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arXiv:2311.12003

Qudits for decomposing multiqubit gates and realizing quantum algorithms

Evgeniy O. Kiktenko, Anastasiia S. Nikolaeva, Aleksey K. Fedorov

Published Nov 20, 2023• 1 citation
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.2311.12003

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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.
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.
Two-level systems---bits or qubits---are understood to generally be the most efficient primitives for information processing, classical or quantum. But this is not to say that there are no roles to be played by multilevel systems. This Colloquium surveys these possible roles for the quantum case. Here we speak of qudits: ddd-level quantum systems. In one interesting example, the use of just one three-level system permits a drastic simplification of the ``Toffoli gate,'' the basic three-qubit primitive of reversible logic. A survey is given of various qudit-qubit embeddings, and the current state of quantum computing experiments using qudits is reviewed.
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.
Qudit is a multi-level computational unit alternative to the conventional 2-level qubit. Compared to qubit, qudit provides a larger state space to store and process information, and thus can provide reduction of the circuit complexity, simplification of the experimental setup and enhancement of the algorithm efficiency. This review provides an overview of qudit-based quantum computing covering a variety of topics ranging from circuit building, algorithm design, to experimental methods. We first discuss the qudit gate universality and a variety of qudit gates including the pi/8 gate, the SWAP gate, and the multi-level-controlled gate. We then present the qudit version of several representative quantum algorithms including the Deutsch-Jozsa algorithm, the quantum Fourier transform, and the phase estimation algorithm. Finally we discuss various physical realizations for qudit computation such as the photonic platform, iron trap, and nuclear magnetic resonance.
Gate-based universal quantum computation is formulated in terms of two types of operations: local single-qubit gates, which are typically easily implementable, and two-qubit entangling gates, whose faithful implementation remains one of the major experimental challenges since it requires controlled interactions between individual systems. To make the most of quantum hardware it is crucial to process information in the most efficient way. One promising avenue is to use higher-dimensional systems, qudits, as the fundamental units of quantum information, in order to replace a fraction of the qubit-entangling gates with qudit-local gates. Here, we show how the complexity of multi-qubit circuits can be lowered significantly by employing qudit encodings, which we quantify by considering exemplary circuits with exactly known (multi-qubit) gate complexity. We discuss general principles for circuit compression, derive upper and lower bounds on the achievable advantage, and highlight the key role played by entanglement and the available gate set. Explicit experimental schemes for photonic as well as for trapped-ion implementations are provided and demonstrate a significant expected gain in circuit performance for both platforms.
Digital-analog quantum computing with two-level systems is a computational paradigm that combines an analog Hamiltonian with single-qubit gates to achieve universality. We extend this framework to ddd-level systems by conjugating an analog Hamiltonian block with single-qudit gates drawn from the Weyl-Heisenberg basis, which provides a natural set of operations for qudit architectures. More specifically, we propose a protocol to simulate arbitrary two-body Hamiltonians with at most O(d4n2)O(d^4 n^2)O(d4n2) analog blocks. The power of this approach is illustrated by the simulation of many-body qudit spin Hamiltonians including magnetic quadrupolar terms.