# Variational Quantum Circuits to Prepare Low Energy Symmetry States

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## Abstract

**:**

## 1. Introduction

## 2. Method

#### 2.1. Method 1—Exact Unitary

#### 2.2. Method 2—Approximate Unitary Construction

## 3. Results

#### 3.1. $XXZ$ Spin Hamiltonian

#### 3.1.1. Reflection Symmetry

#### 3.1.2. Rotation Symmetry

#### 3.2. ${H}_{2}$ Hamiltonian

#### 3.3. Simulation against Noise Models

## 4. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

## Appendix B

## References

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**Figure 1.**Ansatz ($A(\alpha )$) has been used to build the state within a k-dimensional space in Equation (2).

**Figure 4.**Computing the lowest energy state of XXZ Hamiltonian within subspace labelled by the Reflection Symmetry Operator using Unitary constructed from method 1 and method 2. (

**a**) Method 1: Energy error vs. iterations; (

**b**) Method 1: State fidelity vs. iterations; (

**c**) Method 2: Energy error vs. iterations; (

**d**) Method 2: State fidelity vs. iterations; (

**e**) Method 2: Symmetry value vs. iterations; (

**f**) Method 2: Symmetry value vs. iterations.

**Figure 5.**Computing the lowest energy state of XXZ Hamiltonian within subspace labelled by the Rotation Symmetry Operator using Unitary constructed from method 1 and method 2. (

**a**) Method 1: Energy error vs. iterations; (

**b**) Method 1: State fidelity vs. iterations; (

**c**) Method 2: Energy error vs. iterations; (

**d**) Method 2: State fidelity vs. iterations; (

**e**) Method 2: Symmetry value vs. iterations; (

**f**) Method 2: Symmetry value vs. iterations.

**Figure 6.**Computing the lowest energy state of the hydrogen Hamiltonian within ${\sum}_{i}{S}_{z}^{i}=0$ subspace labelled by ${S}^{2}=s(s+1)=0$ using Unitary constructed from method 1 and method 2. (

**a**) Method 1: Energy error vs. iterations; (

**b**) Method 1: State fidelity vs. iterations; (

**c**) Method 2: Energy error vs. iterations; (

**d**) Method 2: State fidelity vs. iterations; (

**e**) Method 2: Symmetry value vs. iterations.

**Figure 7.**Computing the lowest energy state of XXZ Hamiltonian within subspace labelled by the Rotation Symmetry Operator using method 1 on a noisy Qiskit simulator. (

**a**) Method 1: Energy error vs. iterations; (

**b**) Method 1: State fidelity vs. iterations.

**Table 1.**Details of the 4 qubit noise model sampled from Qiskit and used for simulating results shown in Figure 7. The noise model has been restricted to qubit errors only.

Qubit No. | T1 (us) | T2 (us) | Readout-Error (%) | Frequency (GHz) | Anharomonicity (GHz) |
---|---|---|---|---|---|

1 | 121.70 | 17.04 | 7.50 | 4.79 | −0.31 |

2 | 111.68 | 132.02 | 2.24 | 4.94 | −0.30 |

3 | 101.82 | 68.98 | 1.45 | 4.83 | −0.31 |

4 | 116.71 | 85.88 | 2.15 | 4.80 | −0.31 |

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**MDPI and ACS Style**

Selvarajan, R.; Sajjan, M.; Kais, S.
Variational Quantum Circuits to Prepare Low Energy Symmetry States. *Symmetry* **2022**, *14*, 457.
https://doi.org/10.3390/sym14030457

**AMA Style**

Selvarajan R, Sajjan M, Kais S.
Variational Quantum Circuits to Prepare Low Energy Symmetry States. *Symmetry*. 2022; 14(3):457.
https://doi.org/10.3390/sym14030457

**Chicago/Turabian Style**

Selvarajan, Raja, Manas Sajjan, and Sabre Kais.
2022. "Variational Quantum Circuits to Prepare Low Energy Symmetry States" *Symmetry* 14, no. 3: 457.
https://doi.org/10.3390/sym14030457