# Quantum Circuits for the Preparation of Spin Eigenfunctions on Quantum Computers

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Exact Recursive Construction

#### Computational Cost

#### 2.2. Heuristic Variational Construction

#### Ansätze

## 3. Results

- Computational Details.

`statevector_simulator`of Qiskit. We then used the optimized parameters to compute expectation values on quantum hardware. We performed experiments on five-qubit devices available through IBM Quantum Experience, each with a quantum volume [60] of 32, namely,

`ibmq_athens`,

`ibmq_santiago`, and

`ibmq_manila`[61].

- Error mitigation.

- Tomography.

`statevector_simulator`of Qiskit based on the measurement outcomes.

- Target problems.

#### 3.1. Exact Recursive Construction

`qasm_simulator`with noise model from

`ibmq_athens`, and

`ibmq_athens`. We focus on the states $|{\ell}_{L}=0,{\ell}_{LC}=1/2,{\ell}_{R}=0,\ell =1/2,m=-1/2\rangle $ and $|{\ell}_{L}=0,{\ell}_{LC}=1/2,{\ell}_{R}=1\rangle $ $\ell =1/2,m=-1/2$. Although the trends seen for $n=3$ qubits, especially the beneficial impact of EM + RE techniques, are confirmed, a significant difference exists between Figure 7 and Figure 9: in the latter case, error mitigation techniques do not close the gap between exact and simulated expectation values. The same phenomenon is observed in Figure 10, where we compute the fidelities between exact and simulated total spin eigenfunctions, for several representative states, using

`qasm_simulator`and

`ibmq_manila`. The different efficacy of error mitigation techniques for $n=3$ and $n=5$ qubits is an important observation of the present work, and it will be discussed in detail in the conclusions.

#### 3.2. Heuristic Variational Construction

#### 3.2.1. ${R}_{y}$ Ansatz

`qasm_simulator`with noise model from

`ibmq_athens`and on the device

`ibmq_athens`, and the ${R}_{y}$ ansatz with depth $r=3$. Similar trends are seen in both cases, with the combined use of EM + RE systematically improving results. It is clear that the noise models do not faithfully emulate the noise observe in our experiments, as readily seen when comparing the data in Figure 12. Given that the results are highly sensitive to noise affecting the qubits, it will be important to investigate these noise sources further and determine the extent to which they can be mitigated.

`statevector_simulator`(seen in Table 2) and on

`ibmq_athens`. As seen, decoherence significantly affects the fidelities of the obtained wave functions.

#### 3.2.2. Time-Evolution Variational Form

`statevector_simulator`(seen in Table 2) and on

`ibmq_santiago`. The higher computational cost of the time-evolution variational form, compared with the ${R}_{y}$ variational form, results in an increased sensitivity to noise, and ultimately in a worse agreement between exact and simulated quantities, a trend especially pronounced on the actual quantum device.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Structure of Total Spin Eigenfunctions

${\mathit{\ell}}_{01}$ | ℓ | $\phantom{\rule{0.166667em}{0ex}}\mathit{m}\phantom{\rule{0.166667em}{0ex}}$ | $|{\mathit{\ell}}_{01},\mathit{\ell},\mathit{m}\rangle $ |
---|---|---|---|

0 | $1/2$ | $-1/2$ | $\frac{1}{\sqrt{2}}\left(|011\rangle -|101\rangle \right)$ |

0 | $1/2$ | $1/2$ | $\frac{1}{\sqrt{2}}\left(|100\rangle -|010\rangle \right)$ |

1 | $1/2$ | $-1/2$ | $\frac{1}{\sqrt{6}}\left(|011\rangle +|101\rangle \right)-\frac{2}{\sqrt{3}}|110\rangle $ |

1 | $1/2$ | $1/2$ | $\frac{1}{\sqrt{6}}\left(|100\rangle +|010\rangle \right)-\frac{2}{\sqrt{3}}|001\rangle $ |

1 | $3/2$ | $-3/2$ | $|111\rangle $ |

1 | $3/2$ | $-1/2$ | $\frac{1}{\sqrt{3}}\left(|011\rangle +|101\rangle +|110\rangle \right)$ |

1 | $3/2$ | $1/2$ | $\frac{1}{\sqrt{3}}\left(|100\rangle +|010\rangle +|001\rangle \right)$ |

1 | $3/2$ | $3/2$ | $|000\rangle $ |

**Table A2.**Six total spin eigenfunctions for systems of $n=5$ spins with ${\ell}_{LC}=1/2$ and $({\ell}_{L},{\ell}_{R})\in \{(0,0),(0,1),(1,0)\}$.

