### Short description

Scientists at NITU MISiS and the Russian Quantum Center have proposed using extended versions of qubits, known as kudit, to improve the quality and speed of quantum algorithms. Kudites, which can be in three or more states, allow for more densely encoded data and can be used as an auxiliary state to simplify complex logic operations with qubits. By using kudites in the implementation of Grover’s quantum algorithm, the researchers were able to significantly reduce the number of two-part gates required, demonstrating the benefits of using kudites in quantum computing. The results could be applied to quantum processors using ions, neutral atoms, and superconducting circuits.

## Russian scientists have proposed a way to speed up quantum algorithms using kudites

Scientists from NITU MISiS and the Russian Quantum Center have proposed additional levels of the quantum system of algorithms that will improve the final quality of their execution. As a result, the power of the quantum processor will increase, and operations will be performed faster.

The main way to increase the productivity of quantum processors was to increase their qubits. However, the ions or atoms that often act as qubits have more than two levels and can work not only as qubits but also as qubits. A kudit is an extended version of a qubit, which can be in three (kutrite), four (cuquart), five (cuquint) and more states. Such states allow data to be more densely encoded in physical media.

Scientists presented a variant of the use of cuquints and an efficient decomposition model of the generalized Toffoli valve (a universal controlled reversible valve with three inputs and outputs that allows the construction of any reversible logic circuit). As an example, they considered Grover’s quantum algorithm (an algorithm for solving the sorting problem, that is, finding a solution to an equation of the form f(x)=y, where f is a Boolean function of n variables) for searching inside an unordered database.

The space of cuquints can be the space of two qubits with a common additional level. This helps to reduce the number of physical media and use the extra layer as an auxiliary state to simplify the decomposition of multi-qubit gates of complex logic operations with qubits. As a result, it turns out to reduce the number of two-part gates.

The multi-qubit Toffoli gate is a generalized n-qubit version of the universal controlled reversible gate. Its application inverts the state of the n-th qubit if all other n-1 qubits are in the state 1. Having two qubits in each cuquint and using the fifth level as an auxiliary, you can significantly reduce the number of two-part gates in its decomposition.

To perform Grover’s algorithm, it is necessary to repeatedly implement multi-qubit gates. The researchers compared three methods of decomposition of multi-qubit gates within the framework of performing the algorithm on 2-10 qubits, when qubits, cutrits and cuquints are used as information carriers. They demonstrated the reduction of two-part gates.

Implementation on cuquints with a large number (>5) of qubits involved in the algorithm requires much fewer two-part gates. For example, Grover’s 8-qubit algorithm requires more than 1000 two-part gates on qubits, while its implementation on cuquints requires only 88.

This research demonstrates one of the advantages of using kudites for quantum computing. His results can be applied to quantum processors on ions, neutral atoms, superconducting circuits, etc.

In 2021, scientists from the Russian Quantum Center created a prototype computer with a system of four qubits, scaling it with the help of qubits. In 2022, they received a patent for the physical implementation of a quantum computer based on kudites.