# Simulating Static and Dynamic Properties of Magnetic Molecules with Prototype Quantum Computers

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## Abstract

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## 1. Introduction

## 2. Determining the Ground State of Heisenberg Chains by VQE

#### 2.1. Variational Quantum Eigensolver

#### 2.2. Heisenberg Spin Chains and Adapted Ansatz

`statevector_simulator`, i.e., by evaluating the results of quantum circuits directly, via algebraic matrix multiplication and by using the COBYLA [96] classical minimization routine. An ansatz depth $L=2$, corresponding to 12 independent classical parameters and 6 CNOT operations, allows us to obtain good results in the intermediate $B/J$ region. At low field, a more complex heuristic circuit ($L=3$, with 16 free parameters and 9 CNOTs) is required to exactly reconstruct the true ground state. This is a direct consequence of the higher degree of entanglement present in the ground state when the Heisenberg interaction is the dominant contribution. We see here that, when working with heuristic ansatzes, the only natural way to enrich the space of possible trial wavefunctions (e.g., when complex ground states must be approximated) is to increase the number of layers L. However, this strategy may not be optimal in view of scaling the problem size, particularly when the classical cost of the optimization problem associated with VQE implementations is also taken into account. On the one hand, only relatively shallow circuits may be run in practice in a near term scenario, where all quantum operations are subject to environment noise and limited fidelity (see also a discussion on noisy simulations below). On the other hand, the number of classical optimization steps, and hence of circuit executions required to estimate the variational energy ${\langle \mathcal{H}\rangle}_{\theta}$, may become quickly impractical when many classical parameters, intially set at random values, are present in the ansatz.

#### Effect of Noise

- finite relaxation (${T}_{1}$) and coherence (${T}_{2}$) times of the physical qubits, which typically lead to amplitude and phase damping effects;
- single- and 2-qubit gate errors (the latter being usually much higher), acting during the implementation of each quantum operation, due to both imperfections of the coherent qubit manipulation and additional incoherent effects (e.g., depolarizing Pauli noise);
- readout errors, associated with imperfect measurements and erroneous assignment of the outcome, which can be modeled, for example, as bit flip channels.

`qasm_simulator`[99], making use of the

`NoiseModel.from_backend`function (see Materials and Methods, Section 5). The noise parameters are derived from the calibration data of the

`ibmq_kolkata`27-qubit quantum processor, for which a quantum volume of 128 is reported. For simplicity, we construct a uniform noise model by using average parameters across the qubit register (${T}_{1}\sim 135$$\mathsf{\mu}$s, ${T}_{2}\sim 125$$\mathsf{\mu}$s, single-qubit gate error $\sim 2.5\times {10}^{-4}$, 2-qubit gate error $\sim 8\times {10}^{-3}$, readout error $\sim {10}^{-2}$). In all noisy simulations, we use the SPSA classical optimizer [100], which, due to its stochastic nature, is particularly effective in presence of fluctuations and errors affecting the evaluation of the variational energy. Moreover, we increase the stability and accuracy of all energy evaluations and observations made on the noisy quantum circuits by applying the measurement error mitigation protocol provided in the Qiskit Ignis framework [99]. This is based on a calibration matrix, which is first constructed via preparation and measurement experiments on computational basis states and later used as a filter on noisy measurement outcomes [66].

#### 2.3. Finite-Size and Parity Effects

## 3. Dynamical Correlation Functions

`ibmq_bogota`IBM Quantum processor are shown in Figure 5 (light colors), for both real and imaginary parts, compared with exact values (continuous lines). After applying the PaS correction (dark colors), the agreement with experiments on the real quantum hardware using 5 qubits is very good. Notice, in particular, how the error mitigation strategy helps to recover the correct amplitude and phase of the oscillations for ${\mathcal{C}}_{12}$ cross-correlations.

## 4. Discussion and Conclusions

## 5. Materials and Methods

#### Simulations

`statevector_simulator`backend was employed, thus solving for the exact output states and observables via linear algebraic manipulations. Noisy simulations were instead executed with the

`qasm_simulator`backend, which includes the effect of statistical sampling on the observed results, mimicking the real measurement process. In all calculations, each circuit was repeated ${n}_{shot}=8192$ times to reconstruct output averages. The hardware noise was modelled with the

`NoiseModel.from_backend`method [99], which can make use of calibration data from real IBM Quantum devices, implementing thermal relaxation and decoherence effects based on the real physical parameters (${T}_{1}$, ${T}_{2}$, frequency and temperature) found on the processor. Gate errors are associated also with local depolarizing channels, whose strength is tuned in such a way that, when combined with thermal noise and taking into account the gate durations, the effective total noise matches the hardware gate infidelities reported in the calibration data. Moreover, readout errors are included with local bit flip channels. We also notice that fidelity tests were performed, in the case of noisy simulations, by operating the

`qasm_simulator`in density matrix mode, thus extracting the simulated noise-affected quantum state (before measurement) to be compared, e.g., with ideal ground states obtained from exact diagonalization.

