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arXiv:2111.05461

Reflection-Based Adiabatic State Preparation

Jessica Lemieux, Artur Scherer, Pooya Ronagh

Published Nov 10, 2021• 0 citations
arXivHTMLPDF

Abstract

We propose a circuit-model quantum algorithm for eigenpath traversal that is based on a combination of concepts from Grover's search and adiabatic quantum computation. Our algorithm deploys a sequence of reflections determined from eigenspaces of instantaneous Hamiltonians defined along an adiabatic schedule in order to prepare a ground state of a target problem Hamiltonian. We provide numerical evidence suggesting that, for combinatorial search problems, our algorithm can find a solution faster, on average, than Grover's search. We demonstrate our findings by applying both algorithms to solving the NP-hard MAX-2SAT problem.

DOI
10.48550/arxiv.2111.05461

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We propose a circuit-model quantum algorithm for eigenpath traversal that is based on a combination of concepts from Grover's search and adiabatic quantum computation. Our algorithm deploys a sequence of reflections determined from eigenspaces of instantaneous Hamiltonians defined along an adiabatic schedule in order to prepare a ground state of a target problem Hamiltonian. We provide numerical evidence suggesting that, for combinatorial search problems, our algorithm can find a solution faster, on average, than Grover's search. We demonstrate our findings by applying both algorithms to solving the NP-hard MAX-2SAT problem.
Grover's search algorithm was originally proposed for circuit-based quantum computers. A crucial part of it is to query an oracle -- a black-box unitary operation. Generation of this oracle is formally beyond the original algorithm design. Here, we propose a realization of Grover's oracle for a large class of searching problems using a quantum annealing step. The time of this step grows only logarithmically with the size of the database. This suggests an efficient path to application of Grover's algorithm to practically important problems, such as finding the ground state of a Hamiltonian with a spectral gap over its ground state.
We propose a quantum algorithm for solving combinatorial search problems that uses only a sequence of measurements. The algorithm is similar to quantum computation by adiabatic evolution, in that the goal is to remain in the ground state of a time-varying Hamiltonian. Indeed, we show that the running times of the two algorithms are closely related. We also show how to achieve the quadratic speedup for Grover's unstructured search problem with only two measurements. Finally, we discuss some similarities and differences between the adiabatic and measurement algorithms.
Designing proper time-dependent control fields for slowly varying the system to the ground state that encodes the problem solution is crucial for adiabatic quantum computation. However, inevitable perturbations in real applications demand us to accelerate the evolution so that the adiabatic errors can be prevented from accumulation. Here, by treating this trade-off task as a multiobjective optimization problem, we propose a gradient-free learning algorithm with pulse smoothing technique to search optimal adiabatic quantum pathways and apply it to the Landau-Zener Hamiltonian and Grover search Hamiltonian. Numerical comparisons with a linear schedule, local adiabatic theorem induced schedule, and gradient-based algorithm searched schedule reveal that the proposed method can achieve significant performance improvements in terms of the adiabatic time and the instantaneous ground-state population maintenance. The proposed method can be used to solve more complex and real adiabatic quantum computation problems.
The model of adiabatic quantum computation is a new paradigm for designing quantum algorithms. This model is based on the quantum adiabatic theorem, where a quantum computer evolves the quantum system slowly to switch gradually from an initial Hamiltonian with a ground state which is easy to construct, to a final Hamiltonian whose ground state encodes the solution of the problem to be solved. This dissertation presents an experimental study on the discrete quantum adiabatic computation in combinatorial search problems, we take k-Satisfiability problem as target problem to be examined. For k ? 3, k-SAT is NP-complete, i.e., among the most difficult NP problems in the worst case. Here we proposes a new monotonic variation method for the phase shift and mixing functions in the adiabatic quantum algorithm called the Quadric variation method, in order to speed up the algorithm search and decrease the overall resulting search cost. In addition, we present a better parameter configuration for the algorithm to be used with quadric variation as well as the previously proposed methods. The experiment are carried out in solving random instances of 3-SAT problems. The results indicate that the proposed method of using monotonic quadric variation in the phase functions of the adiabatic algorithm greatly improve the search behavior of the adiabatic algorithm and reduces the resulting search cost, also the results indicate that the better configuration of the algorithm parameters could greatly enhance the search behavior for the proposed method as well as the previously proposed methods.
The adiabatic theorem has been recently used to design quantum algorithms of a new kind, where the quantum computer evolves slowly enough so that it remains near its instantaneous ground state, which tends to the solution. We apply this time-dependent Hamiltonian approach to Grover's problem, i.e., searching a marked item in an unstructured database. We find that by adjusting the evolution rate of the Hamiltonian so as to keep the evolution adiabatic on each infinitesimal time interval, the total running time is of order N,\sqrt{N},N​, where N is the number of items in the database. We thus recover the advantage of Grover's standard algorithm as compared to a classical search, scaling as N. This is in contrast with the constant-rate adiabatic approach of Farhi et al. (e-print quant-ph/0001106), where the requirement of adiabaticity is expressed only globally, resulting in a time of order N.
A quantum system will stay near its instantaneous ground state if the Hamiltonian that governs its evolution varies slowly enough. This quantum adiabatic behavior is the basis of a new class of algorithms for quantum computing. We tested one such algorithm by applying it to randomly generated hard instances of an NP-complete problem. For the small examples that we could simulate, the quantum adiabatic algorithm worked well, providing evidence that quantum computers (if large ones can be built) may be able to outperform ordinary computers on hard sets of instances of NP-complete problems.