Simulating Supersymmetric Quantum Mechanics Using Variational Quantum Algorithms

This paper presents a Variational Quantum Eigensolver (VQE) approach for simulating supersymmetric quantum mechanics and studying spontaneous supersymmetry breaking, featuring an adaptive ansatz construction to reduce resource demands and demonstrating preliminary results on real IBM quantum devices with error mitigation.

The Gist

The Big Picture: Finding the "Zero Point" in a Noisy World

Imagine you are trying to find the absolute lowest point in a vast, foggy mountain range. In physics, this "lowest point" is called the ground state, and its energy level tells us if a system is stable or if it's about to break apart.

The scientists in this paper are studying Supersymmetry, a fancy theory that suggests every particle has a "super-partner" (like a boson and a fermion dancing together). They want to know: Do these partners stay in sync, or do they break apart?

• If they stay in sync: The energy is zero. (Supersymmetry is preserved).
• If they break apart: The energy is positive. (Supersymmetry is spontaneously broken).

The Problem: The "Foggy Map" (The Sign Problem)

Traditionally, physicists use supercomputers to simulate these systems using a method called "Monte Carlo." Imagine trying to map a mountain range by taking random steps.

• The Issue: In this specific type of physics, the "map" has a Sign Problem. It's like trying to navigate a foggy mountain where half the steps are marked with a plus sign (+) and half with a minus sign (-). When you add them up, they cancel each other out, leaving you with zero information. It's like trying to hear a whisper in a hurricane.

The Solution: The Quantum Compass

Since classical computers get lost in the fog, the authors use a Quantum Computer. Think of a quantum computer not as a calculator, but as a magical compass that can naturally feel the shape of the mountain without getting confused by the plus/minus signs.

However, current quantum computers are like noisy, old compasses. They are "NISQ" devices (Noisy Intermediate-Scale Quantum). They are small, they make mistakes (noise), and they can't hold complex maps for long.

The Tool: The Adaptive-VQE (The Smart Builder)

To find the lowest energy point on this noisy quantum compass, they use an algorithm called VQE (Variational Quantum Eigensolver).

The Analogy: Building a LEGO Castle
Imagine you need to build a LEGO castle that perfectly matches a specific shape (the ground state).

1. The Old Way: You might try to build a massive, complex castle with thousands of bricks immediately. But on a shaky table (noisy hardware), the whole thing collapses, or it takes too long to build.
2. The New Way (Adaptive-VQE): The authors created a smart builder.
   • It starts with a tiny, simple structure (just a few bricks).
   • It checks: "Does this look like the target shape?"
   • If not, it asks: "Which single brick, if I add it next, will make the shape look most like the target?"
   • It adds that specific brick.
   • It repeats this process, adding only the most helpful bricks, until the shape is good enough.

This "Adaptive" approach is crucial because it avoids building unnecessary, complex structures that would break on the noisy quantum hardware.

The Experiment: Three Different Landscapes

They tested this builder on three different "landscapes" (Superpotentials):

1. Harmonic Oscillator (HO): A smooth, single bowl. (Easy to build, stays stable).
2. Anharmonic Oscillator (AHO): A bowl with a bump in it. (A bit harder, but still stable).
3. Double Well (DW): A landscape with two bowls separated by a hill. (This is the tricky one where the system might break apart).

The Results:

• On Simulators (Perfect World): Their smart builder worked perfectly. It found the exact shape of the ground state for all three landscapes, even as the maps got bigger.
• On Real IBM Quantum Computers (The Noisy World): This is where reality hit.
  • Even for the simplest landscape, the "noisy compass" gave results that were off by a noticeable amount.
  • They tried Error Mitigation (like putting a filter on the compass to reduce static). This improved the accuracy significantly (sometimes by 90x!), but it made the process take 4 times longer and cost much more "computer time."

The Takeaway: "Good Enough" is Better than "Perfect but Broken"

The main lesson of the paper is about truncation.

