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arXiv:1209.4489 · Physical Review A

Time-efficient implementation of quantum search with qudits

Svetoslav S. Ivanov, Hristo Tonchev, Nikolay V. Vitanov

Published Jun 22, 2012• 50 citations
arXivHTMLPDF

Abstract

We propose a simpler and more efficient scheme for the implementation of the multivalued Grover's quantum search. The multivalued search generalizes the original Grover's search by replacing qubits with qudits---quantum systems of $d$ discrete states. The qudit database is exponentially larger than the qubit database and thus it requires fewer particles to control. The Hadamard gate, which is the key elementary gate in the qubit implementation of Grover's search, is replaced by a $d$-dimensional (complex-valued) unitary matrix $\mathbf{F}$, the only condition for which is to have a column of equal moduli elements irrespective of their phases; it can be realized through any physical interaction, which achieves an equal-weight superposition state. An example of such a transformation is the $d$-dimensional discrete Fourier transform, used in earlier proposals; however, its construction is much more costly than that of the far simpler matrix $\mathbf{F}$. We present examples of how such a transform $\mathbf{F}$ can be constructed in realistic qudit systems in a single interaction step.

DOI
10.1103/physreva.85.062321

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We propose a simpler and more efficient scheme for the implementation of the multivalued Grover's quantum search. The multivalued search generalizes the original Grover's search by replacing qubits with qudits---quantum systems of ddd discrete states. The qudit database is exponentially larger than the qubit database and thus it requires fewer particles to control. The Hadamard gate, which is the key elementary gate in the qubit implementation of Grover's search, is replaced by a ddd-dimensional (complex-valued) unitary matrix F\mathbf{F}F, the only condition for which is to have a column of equal moduli elements irrespective of their phases; it can be realized through any physical interaction, which achieves an equal-weight superposition state. An example of such a transformation is the ddd-dimensional discrete Fourier transform, used in earlier proposals; however, its construction is much more costly than that of the far simpler matrix F\mathbf{F}F. We present examples of how such a transform F\mathbf{F}F can be constructed in realistic qudit systems in a single interaction step.
Grover's quantum search algorithm has recently been implemented without entanglement, by replacing multiple particles with a single particle having exponentially many states. We recall that all physical resources must be accounted for to quantify algorithm complexity, and that this scheme typically incurs exponential costs in some other resource(s). In particular, we demonstrate that a recent experimental realization requires exponentially increasing precision. There is, however, a quantum algorithm which searches a "sophisticated" database (not unlike a Web search engine) with a single query, but which we show does not require entanglement even for multiparticle implementations.
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 algorithm provides a quadratic speedup for searching unstructured databases and is traditionally implemented with qubits in Hilbert spaces whose dimensions are powers of two. With the advent of quantum platforms utilizing qudits---quantum systems with more than two levels---there is a need to generalize Grover search to these architectures, including heterogeneous systems with qudits of varying dimensions. Here, we present a unified framework for qudit-based Grover search, detailing the construction of oracles and diffusion operators with and without ancilla qubits and generalizing deterministic and fixed-point search variants that ensure exact or bounded success probabilities. We analyze phase-matching techniques and provide explicit circuit decompositions suitable for diverse hardware platforms. We also compare the corresponding trajectories on the Bloch sphere to provide an intuitive visualization of how the different phase choices amplify the target state. These results facilitate flexible, hardware-oriented protocols for implementing Grover search on qudit processors, potentially reducing circuit depth and enhancing success probabilities, thereby offering a practical toolkit for quantum computation and sensing applications leveraging multilevel quantum systems.
Quantum computing has noteworthy speedup over classical computing by taking advantage of quantum parallelism, i.e., the superposition of states. In particular, quantum search is widely used in various computationally hard problems. Grover's search algorithm finds the target element in an unsorted database with quadratic speedup than classical search and has been proved to be optimal in terms of the number of queries to the database. The challenge, however, is that Grover's search algorithm leads to high numbers of quantum gates, which make it infeasible for the Noise-Intermediate-Scale-Quantum (NISQ) computers. In this paper, we propose a novel hardware efficient quantum search algorithm to overcome this challenge. Our key idea is to replace the global diffusion operation with low-cost local diffusions. Our analysis shows that our algorithm has similar oracle complexity to the original Grover's search algorithm while significantly reduces the circuit depth and gate count. The circuit cost reduction leads to a remarkable improvement in the system success rates, paving the way for quantum search on NISQ machines.
Demonstration of quantum advantage remains challenging due to the increased overhead of controlling large quantum systems. While significant effort has been devoted to qubit-based devices, qudits (ddd-level systems) offer potential advantages in both hardware efficiency and algorithmic performance. In this paper, we demonstrate multi-tone control of a single trapped ion qudit of up to eight levels, as well as the first implementation of Grover's search algorithm on a qudit with dimension five and eight, achieving operation fidelity of 96.8(3)%\%% and 69(6)%\%%, respectively, which correspond to 99.9(1)\% and 97.1(3) \% squared statistical overlap (SSO), respectively, with the expected result for a single iteration of the Grover search algorithm. The performance is competitive when compared to qubit-based systems; moreover, the sequence requires only O(d)\mathcal{O}(d)O(d) single qudit gates and no entangling gates. This work highlights the potential of using qudits for efficient implementations of quantum algorithms.
Demonstration of quantum advantage remains challenging due to the increased overhead of controlling large quantum systems. While significant effort has been devoted to qubit-based devices, qudits (d-level systems) offer potential advantages in both hardware efficiency and algorithmic performance. In this paper, we demonstrate multi-tone control of a single trapped ion qudit of up to eight levels, as well as the implementation of Grover’s search algorithm on a qudit with dimensions five and eight, achieving operation fidelity of 96.8(3)% and 69(6)%, respectively, which correspond to 99.9(1)% and 97.1(3) % squared statistical overlap, respectively, with the expected result for a single iteration of the Grover search algorithm. The performance is competitive when compared to qubit-based systems; moreover, the sequence requires only O(d){{{\mathcal{O}}}}(d)O(d) single-qudit gates and no entangling gates. This work highlights the potential of using qudits for efficient implementations of quantum algorithms. The authors demonstrate high-fidelity multi-tone electronic control of trapped-ion qudits up to d=8 levels, enabling efficient SU(d) operations and showcasing advantage by implementing Grover’s search algorithm using encoding in a single qudit, rather than in multiple qubits.