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arXiv:2509.07855 · Physical Review A

Improving fermionic variational quantum eigensolvers with Majorana swap networks

D. E. Fisher, Suren A. Fldzhyan, Dmitry V Minaev, S. S. Straupe, M. Yu. Saygin

Published Aug 12, 2026• 0 citations
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Abstract

Simulating computationally hard fermionic systems is a promising application of quantum computing. However, mapping nonlocal fermionic operators to qubits often produces deep circuits, rendering such simulations impractical on near-term hardware. We introduce two Majorana swap network compilation strategies for variational quantum eigensolvers that reduce circuit depth and two-qubit gate count. First, we develop a cyclic compilation algorithm that localizes all two-particle interaction terms in a general fermionic Hamiltonian containing up to $\mathcal{O}(M^4)$ such terms using only $\mathcal{O}(M^3)$ auxiliary Majorana-swap transpositions, where $M$ is the number of fermionic modes. Here, the cubic scaling refers to auxiliary routing; a complete UCCGSD ansatz still contains $\mathcal{O}(M^4)$ double-excitation rotations. Second, we design a Majorana swap network for the $k$-UpCCGSD variational ansatz, which is already more compact than UCCGSD. In this setting, our network yields constant-factor reductions of approximately $50$ % in circuit depth and $20$ % in two-qubit gate count under all-to-all connectivity. For the more restricted $2\times N$ connectivity, the reductions are larger --- about $55$ % in circuit depth an $40$ % in gate count. These structural improvements are accompanied by improved robustness in numerical noise simulations on the small molecular instances tested.

