Analytical solution for nonadiabatic quantum annealing to arbitrary Ising spin Hamiltonian

Abstract

We point to the existence of an analytical solution to a general quantum annealing (QA) problem of finding low energy states of an arbitrary Ising spin Hamiltonian HI by implementing time evolution with a Hamiltonian H(t) = HI + g(t)Ht. We will assume that the nonadiabatic annealing protocol is defined by a specific decaying coupling g(t) and a specific mixing Hamiltonian Ht that make the model analytically solvable arbitrarily far from the adiabatic regime. In specific cases of HI, the solution shows the possibility of a considerable quantum speedup of finding the Ising ground state. We then compare predictions of our solution to results of numerical simulations, and argue that the solvable QA protocol produces the optimal performance in the limit of maximal complexity of the computational problem. Our solution demonstrates for the most complex spin glasses a power-law energy relaxation with the annealing time T and uncorrelated from HI annealing schedule. This proves the possibility for spin glasses of a faster than ∼ 1/ log β T energy relaxation.

Application of Quantum Computers in Foreign Exchange Reserves Management

“…where C i• is the i th row of matrix C. For all weights, the derivative can be written as ∂L ∂|w = 2C|w + µ|P + λ|R . Setting all derivatives equal to zero leads to the linear system (46). Note: In the first step, Hadamard gates are applied to t qubits (t depends on the desired precision of the representation of the eigenvalues λ i ).…”

“…Unfortunately, using the solution described above does not recover the whole speed-up. A quick check of system (46) reveals that its main component is a covariance matrix which is hardly sparse. As a result, s = N and the complexity of the HHL algorithm for portfolio optimization increases to O[N log(N )κ/ ].…”

“…For example, the studies [44] concerning the prediction of economic downturns and [45] relating to maritime routing problems, provide empirical evidence that quantum QUBO solvers perform better. Additionally, [46] introduced the non-adiabatic quantum QUBO solver, which should be faster for tasks with no specific internal structure. While the QAOA operates with Hamiltonian H = (1 − t)H 0 + tH Ising , where t is slowly changed from 0 to 1 (the "adiabatic regime"), the non-adiabatic approach uses a Hamiltonian of the form H = H Ising − g t α H init , where g and α are user-defined parameters, H Init is the initial Hamiltonian 41 and t is slowly increased from zero theoretically to infinity.…”

“…However, at the beginning of the millennium, the decoherence times of IBM-like processors were in the tens of nanoseconds. 46 To demonstrate decoherence in practice, we can prepare two simple circuits. The first consists of an X gate applied to a qubit followed by several identity operators I.…”