Dynamic Depth QAOA for Combinatorial Optimization

• DDQAOA is a dynamic-depth extension of QAOA that adapts the circuit depth during execution to address fixed-depth limitations in combinatorial optimization.
• It utilizes methods such as proximal-gradient pruning, discrete adiabatic scheduling, and progressive expansion to balance solution accuracy with circuit resource usage.
• Empirical results demonstrate that DDQAOA achieves competitive approximation ratios while reducing gate counts by up to 40%, improving performance on noisy quantum devices.

Dynamic Depth Quantum Approximate Optimization Algorithm (DDQAOA) designates a family of algorithms for combinatorial optimization that dynamically adapts the quantum circuit depth during execution, removing the need for a priori depth selection. DDQAOA modifies the Quantum Approximate Optimization Algorithm (QAOA) framework by incorporating stepwise or continuous depth-expansion strategies—either through automated proximal pruning, performance-guided layer addition, or analytic scheduling based on adiabatic intuition. The resulting protocols address the practical limitation of fixed-depth QAOA, improving gate efficiency, noise resilience, and optimization success on Noisy Intermediate-Scale Quantum (NISQ) hardware. Approaches subsumed under DDQAOA include proximal-gradient–based pruning schemes, adiabatic-theorem–guided discretization, and adaptive warm-starting with interpolation, as instantiated in recent developments for Max-Cut, constrained shortest path, and general QUBO problems (Pan et al., 2022, Kremenetski et al., 2023, Saini et al., 11 Nov 2025).

1. Foundations and Motivations

The conventional QAOA framework [1] formulates a variational quantum-classical protocol using two non-commuting Hamiltonians. For a problem instance with $N$ qubits:

• Cost Hamiltonian $H_C$ encodes the optimization objective (e.g., Max-Cut uses $$H_C = \sum_{(i,j)\in E} \omega_{ij} \sigma^z_i \sigma^z_j$$; QUBO or Ising forms generalize as $H_C = \sum_{i<j} w_{ij} Z_i Z_j + \sum_i h_i Z_i$).
• Mixer Hamiltonian $H_M = \sum_{n=1}^N X_n$ globally drives state transitions in the computational basis.

The standard p-depth QAOA ansatz is: $$|\psi_p(\boldsymbol\gamma, \boldsymbol\beta)\rangle = \prod_{l=1}^p e^{-i\beta_l H_M} e^{-i\gamma_l H_C} |+\rangle^{\otimes N}$$ with $2p$ variational parameters $(\boldsymbol\gamma, \boldsymbol\beta)$ optimized to extremize the classical objective

$$F_p(\boldsymbol\gamma, \boldsymbol\beta) = \langle \psi_p | H_C | \psi_p \rangle$$

QAOA's practical limitations arise from the need to set the circuit depth p a priori. If $p$ is too low, the variational ansatz lacks expressivity and fails to solve the problem with high accuracy; if $p$ is too high, deep circuits result in prohibitive gate counts, CNOT overhead, decoherence, and noise on NISQ devices. DDQAOA seeks to resolve this issue by adaptive, on-the-fly control of depth informed by algorithmic progress and theoretical structure (Pan et al., 2022, Kremenetski et al., 2023, Saini et al., 11 Nov 2025).

2. Dynamic Depth Selection Strategies

Three principal methodologies for dynamic depth control in DDQAOA have been introduced:

A. Proximal-Gradient Pruning (APG/DDQAOA)

A sparsity-inducing $\ell_1$ penalty is applied to the parameter vector $x=(\boldsymbol\beta, \boldsymbol\gamma)$: $$\min_{x} f(x) + \lambda \|x\|_1 \qquad \text{with} \quad f(x)=\langle\psi(x)|H_C|\psi(x)\rangle$$ Updates use the proximal operator, performing soft thresholding: $$x^{(k+1)} = S_{\lambda\eta} \left( x^{(k)} - \eta\nabla f(x^{(k)}) \right)$$ where $S_{\lambda\eta}$ sets entries with $|x_i| \leq \lambda\eta$ to zero. Layers with both $(\beta_j, \gamma_j) = 0$ are pruned, dynamically reducing circuit depth. Accelerated Proximal Gradient (APG) with extrapolation and nonmonotone line–search provides $O(1/k)$ convergence (to a stationary point in the nonconvex case). Practical convergence and circuit simplification guarantee efficient pruning while preserving or minimally impacting solution quality (Pan et al., 2022).

