⚛️ quantum physics

Improving fermionic variational quantum eigensolvers with Majorana swap networks

This paper introduces two Majorana swap network compilation strategies for variational quantum eigensolvers that significantly reduce circuit depth and two-qubit gate counts for simulating fermionic systems, thereby enhancing their feasibility on near-term quantum hardware.

Original authors: D. E. Fisher, S. A. Fldzhyan, D. V. Minaev, S. S. Straupe, M. Yu. Saygin

Published 2026-08-13

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CC BY 4.0

Original authors: D. E. Fisher, S. A. Fldzhyan, D. V. Minaev, S. S. Straupe, M. Yu. Saygin

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to simulate a complex dance party where the guests are tiny, invisible particles called fermions. These particles have a very strict rule: they hate being too close to their own kind and must swap places in a very specific, chaotic way that creates a "sign" or a mood shift every time they move. In the world of quantum physics, this is how electrons behave in molecules, and understanding this dance is the key to designing new medicines, super-efficient batteries, and stronger materials.

To watch this dance on a computer, scientists use a special tool called a quantum computer. However, these machines speak a different language than the dancing particles. They speak in "qubits," which are like simple light switches that can be on, off, or both at once. The problem is that translating the fermions' complex, mood-swapping dance moves into the qubits' language usually requires a massive, tangled web of instructions. It's like trying to direct a dance party by shouting instructions to every single guest from a megaphone at the same time; the instructions get so long and complicated that the computer gets tired and makes mistakes before the dance even finishes. This is the big hurdle scientists face: how to make the translation short, clean, and fast enough for today's noisy, imperfect quantum machines.

This paper introduces a clever new way to organize that translation, acting like a master choreographer who rearranges the dance floor so the guests can swap places without shouting across the room. The authors, a team from Moscow, propose a strategy using "Majorana swap networks." Think of fermions not as single dancers, but as pairs of dancers holding hands (called Majorana operators). Instead of using the standard, bulky method to swap these pairs, the team uses a new type of "swap gate" that is more precise and requires fewer steps. They developed two main tricks. First, for the most complex dance routines (called UCCGSD), they created a cyclic algorithm that shuffles the dancers into the right positions using far fewer moves than before, specifically reducing the "routing" overhead from a cubic to a cubic scaling (though the total number of dance moves remains high). Second, and perhaps more exciting, they tailored a specific network for a popular, streamlined dance routine called k-UpCCGSD.

When they tested these new networks, the results were a significant improvement. On computers where every qubit can talk to every other qubit (all-to-all connectivity), their new method cut the depth of the instruction circuit by about 50% and reduced the number of two-qubit gates by roughly 20%. On more restricted hardware layouts, which look like a grid of 2 rows by N columns (common in real-world devices), the savings were even more dramatic: about 55% fewer steps in the circuit and a 40% drop in the number of entangling gates. The authors also ran simulations with digital noise to mimic real-world errors, and found that their new method was generally more robust, meaning the simulated "dance" stayed closer to the correct answer even when the computer was glitchy. While this doesn't solve every problem in quantum chemistry, it suggests that by changing how we shuffle the quantum information, we can make these simulations much more practical for the hardware we have today.

Technical Summary: Improving Fermionic Variational Quantum Eigensolvers with Majorana Swap Networks

Problem Statement
Accurate simulation of correlated fermionic systems is a primary application for quantum computing, yet it faces significant hurdles on Noisy Intermediate-Scale Quantum (NISQ) hardware. The core challenge lies in mapping nonlocal fermionic operators to qubits. Standard encodings like Jordan–Wigner (JW) and Bravyi–Kitaev (BK) generate Pauli strings with support that grows with the number of fermionic modes ( $M$ ), leading to deep circuits and high two-qubit gate counts. This nonlocality exacerbates error accumulation. While swap networks (SNs) using fermionic swap gates (FSWAP) have been used to enforce effective locality, the cumulative overhead from nonlocal operators in widely used variational ansatzes, such as the Unitary Coupled Cluster (UCC) family, remains substantial.

Methodology
The authors introduce a framework based on Majorana operators and Majorana swap networks (MSNs) to optimize the compilation of variational quantum eigensolvers (VQE).

