PDF《Finding low-energy conformations of》

81 qubits used in the experiment. Nodes with the same color represent the same logical qubit from the original 19-qubit Ising-like Hamiltonian resulting from the energy function associated with Experiment

4 (see Supplementary material for details). This embedding aims to fulfill the arbitrary connectivity of the Ising expression and allows for the coupling of qubits that are not directly coupled in hardware. www.nature.com/scientificreports SCIENTIFIC REPORTS |

2 :

571 | DOI: 10.1038/srep00571

2 Results The quantum hardware employed consists of

16 units of a recently characterized eightqubit unit cell22,39 . Post-fabrication characteriza- tion determined that only

115 qubits out of the

128 qubit array can be reliably used for computation (see Fig. 1). The array of coupled superconducting flux qubits is, effectively, an artificial Ising spin system with programmable spin-spin couplings and transverse mag- netic fields. It is designed to solve instances of the following (NP- hard40 ) classical optimization problem: Given a set of local longit- udinal fields {hi} and an interaction matrix {Jij}, find the assignment s? ~s? 1s?

2 ? ? ? s? N, that minimizes the objective function E(s), where, E s ? ?~ X 1?i?N hisiz X 1?ivj?N Jijsisj, ?1? jhij # 1, jJijj # 1, and si g {11, -1}. Finding the optimal s* is equivalent to finding the ground state of the corresponding Ising classical Hamiltonian, Hp~ X N 1?i?N hisz i z X N 1?ivj?N Jijsz i sz j ?2? where sz i are Pauli matrices acting on the ith spin. Experimentally, the time-dependent quantum Hamiltonian implemented in the superconductingqubit array is given by, H t ? ?~A t ? ?HbzB t ? ?Hp, t~t=trun, ?3? with Hb~{ P i sx i responsible for quantum tunneling among the localized classical states, which correspond to the eigenstates of Hp (the computational basis). The time-dependent functions A(t) and B(t) are such that A(0) ? B(0) and A(1) = B(1);

in Fig. 2(b), we plot these functions as implemented in the experiment. trun denotes the time elapsed between the preparation of the initial state and the measurement. QA exploits the adiabatic theorem of quantum mechanics, which states that a quantum system initialized in the ground state of a time- dependent Hamiltonian remains in the instantaneous ground state, as long as it is driven sufficiently slowly. Since the ground state of Hp encodes the solution to the optimization problem, the idea behind QA is to adiabatically prepare this ground state by initializing the quantum system in the easy-to-prepare ground state of Hb, which corresponds to a superposition of all 2N states of the computational basis. The system is driven slowly to the problem Hamiltonian, H(t

5 1) <

Hp. Deviations from the ground-state are expected due to deviations from adiabaticity, as well as thermal noise and imperfections in the implementation of the Hamiltonian. The first challenge of the experimental implementation is to map the computational problem of interest into the binary quadratic expression (Eq. 2), which we outline next. In lattice folding, the sequence of amino acids defining the protein is viewed as a sequence of beads (amino acids) connected by strings (peptide bonds). This bead chain occupies points on a two- or three-dimensional lattice. A valid configuration is a self-avoiding walk on the lattice and its energy Figure

2 | Lattice folding mapping for quantum annealing. (a) Step-by-step construction of the binary representation of lattice protein. Two qubits per bond are needed and the bond directions are denoted as '