Algorithmiq Announces Majorana Propagation and Establishes the Case for Smarter Quantum Workflows

Insider Brief:

• Algorithmiq researchers introduced Majorana Propagation, a classical simulation framework for Fermionic circuits used in quantum chemistry and condensed matter physics.

• The method uses a truncation strategy based on Majorana monomial length to simplify simulations while preserving accuracy, enabling efficient evaluation of variational quantum circuits.

• Benchmarking on a clinically relevant molecule showed that MP achieves errors below chemical precision and outperforms tensor network methods in both speed and scalability.

• MP serves as both a classical simulator and a hardware-aware compiler, supporting hybrid quantum-classical workflows and reducing reliance on quantum hardware for optimization.

The quest to bring forward the practicality of quantum computation is anything but straightforward. While hardware improvements such as fidelity, speed, and quantum error correction often dominate the discourse, another strategy can be just as impactful, especially when used in tandem–simplifying quantum computations through better classical algorithms.

In a recently published arXiv preprint, a team of Algorithmiq researchers introduces Majorana Propagation, an algorithmic framework for the classical simulation of Fermionic circuits. Fermionic systems are essential for simulating problems in quantum chemistry and condensed matter physics, where the behavior of electrons—the most common Fermions—plays a central role.

Quantum computers are expected to eventually handle these simulations natively. But until that happens, and even after, accurate classical simulations will remain useful for benchmarking, pre-training, and guiding quantum algorithms. MP provides a general-purpose classical simulator tailored for Fermionic circuits, especially those used in variational algorithms.

Why Simulate Fermions?

Many quantum algorithms, especially those for simulating molecules or materials, ultimately depend on how well we can approximate a system’s ground state. This is often done with variational quantum eigensolvers, which iteratively adjust a quantum circuit to minimize the estimated energy of a target Hamiltonian.

However, these circuits must be initialized with a good reference state, one that overlaps notably with the true ground state. Finding such states is computationally difficult, especially for systems with strong correlations. Some methods run the entire optimization loop on quantum hardware, but this can be costly, error-prone, and inefficient.

That’s where classical simulation steps in. If a classical method can simulate the quantum circuit well enough, the optimization can be carried out entirely on a classical machine, and reserve quantum hardware for execution, not exploration.

Majorana Propagation: Truncation for Simplicity

MP is inspired by an earlier technique called Pauli Propagation, which simulates spin systems by tracking the evolution of Pauli strings (tensor products of Pauli operators) and discarding those that contribute negligibly to final outcomes. MP extends this idea to Fermionic systems, which are described not by Pauli operators but by Majorana operators—a combination of Fermionic creation and annihilation operators with favorable algebraic properties.

In MP, operators like observables are expressed as combinations of Majorana monomials—ordered products of Majorana operators. These monomials evolve under the action of a Fermionic circuit, and during this evolution, new monomials can be generated. To keep the simulation tractable, MP uses a truncation strategy. It discards monomials that are unlikely to contribute to the final expectation value.

The main innovation is truncating based on monomial length, a measure of how many Majorana operators appear in a term. According to the authors, high-length monomials almost never contribute meaningfully to the final result when tracing against a Fock basis state. The paper proves that their contribution is exponentially suppressed, and that once generated, such monomials are unlikely to “flow back” into the important low-length subspace. This makes aggressive early truncation both safe and efficient.

Benchmarking MP on Strongly Correlated Molecular Systems

To benchmark MP, the team applied it to simulate variational circuits for the molecule TLD1433, a clinically relevant, strongly correlated system. Using an adaptive strategy inspired by ADAPT-VQE, they constructed circuits of increasing complexity across three active space sizes: 28, 40, and 52 Fermionic modes.

Results show that MP can simulate expectation values with errors below chemical precision (1.6 millihartree) using only short monomials (e.g., cutoff at length 4). The accuracy improves nearly exponentially with the cutoff length. For small systems, results match exact statevector simulations. For larger systems where exact methods become intractable, MP circuits converge toward high-fidelity approximations of ground states when compared to DMRG benchmarks, which is a state-of-the-art classical method in quantum chemistry.

In terms of run time, MP outperforms existing classical simulation tools. For the 52-mode system, MP completes in a few minutes where tensor network methods (such as MPS) fail to converge within 24 hours. Importantly, most of MP’s time cost is front-loaded in a pre-processing step. Once complete, re-evaluating the circuit at different parameter values is extremely fast—ideal for variational training loops.

Unlike some classical approximations, MP is not limited to low-entanglement or free Fermion models. Because it dynamically builds the relevant algebra on the fly, it can simulate systems in exponentially large operator spaces. And unlike Pauli-based methods, it doesn’t require mapping Fermions to qubits—a process that often introduces non-locality and overhead.

Classical Support for a Practical Quantum Future

While MP is a classical algorithm, its outputs can be translated to quantum hardware through standard Fermion-to-qubit mappings such as Jordan-Wigner or Bravyi-Kitaev. In that sense, it functions as both a simulator and a hardware-aware circuit compiler, capable of preparing circuits that are near-optimal reference states for algorithms like quantum phase estimation.

According to the research team, MP may also be valuable in hybrid quantum-classical workflows, where it offloads part of the workload from the quantum device. This could be especially useful for near-term systems with limited depth and connectivity. Even in the fault-tolerant future, classical methods like MP will remain useful for pre-training, optimization, and analysis.

In the end, we always return to the sum of parts. The path to practicality will not rely on hardware alone. Classical tools that simplify or partially offload quantum computation are useful, both in the near term and beyond. Majorana Propagation exemplifies such a tool, and one that uses the structure of Fermionic systems to perform fast, accurate simulation without heavy quantum resources.

Contributing authors on the study include Aaron Miller, Zoë Holmes, Özlem Salehi, Rahul Chakraborty, Anton
Nykänen, Zoltán Zimborás, Adam Glos, and Guillermo García-Pérez.