QMA-complete problems for stoquastic Hamiltonians and Markov matrices

The Local Consistency Problem for Stoquastic and 1-D Quantum Systems

The Local Hamiltonian problem (finding the ground state energy of a quantum system) is known to be QMA-complete. The Local Consistency problem (deciding whether descriptions of small pieces of a quantum system are consistent) is also known to be QMA-complete. Here we consider special cases of Local Hamiltonian, for ``stoquastic'' and 1-dimensional systems, that seem to be strictly easier than QMA. We show that there exist analogous special cases of Local Consistency, that have equivalent complexity (up to poly-time oracle reductions). Our main technical tool is a new reduction from Local Consi…

Pinned QMA: The power of fixing a few qubits in proofs

What could happen if we pinned a single qubit of a system and fixed it in a particular state? First, we show that this can greatly increase the complexity of static questions -- ground state properties of local Hamiltonian problems with restricted types of terms. In particular, we show that the Pinned commuting and Pinned Stoquastic Local Hamiltonian problems are QMA complete. Second, we show that pinning a single qubit via often repeated measurements also results in universal quantum computation already with commuting and stoquastic Hamiltonians. Finally, we discuss variants of the Ground Sta…

几何局部stoquastic哈密顿量的复杂性

Complexity of geometrically local stoquastic Hamiltonians

本文证明了二维和一维几何局部stoquastic哈密顿量的复杂性为MA难，并展示了相关问题的StoqMA完备性。

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…

拆解随机二分法

Dismantling the Stoquastic Dichotomy

挑战随机二分法控制量子计算等基本计算边界的观点，提出消失几何相位（VGP）能更充分刻画边界。构造特殊哈密顿量，证明相关问题复杂度，确定VGP识别情况，表明计算边界应理解为几何相位结构间的边界。

Rapid mixing of path integral Monte Carlo for 1D stoquastic Hamiltonians

Path integral quantum Monte Carlo (PIMC) is a method for estimating thermal equilibrium properties of stoquastic quantum spin systems by sampling from a classical Gibbs distribution using Markov chain Monte Carlo. The PIMC method has been widely used to study the physics of materials and for simulated quantum annealing, but these successful applications are rarely accompanied by formal proofs that the Markov chains underlying PIMC rapidly converge to the desired equilibrium distribution. In this work we analyze the mixing time of PIMC for 1D stoquastic Hamiltonians, including disordered transv…