Determining the Complexity of the Quantum Adiabatic Algorithm using Monte Carlo Simulations A.P. Young, University of California Santa Cruz Objective Objective Approach Status:  Determine whether the Quantum Adiabatic Algorithm (QAA) shows exponential or polynomial complexity for large sizes when applied to NP-hard problems. (Work in collaboration with V. Smelyanskiy and S. Knysh)  Algorithm has been greatly improved ( “parallel tempering”, checkpointing, increased speed).  Larger sizes have been studied, up to N = 256 compared with N = 128 in preliminary work (Phys. Rev. Lett, 101, (2008)).  For large sizes, some instances have a first order (discontinuous) phase transition, see left figure above (1-q ~ Hamming distance). The fraction with a first order transition increases with increasing size (see right figure above for preliminary data). Instances with a first order transition are very difficult for the QAA.  Use Quantum Monte Carlo simulations, a well established technique in statistical physics, to study large sizes (up to several hundred qubits).  Focus on the minimum energy gap during the evolution of the QAA. A very small gap, e.g. at a quantum phase transition, will be the bottleneck of the QAA.  Initially study the exact cover problem, as in the pioneering work of Farhi et al. but later on will study other problems as well. Example of a 1st order transition Fraction with a 1 st order transition

Research plan for the next 12 months  Determine more precisely, and for a bigger range of sizes, the fraction of instances with a first order transition.  For those instances with a second order (continuous) transition, determine the median complexity for larger sizes (up to N of about 400) than at present.  Study the exact cover problem with random interactions in the quantum Hamiltonian to see if the “crossover” to a first order transition occurs for smaller sizes, where it can be studied more easily.  Do preliminary studies on other NP-hard models, to see if they, too, have a first order quantum phase transition at large sizes. Long term objectives (demonstrations)  Determine whether the fraction of instances with a first order transition tends to 1 for very large sizes for a range of hard optimization problems.  Determine whether the complexity of the QAA is polynomial or exponential for a range of hard optimization problems. Determining the Complexity of the Quantum Adiabatic Algorithm Using Monte Carlo Simulations A.P. Young, University of California Santa Cruz

Quantum Adiabatic Algorithm (Farhi et al.) Exact Cover We studied the same “exact cover” problem as in Farhi et al. There are N Ising spins and M “clauses” each of which involves three spins (chosen at random). A clause is “satisfied” if two of the spins are +1 and the other is -1. To map exact cover onto a Hamiltonian, assign the energy of a clause to be zero if it is satisfied, and a positive integer if it is not. The Hamiltonian of a clause is We chose instances with a “unique satisfying assignment” (USA) and adjusted the ratio M/N to maximize the probability of a USA. Thus we are sitting on the satisfied-unsatisfied phase transition where the problem is particularly difficult. Determining the Complexity of the Quantum Adiabatic Algorithm Using Monte Carlo Simulations A.P. Young, University of California Santa Cruz

Quantum Monte Carlo simulations Determining the Complexity of the Quantum Adiabatic Algorithm Using Monte Carlo Simulations A.P. Young, University of California Santa Cruz

Main New Result For the larger sizes there is an increasing probability that the Quantum Phase Transition is discontinuous (first order). Rounded out for finite-N but over a small range. In the figure, 1-q ~ Hamming distance between different basis states in the ground state. As λ increases through 0.62 the quantum fluctuations suddenly decrease. Energy gap is very small at a first order phase transition. Exactly how small? Needs to be investigated in more detail. Determining the Complexity of the Quantum Adiabatic Algorithm Using Monte Carlo Simulations A.P. Young, University of California Santa Cruz

Fraction of instances with a first order transition (see figure). 50 instances for each size. Future work: * Does the probability tend to 1 at large N? Will study even larger sizes. * Is the 1 st order transition special for the exact cover problem? Will study other problems to find out. * Will make the quantum Hamiltonian disordered to see if the “crossover” occurs at smaller sizes. * Will see if the instances with a continuous (second order) transition have a polynomial complexity. Determining the Complexity of the Quantum Adiabatic Algorithm Using Monte Carlo Simulations A.P. Young, University of California Santa Cruz