Complete 3-Qubit Grover Search on a Programmable Quantum Computer (1703.10535v1)

Abstract: Searching large databases is an important problem with broad applications. The Grover search algorithm provides a powerful method for quantum computers to perform searches with a quadratic speedup in the number of required database queries over classical computers. It is an optimal search algorithm for a quantum computer, and has further applications as a subroutine for other quantum algorithms. Searches with two qubits have been demonstrated on a variety of platforms and proposed for others, but larger search spaces have only been demonstrated on a non-scalable NMR system. Here, we report results for a complete three-qubit Grover search algorithm using the scalable quantum computing technology of trapped atomic ions, with better-than-classical performance. The algorithm is performed for all 8 possible single-result oracles and all 28 possible two-result oracles. Two methods of state marking are used for the oracles: a phase-flip method employed by other experimental demonstrations, and a Boolean method requiring an ancilla qubit that is directly equivalent to the state-marking scheme required to perform a classical search. All quantum solutions are shown to outperform their classical counterparts. We also report the first implementation of a Toffoli-4 gate, which is used along with Toffoli-3 gates to construct the algorithms; these gates have process fidelities of 70.5% and 89.6%, respectively.

• The paper implements Grover's algorithm across 8 single-result and 28 two-result oracles on a trapped ion platform, affirming quantum speedup.
• The study introduces dual oracle formulations—using an ancilla qubit for Boolean and a phase-flip method—to mirror classical search strategies.
• The novel implementation of the Toffoli-4 gate with 11 two-qubit interactions optimizes quantum circuits and enables complex multi-qubit operations.

Overview of "Complete 3-Qubit Grover Search on a Programmable Quantum Computer"

This paper presents an empirical paper focused on implementing a complete 3-qubit Grover search algorithm on a scalable quantum computing platform utilizing trapped atomic ions. The research affirms the expected quantum computational advantage by demonstrating superior performance over classical algorithms in a 3-qubit system, which translates to an 8-element database search.

Key Contributions

1. Implementation of Grover's Algorithm: The research details the execution of Grover's algorithm across all 8 possible single-result oracles and 28 potential two-result oracles using a trapped ion quantum computer. By leveraging scalable quantum technology, the experiment showcases better-than-classical outcomes substantiating quantum computational benefits in practical applications.
2. Oracle Formulation: Two methods of oracle state marking are explored—Boolean oracle using an ancilla qubit and phase-flip oracle. This dual approach extends the versatility of the experimental setup, allowing the oracle to mirror classical searching algorithms in the Boolean formulation.
3. Toffoli-4 Gate Implementation: A significant innovation presented is the first implementation of the Toffoli-4 gate on this platform. The construction is guided by a careful selection of quantum gates, requiring 11 two-qubit interactions, which is an optimization over traditional methods.

Technical Methods

The paper exploits the properties of the Grover algorithm, which operates in four stages: initialization, oracle query, amplitude amplification, and measurement. The initialization creates a superposition state. The oracle flips the amplitude of marked states, and amplification increases their probability of being measured by reflecting around the mean amplitude. The strategic repetition of these operations in quantum computing reduces the search requirement to $O(\sqrt{N})$ queries against $N/2$ on average in classical methods.

The trapped ion approach leverages precise laser control for qubit manipulation and interaction using Ising gates facilitated by collective motional modes of ion chains. Through innovative pulse-segmentation schemes and fine control of optical addressing capabilities, high-fidelity operational gates were achieved.

Results and Performance

• Single-Solution Search: The implementation of Grover's algorithm for single-solution searches reached an algorithmic success probability (ASP) exceeding 43% for phase oracles, against a theoretical maximum of 78.125%. This performance is significantly better than the classical ASP of 25%.
• Two-Solution Search: For two-solution scenarios, experimental ASP values approached 75% for phase oracles, quite close to the theoretical certainty of 100%, remarkably outperforming the classical ASP of approximately 46.4%.

Implications and Future Outlook

The research provides concrete empirical evidence underscoring the advantages of quantum search algorithms, serving as a crucial milestone toward extending such algorithms to larger databases and embedding Grover's search in more complex quantum algorithms as a subroutine. The exploration into gate optimizations and error mitigation strategies, especially around resources conservation in quantum circuit designs, points toward the scalability of these methods for broader adoption.

The demonstrated Toffoli-4 gate also indicates the potential for more sophisticated multi-qubit operations, paving pathways for future explorations into more substantial quantum systems and contributing to the toolkit available for quantum algorithm researchers.

In summary, this paper showcases the iterative and experimental approach necessary for progressing practical quantum computing applications and benchmarks current standards of operation within quantum technologies, emphasizing results that support scalability and efficiency. Future studies may expand on the nuances of noise management and error correction to further align theoretical potential with empirical execution.