Quantum Variational Methods for Supersymmetric Quantum Mechanics

TL;DR

The paper develops quantum variational methods to study a minimal fermion–boson SQM model and detect spontaneous supersymmetry breaking on NISQ hardware. It introduces adaptive AVQE to craft compact, scalable ansätze and employs VQE and VQD to extract ground- and excited-state information, respectively, while carefully managing bosonic truncation artifacts. Results show that problem-tailored, low-gate ansätze can capture SUSY-restoring vs. breaking patterns in statevector simulations, but shot noise and Pauli-string growth pose significant challenges for realistic hardware, motivating future hybrid encodings and more robust excited-state techniques. The study lays groundwork for applying quantum computing to higher-dimensional SUSY QFTs and matrix models, with practical implications for scalable quantum simulations of complex fermion–boson systems.

Abstract

We employ quantum variational methods to investigate a single-site interacting fermion-boson system -- an example of a minimal supersymmetric model that can exhibit spontaneous supersymmetry breaking. Our study addresses the challenges inherent in calculating mixed fermion-boson systems and explores the potential of quantum computing to advance their analysis. By using adaptive variational techniques, we identify optimal ansätze that scale efficiently, allowing for reliable identification of spontaneous supersymmetry breaking. This work lays a foundation for future quantum computing investigations of more complex and physically rich fermion-boson quantum field theories in higher dimensions.

Figures (16)

• Figure 1: An illustration of the system, where the fermionic degree of freedom is represented by $\psi$ and the bosonic degree of freedom by $\phi$. In Eq. (\ref{['eq:H_SQM']}) these correspond to the operators $\hat{q} , \hat{p}$ for the boson, and $\hat{b} , \hat{b}^\dag$ for the fermion.
• Figure 2: An illustration of the digitization of SQM. Left: The starting system, where the fermionic degrees of freedom is represented by $\psi$ and the bosonic one by $\phi$. Right: The digitization, where the symbol consisting of a circle with two lines inside represents a qubit as a two-level system. The fermionic degree of freedom requires a single qubit, while the bosonic degree of freedom is allocated $B$ qubits with $\Lambda =2^B$ bosonic modes.
• Figure 3: An ($n + 1$)-qubit circuit for a generic state $\ket{\Psi} = \ket{f} \ket{b}$. In this representation, the first $B = n$ qubits are used to encode the $2^B$ possible bosonic states, while the last qubit provides the two possible fermionic states.
• Figure 4: The number of Pauli strings, $N_{\text{Paulis}}$, required to encode the Hamiltonian for $N_{\text{qubits}}$ qubits, corresponding to $\Lambda = 2^{N_{\text{qubits}} - 1}$ bosonic modes. The blue points are listed in Table \ref{['tab:full_H_pauli_strings_energybasis']}. The red curve represents the fit function $N_{\text{Paulis}} = 2^{a N_{\text{qubits}} +b}$ for the Harmonic Oscillator. For the Double Well and Anharmonic Oscillator cases, the corresponding fit is $N_{\text{Paulis}} = 2^{a N_{\text{qubits}} + c \log_2(N_{\text{qubits}}) + b}$.
• Figure 5: Comparison of VQE performance for various optimizers, considering 100 independent VQE runs using PennyLane's statevector simulator with a maximum of 10,000 iterations. The top plots show the absolute difference between the ground-state energy from exact diagonalization $E_{\text{exact}}$ and the median energy $E_{\text{median}}$ across the 100 independent runs for an increasing number of bosonic modes $\Lambda$. Plotted data is from converged runs only. The bottom plots show the number of quantum circuit evaluations $N_{\text{evals}}$ aggregated over all 100 VQE runs, including runs that did not converge.