• The smart builder (Adaptive-VQE) showed that the first few bricks it added were the most important. The later bricks added very little value but added a lot of noise.
• The Strategy: Instead of building the full, complex castle, they decided to stop after the first four bricks.
• Why? On a noisy quantum computer, a small, simple structure that is "mostly right" is actually better than a huge, complex structure that is "completely wrong" because it collapsed under the noise.

What's Next?

The authors are now looking at even more complex models (the Wess-Zumino model), which are like trying to map an entire continent instead of a single mountain. Their current quantum compass isn't strong enough yet. They are exploring new methods (like SKQD) that might be more resilient to noise, perhaps by using the "smart builder" just to get a rough sketch, and then using a different technique to refine the details.

In summary: They built a smart, step-by-step LEGO builder that knows how to work on a shaky table. They proved it works in theory, but on real hardware, they learned that sometimes you have to build a smaller, simpler model to get a usable result.

Technical Summary

1. Problem Statement

The paper addresses the challenge of studying Spontaneous Supersymmetry Breaking (SSB) in lattice field theories.

• The Sign Problem: Standard Monte Carlo methods for lattice field theories rely on importance sampling of the Euclidean path integral. However, for real-time dynamics or systems with a vanishing Witten index (a signature of SSB), the path integral weight becomes complex or vanishes, leading to a severe "sign problem" that renders classical Monte Carlo simulations infeasible.
• Hamiltonian Approach: To avoid the sign problem, the authors focus on the Hamiltonian formalism in 0+1D Supersymmetric Quantum Mechanics (SQM). In this framework, SSB is directly linked to the ground-state energy ($E_0$):
  • If $E_0 = 0$, supersymmetry is preserved.
  • If $E_0 > 0$, supersymmetry is spontaneously broken.
• Classical Limitations: While the Hamiltonian formalism avoids the sign problem, the Hilbert space grows exponentially with system size, making exact diagonalization impossible for large systems on classical computers.
• Quantum Opportunity: Quantum computers can naturally represent this Hilbert space with polynomial resources. However, current Noisy Intermediate-Scale Quantum (NISQ) devices are limited by noise, circuit depth, and gate errors, necessitating robust algorithms.

2. Methodology

The authors employ Variational Quantum Eigensolver (VQE) techniques, enhanced by a novel adaptive construction strategy.

A. Theoretical Framework

• Model: SQM with one bosonic and one fermionic degree of freedom.
• Hamiltonian: $H = \frac{1}{2}(\hat{p}^2 + [W'(\hat{q})]^2 - W''(\hat{q})[\hat{b}^\dagger, \hat{b}])$, where $W(\hat{q})$ is the superpotential.
• Superpotentials Tested:
  1. Harmonic Oscillator (HO): $W \propto \hat{q}^2$ (Expected to preserve SUSY).
  2. Anharmonic Oscillator (AHO): $W \propto \hat{q}^2 + \hat{q}^4$ (Expected to preserve SUSY).
  3. Double Well (DW): $W \propto \hat{q}^2 + \hat{q}^3 + \hat{q}$ (Expected to exhibit SSB).
• Regularization & Mapping:
  • The infinite-dimensional bosonic Hilbert space is truncated to $\Lambda$ modes.
  • Bosons are encoded in the Fock basis using $N_b = \log_2(\Lambda)$ qubits.
  • Fermions are mapped to a single qubit using the Jordan-Wigner transformation.

B. Adaptive-VQE (AVQE) Algorithm

To overcome the limitations of fixed ansätze (which are either too shallow to capture the ground state or too deep for noisy hardware), the authors propose an Adaptive-VQE algorithm:

1. Operator Pool: A predefined set of gates $\{R_Y, R_Z, CRY\}$ (single-qubit rotations and controlled rotations).
2. Iterative Construction:
   • Initialize the state in a basis state with non-zero overlap with the expected ground state (e.g., $|10\dots0\rangle$ for HO/AHO, $|00\dots0\rangle$ for DW).
   • Calculate the gradient of the energy expectation value for every operator in the pool.
   • Append the operator with the largest gradient magnitude to the ansatz.
   • Re-optimize parameters (using the previous optimal values as initialization for existing gates).
   • Repeat until the energy change falls below a threshold.
3. Truncation Strategy: Recognizing that NISQ devices suffer from gate noise, the authors observe that the initial gates added by the algorithm have the most significant impact on energy. They propose truncating the ansatz to the first few gates (specifically 4 gates) to balance expressivity with noise resilience.