`statevector_simulator`backend and SPSA (Simultaneous Perturbation Stochastic Approximation algorithm) for noisy simulations. The latter (which is a stochastic gradient-free optimizer) works better in the presence of noise [106]. Executions on real quantum hardware were performed on IBM Quantum devices, made available online for remote access via quantum-computing.ibm.com. These processors are based on superconducting technology, with fixed-frequency transmon qubits, connected to superconducting resonator waveguides for control and readout via microwave pulses [59,107]. The native 2-qubit entangling CNOT operation is obtained via microwave-activated cross resonance interactions [108], mediated by dedicated waveguides connecting neighbouring qubits in linear chains or (portions of) heavy hexagonal lattices.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Statevector simulations on four- (

**a**,

**b**) and six- (

**c**,

**d**) spin rings. (

**a**) Ground state energy VQE results using hardware-heuristic (HA, dark-green points) and physically motivated (PMA, light-green stars) ansatzes for different values of $B/J$. PMA requires only one layer (4 parameters), compared to the 3 or 2 layers needed by HA to converge in the two magnetic-field regions (yielding 16 and 12 parameters, respectively). Each point is the minimal energy solutions of five independents tests with initial parameters randomly set. Inset: corresponding state fidelity. Both ansatz converge to exact solutions (black lines calculated numerically) with high precision. (

**b**) Computed energy as a function of VQE iterations for two different values of $B/J$ belonging to the two distinct regions: low $B/J=0.4$ (blue-tones) and intermediate $B/J=3.2$ (red-tones). COBYLA has been used as classical optimizer. (

**c**,

**d**) As for panel a,b) for a six-spin ring. We used 2 layers (4 parameters) with different initial state for PMA and 5 (24 parameters) or 4 (20 parameters) layers for HA to achieve high precision. For HA, ten independent tests were performed.

**Figure 2.**$N=4$ spin ring noisy simulations, with a custom noise model derived from IBM Quantum chips with Quantum Volume 128. The VQE configuration in terms of maximum number of iterations and number of layers of the ansatzes reflects the configuration used for noiseless statevector calculations of Figure 1. (

**a**) Ground state energy VQE results using heuristic (HA, dark-green) and Heisenberg (PMA, light-green) ansatzes for different values of $B/J$. Inset: related ground-state fidelity. (

**b**) Observables computed on the ground state computed by VQE: longitudinal magnetization $M={\sum}_{i}{s}_{i}^{z}$ and total spin ${S}^{2}$ expectation value. Both ansatzes perform similarly in presence of noise. (

**c**) Ground state energy convergence as a function of VQE iterations for two different values of $B/J$, belonging to the two distinct regions: low $B/J=0.1$ (blue-tones) and intermediate $B/J=2.7$ (red-tones). SPSA has been used as classical optimizer. The horizontal lines indicate the exact solutions. Remarkably, PMA needs almost one order of magnitude less iterations to converge. (

**d**) Expectation value of local spin observables ${s}_{i}^{z}$ and ${s}_{i}^{z}{s}_{i+1}^{z}$. The black dots indicate the exact solution. Radar plots highlight that both ansatzes reconstruct the correct translational invariance of the target model.

**Figure 3.**$N=6$ spin ring noisy simulations, with a custom noise model derived by IBM Quantum chips with Quantum Volume 128, analogous to the $N=4$ case reported in Figure 2. Ground state energy VQE results with corresponding fidelity (

**a**), observables (

**b**,

**d**) and energy convergence (

**c**). Results are generally worse compared to Figure 2, due to the increased complexity of the target model, but the PMA better reconstructs the ground state, with the correct symmetry (see observables in panels (

**b**,

**d**)).

**Figure 4.**Finite size and parity effects. Expectation value of local spin operators $\langle {s}_{i}^{z}\rangle $ computed on the ground state obtained from noisy VQE simulations at low (

**a**) and intermediate magnetic field (

**b**) for a different number of sites (N) and topology (open vs closed chains). Dashed lines indicate exact solutions, generally in good agreement with simulations.

**Figure 5.**Real (

**a**–

**c**) and Imaginary (

**d**–

**f**) parts of the dynamical correlation functions ${\mathcal{C}}_{ij}^{xx}$ for $i=j=1,2$ (

**top**panels) and $i=2,j=1$ (

**bottom**) on a $N=4$ closed Heisenberg ring, in the large-field region. Phase and scale correction, as in Ref. [89], has been applied to the raw results computed on the

`ibmq_bogota`IBM Quantum processor, obtaining a very good agreement with exact results.

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**MDPI and ACS Style**

Crippa, L.; Tacchino, F.; Chizzini, M.; Aita, A.; Grossi, M.; Chiesa, A.; Santini, P.; Tavernelli, I.; Carretta, S.
Simulating Static and Dynamic Properties of Magnetic Molecules with Prototype Quantum Computers. *Magnetochemistry* **2021**, *7*, 117.
https://doi.org/10.3390/magnetochemistry7080117

**AMA Style**

Crippa L, Tacchino F, Chizzini M, Aita A, Grossi M, Chiesa A, Santini P, Tavernelli I, Carretta S.
Simulating Static and Dynamic Properties of Magnetic Molecules with Prototype Quantum Computers. *Magnetochemistry*. 2021; 7(8):117.
https://doi.org/10.3390/magnetochemistry7080117

**Chicago/Turabian Style**

Crippa, Luca, Francesco Tacchino, Mario Chizzini, Antonello Aita, Michele Grossi, Alessandro Chiesa, Paolo Santini, Ivano Tavernelli, and Stefano Carretta.
2021. "Simulating Static and Dynamic Properties of Magnetic Molecules with Prototype Quantum Computers" *Magnetochemistry* 7, no. 8: 117.
https://doi.org/10.3390/magnetochemistry7080117