[Paper Review] Local, Expressive, Quantum-Number-Preserving VQE Ansatze for Fermionic Systems

Gian-Luca Anselmetti, David Wierichs+2 more|arXiv (Cornell University)|Apr 12, 2021

Quantum Computing Algorithms and Architecture54 references94 citations

TL;DR

This paper proposes a novel, local, and expressive Variational Quantum Eigensolver (VQE) ansatz for fermionic systems that exactly preserves particle and spin quantum numbers—specifically the number of α and β electrons and total spin squared—under the Jordan-Wigner mapping. The ansatz uses a gate fabric composed of orbital rotations and pair exchange gates, achieving universality in quantum state representation at low circuit depth, with numerical evidence showing robust convergence and avoidance of barren plateaus when parameters are sufficiently numerous and well-initialized.

ABSTRACT

We propose VQE circuit fabrics with advantageous properties for the simulation of strongly correlated ground and excited states of molecules and materials under the Jordan-Wigner mapping that can be implemented linearly locally and preserve all relevant quantum numbers: the number of spin up ($\alpha$) and down ($\beta$) electrons and the total spin squared. We demonstrate that our entangler circuits are expressive already at low depth and parameter count, appear to become universal, and may be trainable without having to cross regions of vanishing gradient, when the number of parameters becomes sufficiently large and when these parameters are suitably initialized. One particularly appealing construction achieves this with just orbital rotations and pair exchange gates. We derive optimal four-term parameter shift rules for and provide explicit decompositions of our quantum number preserving gates and perform numerical demonstrations on highly correlated molecules on up to 20 qubits.

Motivation & Objective

• To develop a VQE ansatz that preserves all relevant quantum numbers (α/β electron counts and total spin) in fermionic systems under Jordan-Wigner mapping.
• To design a gate fabric with linear locality, low gate count, and compatibility with NISQ device connectivity.
• To achieve high expressiveness and robust optimization behavior, avoiding barren plateaus, through a structured, symmetry-preserving circuit design.
• To demonstrate numerical universality and convergence robustness across diverse quantum number irreps, including highly correlated molecules.

Proposed method

• Designs a gate fabric using 2-qubit SU(4) gates decomposed into orbital rotations and pair exchange operations, forming a tessellation of alternating layers.
• Constructs quantum number-preserving (QNP) gates—specifically ˆQ-type and ˆF-type—by ensuring unitary evolution commutes with particle and spin symmetry operators.
• Derives optimal four-term parameter shift rules for gradient computation, enabling noise-free analytical gradients in optimization.
• Employs L-BFGS optimization with analytical gradients to test convergence and universality on Haar-random states and molecular systems.
• Validates the ansatz on up to 20-qubit systems, including highly correlated molecules, using exact diagonalization and state fidelity metrics.
• Introduces a systematic method to identify non-universal edge cases (e.g., fully occupied or empty subspaces) and provides a more robust 5-parameter ˆF-type gate as a universal alternative.

Experimental results

Research questions

• RQ1 Can a VQE ansatz be constructed that preserves all relevant quantum numbers (α/β electron counts and total spin) while maintaining low circuit depth and local connectivity?
• RQ2 Does the proposed gate fabric achieve universal state preparation within a polynomial-depth circuit, even in the presence of strong electron correlation?
• RQ3 Can the ansatz avoid barren plateaus and exhibit robust, geometric convergence during optimization, especially as parameter count increases?
• RQ4 Are there specific quantum number irreps where the ansatz fails to be universal, and if so, can they be systematically identified and mitigated?
• RQ5 How does the performance of the ˆQ-type QNP gate fabric compare to the more complex ˆF-type fabric in terms of universality and convergence?

Key findings

• The ˆQ-type QNP gate fabric achieves universal state preparation for the vast majority of quantum number irreps in the Hilbert space, with numerical fidelity discrepancies below 10−13 for universal cases.
• Non-universal behavior occurs only in edge cases with high-spin constraints and fully occupied or empty orbital subspaces, such as (Nα=0, Nβ=2, S=2) for M=4, where the overlap discrepancy reaches ~10−2.
• The ˆF-type QNP gate fabric, a 5-parameter alternative, shows no edge-case non-universality and is numerically universal across all tested irreps.
• Convergence of the ansatz under L-BFGS optimization exhibits a distinct phase change at the universality threshold: circuits with sufficient parameters show strongly geometric convergence down to machine epsilon.
• The ansatz demonstrates robustness to barren plateaus, with no vanishing gradient issues when parameter count exceeds the universality threshold and parameters are suitably initialized.
• The proposed gate fabric, composed of orbital rotations and pair exchange gates, achieves high expressiveness even at low depth (D=1 to D=18), with fidelity to target states approaching unity.