3. Key Contributions

• Adaptive Ansatz Construction: Development of an AVQE algorithm tailored for SQM that builds problem-specific circuits iteratively, reducing the number of variational parameters compared to generic hardware-efficient ansätze.
• Systematic Pattern Discovery: Identification of consistent patterns in the optimal gate sequences as the bosonic cutoff $\Lambda$ increases. This allows for the extrapolation of ansätze to larger system sizes.
• Truncation for NISQ: Demonstration that truncating the adaptive ansatz to a small number of gates (e.g., 4) provides the optimal trade-off between accuracy and noise susceptibility on real hardware.
• Real Hardware Benchmarking: Execution of VQE circuits on IBM Quantum devices (ibm_torino and ibm_kingston) with and without error mitigation, providing a realistic assessment of resource costs and accuracy.

4. Results

Classical Simulations (Statevector)

• The AVQE successfully reconstructed ground states for HO, AHO, and DW potentials across various $\Lambda$ (up to 64).
• Accuracy: For small $\Lambda$, the VQE energy matched exact diagonalization results to machine precision.
• Scaling: As $\Lambda$ increased, the number of gates ($N_{gates}$) increased, but the structure of the initial gates remained consistent.
• Optimization Challenges: For large $\Lambda$, the classical optimizer struggled to converge due to the increasing number of parameters, causing a slight decline in precision.

Real Hardware Results (IBM QPUs)

• Accuracy Limits: Without error mitigation, the energy error $|\Delta E|$ was roughly $10^{-2}$ to $10^{-1}$, significantly worse than classical simulations ($10^{-3}$ or better).
• Error Mitigation:
  • Resilience Level 1 (TREX): Improved accuracy by $\approx 7\times$ for the AHO case.
  • Resilience Level 2 (Zero-Noise Extrapolation + Gate Twirling): Improved accuracy by $>90\times$ for AHO.
  • Cost: Error mitigation increased QPU usage per job by $2.5\times$ (Level 1) and $4\times$ (Level 2).
• Resource Scaling: Increasing $\Lambda$ increased the number of Pauli terms ($N_P$) and variational parameters ($N_\theta$). The number of parameters ($N_\theta$) had a more significant impact on QPU usage than the number of Pauli terms.
• Overhead: Significant overhead (approx. 3 seconds per job) was observed even for trivial circuits, highlighting the current limitations of cloud-based QPU access.

5. Significance and Future Directions

• Feasibility in NISQ Era: The work demonstrates that while full-scale SQM simulations are currently limited by noise and resource costs, truncated adaptive ansätze offer a viable path forward for near-term devices.
• Diagnostic Power: The method provides a clear, noise-resilient diagnostic for SSB by measuring ground-state energy, bypassing the sign problem inherent in Monte Carlo methods.
• Future Work (Wess-Zumino Model): The authors are extending this work to the 1+1D Wess-Zumino (WZ) model, which is significantly more complex.
  • Current VQE approaches struggle with the WZ model due to the explosion in qubits and Pauli strings.
  • New Direction: The team is exploring the Sample-based Krylov Quantum Diagonalization (SKQD) algorithm. This hybrid approach performs pre/post-processing classically and only samples basis states on the QPU, potentially offering better noise resilience.
  • Hybrid Strategy: They propose using the AVQE to generate a good initial state (even if truncated) to ensure non-zero overlap with the ground state for the SKQD algorithm.

Conclusion

The paper successfully bridges the gap between theoretical lattice field theory and practical quantum computing. By combining an adaptive ansatz construction with strategic truncation and error mitigation, the authors demonstrate a pathway to simulate supersymmetric systems on current hardware, laying the groundwork for more complex 1+1D field theory simulations in the future.