${\mathit{\ell}}_{\mathit{L}}$ | ${\mathit{\ell}}_{\mathbf{LC}}$ | ${\mathit{\ell}}_{\mathit{R}}$ | ℓ | $\phantom{\rule{0.166667em}{0ex}}\mathit{m}\phantom{\rule{0.166667em}{0ex}}$ | $|{\mathit{\ell}}_{\mathit{L}},{\mathit{\ell}}_{\mathbf{LC}},{\mathit{\ell}}_{\mathit{R}},\mathit{\ell},\mathit{m}\rangle $ |
---|---|---|---|---|---|

0 | $1/2$ | 0 | $1/2$ | $-1/2$ | $\frac{1}{2}\left(|0\phantom{\rule{0.166667em}{0ex}}1\rangle -|1\phantom{\rule{0.166667em}{0ex}}0\rangle \right)|1\rangle \left(|0\phantom{\rule{0.166667em}{0ex}}1\rangle -|1\phantom{\rule{0.166667em}{0ex}}0\rangle \right)$ |

0 | $1/2$ | 0 | $1/2$ | $1/2$ | $\frac{1}{2}\left(|0\phantom{\rule{0.166667em}{0ex}}1\rangle -|1\phantom{\rule{0.166667em}{0ex}}0\rangle \right)|0\rangle \left(|0\phantom{\rule{0.166667em}{0ex}}1\rangle -|1\phantom{\rule{0.166667em}{0ex}}0\rangle \right)$ |

0 | $1/2$ | 1 | $1/2$ | $-1/2$ | $\sqrt{\frac{1}{3}}\left(|0\phantom{\rule{0.166667em}{0ex}}1\rangle -|1\phantom{\rule{0.166667em}{0ex}}0\rangle \right)\left[\frac{1}{2}|1\rangle \left(|0\phantom{\rule{0.166667em}{0ex}}1\rangle +|1\phantom{\rule{0.166667em}{0ex}}0\rangle \right)-|0\phantom{\rule{0.166667em}{0ex}}1\phantom{\rule{0.166667em}{0ex}}1\rangle \right]$ |

0 | $1/2$ | 1 | $1/2$ | $1/2$ | $\sqrt{\frac{1}{3}}\left(|0\phantom{\rule{0.166667em}{0ex}}1\rangle -|1\phantom{\rule{0.166667em}{0ex}}0\rangle \right)\left[|1\phantom{\rule{0.166667em}{0ex}}0\phantom{\rule{0.166667em}{0ex}}0\rangle -\frac{1}{2}|0\rangle \left(|0\phantom{\rule{0.166667em}{0ex}}1\rangle +|1\phantom{\rule{0.166667em}{0ex}}0\rangle \right)\right]$ |

1 | $1/2$ | 0 | $1/2$ | $-1/2$ | $\sqrt{\frac{1}{3}}\left[\frac{1}{2}\left(|0\phantom{\rule{0.166667em}{0ex}}1\rangle +|1\phantom{\rule{0.166667em}{0ex}}0\rangle \right)|1\rangle -|1\phantom{\rule{0.166667em}{0ex}}1\phantom{\rule{0.166667em}{0ex}}0\rangle \right]\left(|0\phantom{\rule{0.166667em}{0ex}}1\rangle -|1\phantom{\rule{0.166667em}{0ex}}0\rangle \right)$ |

1 | $1/2$ | 0 | $1/2$ | $1/2$ | $\sqrt{\frac{1}{3}}\left[|0\phantom{\rule{0.166667em}{0ex}}0\phantom{\rule{0.166667em}{0ex}}1\rangle -\frac{1}{2}\left(|0\phantom{\rule{0.166667em}{0ex}}1\rangle +|1\phantom{\rule{0.166667em}{0ex}}0\rangle \right)|0\rangle \right]\left(|0\phantom{\rule{0.166667em}{0ex}}1\rangle -|1\phantom{\rule{0.166667em}{0ex}}0\rangle \right)$ |

## Appendix B. Quantum Computing Terms and Symbols

**Figure A1.**Examples of quantum gates and circuit elements. From top to bottom: single-qubit rotations, single-qubit Pauli operators, single-qubit operations in the Clifford group, measurements, and two-qubit gates. Adapted from reference [34].