DOI
10.1103/kpcf-b3q7

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Simulating computationally hard fermionic systems is a promising application of quantum computing. However, mapping nonlocal fermionic operators to qubits often produces deep circuits, rendering such simulations impractical on near-term hardware. We introduce two Majorana swap network compilation strategies for variational quantum eigensolvers that reduce circuit depth and two-qubit gate count. First, we develop a cyclic compilation algorithm that localizes all two-particle interaction terms in a general fermionic Hamiltonian containing up to O(M4)\mathcal{O}(M^4)O(M4) such terms using only O(M3)\mathcal{O}(M^3)O(M3) auxiliary Majorana-swap transpositions, where MMM is the number of fermionic modes. Here, the cubic scaling refers to auxiliary routing; a complete UCCGSD ansatz still contains O(M4)\mathcal{O}(M^4)O(M4) double-excitation rotations. Second, we design a Majorana swap network for the kkk-UpCCGSD variational ansatz, which is already more compact than UCCGSD. In this setting, our network yields constant-factor reductions of approximately 505050 % in circuit depth and 202020 % in two-qubit gate count under all-to-all connectivity. For the more restricted 2×N2\times N2×N connectivity, the reductions are larger --- about 555555 % in circuit depth an 404040 % in gate count. These structural improvements are accompanied by improved robustness in numerical noise simulations on the small molecular instances tested.
Molecular quantum simulations with the variational quantum eigensolver (VQE) rely on ansatz states that approximate the molecular ground states. These ansatz states are generally defined by parametrized fermionic excitation operators and an initial reference state. Efficient ways to perform fermionic excitations are vital for the realization of the VQE on noisy intermediate-scale quantum computers. Here, we address this issue by first demonstrating circuits that perform qubit excitations, excitations that do not account for fermionic anticommutation relations. We then extend the functionality of these circuits to perform fermionic excitations. Compared to circuits constructed with the standard use of ``CNOTCNOTCNOT staircases'', our circuits offer a linear reduction in the number of CNOTCNOTCNOT gates, by a factor of 2 (8) per single (double) qubit or fermionic excitation, respectively. Our results reduce the requirements for near-term realizations of quantum molecular simulations.
The practical use of many types of near-term quantum computers requires accounting for their limited connectivity. One way of overcoming limited connectivity is to insert swaps in the circuit so that logical operations can be performed on physically adjacent qubits, which we refer to as solving the `routing via matchings' problem. We address the routing problem for families of quantum circuits defined by a hypergraph wherein each hyperedge corresponds to a potential gate. Our main result is that any unordered set of kkk-qubit gates on distinct kkk-qubit subsets of nnn logical qubits can be ordered and parallelized in O(nk−1)O(n^{k-1})O(nk−1) depth using a linear arrangement of nnn physical qubits; the construction is completely general and achieves optimal scaling in the case where gates acting on all (nk)\binom{n}{k}(kn​) sets of kkk qubits are desired. We highlight two classes of problems for which our method is particularly useful. First, it applies to sets of mutually commuting gates, as in the (diagonal) phase separators of Quantum Alternating Operator Ansatz (Quantum Approximate Optimization Algorithm) circuits. For example, a single level of a QAOA circuit for Maximum Cut can be implemented in linear depth, and a single level for 333-SAT in quadratic depth. Second, it applies to sets of gates that do not commute but for which compilation efficiency is the dominant criterion in their ordering. In particular, it can be adapted to Trotterized time-evolution of fermionic Hamiltonians under the Jordan-Wigner transformation, and also to non-standard mixers in QAOA. Using our method, a single Trotter step of the electronic structure Hamiltonian in an arbitrary basis of nnn orbitals can be done in O(n3)O(n^3)O(n3) depth while a Trotter step of the unitary coupled cluster singles and doubles method can be implemented in O(n2η)O(n^2 η)O(n2η) depth, where ηηη is the number of electrons.
The fermionic SWAP network is a qubit routing sequence that can be used to efficiently execute the Quantum Approximate Optimization Algorithm (QAOA). Even with a minimally-connected topology on an n-qubit processor, this routing sequence enables O(n^2) operations to execute in O(n) steps. In this work, we optimize the execution of fermionic SWAP networks for QAOA through two techniques. First, we take advantage of an overcomplete set of native hardware operations [including 150 ns controlled-pi/2 phase gates with up to 99.67(1)% fidelity] in order to decompose the relevant quantum gates and SWAP networks in a manner which minimizes circuit depth and maximizes gate cancellation. Second, we introduce Equivalent Circuit Averaging, which randomizes over degrees of freedom in the quantum circuit compilation to reduce the impact of systematic coherent errors. Our techniques are experimentally validated on the Advanced Quantum Testbed through the execution of QAOA circuits for finding the ground state of two- and four-node Sherrington-Kirkpatrick spin-glass models with various randomly sampled parameters. We observe a ~60% average reduction in error (total variation distance) for QAOA of depth p = 1 on four transmon qubits on a superconducting quantum processor.
Simulating fermionic systems on quantum hardware requires compiling fermionic Hamiltonians into executable quantum circuits. Existing approaches treat each compilation stage independently, applying heuristics with localized objectives that produce circuits with superquartic gate count and depth scaling and compilation times reaching several hours for large instances. We present Accordion, an end-to-end framework that co-designs the fermion-to-qubit mapping with circuit synthesis and hardware routing. Accordion fixes the Jordan Wigner mapping, which despite its higher Pauli weight produces Pauli operators with structural regularity that enables provably efficient circuit generation. For full-rank all-to-all electronic structure Hamiltonians, we prove O(N^4) gate count and circuit depth, matching the information-theoretic lower bound imposed by the Theta(N^4) second excitation terms. On linear, IBM heavy-hex, and square-grid architectures, Accordion reduces gate count by up to 79% and circuit depth by up to 77% relative to the best baseline.
Variational quantum algorithms are promising applications of noisy intermediate-scale quantum (NISQ) computers. These algorithms consist of a number of separate prepare-and-measure experiments that estimate terms in a Hamiltonian. The number of terms can become overwhelmingly large for problems at the scale of NISQ hardware that may soon be available. We use unitary partitioning (developed independently by Izmaylov et al. [J. Chem. Theory Comput. 16, 190 (2020)]) to define variational quantum eigensolver procedures in which additional unitary operations are appended to the ansatz preparation to reduce the number of terms. This approach may be scaled to use all coherent resources available after ansatz preparation. We also study the use of asymmetric qubitization to implement the additional coherent operations with lower circuit depth. Using this technique, we find a constant factor speedup for lattice and random Pauli Hamiltonians. For electronic structure Hamiltonians, we prove that linear term reduction with respect to the number of orbitals, which has been previously observed in numerical studies, is always achievable. For systems represented on 10--30 qubits, we find that there is a reduction in the number of terms by approximately an order of magnitude. Applied to the plane-wave dual-basis representation of fermionic Hamiltonians, however, unitary partitioning offers only a constant factor reduction. Finally, we show that noncontextual Hamiltonians may be reduced to effective commuting Hamiltonians using unitary partitioning.
Fermionic Ansatz state preparation is a critical subroutine in many quantum algorithms such as the variational quantum eigensolver for quantum chemistry and condensed-matter applications. The shallowest circuit depth needed to prepare Slater determinants and correlated states to date scales at least linearly with respect to the system size NNN. Inspired by data-loading circuits developed for quantum machine learning, we propose an alternate paradigm that provides shallower, yet scalable, O(dlog22N)O(d{log}_{2}^{2}N)O(dlog22​N) two-qubit gate-depth circuits to prepare such states with ddd fermions, offering a subexponential reduction in NNN over existing approaches in second quantization, enabling high-accuracy studies of $d\ensuremath{\ll}O(N/{log}_{2}^{2}N)$ fermionic systems with larger basis sets on near-term quantum devices.