Quantum search with hybrid adiabatic–quantum-walk algorithms and realistic noise

Abstract

Computing using a continuous-time evolution, based on the natural interaction Hamiltonian of the quantum computer hardware, is a promising route to building useful quantum computers in the near-term. Adiabatic quantum computing, quantum annealing, computation by continuous-time quantum walk, and special purpose quantum simulators all use this strategy. In this work, we carry out a detailed examination of adiabatic and quantum walk implementation of the quantum search algorithm, using the more physically realistic hypercube connectivity, rather than the complete graph, for our base Hamiltonian. We calculate optimal adiabatic schedules both analytically and numerically for the hypercube, and then interpolate between adiabatic and quantum walk searching, obtaining a family of hybrid algorithms. We show that all of these hybrid algorithms provide the quadratic quantum speed up when run with optimal parameter settings, which we determine and discuss in detail. We incorporate the effects of multiple runs of the same algorithm, noise applied to the qubits, and two types of problem misspecification, determining the optimal hybrid algorithm for each case. Our results reveal a rich structure of how these different computational mechanisms operate and should be balanced in different scenarios. For large systems with low noise and good control, quantum walk is the best choice, while hybrid strategies can mitigate the effects of many shortcomings in hardware and problem misspecification. arXiv:1709.00371v3 [quant-ph]

Robust Diabatic Grover Search by Landau–Zener–Stückelberg Oscillations

“…Note that none of these constraints requires prior knowledge of the gap , other than an upper bound. Hence, the algorithm is robust to a multiplicative error of the Hamiltonian due to calibration errors, similar to previous approaches [ 25 ]. A different choice of parameters may improve the algorithm’s robustness to the variations in the total evolution time, as demonstrated in [ 26 ].…”

“…The limit yields maximal , and corresponds to evolving by the time-independent Hamiltonian , which we denote . This is exactly the “analog” algorithm for the search problem by Farhi and Gutmann [ 24 ], and, more generally, a search by a quantum walk [ 25 , 27 , 28 , 29 , 30 , 31 , 32 , 33 ]. The Hamiltonian is the core of algorithms for the search problem: in the adiabatic algorithm [ 12 ], the Hamiltonian spends most of the time close to , where the gap is minimal [ 25 ], while the original gate-model algorithm by Grover [ 5 ] can be seen as a simulation (or an approximation by Trotter formula [ 34 ]) of [ 24 ].…”

How to Compute Using Quantum Walks

“…As for the quantum walk, the qubits start in the initial state in equation (8), the ground state ofĤ h , and are measured after a time t f ∝ √ N to obtain a quadratic quantum speed up. Both quantum walk and adiabatic quantum search algorithms are analytically solvable [14,43] in the large N limit, because they reduce to a two state single avoided crossing model [14,39]. There are many subtleties involved in setting the parameters A(t), B(t), γ, to achieve a quantum speed up, for detailed discussion and results, see [39].…”

“…For quantum walk, the system oscillates between the equal superposition initial state, and the marked state, while for adiabatic, the system transitions steadily from the initial state to the marked state. A smooth interpolation between adiabatic and quantum walk can be done, and optimal contributions from both mechanisms employed to improve performance on imperfect hardware [39].…”

“…They are effective when used with many short runs and repeats, which suits early, noisy quantum hardware. The benefits of combining both quantum walk and adiabatic techniques can be demonstrated in a range of settings, especially in the presence of hardware imperfections [39]. However, it is important to use quantum algorithms where they will be most effective, to obtain a practical quantum advantage.…”