# Spin Foam Vertex Amplitudes on Quantum Computer—Preliminary Results

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## Abstract

**:**

## 1. Introduction

## 2. Intertwiner Qubit

## 3. Vertex Amplitude

## 4. A Quantum Algorithm

#### 4.1. Example 1—Single Tetrahedron

#### 4.2. Example 2—Two Tetrahedra

## 5. Evaluation of Vertex Amplitude

## 6. Summary

- Introduction of a quantum circuit for the general intertwiner qubit $|\mathcal{I}\rangle $ (Equation (11)).
- Determination of the phase of vertex amplitude with the use of quantum algorithms (e.g., Quantum Phase Estimation Algorithm [56]).
- Investigation of different types of the state $|W\rangle $.
- Analysis of spin networks with up to 10 nodes on quantum computers simulators.
- A possibility of solving quantum constraints with the use of quantum circuits.
- Investigation of the architectures of forthcoming quantum processors (with the $N>100$ number of qubits) in terms of application to spin foam transition amplitudes.

## Funding

## Conflicts of Interest

## Appendix A. Quantum Computing Technologies

## Appendix B. Basics of Quantum Computing

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1. | In general, quantum variables associated with higher dimensional Hilbert spaces may be considered. |

2. | In order to prove it let us consider an asymmetric 1D random walk with probabilities $p=\frac{1}{16}$ (one of the basis states) and $q=1-p=\frac{15}{16}$ (rest of the 15 basis sates), for which $\frac{s}{n}=\sqrt{\frac{1}{16}\frac{15}{16}\frac{1}{n}}\approx \frac{1}{4\sqrt{n}}$. |

3. | Performing an inverse mapping ${\widehat{h}}_{x}^{\u2020}(\pi /2)$ we can map the state $i|1\rangle $ back to ${\widehat{h}}_{x}^{\u2020}(\pi /2)i|1\rangle =-i\widehat{X}i|1\rangle =|0\rangle $. |

4. | This statement is made under assumption that no ancilla qubits are required. |

**Figure 2.**A single 4-valent node together with the entering links with spin labels $j=1/2$. The intertwiner qubit $|\mathcal{I}\rangle $ is a degree of freedom defined at the node. The node is dual to the tetrahedron (3-symplex) as represented in the picture.

**Figure 3.**Quantum circuit used to generate $|{0}_{s}\rangle $ state from the initial state $|0000\rangle $. The (green) boxes with letter X represent the bit-flip gates, the (blue) boxes with letter H represents the Hadamard gates, while the next operations from the left are the CNOT 2-qubit gates. Finally, the (pink) boxes on the right represent measurements performed at every one of the four involved qubits.

**Figure 4.**Pentagram spin network corresponding to boundary of a single vertex of spin foam. The boundary geometry has topology of three-sphere.

**Figure 5.**A single-node spin network associated with identification of the pairs of face of a tetrahedron.

**Figure 6.**Quantum circuit used to evaluate ${\left|\langle W|\Psi \rangle \right|}^{2}$ the boundary transition amplitude of the spin network presented in Figure 5.

**Figure 8.**Quantum circuit used to evaluate ${\left|\langle W|\Psi \rangle \right|}^{2}$ the boundary transition amplitude of the spin network presented in Figure 7. The qubits $\{0,1,2,3\}$ belong to the one node while the qubits $\{4,5,6,7\}$ belong to the another. The links are between the pairs of qubits: $\{0,4\}$, $\{1,5\}$, $\{2,6\}$ and $\{3,7\}$.

**Figure 9.**Quantum circuit used to determine $|A({0}_{s},{0}_{s},{0}_{s},{0}_{s},{0}_{s}){|}^{2}$. Nodes of the spin network correspond to the following sets of qubits: $\{0,1,2,3\}$, $\{4,5,6,7\}$, $\{8,9,10,11\}$, $\{12,13,14,15\}$, $\{16,17,18,19\}$. The links are between the pairs of qubits: $\{0,19\}$, $\{1,14\}$, $\{2,9\}$, $\{3,4\}$, $\{5,18\}$, $\{6,13\}$, $\{7,8\}$, $\{10,17\}$, $\{11,12\}$ and $\{15,16\}$.

**Table 1.**Results of measurements of $P\left(i\right)=|{a}_{i}{|}^{2}$ for the quantum circuit presented in Figure 3.

No. | Probability | Theory | IBM Simulator | IBM Q ibmqx4 | QX Simulator |
---|---|---|---|---|---|

