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 similarly, but the Hamiltonian is not necessarily local, but rather sparse. We show that the Separable-Sparse-Hamiltonian problem is QMA(2)-Complete, while Separable-Local-Hamiltonian is in QMA. This should be compared to the Local-Hamiltonian problem, and the Sparse-Hamiltonian problem which are both QMA-Complete. To the best of our knowledge, Separable-SPARSE-Hamiltonian is the first non-trivial problem shown to be QMA(2)-Complete.
It is shown that it is possible to construct the quantum wave functions for non-separable but integrable two-dimensional Hamiltonian systems, by solving suitable Dirichlet boundary values problems inside and outside the regions spanned by particular families of classical trajectories, in one-to-one correspondence with the quantum state. The method is applied both to the Schrodinger equation, and to the quantum Hamilton-Jacobi equation. The boundary values are obtained by integrating the one-dim equations on the caustics arcs enveloping the classical trajectories. This approach gives the same r…
Efficient fourth order symplectic integrators for near-harmonic separable Hamiltonian systems
提出了可分Hamilton系统H（p，q）=T（p）+V（q）的数值积分的有效四阶辛积分器.通过对简谐振子的高阶分解，得到了5 ~ 9级的对称分裂系数。对各种哈密顿系统的性能进行了评估：积分误差与S。Blanes et al.（2013），W. Kahan et al.（1999）和H. Yoshida（1990）.数值试验表明，对于一般的Hamilton系统，本文所得到的积分器的性能明显优于以往的积分器。
Supersymmetry is a technique that allows us to extract information about the states and spectra of quantum mechanical systems which may otherwise be unsolvable. In this paper we reconstruct Ioffe's set of states for the singular non-separable two-dimensional Morse potential using supersymmetry from a non-degenerate set of states constructed for the initial separable Morse Hamiltonian. We define generalised coherent states, compute their uncertainty relations, and we find that the singularity in the partner Hamiltonian significantly affects the localisation of the coherent state wavefunction.
The quantization method based on the quantum Hamiltonian Jacobi equation, is extended to two-dimensional non-separable but integrable Hamiltonians. It is shown that each wave function for those systems corresponds to a well-defined family of classical trajectories, enveloped by a caustic. The energy eigenvalues and the values of the wave functions on the caustic are obtained by solving the 1-dim quantum Hamilton Jacobi equation of the caustic' arcs. Results are in good agreement with those obtained by usual methods.
General method for obtaining the energy minimum of spin Hamiltonians for separable states
University of the Basque Country UPV/EHU（巴斯克国立大学）；Donostia International Physics Center DIPC（多诺西亚国际物理中心）；IKERBASQUE, Basque Foundation for Science（伊克尔巴斯基奎，巴斯克科学基金会）；HUN-REN Wigner Research Centre for Physics（匈牙利研究与教育网络维格纳物理研究中心）；Budapest University of Technology and Economics（布达佩斯技术与经济大学）；Institute of Mathematics, Budapest University of Technology and Economics（布达佩斯技术与经济大学数学研究所）；HUN-REN Rényi Institute of Mathematics（匈牙利研究与教育网络雷尼数学研究所）
We consider bipartite entangled states that cannot outperform separable states in any linear interferometer. Then, we show that these states can still be more useful metrologically than separable states if several copies of the state are provided or an ancilla is added to the quantum system. We present a general method to find the local Hamiltonian for which a given quantum state performs the best compared to separable states. We obtain analytically the optimal Hamiltonian for some quantum states with a high symmetry. We show that all bipartite entangled pure states outperform separable states…
We elucidate the distinction between global and termwise stoquasticity for local Hamiltonians and prove several complexity results. We show that the stoquastic local Hamiltonian problem is $\textbf{StoqMA}$-complete even for globally stoquastic Hamiltonians. We study the complexity of deciding whether a local Hamiltonian is globally stoquastic or not. In particular, we prove $\textbf{coNP}$-hardness of deciding global stoquasticity in a fixed basis and $Σ_2^p$-hardness of deciding global stoquasticity under single-qubit transformations. As a last result, we expand the class of sign-curing tran…
We prove that 2-Local Hamiltonian (2-LH) with Low Complexity problem is QCMA-complete by combining the results from the QMA-completeness[4] of 2-LH and QCMA-completeness of 3-LH with Low Complexity[6]. The idea is straightforward. It has been known that 2-LH is QMA-complete. By putting a low complexity constraint on the input state, we make the problem QCMA. Finally, we use similar arguments as in [4] to show that all QCMA problems can be reduced to our proposed problem.
Using a separable many-body variational wavefunction, we formulate a self-consistent effective Hamiltonian theory for fermionic many-body system. The theory is applied to the two-dimensional Hubbard model as an example to demonstrate its capability and computational effectiveness. Most remarkably for the Hubbard model in 2-d, a highly unconventional quadruple-fermion non-Cooper-pair order parameter is discovered.
Epsilon-net method for optimizations over separable states
我们给出了优化问题的算法：$\max_ρ\ip{Q}ρ$，其中$Q$是一个厄米矩阵，变量$ρ$是一个二分{ \em可分}量子态。这个问题是量子计算和信息中几个问题的核心，例如QMA的复杂性。虽然这个问题是NP难的，但我们的算法在几个感兴趣的实例中比蛮力更好。特别是，他们给出了承诺问题的PSPACE上界，该承诺问题允许QMA（2）协议，其中验证者在两个证明上仅执行对数数量的基本门，以及决定二分局部哈密顿量是否具有大或小的基态能量的承诺问题。对于$Q\ge0$，我们的算法在$\|Q\|_F$中按时间指数运行。虽然这种算法的存在性最近由Brand{ \em Proceedings of the 43 rd annual ACM Symposium on Theory of Computation}，343- 352，2011年首次证明，但我们的算法在概念上更简单。
We propose how to compute the complexity of operators generated by Hamiltonians in quantum field theory (QFT) and quantum mechanics (QM). The Hamiltonians in QFT/QM and quantum circuit have a few essential differences, for which we introduce new principles and methods for complexity. We show that the complexity geometry corresponding to one-dimensional quadratic Hamiltonians is equivalent to AdS$_3$ spacetime. Here, the requirement that the complexity is nonnegative corresponds to the fact that the Hamiltonian is lower bounded and the speed of a particle is not superluminal. Our proposal prove…
