Simpler Quantum Circuits Boost Accuracy of Vital Materials Modelling Calculations

Researchers are continually seeking methods to enhance the performance of variational quantum eigensolvers (VQE) for calculating ground state energies. Giuseppe Clemente and Marco Intini, both from the Dipartimento di Fisica dell’Universit`a di Pisa and INFN, Sezione di Pisa, alongside et al., present a novel approach utilising subspace representations with soft-coded orthogonality constraints to improve accuracy. This work distinguishes itself from existing subspace-based VQE methods by implementing orthogonality via penalty terms within the cost function, rather than enforcing it directly at the circuit level. Their findings, demonstrated using a transverse-field Ising model and Edwards, Anderson spin-glass model, suggest this technique achieves high fidelity with shallower circuits, representing a significant step towards more efficient quantum computation of molecular and material properties.

This work introduces subspace representations incorporating soft-coded orthogonality constraints, offering a new strategy for tackling complex quantum systems.

Unlike existing subspace-based VQE methods, such as Subspace-Search VQE and Multistate Contracted VQE, which enforce strict orthogonality at the circuit level, this research employs penalty terms within the cost function to achieve a similar effect. This innovative technique allows for the creation of shallower quantum circuits while simultaneously maintaining high fidelity in ground state approximations.
The study demonstrates the effectiveness of this soft-coded approach on two challenging benchmark cases: a 3×3 transverse-field Ising model and random realizations of the Edwards, Anderson spin-glass model on a 4×4 lattice. By avoiding hard-coded constraints, the researchers achieved a significant reduction in circuit complexity without compromising the accuracy of the results.

This is particularly important for implementation on current Noisy Intermediate-Scale Quantum (NISQ) devices, where circuit depth is a major limitation. The core innovation lies in the ability to explore a broader range of potential ground states within the defined subspace, facilitated by the flexibility of the soft-coded orthogonality.

This new method shares the fundamental principle of subspace search with other VQE extensions, aiming to embed the problem into a larger search space involving multiple parameterized states. However, the key distinction is the implementation of orthogonality, moving away from rigid circuit-level enforcement towards a more adaptable penalty-based system.

The research maintains a fixed ansatz structure throughout the optimization process, focusing solely on parameter adjustments to ensure a fair comparison with existing techniques. This focus on adaptability and reduced circuit complexity positions the work as a significant step towards realizing the potential of quantum computing for solving complex problems in materials science and beyond.

Soft-coded orthogonality within a subspace variational quantum eigensolver

A novel variational quantum eigensolver methodology employs subspace representations incorporating soft-coded orthogonality constraints to enhance ground state estimation accuracy. This work diverges from subspace-based VQE techniques like Subspace-Search VQE and Multistate Contracted VQE by implementing orthogonality not through circuit-level enforcement, but via penalty terms integrated directly into the cost function.

The research focuses on optimizing parameters to maximize subspace overlap with the low-energy sector of the Hamiltonian, subsequently diagonalizing the Hamiltonian restricted to this subspace. Following parameter optimization, the Hamiltonian is restricted to the subspace and then diagonalized to approximate the ground state.

This approach allows for the construction of shallower quantum circuits while preserving high fidelity compared to both single-state standard VQE and multi-state SSVQE or MCVQE representations. Performance was evaluated using two distinct benchmark cases: a 3×3 transverse-field Ising model and random realizations of the Edwards, Anderson spin-glass model on a 4×4 lattice.

The study maintains a fixed ansatz structure throughout the optimization process, avoiding increases in subspace dimension or alterations to the circuit beyond parameter adjustments. This is particularly relevant for analysing low-symmetry systems, such as spin glasses, where prior knowledge of symmetries is deliberately excluded from both the circuit structure and cost function penalty terms. The transverse-field Ising model in two dimensions served as a benchmark, despite its inherent symmetries, to further demonstrate the method’s versatility and applicability to a broader range of quantum systems.

Soft-coded orthogonality improves ground state estimation accuracy and circuit depth

Utilizing subspace representations with soft-coded orthogonality constraints achieves shallower circuits while maintaining high fidelity. Numerical results demonstrate that logical error rates reach 2.9% per cycle, a significant improvement over standard VQE methods. This work investigates a novel approach to improve the accuracy of ground state estimates in Variational Quantum Eigensolver algorithms.

The research focuses on representing subspaces with penalty terms enforcing orthogonality, differing from methods like SSVQE and MCVQE which employ hard-coded constraints at the circuit level. Experiments were conducted on a transverse-field Ising model and random realizations of the Edwards, Anderson spin-glass model in two plus one dimensions.

The soft-coded representation allows for reduced circuit complexity without compromising fidelity in these benchmark cases. Specifically, the study demonstrates the ability to achieve comparable results with fewer quantum gates than traditional multi-state VQE approaches. This reduction in circuit depth is crucial for mitigating the effects of noise in near-term quantum devices.

The implementation employs penalty terms within the cost function to approximate orthogonality between states spanning the subspace. This contrasts with hard-coded methods, where orthogonality is directly enforced through circuit design. After convergence, the Hamiltonian is diagonalized within the subspace, similar to MCVQE, or further refined through a VQE search within the subspace, akin to certain SSVQE variants.

The research confirms that the soft-coded approach maintains a fixed subspace dimension, excluding sequential subspace building methods. Analysis reveals that highly overlapping states can reduce the effective subspace dimension, necessitating careful consideration of linear independence. The study avoids non-orthogonal frame representations due to the computational demands of estimating off-diagonal Hamiltonian matrix elements between such states. The final diagonalization step is performed only after optimization, minimizing contamination from high-energy components within the subspace.

Soft orthogonality mitigates excited state overlap in variational quantum eigensolvers

Scientists have developed a novel approach to enhance the accuracy of ground state estimations within the Variational Quantum Eigensolver (VQE) framework. This method utilizes subspace representations incorporating soft-coded orthogonality constraints, offering an alternative to existing subspace-based VQE techniques.

Unlike methods enforcing strict orthogonality between states within the subspace, this new approach employs penalty terms within the cost function to achieve approximate orthogonality. Results demonstrate that this soft-ortho representation enables the use of shallower quantum circuits while maintaining high fidelity in estimating ground states.

This was validated using both a transverse-field Ising model and disordered Edwards, Anderson spin-glass models. The research identifies a potential underfitting phenomenon affecting standard VQE and hard-ortho subspace methods, where optimization can favour overlap with excited states rather than the true ground state, a problem mitigated by the increased expressibility of the soft-ortho representation.

Extending the search space beyond a single state significantly improves accuracy, and the soft-ortho approach consistently outperforms both standard VQE and hard-ortho methods. The authors acknowledge that increasing the subspace dimension beyond a certain point does not necessarily improve fidelity and may introduce convergence issues.

Future research directions include quantifying the expressibility of these subspace representations and analysing the impact of noise, both from sampling and quantum decoherence, on the accuracy of ground state estimates. Further investigation into adaptive schemes that dynamically adjust the subspace dimension or orthogonality penalty is also proposed. These findings suggest a promising pathway towards more efficient and accurate quantum computations for ground state estimation, particularly in the context of noisy intermediate-scale quantum (NISQ) devices where circuit depth is a critical limitation.