Abstract The development of a universal fault-tolerant quantum computer that can solve efficiently various difficult computational problems is an outstanding challenge for science and technology. In this work, we propose a technique for an efficient implementation of quantum algorithms with multilevel quantum systems (qudits). Our method uses a transpilation of a circuit in the standard qubit form, which depends on the characteristics of a qudit-based processor, such as the number of available qudits and the number of accessible levels. This approach provides a qubit-to-qudit mapping and comparison to a standard realization of quantum algorithms highlighting potential advantages of qudits. We provide an explicit scheme of transpiling qubit circuits into sequences of single-qudit and two-qudit gates taken from a particular universal set. We then illustrate our method by considering an example of an efficient implementation of a 6-qubit quantum algorithm with qudits. In this particular example, we demonstrate how using qudits allows a decreasing amount of two-body interactions in the qubit circuit implementation. We expect that our findings are of relevance for ongoing experiments with noisy intermediate-scale quantum devices that operate with information carriers allowing qudit encodings, such as trapped ions and neutral atoms, as well as optical and solid-state systems.
Abstract The development of a universal fault-tolerant quantum computer that can solve efficiently various difficult computational problems is an outstanding challenge for science and technology. In this work, we propose a technique for an efficient implementation of quantum algorithms with multilevel quantum systems (qudits). Our method uses a transpilation of a circuit in the standard qubit form, which depends on the characteristics of a qudit-based processor, such as the number of available qudits and the number of accessible levels. This approach provides a qubit-to-qudit mapping and comparison to a standard realization of quantum algorithms highlighting potential advantages of qudits. We provide an explicit scheme of transpiling qubit circuits into sequences of single-qudit and two-qudit gates taken from a particular universal set. We then illustrate our method by considering an example of an efficient implementation of a 6-qubit quantum algorithm with qudits. In this particular example, we demonstrate how using qudits allows a decreasing amount of two-body interactions in the qubit circuit implementation. We expect that our findings are of relevance for ongoing experiments with noisy intermediate-scale quantum devices that operate with information carriers allowing qudit encodings, such as trapped ions and neutral atoms, as well as optical and solid-state systems.
The paradigm behind digital quantum computing inherits the idea of using binary information processing. Nature in fact gives much more rich structures of physical objects that can be used for encoding information, which is especially interesting in the quantum-mechanical domain. In this Colloquium several ideas are reviewed that indicate how multilevel quantum systems, also known as qudits, can be used for efficient realization of quantum algorithms, which are represented via standard qubit circuits. The focus in the Colloquium is on techniques for leveraging qudits for simplifying decomposition of multiqubit gates and for compressing quantum information by encoding multiple qubits in a single qudit. As discussed in the Colloquium, these approaches can be efficiently combined. This allows a reduction in the number of entangling (two-body) operations and the number of quantum information carriers used compared to straightforward qubit realizations. These theoretical schemes can be implemented with quantum computing platforms of various natures, such as trapped ions, neutral atoms, superconducting junctions, quantum light, spin systems, and molecules. The Colloquium concludes by summarizing a set of open problems whose resolution will be an important further step toward employing universal qudit-based processors for running qubit algorithms.
Recent developments in qudit-based quantum computing, in particular with trapped ions, open interesting possibilities for scaling quantum processors without increasing the number of physical information carriers. Here, we propose a method for compiling quantum circuits in the case, where qubits are embedded into qudits of experimentally relevant dimensionalities, d=3,...,8, for the trapped-ion platform. In particular, we demonstrate how single-qubit, two-qubit, and multiqubit gates can be realized using single-qudit operations and the $\text{M}\o{}\text{lmer-S}\o{}\text{rensen}$ (MS) gate as a basic two-particle operation. We expect that our findings are directly applicable to the current generation of trapped-ion-based qudit processors.
Qubits, which are the quantum counterparts of classical bits, are used as basic information units for quantum information processing, whereas underlying physical information carriers, e.g., (artificial) atoms or ions, admit encoding of more complex multilevel states—qudits. Recently, significant attention has been paid to the idea of using qudit encoding as a way for further scaling quantum processors. In this work, we present an efficient decomposition of the generalized Toffoli gate on five-level quantum systems—so-called ququints—that use ququints’ space as the space of two qubits with a joint ancillary state. The basic two-qubit operation we use is a version of the controlled-phase gate. The proposed N-qubit Toffoli gate decomposition has O(N) asymptotic depth and does not use ancillary qubits. We then apply our results for Grover’s algorithm, where we indicate on the sizable advantage of using the qudit-based approach with the proposed decomposition in comparison to the standard qubit case. We expect that our results are applicable for quantum processors based on various physical platforms, such as trapped ions, neutral atoms, protonic systems, superconducting circuits, and others.
Universal quantum computers require fault-tolerant logical qudits, as qudits naturally align with the simulation of multi-level physical systems. Here, we present a general framework and working examples for encoding fault-tolerant logical qudits in finite-dimensional spin systems. We construct distance-3, distance-5 codewords, and general 2t+1-distance codes that can be implemented using a single physical qudit or a small number of coupled qudits for higher distances, while requiring a Hilbert space dimension significantly smaller than conventional constructions based on multiple logical qubits. Logical operations and error correction protocols can be implemented with polynomial scaling in the number of elementary operations. We further discuss schematic designs for physical implementation and required single-gate fidelities, which are compatible with current spin qudit platforms. This strategy provides a resource-efficient path toward realizing fault-tolerant logical qudits in finite multi-level physical systems.
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) 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.
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 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) single qudit gates and no entangling gates. This work highlights the potential of using qudits for efficient implementations of quantum algorithms.