# A Quantum Cellular Automata Type Architecture with Quantum Teleportation for Quantum Computing

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theory

#### 2.1. Quantum Cellular Automata

- V forming a grid ($V={\mathbb{Z}}^{n}$)
- E connecting nodes x to $x+z$, with $x\in {\mathbb{Z}}^{n}$ and $z\in {\{0,1\}}^{n}$
- H being of finite dimension

#### 2.2. Single Gate Operations

#### 2.3. Nearest Neighbor Circuits

- The qubit partakes in a single-qubit operation. In this case, the qubit is teleported horizontally to position. Position is decreased by one and the qubit is marked as teleported. The case where the qubit partakes in no operation at all is handled the same way.
- The qubit partakes in a two-qubit operation. In that case the qubit is not marked as teleported and it is teleported horizontally to position. The qubit it interacts with is then teleported to position-1. Position is decreased by two and the two qubits are marked as teleported.

Algorithm 1: Algorithm for grid rearrangement |

## 3. Arbitrary Quantum Operations

- Each operation is to be performed only by neighboring qubits.
- In each step of the computation, a single two-qubit gate is to be applied to the whole register holding the current state of the system.

#### 3.1. Single-Qubit Operations

#### 3.2. Two-Qubit Quantum Operations

## 4. General Algorithm Flowchart

- If the quantum gate is a single-qubit gate, the transformation presented in Section 3.1 (Figure 5) is applied
- If the quantum gate is a two-qubit gate, the transformation presented in Section 3.2 (Figure 6) is applied

## 5. Application–Quantum Fourier Transform

^{n}consists of $n-1$ controlled-phase operations plus a Hadamard operation.

- The grid needs ${N}^{2}$ working qubits plus $4N$ qubits for the row of the ancila qubits. An additional $2N$ qubits holding prepared states of |0〉 and |1〉 are needed if we wish to perform all the ancila teleportation operations in parallel.
- Each computational step needs N horizontal teleportation operations plus N vertical (the extra two teleportation in the case of two-qubit gates can be ignored). There are, also, at max N teleportation operations for the ancila qubits plus at max N initialization operations, since we need the qubits holding the prepared states to be returned to their original states after the teleportation operations. Finally, there are N controlled operations in each phase of the computations.

#### Flowchart of Quantum Fourier Transform Operations

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

QCA | Quantum Cellular Automata |

QTM | Quantum Circuit Model |

QFT | Quantum Fourier Transform |

QLG | Quantum Labeled Graph |

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**Figure 1.**Quantum Cellular Automata (QCA) architecture, as proposed in Reference [7].

**Figure 3.**Nearest Neighbor interactions in a two dimensional grid. Initial configuration. Qubits of the same color are to interact with each other while qubits having a unique color will be acted upon by a single-qubit gate.

**Figure 4.**Applying the algorithm for grid rearrangement in an example configuration. Qubits of the same color will partake in a single two-qubit operation. The leftmost column is the initial configuration, after the transposition the qubits are teleported to the right in the position indicated by the corresponding color.

**Figure 5.**Initial configuration of the computational grid. The horizontal line separates the qubits partaking to the main computation from the ancilla qubits.

**Figure 10.**Controlled ${R}_{\pi /2}$. From initial configuration (left side) to the first horizontal teleportation. Qubits 1 and 2 will interact so they are teleported to the same column.

**Figure 11.**Controlled ${R}_{\pi /2}$. Vertical teleportation and the application of the two-qubit quantum operation.

**Figure 13.**Hadamard gate. Vertical teleportation and applying the gate using control qubits in pre-prepared states.

**Figure 14.**Stages for the execution of the Hadamard gate. It is implied that the operation performed in the last stage is the controlled Hadamard operation.

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**MDPI and ACS Style**

Ntalaperas, D.; Giannakis, K.; Konofaos, N. A Quantum Cellular Automata Type Architecture with Quantum Teleportation for Quantum Computing. *Entropy* **2019**, *21*, 1235.
https://doi.org/10.3390/e21121235

**AMA Style**

Ntalaperas D, Giannakis K, Konofaos N. A Quantum Cellular Automata Type Architecture with Quantum Teleportation for Quantum Computing. *Entropy*. 2019; 21(12):1235.
https://doi.org/10.3390/e21121235

**Chicago/Turabian Style**

Ntalaperas, Dimitrios, Konstantinos Giannakis, and Nikos Konofaos. 2019. "A Quantum Cellular Automata Type Architecture with Quantum Teleportation for Quantum Computing" *Entropy* 21, no. 12: 1235.
https://doi.org/10.3390/e21121235