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**Figure 1.**(

**Left**): exact construction of total spin eigenfunctions for $n=2$ qubits. (

**Right**): qubit representation of the total spin eigenfunction $|\ell ,m\rangle $ for $n=2$ qubits, and gates ${G}_{\ell ,m}$ used to encode it.

**Figure 2.**Quantum circuits for exact recursive construction of total spin eigenfunctions, with ${\widehat{U}}_{0}={\widehat{U}}_{{\ell}_{1},m-1/2}$ and ${\widehat{U}}_{1}={\widehat{U}}_{{\ell}_{1},m+1/2}$ to avoid clutter. The two circuits (

**a**,

**b**) are equivalent to each other, and Equations (6) and (7), respectively.

**Figure 3.**Illustration of the exact recursive construction for $n=3$ spins. The single-qubit gates ${G}_{k}$ and ${G}_{k}^{\prime}$ are as listed in the right portion of Figure 1.

**Figure 4.**Illustration of the ${R}_{y}$ ansatz, with $\mathsf{CNOT}$ gates acting on qubits (0,1) and (1,2), and initial state $|{\mathrm{\Psi}}_{0}\rangle =|0,0,0\rangle $.

**Figure 5.**Time evolution-inspired variational form. (

**a**) Time evolution under the Heisenberg Hamiltonian for $r=2$ steps and $n=3$ qubits is approximated with a primitive Trotter formula; in the first step of time evolution, the unitary ${e}^{-it{\widehat{H}}_{0}}$ is not applied in the first step of time evolution, since the top $n-1$ qubits are prepared in an eigenstate of ${\widehat{H}}_{0}$. (

**b**) Each unitary ${e}^{-it{\widehat{H}}_{0}}$ and ${e}^{-it\widehat{V}}$ (blue, red blocks in the left panel) is approximated by a product of unitary ${u}_{ij}\left(tJ\right)$ (rectangles connected by a vertical line). (

**c**) Each unitary ${u}_{ij}\left(tJ\right)$ is replaced by a parametrized two-qubit gate.

**Figure 6.**Representation of a set of three (

**a**) and five (

**b**) spins on graphs with triangle connectivity. Black circles represent spins, labeled from 0 to 4, and red lines connect neighboring qubits. Black dashes lines represent subsets ${\mathrm{\Omega}}_{01}$ (in panel (

**a**)) and ${\mathrm{\Omega}}_{L}={\mathrm{\Omega}}_{01}$, ${\mathrm{\Omega}}_{LC}={\mathrm{\Omega}}_{012}$, ${\mathrm{\Omega}}_{R}={\mathrm{\Omega}}_{34}$ (in panel (

**b**)).

**Figure 7.**Results from the exact recursive construction, for systems of $n=3$ spins prepared in $|{\ell}_{01}=0,\ell =1/2,m=-1/2\rangle $ and $|{\ell}_{01}=1,\ell =3/2,m=-1/2\rangle $ ((

**a**–

**f**), respectively), on the

`ibmq`_

`athens`device. (

**a**,

**d**): Quantum circuits to encode the target states. (

**b**,

**e**): Expectation values of the total spin operators ${\widehat{S}}_{01}^{2}$, ${\widehat{S}}^{2}$, and ${\widehat{S}}_{z}$ from raw hardware simulations (light green, leftmost bar), simulations employing readout error mitigation (EM, teal, center bar), and simulations employing readout error mitigation and Richardson extrapolation (EM + RE, light blue, rightmost bar). Exact values are marked by black lines. (

**c**,

**f**): Extrapolation of the expectation values in the middle panels versus number of CNOT gates. Crosses (r = 1), circles (r = 3.5), and diamonds (r = 0) denote error-mitigated, noise-amplified, and extrapolated quantities, respectively. Red, yellow, and blue symbols indicate ${\widehat{S}}_{01}^{2}$, ${\widehat{S}}^{2}$, and ${\widehat{S}}_{z}$, respectively.