B. Discrete Adiabatic Scheduling

Under analytic control, gradually-varying angle schedules $\gamma(f), \beta(f)$ (for $f \in [0,1]$) are discretized into QAOA layers according to the discrete adiabatic theorem (DAT). For small increments, the dynamics track the continuous adiabatic path. However, above a threshold $\Delta^* = 2\pi/\Delta E_{\text{max}}$ (with $\Delta E_{\text{max}}$ the largest cost eigenvalue gap), "wrap-around" and eigenvector exchange can cause abrupt performance loss. The DDQAOA strategy leverages this by adaptively choosing $p$ and step size $\Delta$ to either (i) remain adiabatic below $\Delta^*$, or (ii) intentionally step over narrow avoided crossings diabatically, achieving high performance with minimal depth (Kremenetski et al., 2023).

C. Progressive Depth Expansion with Parameter Transfer

A practical expansion protocol starts with $p=1$ and increments $p$ only when classical convergence stalls (detected by cost improvement/variance thresholds). Upon expanding to $p+1$, learned parameters are interpolated (linear for $p < 4$, cubic for $p \geq 4$) and warm-started. The optimization thus proceeds with increasing expressivity only when justified by algorithmic progress, reducing total resource use (Saini et al., 11 Nov 2025).

Approach	Depth Control Mechanism	Pruning/Expansion Rule
Proximal-gradient (Pan et al., 2022)	$\ell_1$ penalty $\rightarrow$ sparsity	Threshold small angles, remove layer
Discrete adiabatic (Kremenetski et al., 2023)	Analytic eigenvalue/gap analysis	Analytical estimate of $p$ for given schedule
Progressive expansion (Saini et al., 11 Nov 2025)	Empirical convergence + interpolation	Add layer when progress stalls, parameter interpolation

3. Mathematical Analysis and Convergence

Proximal gradient-based DDQAOA offers formal convergence guarantees. If $f(x)$ is L-Lipschitz smooth and optionally convex, the update

$$x_{k+1} = S_{\lambda\eta} \left( x_k - \eta\nabla f(x_k) \right)$$

with step size $\eta < 1/L$ satisfies

$$F(x_{k+1}) \leq F(x_k) - \left( \frac{1}{2\eta} - \frac{L}{2} \right) \| x_{k+1} - x_k \|^2$$

yielding $O(1/k)$ convergence. In the general (non-convex) QAOA objective, the APG framework retains $O(1/k)$ convergence to stationary points per [IJCAI '17, Li & Lin 2015]. In discrete adiabatic scheduling, the DAT bounds the error of the state transfer as $O(1/L)$ in the depth $L=p$, provided the minimum gap $\Delta_{\min}$ remains bounded (Kremenetski et al., 2023). Ensuring depth does not cross wrap-around points (where the eigenstate swap occurs) is essential for securing convergence to the target ground state.

4. Algorithmic Implementation and Pseudocode

An exemplar pseudocode for dynamic-depth pruning via proximal updates proceeds as follows (Pan et al., 2022):

Algorithm: Dynamic-Depth QAOA via Proximal-Gradient Inputs: Initial depth p0, x0 ∈ (−π,π]^{2p0}; step size η < 1/L; regularizer λ > 0; max iters K; tolerance tol; APG window q. Initialize: x_{–1} = x0, x_0 = x0. For k = 0 ... K–1 do 1. Extrapolate (if APG): y = x_k + ((k–1)/(k+2))(x_k – x_{k–1}) 2. Nonmonotone check: F_max = max_{t in [max(0,k–q) ... k]} [f(x_t)+λ∥x_t∥_1]; set v = y if f(y)+λ∥y∥_1 ≤ F_max else x_k. 3. Compute gradient ∇f(v) by finite differences or parameter-shift. 4. Proximal update: x_{k+1} = S_{λη}(v − η∇f(v)), apply soft threshold elementwise. 5. Prune angles: remove layer j for which both β_j, γ_j = 0. 6. Terminate if |F(x_{k+1}) – F_max| < tol. End for Output: Final depth p_final, optimized parameters x_final.

