Quantum Algorithm Tames Complex 27-Qubit Systems

Scientists are increasingly focused on optimising the Variational Quantum Eigensolver (VQE) to accurately simulate complex quantum systems, particularly given the limitations of current noisy intermediate-scale quantum (NISQ) technology. Ashutosh P. Tripathi and Nilmani Mathur, from the Department of Theoretical Physics at the Tata Institute of Fundamental Research, alongside Vikram Tripathi and colleagues, have investigated the performance of various quantum ansatzes when applied to the transverse-field Ising model (TFIM) across one, two, and three dimensions, utilising systems of up to 27 qubits. This research is significant because it benchmarks the EfficientSU2 and Hamiltonian Variational Ansatz (HVA) methods, including a symmetry-broken variant, using metrics such as energy variance and spin correlations, offering crucial insights into ansatz expressivity and fidelity for tackling strongly correlated systems and advancing the field of quantum simulation.

For decades, accurately modelling complex materials has remained a major challenge for physicists and computer scientists. This to better use the power of emerging quantum computers for materials discovery. However, the effectiveness of VQE hinges on its ability to accurately prepare the ground state of a system, a task that becomes exceptionally difficult when dealing with highly entangled or degenerate states.

Recent work addresses this challenge by rigorously testing VQE’s performance on the Transverse-Field Ising Model (TFIM), a widely used benchmark in condensed matter physics. To achieve reliable results requires careful consideration of the quantum circuit used within the VQE framework, known as the ansatz. A key focus lies in characterising the entanglement properties of the system, which are known to change dramatically as the strength of the transverse field is varied.

By examining quantities like entanglement entropy and spin correlations, scientists gain insight into the underlying quantum behaviour of the model. Simulations were performed on systems of up to 27 qubits. Representing a substantial step forward in the scale of problems that can be addressed with VQE. The ability to accurately capture the highly entangled ground state when the transverse field is weak is vital for understanding quantum magnetism and related phenomena.

VQE offers a potential pathway to simulate these systems on future quantum computers, opening up possibilities for designing new materials with tailored properties. At the heart of this progress is a quantitative measure of ansatz expressivity, the frame potential. This assesses how well a circuit can generate a diverse set of quantum states. Simulations were conducted to explore the transverse-field Ising model (TFIM) across one, two. Three dimensions, allowing for detailed analysis of entanglement properties. To refine the methodology, a symmetry-breaking layer incorporating Rz gates was added to the HVA, creating the HVA with Symmetry Breaking (HVA-SB) circuit.

This addition aimed to improve the overlap state for the Hamiltonian, potentially enhancing the accuracy of ground state energy calculations. Expressivity, defined as the ability of a quantum circuit to approximate random states via parameter variation, was assessed using the frame potential, a measure quantifying state overlap under uniform parameter sampling.

Histograms generated from 10 4 independent samples revealed that the HEA consistently exhibited the lowest frame potential values, suggesting it possessed the highest expressivity among the tested ansatzes and a greater capacity to represent a diverse range of quantum states. By systematically comparing these distinct ansatzes, researchers aimed to highlight the interaction between circuit architecture and the classical optimisation process within the VQE framework.

Each circuit was designed to address specific challenges in representing the highly entangled ground state of the TFIM. These parametric quantum circuits (PQCs) allowed for tunable parameters. Meanwhile, the potential to achieve the ground state through optimised parameter values — the HVA’s construction directly reflects the Hamiltonian terms, offering a physics-inspired approach to wavefunction preparation. Meanwhile, the HEA prioritized ease of implementation on existing hardware, balancing expressivity with practical considerations.

At the same time, the inclusion of the symmetry-breaking layer in HVA-SB was a deliberate attempt to overcome limitations associated with degenerate ground states, potentially improving the convergence of the classical optimisation algorithm. Through carefully controlling these methodological aspects, The project team sought to establish a strong and reliable platform for benchmarking ansatz performance.

Variational layer scaling and energy variance behaviours in the Transverse-Field Ising Model

Simulations utilising the Transverse-Field Ising Model reached a maximum qubit count of 27 — demonstrating the capacity of this effort to explore relatively complex quantum systems. Observed in both one- and two-dimensional lattices.

Specifically, the energy variance for the 1D TFIM exhibited distinct behaviours among the tested ansatzes, hardware-efficient (HEA). HEA, with a parameter count proportional to system size, showed gradual performance improvement with increasing circuit depth. Conversely, HVA and HVA-SB displayed a sharp transition from inaccurate to accurate energy estimation, stemming from the restricted Hilbert space explored by Hamiltonian-based circuits.

Inclusion of symmetry-breaking layers in HVA-SB partially addressed this limitation, enabling access to parity-violating states and smoother improvements. For the 1D TFIM, spin correlation exhibited a pronounced change near the critical point. In turn, the large single-site entanglement entropy of 1, measured in units of ln 2 for HVA. Arises from finite system sizes where the ground state lacks spontaneous symmetry breaking.

Both HEA and HVA-SB captured qualitative changes in curvature near criticality, underlining the importance of ansatz selection for resolving phase-transition signatures. Meanwhile, to extend to a two-dimensional 4×4 TFIM lattice, optimisation became more unstable, with HVA performing poorly in low-entanglement regimes. At this scale, HEA underestimated entropy in highly entangled, low-field regions, tending towards spontaneously symmetry-broken states.

Analysis of the three-dimensional TFIM involved restricting the ansatz to real amplitudes by removing Rz rotations, reducing parameters while maintaining a smooth optimisation field. Initialising parameters using optimised values from nearby transverse-field points also reduced optimisation time. As a result, ground-state energies remained approximately independent of lattice size. Meanwhile, entanglement entropy continued to exhibit clear signatures of critical behaviour.

Quantum simulation fidelity constrained by optimisation challenges

At the same time, scientists are beginning to map the limits of what is currently achievable with near-term quantum computers, not by chasing ever-larger qubit counts, but by carefully examining how well existing algorithms perform as complexity increases. For years, the field has been driven by the promise of exponential speedups. Often overlooking the very real difficulties of building and controlling even a modest number of qubits. Scientists are acknowledging that simply adding more qubits doesn’t automatically translate into better results if the underlying algorithms struggle with noise and optimisation challenges.

Attention is turning to refining existing algorithms and developing strategies to mitigate these issues — a shift that feels less like a revolution and more like a necessary course correction. The challenge lies in the fact that many materials exhibit “strong correlation”, where electrons interact in ways that defy simple approximations, and accurately modelling these systems requires quantum computers to represent the entanglement between particles. A task that demands both expressive power from the quantum circuit and a reliable method for finding the lowest energy state.

By testing different circuit designs, this effort highlights the trade-offs involved, showing that increased expressivity can come at the cost of optimisation stability. Once more sophisticated algorithms and optimisation techniques emerge, the potential for designing new materials with tailored properties will become more realistic. This effort demonstrates the ability to handle relatively complex simulations, offering a glimpse of what might be possible with near-term devices.

Significant hurdles remain, including the need for better error correction and more efficient ways to scale up these calculations. Beyond this specific model, the lessons learned will be applicable to a wide range of quantum simulations, from drug discovery to fundamental physics.