Finite-Depth, Finite-Shot Guarantees for Constrained Quantum Optimization via Fejér Filtering

TL;DR

Finite-layer alternations of the CE--QAOA, a constraint-aware ansatz that operates natively on block one-hot manifolds, are studied, finding coherent realizations of hardware-efficient positive spectral filters as a main open direction.

Abstract

We study finite-layer alternations of the \emph{Constraint--Enhanced Quantum Approximate Optimization Algorithm} (CE--QAOA), a constraint-aware ansatz that operates natively on block one-hot manifolds. Our focus is on feasibility and optimality guarantees. We show that restricting cost angles to a harmonic lattice exposes a positive Fejér filter acting on the cost-phase unitary $U_C(γ)=e^{-iγH_C}$ \emph{in a cost-dephased reference model (used only for analysis)}. Under a wrapped phase-separation condition, this yields \emph{dimension-free} finite-depth and finite-shot lower bounds on the success probability of sampling an optimal solution. In particular, we obtain a ratio-form guarantee \[ q_0 \;\ge\; \frac{x}{1+x}, \qquad x \;=\; (p{+}1)^2 \sin^2(δ/2)\,C_β, \] where $q_0$ is the single-shot success probability, $C_β$ is the mixer-envelope mass on the optimal set, $δ$ is a phase-gap proxy, and $p$ is the number of layers. Riemann--Lebesgue averaging extends the discussion beyond exact lattice normalization. We conclude by outlining coherent realizations of hardware-efficient positive spectral filters as a main open direction.

Figures (3)

• Figure 1: Depth-$p=3$ CE-QAOA for $m=3$ blocks of $n=4$ qubits. Each layer applies a global cost $U_C(\gamma_\ell)$ over all $mn$ wires, followed by parallel block-local XY mixers $U_M^{(j)}(\beta_\ell)$.
• Figure 2: Fixed-$p$ certification curves for the sufficient Fejér peaking bound (target $1-\varepsilon$ with $\varepsilon=0.1$). For each depth $p$, the bound certifies $\Pr[x^\star]\ge 1-\varepsilon$ whenever $C_\beta \ge C_{\min}(\delta;\varepsilon,p)$, i.e. in the region above the corresponding curve. The vertical colorbar encodes $p$ from $10^1$ (blue) to $10^{10}$ (red). Monotonicity: increasing either the envelope mass $C_\beta$ or the phase-gap proxy $\delta$reduce the certified depth. Conservatism: because $\delta$ is a wrapped phase-separation proxy, very small values (e.g. $\delta\sim 10^{-2}$) already correspond to near phase-collisions. In this regime the bound pessimistic and can predict large orders. Optimistic (too-large) estimates of $C_{\beta}$ or $\delta$ can underpredict the required depth and risk missing the optimum, while conservative (smaller) estimates only inflate the certified depth. Even with tiny $C_\beta$, the extreme dept region is not approached as long as $\delta$ remains under control.
• Figure 3: Phase diagram for the Fejér-based lower bound. Deeper green indicates higher single-shot success $q_0$; pale green indicates weaker performance.

Theorems & Definitions (39)

• Definition 1: CE--QAOA kernel
• Proposition 2: Spectral gap of one-block XY mixer
• Proposition 3: Invariance and quditization of the block–XY mixer
• Proposition 4: Ergodicity of the angle-averaged XY mixer on $\mathcal{H}_1$
• Lemma 5: Single--block primitivity
• proof
• Lemma 6: Global primitivity for one mixer layer
• proof
• Corollary 7: Perron--Frobenius for the global mixer
• Definition 8: Penalty level-set states