In this work, we present a multi-layer quantum search method that generates an exponential speedup of the standard Grover's algorithm. As direct applications, any NP problems can be solved efficiently on a quantum circuit with only polynomial gate complexity. In particular, such multi-layer search can solve the factoring problem with an exponential speedup, providing an alternative to Shor's algorithm. Our results show that the exponential speedup of quantum circuits is ubiquitous, and Grover's search is much more powerful than that has been demonstrated. With no contradiction to the quadratic optimality of single-layer query complexity, the great potential of Grover's search is fully released by such multi-layer search design.
In this work, we present a multi-layer quantum search method that generates an exponential speedup of the standard Grover's algorithm. As direct applications, any NP problems can be solved efficiently on a quantum circuit with only polynomial gate complexity. In particular, such multi-layer search can solve the factoring problem with an exponential speedup, providing an alternative to Shor's algorithm. Our results show that the exponential speedup of quantum circuits is ubiquitous, and Grover's search is much more powerful than that has been demonstrated. With no contradiction to the quadratic optimality of single-layer query complexity, the great potential of Grover's search is fully released by such multi-layer search design.
Grover's search algorithm is renowned for its dramatic speedup in solving many important scientific problems. The recently proposed Variational Quantum Search (VQS) algorithm has shown an exponential advantage over Grover's algorithm for up to 26 qubits. However, its advantage for larger numbers of qubits has not yet been proven. Here we show that the exponentially deep circuit required by Grover's algorithm can be replaced by a multi-controlled NOT gate together with either a single layer of Ry gates or two layers of circuits consisting of Hadamard and NOT gates, which is valid for any number of qubits greater than five. We prove that the VQS, with a single layer of Ry gates as its Ansatz, has near-perfect reachability in finding the good element of an arbitrarily large unstructured data set, and its reachability exponentially improves with the number of qubits, where the reachability is defined to quantify the ability of a given Ansatz to generate an optimal quantum state. Numerical studies further validate the excellent reachability of the VQS. Proving the near-perfect reachability of the VQS, with a depth-1 Ansatz, for any number of qubits completes an essential step in proving its exponential advantage over Grover's algorithm for any number of qubits, and the latter proving is significant as it means that the VQS can efficiently solve NP-complete problems.
Abstract With powerful quantum computers already built1,2, we need more efficient quantum algorithms to achieve quantum supremacy over classical computers in the noisy intermediate-scale quantum (NISQ) era2,3. Grover's search algorithm4,5 and its generalization, quantum amplitude amplification6, provide quadratic speedup in solving many important scientific problems7. However, they still have exponential time complexity as the depths of their quantum circuits increase exponentially with the number of qubits6,7. To address this problem, we propose a new algorithm, Variational Quantum Search (VQS), which is based on the celebrated variational quantum algorithms8,9 and includes a parameterized quantum circuit, known as Ansatz. We show that a depth-10 Ansatz can amplify the total probability of k (k≥1) good elements, out of 2n elements represented by n+1 qubits, from k/2n to nearly 1, as verified for n up to 26, and that the maximum depth of quantum circuits in the VQS increases linearly with the number of qubits. We demonstrate that a depth-56 circuit in VQS can replace a depth-270,989 circuit in Grover's algorithm, and thus VQS is more suitable for NISQ computers. We envisage our VQS could exponentially speed up the solutions to many important problems, including the NP-complete problems, which is widely considered impossible.
Quantum computing introduces an efficient way for unfolding complex problems on computing systems. It uses concept of superposition and entanglement, & hence supports intrinsic parallelism for obtaining the speedup in context of time over classical computing. Quantum algorithms can be efficiently implemented over quantum computing system. Most popular quantum algorithms like Grover's and Shor's provide a great foundation to specifically design the algorithms. These algorithms work on improving the lower runtime bounds of classical algorithms. Grover's algorithm exhibits a quadratic polynomial speedup while searching an element among `N' elements unstructured database. The algorithmic strategy can be applied over various computer science related applications and some important are being considered in this article i.e. pattern matching, data mining and machine learning. This article explores quantum fundamentals; history based evolution followed by Grover's search algorithm in detail with cascading advantages, limitations & applications and also finally concludes with the significance of quantum computing & quantum algorithmic domains.
Grover's search algorithm is the cornerstone of many applications of quantum computing, providing a quadratic speed-up over classical methods. One limitation of the algorithm is that it requires knowledge of the number of solutions to obtain an optimal success probability, due to the oscillatory dynamics between the initial and solution states (the ``souffl{é} problem''). While various methods have been proposed to solve this problem, each has its drawbacks in terms of inefficiency or sensitivity to control errors. Here, we modify Grover's algorithm so that, for suitably chosen parameters, the usual oscillatory dynamics are replaced by an approximately exponential convergence into the solution subspace. The basic idea is to couple the solution states to an engineered ancilla reservoir such that the initial state is nonreflectively absorbed. Trotterizing the continuous algorithm yields a quantum circuit that gives equivalent performance, while preserving the same quadratic quantum speedup as the original algorithm.
Grover's algorithm can solve NP-complete problems on quantum computers faster than all the known algorithms on classical computers. However, Grover's algorithm still needs exponential time. Due to the BBBV theorem, Grover's algorithm is optimal for searches in the domain of a function, when the function is used as a black box. We analyze the NP-complete set {(⟨M⟩,1n,1t)∣ TM M accepts an x∈{0,1}n within t steps}. If t is large enough, then M accepts each word in L(M) with length n within t steps. So, one can use methods from computability theory to show that black box searching is the fastest way to find a solution. Therefore, Grover's algorithm is optimal for NP-complete problems.
The power of quantum mechanics, that is too complex for conventional computers, can be solved by an innovative model of computing known as quantum computing. Quantum algorithms can provide exponential speedups for some types of problems, such as many difficult mathematical ones. In this paper, we review some of the most important quantum algorithms for hard mathematical problems. When factoring large numbers, Shor's algorithm is orders of magnitude faster than any other known classical algorithm. The Grover's algorithm, which searches unsorted databases much more quickly than conventional algorithms, is then discussed.