Convergence to the ground-state energy in the thermodynamic limit of the Ising model in a strong transverse field

1993 ◽  
Vol 73 (1-2) ◽  
pp. 345-360
Author(s):  
Martin Pokorny
Keyword(s):  
Ising Model ◽  
Thermodynamic Limit ◽  
Ground State Energy ◽  
Ground State ◽  
State Energy ◽  
Transverse Field ◽  
Strong Transverse

scholarly journals Monte Carlo simulation of stoquastic Hamiltonians

2015 ◽  
Vol 15 (13&14) ◽  
pp. 1122-1140
Author(s):  
Sergey Bravyi
Keyword(s):  
Monte Carlo ◽  
Ising Model ◽  
Ground State ◽  
State Energy ◽  
Transverse Field ◽  
Variable Number ◽  
Monte Carlo Algorithm ◽  
Standard Product ◽  
Ferromagnetic Ising Model ◽  
Transverse Field Ising Model

Stoquastic Hamiltonians are characterized by the property that their off-diagonal matrix elements in the standard product basis are real and non-positive. Many interesting quantum models fall into this class including the Transverse field Ising Model (TIM), the Heisenberg model on bipartite graphs, and the bosonic Hubbard model. Here we consider the problem of estimating the ground state energy of a local stoquastic Hamiltonian $H$ with a promise that the ground state of $H$ has a non-negligible correlation with some `guiding' state that admits a concise classical description. A formalized version of this problem called Guided Stoquastic Hamiltonian is shown to be complete for the complexity class $\MA$ (a probabilistic analogue of $\NP$). To prove this result we employ the Projection Monte Carlo algorithm with a variable number of walkers. Secondly, we show that the ground state and thermal equilibrium properties of the ferromagnetic TIM can be simulated in polynomial time on a classical probabilistic computer. This result is based on the approximation algorithm for the classical ferromagnetic Ising model due to Jerrum and Sinclair (1993).

scholarly journals Thermodynamic limit of the spin-$$ \frac{1}{2} $$ XYZ spin chain with the antiperiodic boundary condition

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Zhirong Xin ◽  
Yusong Cao ◽  
Xiaotian Xu ◽  
Tao Yang ◽  
Junpeng Cao ◽  
...  
Keyword(s):  
Boundary Condition ◽  
Bethe Ansatz ◽  
Thermodynamic Limit ◽  
Ground State Energy ◽  
Ground State ◽  
Spin Chain ◽  
State Energy ◽  
Elementary Excitations ◽  
Antiperiodic Boundary Condition ◽  
Bethe Ansatz Solution

Abstract Based on its off-diagonal Bethe ansatz solution, we study the thermodynamic limit of the spin-$$ \frac{1}{2} $$ 1 2 XYZ spin chain with the antiperiodic boundary condition. The key point of our method is that there exist some degenerate points of the crossing parameter ηm,l, at which the associated inhomogeneous T − Q relation becomes a homogeneous one. This makes extrapolating the formulae deriving from the homogeneous one to an arbitrary η with O(N−2) corrections for a large N possible. The ground state energy and elementary excitations of the system are obtained. By taking the trigonometric limit, we also give the results of antiperiodic XXZ spin chain within the gapless region in the thermodynamic limit, which does not have any degenerate points.

scholarly journals Quantum approximate optimization of the long-range Ising model with a trapped-ion quantum simulator

2020 ◽  
Vol 117 (41) ◽  
pp. 25396-25401 ◽  
Author(s):  
Guido Pagano ◽  
Aniruddha Bapat ◽  
Patrick Becker ◽  
Katherine S. Collins ◽  
Arinjoy De ◽  
...  
Keyword(s):  
Ising Model ◽  
Long Range ◽  
Ground State Energy ◽  
Ground State ◽  
State Energy ◽  
System Size ◽  
Single Shot ◽  
Trapped Ion ◽  
Approximate Optimization ◽  
Quantum Simulator

Quantum computers and simulators may offer significant advantages over their classical counterparts, providing insights into quantum many-body systems and possibly improving performance for solving exponentially hard problems, such as optimization and satisfiability. Here, we report the implementation of a low-depth Quantum Approximate Optimization Algorithm (QAOA) using an analog quantum simulator. We estimate the ground-state energy of the Transverse Field Ising Model with long-range interactions with tunable range, and we optimize the corresponding combinatorial classical problem by sampling the QAOA output with high-fidelity, single-shot, individual qubit measurements. We execute the algorithm with both an exhaustive search and closed-loop optimization of the variational parameters, approximating the ground-state energy with up to 40 trapped-ion qubits. We benchmark the experiment with bootstrapping heuristic methods scaling polynomially with the system size. We observe, in agreement with numerics, that the QAOA performance does not degrade significantly as we scale up the system size and that the runtime is approximately independent from the number of qubits. We finally give a comprehensive analysis of the errors occurring in our system, a crucial step in the path forward toward the application of the QAOA to more general problem instances.

Sign in / Sign up

Export Citation Format

Share Document