[Paper Review] Behavior of Analog Quantum Algorithms

Lucas T. Brady, Lucas Kocia+4 more|ArXiv.org|Jul 2, 2021

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper reveals that the Quantum Approximate Optimization Algorithm (QAOA) emulates an optimal analog quantum protocol characterized by a smooth adiabatic path with superposed oscillations, where QAOA layer durations match the oscillation period and bang ratios reflect the average annealing curve. The optimal protocol achieves coherent error cancellation via ground and first excited state interference, and a new algorithm based on this insight outperforms standard QAOA and quantum annealing by using few variational parameters to capture the oscillatory structure.

ABSTRACT

Analog quantum algorithms are formulated in terms of Hamiltonians rather than unitary gates and include quantum adiabatic computing, quantum annealing, and the quantum approximate optimization algorithm (QAOA). These algorithms are promising candidates for near-term quantum applications, but they often require fine tuning via the annealing schedule or variational parameters. In this work, we explore connections between these analog algorithms, as well as limits in which they become approximations of the optimal procedure.Notably, we explore how the optimal procedure approaches a smooth adiabatic procedure but with a superposed oscillatory pattern that can be explained in terms of the interactions between the ground state and first excited state that effect the coherent error cancellation of diabatic transitions. Furthermore, we provide numeric and analytic evidence that QAOA emulates this optimal procedure with the length of each QAOA layer equal to the period of the oscillatory pattern. Additionally, the ratios of the QAOA bangs are determined by the smooth, non-oscillatory part of the optimal procedure. We provide arguments for these phenomena in terms of the product formula expansion of the optimal procedure. With these arguments, we conclude that different analog algorithms can emulate the optimal protocol under different limits and approximations. Finally, we present a new algorithm for better approximating the optimal protocol using the analytic and numeric insights from the rest of the paper. In practice, numerically, we find that this algorithm outperforms standard QAOA and naive quantum annealing procedures.

Motivation & Objective

• To understand the fundamental connections between analog quantum algorithms—quantum adiabatic computing, quantum annealing, and QAOA—by analyzing their shared optimal control protocols.
• To explain the asymptotic behavior of QAOA at large circuit depth, particularly the emergence of smooth, curve-like variational parameter profiles observed numerically.
• To identify the physical origin of oscillations in optimal annealing curves and show their role in coherent error cancellation between ground and first excited states.
• To develop a new, practical quantum algorithm that approximates the optimal protocol using minimal variational parameters while outperforming standard QAOA and quantum annealing.

Proposed method

• The authors use optimal control theory to derive the optimal protocol for analog quantum algorithms, which exhibits a bang-anneal-bang structure with a smooth, oscillatory middle region.
• They analyze the near-adiabatic limit of the optimal curve, deriving the form of oscillations as a function of phase differences between ground and first excited state amplitudes.
• Using product formula expansions, they show that Trotterization error is minimized when the step size matches the oscillation period of the optimal curve.
• They establish a correspondence between QAOA layer durations and the oscillation period of the optimal annealing curve, and between QAOA bang ratios and the average value of the curve over each period.
• They construct a new ansatz algorithm that uses QAOA to extract the structure of the optimal annealing region and then treats key features—initial/final bangs and oscillatory pattern—as variational parameters.
• Numerical simulations validate that the new algorithm outperforms standard QAOA and naive quantum annealing, approaching the performance of the full optimal protocol.

Experimental results

Research questions

• RQ1 How do quantum adiabatic computing, quantum annealing, and QAOA relate to one another in terms of their underlying optimal control protocols?
• RQ2 Why do QAOA variational parameters at large depth converge to smooth, curve-like profiles resembling annealing paths, despite being discrete and bang-bang?
• RQ3 What is the physical origin and role of oscillations in the optimal annealing curve, particularly in the near-adiabatic regime?
• RQ4 Can the structure of the optimal protocol be approximated with a minimal number of variational parameters to create a practical, high-performing algorithm?

Key findings

• The optimal protocol for analog quantum algorithms features a smooth adiabatic path with superposed oscillations that arise from coherent interference between the ground and first excited states, enabling error cancellation in diabatic transitions.
• QAOA emulates this optimal protocol by using layer durations that match the period of the oscillatory pattern in the optimal curve, explaining its asymptotic behavior at large circuit depth.
• The ratios of QAOA bangs correspond to the average value of the optimal annealing curve over each oscillation period, linking the discrete QAOA structure to the continuous optimal control function.
• The near-adiabatic limit of the optimal curve is analytically derived, showing that oscillations are essential for managing leakage and improving performance beyond monotonic adiabatic evolution.
• A new algorithm is proposed that uses QAOA to extract the structure of the optimal annealing region and then parameterizes the initial and final bangs and oscillatory pattern, outperforming standard QAOA and quantum annealing in numerical tests.
• The new algorithm achieves performance close to the optimal protocol while using far fewer variational parameters, making it feasible for near-term quantum devices.