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Physical Review A

Implementing a quantum search algorithm with nonorthogonal states

Shi-Chao Wang, Yan Chi, Li Zhi Yu, Zhe Sun, Qi-Ping Su, Chui‐Ping Yang

Published Mar 15, 2021• 7 citations
Publisher

Abstract

The search algorithm is one of the most widely studied quantum algorithms, while its complete implementation in experiments is difficult and only a few experimental works have been reported. Based on Grover's algorithm, we propose a scheme of quantum search using nonorthogonal states and experimentally implement the entire scheme in a linear optical system. With this scheme, a large-sized search space can be encoded using only one qubit and the search probability of the target state is $100%$ in theory. The experimental setup and the operations for implementing this scheme are extremely simple. In our linear optical experiment, different target states are successfully found from a database of nine elements with all fidelities $>0.99$, the state evolution from the initial state to a target state is studied, and the medium states and the final state are shown in a Bloch sphere. The scheme is general and can be easily realized in a wide range of quantum systems. This work may pave an avenue to implement quantum algorithms with nonorthogonal states.

DOI
10.1103/physreva.103.032413

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The search algorithm is one of the most widely studied quantum algorithms, while its complete implementation in experiments is difficult and only a few experimental works have been reported. Based on Grover's algorithm, we propose a scheme of quantum search using nonorthogonal states and experimentally implement the entire scheme in a linear optical system. With this scheme, a large-sized search space can be encoded using only one qubit and the search probability of the target state is 100100%100 in theory. The experimental setup and the operations for implementing this scheme are extremely simple. In our linear optical experiment, different target states are successfully found from a database of nine elements with all fidelities >0.99>0.99>0.99, the state evolution from the initial state to a target state is studied, and the medium states and the final state are shown in a Bloch sphere. The scheme is general and can be easily realized in a wide range of quantum systems. This work may pave an avenue to implement quantum algorithms with nonorthogonal states.
Abstract Quantum search algorithms have important applications in quantum information, which have been experimentally demonstrated in several quantum systems with database sizes up to 16. We first propose an efficient quantum search scheme based on an optimized Grover’s algorithm and a novel encoding method. With this scheme, a target element can be deterministically searched in a large database using only a single qubit. Then we implement this search scheme in linear optics with database sizes up to 40. Experimental results fit well with the theoretical results, with all fidelities greater than 99.7%. The limitation of the number of elements encoded in a single qubit is also investigated in the experiment. The experimental results indicate that a target state can be successfully searched from a database of size 36 using a single qubit in our laboratory. Since only one qubit and one-qubit gates are required, this search scheme is quite efficient and general, and can be adopted to implement the same task in most quantum systems.
Grover's search algorithm is a well-known quantum algorithm that has been extensively studied and improved to increase its success rate and enhance its flexibility. However, most improved search algorithms require an adjustment of the oracle, which may not be feasible in practical problem-solving scenarios. In this work, we report an experimental demonstration of a deterministic quantum search for multiple marked states without adjusting the oracle. A linear optical setup is designed to search for two marked states, one in a 16-state database with an initial equal-superposition state and the other in an 8-state database with different initial nonequal-superposition states. The evolution of the probability of finding each state in the database is also measured and displayed. Our experimental results agree well with the theoretical predictions, thereby proving the feasibility of the search protocol and the implementation scheme. This work is a pioneering experimental demonstration of deterministic quantum search for multiple marked states without adjusting the oracle.
The essential operations of a quantum computer can all be accomplished using only standard linear optical elements (e.g., beamsplitters, waveplates, polarizers, etc.), with the individual bits represented by different spatial orpolarization degrees of freedom. The difference from a genuine quantum computer with distinct entangleable registers is that the optical implementation requires a number of elements which grows exponentially with the number of bits. However, by “compiling” the code, and making multiple use of redundant components, the required number of elements can be substantially reduced, allowing the realization of quantum algorithms involving several bits. Here we present a simple optical implementation of Grover’s algorithm for efficiently searching a database. In our example a database of four elements is searched with a single query, in contrast to the classical expected value of 2.25 queries. It is seen that the “quantum” computer in this case is no more than an interferometer, albeit a complicated one. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Grover’s algorithm harnesses the power of quantum computing to swiftly locate specific elements in an unstructured database, outperforming classical computers in tasks like database searching. This algorithm capitalizes on the unique ability of qubits to be in both 0 and 1 states simultaneously (superposition), allowing it to scan the entire search space at once. It then boosts the probability of the target element, making it more prominent. While the foundational concepts of Grover’s algorithm are well-documented, practical implementation using quantum operators, especially for large search spaces, remains less explored beyond basic examples with a small number of qubits. Existing general synthesis techniques often involve numerous operators or are time-consuming. Our proposed methods specifically address the amplitude-amplification component of Grover’s algorithm for any size of search space. These methods detail the required types and quantities of qubits and operators, emphasizing minimal usage and efficient assembly. Developed and evaluated in Python, our methods consistently identified target elements with over 95% accuracy and achieved configurations comparably compact as those in the existing literature but at a faster pace. We anticipate that these methods facilitate practical implementations of Grover’s algorithm across various domains.
Grover's quantum search algorithm is an important landmark discovery in the area of quantum computing that drives a quantum computer from a known initial state(source state) to an unknown final state(target state) by using selective phase inversions of these states. The algorithm is simply a successive iteration of the Grover's search operator, which is a product of selective inversion operators of source and target states. In its original form, Grover's operator performs the selective inversions only on a unique source state and a unique target state. In the generalized version, we make our search operator to perform selective inversions on more than one source and target states. First, we find that the case of selective inversions of two source and two target states can be easily analysed somewhat similar to the case of Grover's algorithm. But unlike Grover's operator, the selective inversions of three source and three target states is not so easy to study analytically and hence we perform numerical calculations on some specific cases. We find that these generalized search operators also give us a successful quantum search as expected.