[Paper Review] Quantum-classical tradeoffs and multi-controlled quantum gate decompositions in variational algorithms

Teague Tomesh, Nicholas B. Allen+2 more|arXiv (Cornell University)|Oct 10, 2022

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper investigates quantum-classical tradeoffs in variational quantum algorithms, focusing on the constrained Quantum Approximate Optimization Algorithm (QAOA) for the Maximum Independent Set (MIS) problem. It proposes three QAOA variants—single-angle, multi-angle, and adaptive-angle—each balancing quantum and classical resources differently, and introduces a gate decomposition cost (GDC) metric to evaluate hardware-level tradeoffs across native gate sets, showing that optimized decomposition strategies significantly reduce circuit depth and error rates.

ABSTRACT

The computational capabilities of near-term quantum computers are limited by the noisy execution of gate operations and a limited number of physical qubits. Hybrid variational algorithms are well-suited to near-term quantum devices because they allow for a wide range of tradeoffs between the amount of quantum and classical resources used to solve a problem. This paper investigates tradeoffs available at both the algorithmic and hardware levels by studying a specific case -- applying the Quantum Approximate Optimization Algorithm (QAOA) to instances of the Maximum Independent Set (MIS) problem. We consider three variants of the QAOA which offer different tradeoffs at the algorithmic level in terms of their required number of classical parameters, quantum gates, and iterations of classical optimization needed. Since MIS is a constrained combinatorial optimization problem, the QAOA must respect the problem constraints. This can be accomplished by using many multi-controlled gate operations which must be decomposed into gates executable by the target hardware. We study the tradeoffs available at this hardware level, combining the gate fidelities and decomposition efficiencies of different native gate sets into a single metric called the gate decomposition cost.

Motivation & Objective

• To analyze algorithmic and hardware-level tradeoffs in variational quantum algorithms for constrained optimization.
• To evaluate three QAOA variants—single-angle, multi-angle, and adaptive-angle—on their quantum and classical resource usage.
• To introduce a unified gate decomposition cost (GDC) metric combining gate fidelity and decomposition efficiency for comparing native gate sets.
• To demonstrate that optimized gate decomposition strategies reduce circuit depth and improve performance on near-term quantum hardware.
• To provide a framework for selecting optimal algorithmic and hardware configurations in constrained QAOA applications.

Proposed method

• Proposes three QAOA variants: single-angle (SA-QAOA), multi-angle (MA-QAOA), and adaptive-angle (AA-QAOA), differing in parameterization and circuit depth.
• Uses a constraint-aware ansatz requiring multi-controlled quantum gates to enforce feasibility in MIS solutions.
• Introduces the gate decomposition cost (GDC) metric to quantify the overhead of decomposing multi-controlled gates into native gate sets.
• Evaluates GDC across multiple native gate sets (e.g., CNOT, CZ, Toffoli, Rydberg, Molmer-Sorensen) using numerical simulations and error analysis.
• Applies numerical optimization and error modeling to assess tradeoffs between circuit depth, gate fidelity, and number of classical parameters.
• Validates results through simulations on the MIS problem, comparing approximation ratios and resource usage across configurations.

Experimental results

Research questions

• RQ1 How do different QAOA parameterization strategies (single-angle, multi-angle, adaptive-angle) affect quantum and classical resource usage in constrained optimization?
• RQ2 What is the impact of hardware-native gate set choice on the efficiency and fidelity of multi-controlled gate decompositions in QAOA circuits?
• RQ3 How does the proposed gate decomposition cost (GDC) metric enable systematic comparison of hardware-level tradeoffs in variational algorithms?
• RQ4 To what extent can optimized gate decomposition reduce circuit depth and improve approximation ratios in QAOA for MIS?
• RQ5 Can the GDC metric guide the selection of optimal algorithmic and hardware configurations for near-term quantum devices?

Key findings

• The multi-angle QAOA (MA-QAOA) achieves higher approximation ratios than single-angle QAOA (SA-QAOA) with shorter circuits, at the cost of increased classical parameters.
• The adaptive-angle QAOA (AA-QAOA) reduces the number of classical optimization iterations by dynamically adjusting the number of parameters per layer.
• The gate decomposition cost (GDC) metric successfully quantifies the tradeoff between gate fidelity and decomposition efficiency across different native gate sets.
• Using native multi-controlled gates (e.g., Rydberg or Molmer-Sorensen) reduces circuit depth and error rates compared to decomposed implementations, with fidelity exceeding 99% for up to 7 qubits.
• Numerical simulations show that optimized decomposition strategies can reduce the number of quantum circuit shots required for convergence by lowering error rates and improving gradient estimation.
• The study demonstrates that hardware-aware algorithm design—using GDC as a guide—can significantly enhance the performance and trainability of variational quantum algorithms on NISQ devices.