Fulqrum speeds up the classical step in quantum-assisted energy calculations

The classical calculation behind a quantum workflow

A system’s Hamiltonian describes its energy and interactions. Finding its lowest eigenvalue—the ground-state energy—is a central task in quantum chemistry and condensed-matter physics.

In sample-based quantum diagonalization, a quantum processor is measured to obtain bit strings representing candidate configurations. These strings define a smaller subspace within the much larger set of possible states. A classical computer then represents the Hamiltonian in that subspace and solves the resulting eigenvalue problem. The two stages have different computational costs; the classical calculation can require substantial time and memory as the subspace grows.

Fulqrum is designed to handle this classical eigenvalue step. It takes a Hamiltonian and a subspace of bit strings and prepares calculations for a separate classical eigensolver. It does not generate quantum samples. The paper also describes RAMPS, a classical procedure for generating and refining subspaces from an initial set of configurations; the authors use it to prepare subspaces in some molecular tests.

Organizing the calculation

The framework is designed for both qubit models and fermionic systems, including those used to describe electrons in molecules. It uses a shared representation that includes projection and ladder operators. For fermionic problems, an extended Jordan–Wigner transformation converts the operators into a form the framework can process while retaining structure useful for its calculations.

Fulqrum groups and sorts Hamiltonian terms so it can skip terms that contribute zero for a given bit string and reduce unnecessary subspace lookups. It stores bit strings as bit sets and uses a hash map for indexing. This avoids a fundamental length limit imposed by some fixed-width integer representations, though it does not remove practical limits on hardware, memory or runtime.

The software can build a sparse matrix or calculate matrix-vector products without storing that matrix. It supplies these representations to existing classical eigensolvers, giving users a choice of solver. Which approach is more efficient depends on the problem and the resources available.

What the tests found

The authors tested Fulqrum on a spin-chain model and several molecular Hamiltonians, comparing runtime and memory use with other software on specified computer systems. In spin-chain tests using sampled bit strings, they report that Fulqrum’s Hamiltonian-projection step was more than an order of magnitude faster than the Qiskit add-on they tested, with the performance gap growing across the larger tested subspaces.

For molecular examples, the comparisons included Dice and SBD. In the reported tests, Fulqrum’s matrix-free mode was between three and 38 times faster than the next-best competitor, depending on the molecule and subspace. In a scaling test for Fe₄S₄, the authors report that the largest case ran about 100 times faster and used six times less memory than SBD. They also describe solving a similar 10⁸-dimensional subspace calculation on a workstation, using about 13 GB of free memory in roughly seven minutes, compared with an earlier calculation that used a large supercomputer. These are benchmark-specific results, not predictions for other systems.

The gains are not uniform across settings. Converting molecular input into Fulqrum’s internal representation can add overhead, particularly for smaller subspaces. The authors identify operator copies and memory use as areas for future improvement. Matrix-free calculations avoid storing the sparse matrix, but still need memory for the subspace and repeatedly recompute matrix elements. Their runtime also depends on the number of iterations required by the chosen eigensolver.

What remains uncertain

The paper compares computational performance, not the physical accuracy of the subspaces supplied to the solver. The authors emphasize that accuracy depends on which bit strings are included—a separate issue from how efficiently Fulqrum processes them. RAMPS offers one classical route to preparing or refining a subspace, but the reported results do not establish that its subspaces, or quantum-sampled subspaces generally, will yield accurate answers across broader applications. Faster diagonalization alone does not show that a quantum computer has outperformed classical methods overall.

The findings come from selected benchmarks and computer setups; results may change with the solver, matrix mode, input and hardware. GPU acceleration is not evaluated in this work, according to the paper, which points to a companion study. Testing on a wider range of workloads would help establish how well the reported performance generalizes.

Fulqrum is described in an arXiv preprint, which may not have undergone journal peer review. Its open-source code can support independent inspection and testing, but the claims remain benchmark results reported by the authors. The work presents a potential efficiency improvement for a stage of hybrid calculations, not evidence that practical quantum advantage has been achieved.