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Quantum Physics Letters

A Simple Quantum Algorithm for Exponentially Fast Target Searching in the Unstructured Database

Dhananjay P. Mehendale

Published Aug 1, 2017• 3 citations
Publisher
DOI
10.18576/qpl/060203

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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!

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Grover’s search algorithm (GSA) offers quadratic speedup in searching unstructured databases but suffers from exponential circuit depth complexity. Here, we present two quantum circuits called HX and Ry layers for the searching problem. Remarkably, both circuits maintain a fixed circuit depth of two and one, respectively, irrespective of the number of qubits used. When the target element’s position index is known, we prove that either circuit, combined with a single multi-controlled X gate, effectively amplifies the target element’s probability to over 0.99 for any qubit number greater than seven. To search unknown databases, we use the depth-1 Ry layer as the ansatz in the Variational Quantum Search (VQS), whose efficacy is validated through numerical experiments on databases with up to 26 qubits. The VQS with the Ry layer exhibits an exponential advantage, in circuit depth, over the GSA for databases of up to 26 qubits.
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!