Progressive depth-expansion DDQAOA (Saini et al., 11 Nov 2025) implements this as an outer loop: at each $p$, optimize via standard variational techniques (e.g., Adam, parameter-shift gradient), monitor for convergence (cost improvement/variance stall), then transfer and interpolate angles to the next $p$ when increasing expressivity is warranted.

5. Performance Benchmarks and Resource Usage

Empirical validations for DDQAOA demonstrate both competitive or superior approximation performance and substantial resource (gate count) savings:

Max-Cut (Proximal DDQAOA (Pan et al., 2022)):

• On 7-node graphs, regularization $\lambda=0.72$ prunes to $p_\text{final}=11$ with $\approx 0.90$ approximation ratio after $\sim$43 iterations. Further unconstrained optimization delivers $r \approx 0.927$, with circuit depth reduced from 14 → 8 layers (≈40% reduction).
• Fixed-depth QAOA at $p=7$ achieves $r \approx 0.975$ but uses 14 layers; DDQAOA achieves $>30\%$ depth reduction for $r \geq 0.9$.
• Proximal DDQAOA requires only $O(\log p)$ sweeps over $\lambda$ for depth selection, compared to $O(p)$ for grid search.

Constrained Shortest Path (Expansion DDQAOA (Saini et al., 11 Nov 2025)):

• 10-qubit graphs: DDQAOA achieves $\bar r = 0.969\,(0.011)$; fixed $p=15$ QAOA achieves $\bar r = 0.953$. DDQAOA uses 217% fewer total CNOTs than $p=15$ QAOA.
• 16-qubit graphs: DDQAOA $\bar r = 0.990\,(0.003)$ vs.\ fixed $p=15$ $\bar r = 0.985$; CNOT savings reach 159.3%.
• Gate cost grows stepwise from minimal (e.g., 90 → 900 CNOTs for $p=1$ to $p=10$ at 10 qubits), but total cumulative gate cost remains below or competitive with fixed-$p$ QAOA.

Benchmark	DDQAOA Approx. Ratio (10q, 16q)	CNOT Reduction vs $p$=15 (%)
Max-Cut [7q]	$r=0.90$ ($p=8$)	$\sim$40%
CSPP [10q/16q]	$0.969/0.990$	$217/159.3$ (cum.)

A plausible implication is that DDQAOA adapts favorably to larger instance sizes, systematically reducing required circuit depth and aggregate gate count for targeted approximation ratios.

6. Practical Guidelines for NISQ Implementation

Key heuristics emerge from numerical and analytic studies:

• Starting Depth ($p_0$): For proximal-pruning schemes, $p_0$ should slightly exceed the anticipated optimal value based on problem size.
• Gradient Estimation: Use parameter-shift rules; total measurement cost $\sim O(2 p_\text{final} \times \text{iters})$.
• Hyperparameters: Initial $\lambda$ set based on objective magnitude, decayed geometrically. Learning rates in the $0.003$–$0.01$ range proved efficient for Max-Cut; Adam used for CSPP.
• Layer Merge: When pruning, neighboring gates may be collapsed for further simplification.
• Post-Pruning Refinement: After depth reduction, one may switch off the regularizer ($\lambda\to 0$) and run unconstrained gradient descent to maximize solution quality.
• Adiabatic Scheduling: Analytical estimation of the eigenvalue spectrum ($\Delta E_\text{max}$) and gap structure enables depth tuning to anticipate large-angle failures.
• Noise-Awareness: DDQAOA halts depth expansion when noise-induced performance saturates, limiting decoherence exposure (empirically observed as the peak $r$ vs.\ $p$ curve on noisy devices).

7. Limitations and Future Directions

Current DDQAOA protocols have been validated up to 16 qubits on noise-free simulators. The practical impact on real quantum hardware—including noise resilience, calibration for device-specific connectivity, and CNOT error accumulation—remains to be characterized. Further research is anticipated in:

• Alternative convergence criteria (e.g., gradient norm thresholds)
• Adaptive interpolation kernels for parameter transfer
• Performance tuning under realistic gate noise and device-specific constraints
• Broader benchmarking on a variety of QUBO-based problems (MaxCut, graph coloring, scheduling)

This suggests that dynamic-depth strategies will play a key role in rendering QAOA and related variational quantum algorithms viable on NISQ-era and early fault-tolerant hardware.