Classical and quantum annealing in the median of three-satisfiability

Abstract

We determine the classical and quantum complexities of a specific ensemble of three-satisfiability problems with a unique satisfying assignment for up to N = 100 and 80 variables, respectively. In the classical limit, we employ generalized ensemble techniques and measure the time that a Markovian Monte Carlo process spends in searching classical ground states. In the quantum limit, we determine the maximum finite correlation length along a quantum adiabatic trajectory determined by the linear sweep of the adiabatic control parameter in the Hamiltonian composed of the problem Hamiltonian and the constant transverse field Hamiltonian. In the median of our ensemble, both complexities diverge exponentially with the number of variables. Hence, standard, conventional adiabatic quantum computation fails to reduce the computational complexity to polynomial. Moreover, the growth-rate constant in the quantum limit is 3.8 times as large as the one in the classical limit, making classical fluctuations more beneficial than quantum fluctuations in ground-state searches.

Assessment of Quantum Annealing for the Construction of Satisfiability Filters

“…Consistent with the findings of Ref. [34,35,36,37], Fig. 6 provides no evidence that SQA requires a lower computational effort to find one solution for all tested problem sizes.…”

Adiabatic Quantum Simulation of Quantum Chemistry

“…AQC has been applied to classical optimization problems that lie in the complexity class NP. For example, studies have been performed on satisfiability 11 12 13 , Exact Cover 7 8 , 3-regular 3-XORSAT and 3-regular Max-Cut 14 , random instances of classical Ising spin glasses 15 , protein folding 16 17 and machine learning 18 19 . AQC has also been applied to structured and unstructured search 20 21 , search engine ranking 22 and artificial intelligence problems arising in space exploration 23 .…”

Quantum annealing with trigger Hamiltonians: Application to 2-satisfiability and nonstoquastic problems

“…Furthermore, previous studies have indicated an exponential scaling of quantum annealing for solving both 2-SAT and NP-Complete 3-SAT problems based on the (extended) mapping given in Eq. ( 9) [13,29]. We have chosen number of clauses, M = N + 1, as the satisfiability threshold for a random set of 2-SAT problems, defined as the threshold at which the cost functions become unsatisfiable in the mean with a high probability, occurs approximately at M/N ≈ 1 [28].…”

“…For quantum annealing, the minimum energy gap that occurs during the annealing process between the ground state and the first-excited state of the Hamiltonian is of great importance, in particular in the adiabatic limit of quantum annealing. However, it has been found that, for the Ising Hamiltonian, which is used as a standard model Hamiltonian for encoding optimization problems, the minimum energy gap may be accompanied by a quantum phase transition, and for difficult problems it can close exponentially fast as the system size grows [12][13][14][15]. Therefore, the time required to ascertain an adiabatic evolution of the state also increases exponentially for these problems.…”