**Figure 8.**

**Left**: fidelities of $n=3$ qubits prepared in the total spin eigenstates $|{\ell}_{01},\ell ,m\rangle $. Fidelities are computed through qubit state tomography employing readout error mitigation (EM), and gate noise is mitigated by a Richardson extrapolation (RE).

**Right**: Richardson extrapolation of the error-mitigated fidelities for systems of $n=3$ qubits prepared in the states $|{\ell}_{01}=0,\ell =1/2,m=-1/2\rangle $ (yellow symbols) and $|{\ell}_{01}=1,\ell =3/2,m=-1/2\rangle $ (red symbols) with the exact recursive procedure. Circles and crosses denote measured an extrapolated quantities, respectively.

**Figure 9.**Results from the exact recursive construction, for systems of $n=5$ spins prepared in the wave functions $|{s}_{L}=0,{s}_{LC}=1/2,{s}_{R}=0,s=1/2,{s}_{z}=-1/2\rangle $ and $|{s}_{L}=0,{s}_{LC}=1/2,{s}_{R}=1,s=1/2,{s}_{z}=-1/2\rangle $ ((

**a**–

**f**), respectively) on the

`ibmq`_

`athens`device. (

**a**,

**d**): Quantum circuits to encode the target states. (

**b**,

**e**): Expectation values of the total spin operators ${\widehat{S}}_{01}^{2}$, ${\widehat{S}}^{2}$, and ${\widehat{S}}_{z}$ from raw hardware simulations (light green, leftmost bar), simulations employing readout error mitigation (EM, teal, center bar), and simulations employing readout error mitigation and Richardson extrapolation (EM + RE, light blue, rightmost bar). Exact values are marked by black lines. (

**c**,

**f**): Extrapolation of the expectation values in the middle panels versus number of CNOT gates. Crosses (r = 1), circles (r = 3.5), and diamonds (r = 0) denote error-mitigated, noise-amplified, and extrapolated quantities, respectively. Red, yellow, and blue symbols indicate ${\widehat{S}}_{01}^{2}$, ${\widehat{S}}^{2}$, and ${\widehat{S}}_{z}$, respectively.

**Figure 10.**Fidelities between exact and simulated total spin eigenfunctions for $n=5$ qubits, computed using

`qasm_simulator`and

`ibmq_manila`.

**Figure 11.**Cost function values ${C}_{\mathrm{opt}}$ of optimal quantum circuits for $n=3$ qubits using the ${R}_{y}$ ansatz with depth $r=3$, computed with

`statevector_simulator`. The ideal value of the cost function is zero.

**Figure 12.**Bar charts: expectation values of the spin operators ${\widehat{S}}^{2}$, ${\widehat{S}}_{z}$, and ${\widehat{S}}_{01}^{2}$ for systems of $n=3$ qubits prepared in $|0,1/2,-1/2\rangle $ (12

**a**,

**c**) and $|1,1/2,-1/2\rangle $ (

**e**,

**g**) using

`qasm_simulator`with noise model from

`ibmq_athens`(

**a**,

**e**) and on the device

`ibmq_athens`(

**c**,

**g**) and the ${R}_{y}$ ansatz with depth $r=3$. (

**b**,

**d**,

**f**,

**h**): Richardson extrapolation analysis for the expectation values shown in the bar charts. The vertical scales are matched to the scales in the accompanying bar chart.

**Figure 13.**Fidelities for $n=3$ between exact total spin eigenstates and VQE wave functions computed with

`statevector_simulator`and

`ibmq_athens`(light, dark blue columns).

**Figure 14.**Bar charts: expectation values of the spin operators ${\widehat{S}}^{2}$, ${\widehat{S}}_{z}$, and ${\widehat{S}}_{01}^{2}$ for systems of $n=3$ qubits prepared in $|0,1/2,-1/2\rangle $ (

**a**,

**c**) and $|1,1/2,-1/2\rangle $ (

**e**,

**g**) using the

`ibmq_santiago`with noise model from

`ibmq_santiago`(

**a**,

**e**) and on the device

`ibmq_santiago`(

**c**,

**g**) and the time-evolution variational form. (

**b**,

**d**,

**f**,

**h**): Richardson extrapolation analysis for the expectation values shown in the bar charts.The vertical scales are matched to the scales in the accompanying bar chart.

