The Lie Algebra of XY-mixer Topologies and Warm Starting QAOA for Constrained Optimization

TL;DR

This work analyzes XY-mixer-based variational quantum algorithms for constrained optimization by explicitly decomposing the dynamical Lie algebras (DLAs) arising from different XY-topologies. It shows that 1D path/ring topologies with $R_Z$ gates yield polynomial-sized DLAs, while all-to-all or $R_{ZZ}$-augmented topologies yield exponentially-large DLAs, and it uses this insight to implement a warm-start strategy: pretrain on a polynomial-DLA circuit and transfer parameters to the full, exponential-DLA circuit. The paper provides detailed DLA decompositions (e.g., $\mathfrak{g}_{XY}^{P}\cong \mathfrak{so}(n)$, $\mathfrak{g}_{XY,Z}^{C}\cong \mathfrak{u}(1)\oplus\mathfrak{su}(n)\oplus\mathfrak{su}(n)$) and conjectures for exponential cases, supported by small-$n$ numerics. It demonstrates the practical payoff by warm-starting QAOA for Portfolio Optimization, Sparsest $k$-Subgraph, and Graph Partitioning, including SP500 benchmarks, where warm-started MA-QAOA delivers dramatic improvements in approximation ratio and success probability over random starts, and outperforms shared-angle variants. The work highlights that structured, symmetry-preserving mixer designs can enable strong performance on NP-hard constrained problems and points to future directions in proving DLA conjectures and extending warm-starting to broader quantum optimization tasks.

Abstract

The XY-mixer has widespread utilization in modern quantum computing, including in variational quantum algorithms, such as Quantum Alternating Operator Ansatz (QAOA). The XY ansatz is particularly useful for solving Cardinality Constrained Optimization tasks, a large class of important NP-hard problems. First, we give explicit decompositions of the dynamical Lie algebras (DLAs) associated with a variety of $XY$-mixer topologies. When these DLAs admit simple Lie algebra decompositions, they are efficiently trainable. An example of this scenario is a ring $XY$-mixer with arbitrary $R_Z$ gates. Conversely, when we allow for all-to-all $XY$-mixers or include $R_{ZZ}$ gates, the DLAs grow exponentially and are no longer efficiently trainable. We provide numerical simulations showcasing these concepts on Portfolio Optimization, Sparsest $k$-Subgraph, and Graph Partitioning problems. These problems correspond to exponentially-large DLAs and we are able to warm-start these optimizations by pre-training on polynomial-sized DLAs by restricting the gate generators. This results in improved convergence to high quality optima of the original task, providing dramatic performance benefits in terms of solution sampling and approximation ratio on optimization tasks for both shared angle and multi-angle QAOA.

Figures (13)

• Figure 1: Sub-algebras of XY Topologies. A Venn Diagram comparing the space of linear operators of different XY topologies studied in this manuscript, summarized in Sec. \ref{['sec:liealg']}.
• Figure 2: Warm starting QAOA for Constrained Optimization. Given a constrained optimization problem, such as Portfolio Optimization, we (1) tailor a collection of mixing and phase-separating operators, (2) restrict the ansatz and pretrain, (3) transfer the learned parameters for the full circuit, and (4) refine through training with the full parameter set.
• Figure 3: Warm starting QAOA. The figure on the top shows the circuit utilized to warm start QAOA without the $R_{ZZ}$ terms. The figure on the bottom shows the full circuit for QAOA. These corresponds to a polynomial and exponential sized DLA, respectively.
• Figure 4: Depictions of XY-mixer Topologies. Diagrammatic representations of different XY-mixer associated generators and their associated Lie algebra. Red lines connecting qubits corresponding to 2-body XY-mixer terms. Blue circles correspond to single $R_Z$ interactions whereas gray circles are identity. Blue lines between nodes correspond to 2-body $R_{ZZ}$ interactions. The superscript on $\mathfrak{g}$ corresponds to the XY-mixer connectivity.
• Figure 5: Containment diagram for the polynomial-sized dynamical XY Lie algebras. An arrow indicates a Lie sub-algebra. The dashed arrow indicates that $\mathfrak{g}_{XY}^{C,O}(n)$ is isomorphic to the $\mathfrak{su}\space\left(n\right)$ component of $\mathfrak{g}_{XY,Z}^{P}(n)$ but the basis elements and therefore associated subspaces in $\mathfrak{su}\space\left(2^n\right)$ are different.

Theorems & Definitions (73)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Conjecture 1
• Lemma 1
• Theorem 5
• Definition S.2.1: Dynamical Lie algebra
• Remark 1
• Remark 2