Majorana Representation: Instead of standard fermionic modes, the approach represents each mode using two Majorana operators ( $\hat{\gamma}$ ), which obey simple anticommutation relations and map naturally to the Pauli group.
Majorana Swap Gates (MSWAP): The authors define MSWAP gates as operations that implement signed permutations of two Majorana operators. Unlike FSWAPs, which exchange entire fermionic modes, MSWAPs act on individual Majorana components. This allows for finer-grained control over operator support and admits compact decompositions (e.g., a single two-qubit Clifford operation for adjacent modes in JW encoding).
Modified Encodings: By applying layers of MSWAP gates, the authors create "modified JW encodings." In these intermediate states, excitation operators (single and double) can be represented by shorter Pauli strings, reducing the number of required entangling gates.
Ansatz Targets: The work focuses on two specific UCC variants:
- UCCGSD: The generalized singles-and-doubles ansatz, containing $O(M^4)$ double-excitation terms.
- k-UpCCGSD: A more compact ansatz restricting double excitations to spatially paired orbitals, often repeated in $k$ layers.

Key Contributions

Optimized Excitation Decompositions:
The authors provide gate constructions for double-excitation rotations that integrate directly with MSWAP layers. By synthesizing rotations in a modified encoding where the Pauli support is compact, they achieve a 40% reduction in circuit depth for the four-qubit core of double excitations compared to the standard Yordanov decomposition (12 CX gates vs. 13 CX gates).
Cyclic Permutation Algorithm for UCCGSD:
A new algorithm is developed to localize all $O(M^4)$ double-excitation terms in the general UCCGSD ansatz.
- Mechanism: It uses a cyclic sweep procedure to move selected modes through an "active list" of fermionic pairs.
- Scaling: The algorithm exposes all required four-mode supports using only $O(M^3)$ auxiliary MSWAP transpositions. This improves upon the $O(M^4)$ auxiliary swap overhead of prior FSN-based approaches.
- Clarification: The authors note that while the routing overhead is reduced to cubic, the total number of variational double-excitation rotations remains $O(M^4)$ . The improvement lies in the efficiency of the routing required to make these terms local.
MSN Circuits for k-UpCCGSD:
For the k-UpCCGSD ansatz, the authors design MSN circuits tailored to a $2 \times N$ qubit layout (separate rows for $\alpha$ and $\beta$ spin orbitals).
- Fusion: Excitation rotations are fused directly into the MSWAP layers, eliminating the need for separate swap and rotation phases.
- Comparison: Compared to the FSN baseline, the MSN approach avoids unnecessary permutations between opposite-spin orbitals.
Noise-Aware Evaluation:
The study evaluates these compilation strategies using depolarizing and correlated error channels, as well as noise models for superconducting and ion-trap devices.

Results
Numerical benchmarks on small molecular instances yield the following performance metrics:

All-to-All Connectivity:
- Circuit Depth: Reduced by approximately 50%.
- Two-Qubit Gate Count: Reduced by approximately 20%.
Restricted $2 \times N$ Connectivity:
- Circuit Depth: Reduced by approximately 55%.
- Two-Qubit Gate Count: Reduced by approximately 40%.
Noise Robustness: The MSN-based compilations generally demonstrate lower or near-lowest energy susceptibility to noise compared to fixed-baseline alternatives (such as standard FSN or standard decompositions), though this is channel-dependent.

Significance and Claims
The paper claims that structural improvements in circuit compilation via Majorana swap networks can significantly mitigate the resource overhead of fermionic simulations on NISQ hardware. The significance is twofold:

Efficiency: The proposed methods reduce the constant factors in circuit depth and gate counts, which is critical when swap and measurement overheads are decisive for algorithm success.
Scalability: The $O(M^3)$ routing overhead for UCCGSD represents an asymptotic improvement over previous $O(M^4)$ routing strategies, even if the total rotation count remains high.
Hardware Compatibility: The specific design of the MSN for $2 \times N$ layouts aligns naturally with existing hardware topologies (e.g., superconducting qubits), offering a practical path to more robust VQE implementations.

The authors conclude that these structural optimizations, combined with noise-aware design, offer a viable pathway to improving the performance of variational quantum eigensolvers on current and near-future quantum devices.

Technical Summary: Improving Fermionic Variational Quantum Eigensolvers with Majorana Swap Networks

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