an n-qubit system, decide whether these are consistent with a single global state. This problem was first suggested by Aharonov. We show that it is QMA-complete, via an oracle reduction from Local Hamiltonian. This uses alg…

Complexity classification of local Hamiltonian problems

The calculation of ground-state energies of physical systems can be formalised as the k-local Hamiltonian problem, which is the natural quantum analogue of classical constraint satisfaction problems. One way of making the problem more physically meaningful is to restrict the Hamiltonian in question by picking its terms from a fixed set S. Examples of such special cases are the Heisenberg and Ising models from condensed-matter physics. In this work we characterise the complexity of this problem for all 2-local qubit Hamiltonians. Depending on the subset S, the problem falls into one of the foll…

关注长程电子相关性：一种不依赖尺寸的深度学习方法，用于从单电子和双电子积分预测分子的电子能

Paying attention to long-range electron correlation: a size-independent deep-learning approach to predicting molecules' electronic energies from one- and two-electron integrals

本文提出了一种不依赖尺寸的深度学习方法，通过单电子和双电子积分预测分子电子能，利用注意力机制保持尺寸一致性，提高预测准确性。

Detailed Analysis of Circuit-to-Hamiltonian Mappings

The circuit-to-Hamiltonian construction has found widespread use within the field of Hamiltonian complexity, particularly for proving QMA-hardness results. In this work we examine the ground state energies of the Hamiltonian for standard clock constructions and those which require dynamic initialisation. We put exponentially tight bounds on these ground state energies and also determine improved scaling bounds in the case where there is a constant probability of the computation being rejected. Furthermore, we prove a collection of results concerning the low-energy subspace of quantum walks on …

Importance of the spectral gap in estimating ground-state energies

The field of quantum Hamiltonian complexity lies at the intersection of quantum many-body physics and computational complexity theory, with deep implications to both fields. The main object of study is the LocalHamiltonian problem, which is concerned with estimating the ground-state energy of a local Hamiltonian and is complete for the class QMA, a quantum generalization of the class NP. A major challenge in the field is to understand the complexity of the LocalHamiltonian problem in more physically natural parameter regimes. One crucial parameter in understanding the ground space of any Hamil…

Computational Complexity of the Ground State Energy Density Problem

We study the complexity of finding the ground state energy density of a local Hamiltonian on a lattice in the thermodynamic limit of infinite lattice size. We formulate this rigorously as a function problem, in which we request an estimate of the ground state energy density to some specified precision; and as an equivalent promise problem, $\mathsf{GSED}$, in which we ask whether the ground state energy density is above or below specified thresholds. The ground state energy density problem is unusual, in that it concerns a single, fixed Hamiltonian in the thermodynamic limit, whose ground stat…

基态问题的近似难解性

Hardness of approximation for ground state problems

该研究针对量子CSP近似难解性问题，证明基态连通性问题近似为QCMA完全、基态纠缠估计近似为QCMA难解，还得到k-SAT重构问题近似的NP完全性。