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arXiv:1603.05826

Multi-step quantum algorithm for solving the 3-bit exact cover problem

Hefeng Wang

Published Mar 18, 2016• 0 citations
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Abstract

We present a multi-step quantum algorithm for solving the $3$-bit exact cover problem, which is one of the NP-complete problems. Unlike the brute force methods have been tried before, in this algorithm, we showed that by applying the clauses of the Boolean formula sequentially and introducing non-unitary operations, the state that satisfies all of the clauses can be projected out from an equal superposition of all computational basis states step by step, and the search space is reduced exponentially. The runtime of the algorithm is proportional to the number of clauses, therefore scales polynomial to the size of the problem. Our results indicate that quantum computers may be able to outperform classical computers in solving NP-complete problems.

DOI
10.48550/arxiv.1603.05826

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An adiabatic quantum algorithm may lose quantumness such as quantum coherence entirely in its long runtime, and consequently the expected quantum speedup of the algorithm does not show up. Here we present a general ultrafast adiabatic quantum algorithm. We show that by applying a sequence of fast random or regular signals during evolution, the runtime can be reduced substantially, whereas advantages of the adiabatic algorithm remain intact. We also propose a randomized Trotter formula and show that the driving Hamiltonian and the proposed sequence of fast signals can be implemented simultaneously. We illustrate the algorithm by solving the NP-complete 3-bit exact cover problem (EC3), where NP stands for nondeterministic polynomial time, and put forward an approach to implementing the problem with trapped ions.
We develop three new quantum algorithms for searching the desired target state in the unstructured database of size N. The first algorithm requires Log N iterative steps. It constructs two quantum bags of equal size in terms of two quantum states, out of which exactly one quantum state will have nonzero overlap with the target state. This determination of overlap is done by taking the inner product, in Log N time [2], of the implicitly known target state with any one of these two quantum states. The second algorithm requires just one single step which uses a new suitable operator and the choice of this operator is problem dependent, i.e. it depends upon the number of qubits required to be used to represent an element in the index set. The third algorithm again requires only a single step and this algorithm makes use of a fixed (same) operator. It is known that algorithm for unstructured database search can be easily adaptable for solving NP-Complete problems. However, the computational complexity of NP-Complete problems after the adaptations of both the classical as well as quantum [1] search algorithms remains of the exponential order as the exponent for quantum [1] algorithm changes only to one-half times the exponent for classical algorithm. But for our quantum algorithms the exponent falls substantially so that our new quantum algorithms for unstructured search are capable if reducing the computational complexity of NP-Complete problems to polynomial order!
The exact cover problem is an NP-complete problem with broad applications. Studies show that although applying the Quantum Approximate Optimization Algorithm (QAOA) to this problem can yield improved solution quality with deeper circuit depth, it can limit the algorithm's applicability on noisy intermediate-scale quantum devices. To improve solution quality at shallow depth, we propose a Quantum-Assisted Recursive Algorithm (QARA) for solving the exact cover problem. QARA addresses the problem by alternately applying classical and quantum pruning. Classical pruning is a repeatable pre-processing step to simplify the problem. When the classical pruning cannot promote the problem simplification, quantum pruning is invoked. During quantum pruning, QARA extracts information from the QAOA's output state to identify the subset with the strongest selection bias. This subset is then used to prune the problem based on our problem-tailored reduction rules. Furthermore, QARA incorporates a local verification and rollback mechanism to assistively judge the effectiveness of the quantum simplification. After quantum pruning, classical pruning is applied again to the reduced problem if the remaining subsets and element set are not null. This alternating process repeats until the original problem is fully resolved. In our numerical simulations, we evaluate the performance of QARA at one-layer depth on 140 instances with subset sizes ranging from 8 to 20. Numerical results show that the probability of QARA in finding an exact solution is approximately 60\% higher than that of both QAOA and Recursive QAOA, highlighting its efficiency.
A previously developed quantum search algorithm for solving 1-SAT problems in a single step is generalized to apply to a range of highly constrained k-SAT problems. We identify a bound on the number of clauses in satisfiability problems for which the generalized algorithm can find a solution in a constant number of steps as the number of variables increases. This performance contrasts with the linear growth in the number of steps required by the best classical algorithms, and the exponential number required by classical and quantum methods that ignore the problem structure. In some cases, the algorithm can also guarantee that insoluble problems in fact have no solutions, unlike previously proposed quantum search algorithms. 1. Introduction Quantum computers (Benioff, 1982; Bernstein & Vazirani, 1993; Deutsch, 1985, 1989; DiVincenzo, 1995; Feynman, 1986; Lloyd, 1993) offer a new approach to combinatorial search problems (Garey & Johnson, 1979) with quantum parallelism, i.e., the abilit...