# Generalized Toffoli Gate Decomposition Using Ququints: Towards Realizing Grover’s Algorithm with Qudits

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## Abstract

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## 1. Introduction

## 2. Ququint Processor

#### Ququint as Two Qubits and Ancillary State

## 3. Toffoli Gate Implementation

## 4. Application to Grover’s Algorithm

## 5. Conclusions and Outlook

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Realization of a generalized controlled inversion ${\mathsf{CX}}_{{Q}_{1}{Q}_{2}}^{i\to k,\ell}$ gate via the generalized controlled-phase ${\mathsf{CZ}}_{{Q}_{1}{Q}_{2}}^{i\leftrightarrow j}$ and ${H}_{{Q}_{2}}^{(k,\ell )}$ gate on two ququints. On the left-hand side of the image, the black-painted circle with white i denotes a control qudit and the control state $|i\rangle $ for the ${\mathsf{CX}}_{{Q}_{1}{Q}_{2}}^{i\to k,\ell}$ gate. The corresponding target qudit is denoted by the white circle with an arrow between k and ℓ. The ${\mathsf{CX}}_{{Q}_{1}{Q}_{2}}^{i\to k,\ell}$ gate exchanges populations between levels of the $|k\rangle $ and $|\ell \rangle $ states of the target qudit, given that the control qudit is in the state $|i\rangle $. On the right-hand side of the image, the gate with two connected black-painted circles corresponds to the controlled-phase ${\mathsf{CZ}}_{{Q}_{1}{Q}_{2}}^{i\leftrightarrow \ell}$ operation, which applies a phase factor $-1$ to the state of two ququints ${|i\ell \rangle}_{{Q}_{1}{Q}_{2}}$ and leaves other states unchanged. Single-qudit ${H}^{(k,\ell )}$ gates denote two-dimensional Hadamard transformations realized at levels $|k\rangle $ and $|\ell \rangle $ of ${Q}_{2}$.

**Figure 2.**${\mathsf{C}}^{N-1}\mathsf{Z}$ gate decomposition on ququints with ${\mathsf{CX}}^{i\to k,\ell}$ gates for $N\ge 6$. In the central part of the circuit, we apply a controlled-phase gate ${\mathsf{CZ}}_{{Q}_{{N}^{\prime}-1}{Q}_{{N}^{\prime}}}^{4\leftrightarrow 1}$ if N is odd and mapping (8) is used, two gates ${\mathsf{CZ}}_{{Q}_{{N}^{\prime}-1}{Q}_{{N}^{\prime}}}^{4\leftrightarrow 2}$ and ${\mathsf{CZ}}_{{Q}_{{N}^{\prime}-1}{Q}_{{N}^{\prime}}}^{4\leftrightarrow 3}$ if N is odd and mapping (1) is used, or a controlled-phase gate ${\mathsf{CZ}}_{{Q}_{{N}^{\prime}-1}{Q}_{{N}^{\prime}}}^{4\leftrightarrow 3}$ if N is even. Labeling of gates is the same as in Figure 1.

**Figure 3.**Grover’s algorithm for search item $\omega =10,101$ over ${2}^{5}=32$ items. Each of the four iterations has two multiply-controlled gates: one in the oracle and one in the diffusion operator. Both these multiply-controlled gates can be efficiently decomposed into two-qudit gates with ququints.

**Figure 4.**Two-qudit gate counts for implementations of n-qubit Grover’s algorithm (n is from 2 to 10) with the qubit-based decomposition method [36], which requires $n-2$ ancillary qubits for n-qubit gate decomposition and has linear scaling; qutrit-based decomposition method [73]; and the proposed ququint-based decomposition method. Plotted data take into account an increase in the number of Grover’s steps in quantum circuits with an increase in the number of involved qubits.

**Table 1.**${\mathsf{C}}^{N-1}\mathsf{Z}$ gate implementation on ququints for $N=3,\cdots ,6$ with ${\mathsf{CZ}}^{i\leftrightarrow j}$ and ${\mathsf{CX}}^{i\to k,\ell}$ gates for two possible variants of mapping for the ‘bottom’ ququint. Labeling of gates is the same as in Figure 1.

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**MDPI and ACS Style**

Nikolaeva, A.S.; Kiktenko, E.O.; Fedorov, A.K.
Generalized Toffoli Gate Decomposition Using Ququints: Towards Realizing Grover’s Algorithm with Qudits. *Entropy* **2023**, *25*, 387.
https://doi.org/10.3390/e25020387

**AMA Style**

Nikolaeva AS, Kiktenko EO, Fedorov AK.
Generalized Toffoli Gate Decomposition Using Ququints: Towards Realizing Grover’s Algorithm with Qudits. *Entropy*. 2023; 25(2):387.
https://doi.org/10.3390/e25020387

**Chicago/Turabian Style**

Nikolaeva, Anstasiia S., Evgeniy O. Kiktenko, and Aleksey K. Fedorov.
2023. "Generalized Toffoli Gate Decomposition Using Ququints: Towards Realizing Grover’s Algorithm with Qudits" *Entropy* 25, no. 2: 387.
https://doi.org/10.3390/e25020387