**Table 1.**Fidelity and purity of the state ${\rho}_{\mathrm{circuit}}$, for systems of $n=3$ spins prepared in the states $|{\ell}_{01}=0,\ell =1/2,m=-1/2\rangle $ and $|{\ell}_{01}=1,\ell =3/2,m=-1/2\rangle $ (top, bottom, respectively) using exact recursive construction. The fidelity is extrapolated as shown in the right panel of Figure 8.

$({\mathit{\ell}}_{01},\mathit{\ell},\mathit{m})$ | $\mathit{F}({\mathit{\rho}}_{\mathbf{circuit}},{\mathit{\psi}}_{\mathbf{ideal}})$ (Raw) | $\mathit{P}\left({\mathit{\rho}}_{\mathbf{circuit}}\right)$ (Raw) | $\mathit{F}({\mathit{\rho}}_{\mathbf{circuit}},{\mathit{\psi}}_{\mathbf{ideal}})$ (EM + RE) |
---|---|---|---|

$(0,1/2,-1/2)$ | $0.78708$ | $0.79982$ | $0.9720\pm 0.0306$ |

$(1,3/2,-1/2)$ | $0.86417$ | $0.90659$ | $0.9944\pm 0.0057$ |

**Table 2.**Fidelities of optimal quantum circuits for $n=3$ qubits using the ${R}_{y}$ ansatz with depth $r=3$, computed with

`statevector_simulator`.

$({\mathit{\ell}}_{01},\mathit{\ell},\mathit{m})$ | ${\left|\langle {\mathit{\psi}}_{\mathbf{ideal}}|{\mathrm{\Psi}}_{\mathbf{opt}}\rangle \right|}^{2}$ |
---|---|

$(0,1/2,-1/2)$ | $1.000000$ |

$(0,1/2,\phantom{-}1/2)$ | $1.000000$ |

$(1,1/2,-1/2)$ | $1.000000$ |

$(1,1/2,\phantom{-}1/2)$ | $1.000000$ |

$(1,3/2,-3/2)$ | $1.000000$ |

$(1,3/2,-1/2)$ | $0.396621$ |

$(1,3/2,\phantom{-}1/2)$ | $0.024675$ |

$(1,3/2,\phantom{-}3/2)$ | $1.000000$ |

**Table 3.**Circuit depth required to approximate three-qubit total spin eigenstates using the time-evolution variational form with $r=2$ repetitions from

`statevector_simulator`(full connectivity, single-qubit, and CNOT native gates) and

`ibmq_santiago`(linear connectivity, device-specific native gates).

$({\mathit{\ell}}_{01},\mathit{\ell},\mathit{m})$ | ${\mathit{d}}_{\mathbf{statevector}\_\mathtt{simulator}}$ | ${\mathit{d}}_{\mathtt{ibmq}\_\mathbf{santiago}}$ |
---|---|---|

$(0,1/2,-1/2)$ | 76 | 117 |

$(0,1/2,\phantom{-}1/2)$ | 76 | 118 |

$(1,1/2,-1/2)$ | 76 | 120 |

$(1,1/2,\phantom{-}1/2)$ | 76 | 118 |

$(1,3/2,-3/2)$ | 76 | 117 |

$(1,3/2,-1/2)$ | 76 | 119 |

$(1,3/2,\phantom{-}1/2)$ | 76 | 126 |

$(1,3/2,\phantom{-}3/2)$ | 65 | 94 |

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**MDPI and ACS Style**

Carbone, A.; Galli, D.E.; Motta, M.; Jones, B. Quantum Circuits for the Preparation of Spin Eigenfunctions on Quantum Computers. *Symmetry* **2022**, *14*, 624.
https://doi.org/10.3390/sym14030624

**AMA Style**

Carbone A, Galli DE, Motta M, Jones B. Quantum Circuits for the Preparation of Spin Eigenfunctions on Quantum Computers. *Symmetry*. 2022; 14(3):624.
https://doi.org/10.3390/sym14030624

**Chicago/Turabian Style**

Carbone, Alessandro, Davide Emilio Galli, Mario Motta, and Barbara Jones. 2022. "Quantum Circuits for the Preparation of Spin Eigenfunctions on Quantum Computers" *Symmetry* 14, no. 3: 624.
https://doi.org/10.3390/sym14030624