1 | $|{a}_{0000}{|}^{2}$ | 0 | 0 | 0.014 | 0 |

2 | $|{a}_{0001}{|}^{2}$ | 0 | 0 | 0.058 | 0 |

3 | $|{a}_{0010}{|}^{2}$ | 0 | 0 | 0.050 | 0 |

4 | $|{a}_{0011}{|}^{2}$ | 0 | 0 | 0.004 | 0 |

5 | $|{a}_{0100}{|}^{2}$ | 0 | 0 | 0.023 | 0 |

6 | $|{a}_{0101}{|}^{2}$ | 0.25 | 0.264 | 0.109 | 0.252 |

7 | $|{a}_{0110}{|}^{2}$ | 0.25 | 0.232 | 0.091 | 0.241 |

8 | $|{a}_{0111}{|}^{2}$ | 0 | 0 | 0.009 | 0 |

9 | $|{a}_{1000}{|}^{2}$ | 0 | 0 | 0.034 | 0 |

10 | $|{a}_{1001}{|}^{2}$ | 0.25 | 0.248 | 0.159 | 0.230 |

11 | $|{a}_{1010}{|}^{2}$ | 0.25 | 0.256 | 0.158 | 0.276 |

12 | $|{a}_{1011}{|}^{2}$ | 0 | 0 | 0.012 | 0 |

13 | $|{a}_{1100}{|}^{2}$ | 0 | 0 | 0.034 | 0 |

14 | $|{a}_{1101}{|}^{2}$ | 0 | 0 | 0.132 | 0 |

15 | $|{a}_{1110}{|}^{2}$ | 0 | 0 | 0.110 | 0 |

16 | $|{a}_{1111}{|}^{2}$ | 0 | 0 | 0.003 | 0 |

**Table 2.**Results of measurements of $P\left(0\right)=|{a}_{0}{|}^{2}$ for the quantum circuit presented in Figure 6, using both the IBM simulator and the QX simulator. Each measurement corresponds to the number of shots equal to 1024.

No. | ${\mathit{P}}_{0}$ (QX) | Hits of $|0\rangle $ (QX) | ${\mathit{P}}_{0}$ (IBM) | Hits of $|0\rangle $ (IBM) |
---|---|---|---|---|

1 | 0.255859375 | 262 | 0.263671875 | 270 |

2 | 0.248046875 | 254 | 0.2529296875 | 259 |

3 | 0.267578125 | 274 | 0.2578125 | 264 |

4 | 0.2568359375 | 263 | 0.27734375 | 284 |

5 | 0.2568359375 | 263 | 0.232421875 | 238 |

6 | 0.25 | 256 | 0.263671875 | 270 |

7 | 0.2578125 | 264 | 0.25 | 256 |

8 | 0.23828125 | 244 | 0.244140625 | 250 |

9 | 0.240234375 | 246 | 0.2607421875 | 267 |

10 | 0.25 | 256 | 0.2763671875 | 283 |

**Table 3.**Results of measurements of $P\left(0\right)=|{a}_{0}{|}^{2}$ for the quantum circuit presented in Figure 8 using both the IBM simulator and the QX simulator. Each measurement corresponds to the number of shots equal 1024.

No. | ${\mathit{P}}_{0}$ (QX) | Hits of $|0\rangle $ (QX) | ${\mathit{P}}_{0}$ (IBM) | Hits of $|0\rangle $ (IBM) |
---|---|---|---|---|

1 | 0.0595703125 | 61 | 0.0556640625 | 57 |

2 | 0.0595703125 | 61 | 0.06640625 | 68 |

3 | 0.06640625 | 68 | 0.060546875 | 62 |

4 | 0.0615234375 | 63 | 0.064453125 | 66 |

5 | 0.080078125 | 82 | 0.0634765625 | 65 |

6 | 0.0498046875 | 51 | 0.0595703125 | 61 |

7 | 0.052734375 | 54 | 0.0615234375 | 63 |

8 | 0.072265625 | 74 | 0.0537109375 | 55 |

9 | 0.0673828125 | 69 | 0.052734375 | 54 |

10 | 0.0634765625 | 65 | 0.0654296875 | 67 |

**Table 4.**Results of measurements of $P\left(0\right)=|{a}_{0}{|}^{2}$ for the quantum circuit presented in Figure 9 using the QX simulator. Each measurement corresponds to the number of shots equal to 1024.

No. | ${\mathit{P}}_{0}$ | Hits of $|0\rangle $ |
---|---|---|

1 | 0.0009765625 | 1 |

2 | 0.0029296875 | 3 |

3 | 0 | 0 |

4 | 0.0009765625 | 1 |

5 | 0.001953125 | 2 |

6 | 0.0009765625 | 1 |

7 | 0.001953125 | 2 |

8 | 0.0009765625 | 1 |

9 | 0.0029296875 | 3 |

10 | 0.0009765625 | 1 |

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**MDPI and ACS Style**

Mielczarek, J.
Spin Foam Vertex Amplitudes on Quantum Computer—Preliminary Results. *Universe* **2019**, *5*, 179.
https://doi.org/10.3390/universe5080179

**AMA Style**

Mielczarek J.
Spin Foam Vertex Amplitudes on Quantum Computer—Preliminary Results. *Universe*. 2019; 5(8):179.
https://doi.org/10.3390/universe5080179

**Chicago/Turabian Style**

Mielczarek, Jakub.
2019. "Spin Foam Vertex Amplitudes on Quantum Computer—Preliminary Results" *Universe* 5, no. 8: 179.
https://doi.org/10.3390/universe5080179