[Paper Review] Lower bounds on the number of rounds of the quantum approximate optimization algorithm required for guaranteed approximation ratios

Naphan Benchasattabuse, Andreas Bärtschi+4 more|arXiv (Cornell University)|Aug 29, 2023

Quantum Computing Algorithms and ArchitectureComputer Science3 citations

TL;DR

This paper establishes the first theoretical lower bounds on the number of QAOA rounds required to guarantee a constant approximation ratio for combinatorial optimization problems. By linking QAOA to quantum annealing and analyzing the spectral norm of the commutator between problem and mixer Hamiltonians, it proves that Grover-style mixing requires polynomial rounds for most problems, while recovering the Grover lower bound for unstructured search.

ABSTRACT

The quantum approximate optimization algorithm, also known in its generalization as the quantum alternating operator ansatz, (QAOA) is a heuristic hybrid quantum-classical algorithm for finding high-quality approximate solutions to combinatorial optimization problems, such as maximum satisfiability. While the QAOA is well studied, theoretical results as to its runtime or approximation ratio guarantees are still relatively sparse. We provide some of the first lower bounds for the number of rounds (the dominant component of QAOA runtimes) required for the QAOA. For our main result, we (i) leverage a connection between quantum annealing times and the angles of the QAOA to derive a lower bound on the number of rounds of the QAOA with respect to the guaranteed approximation ratio. We apply and calculate this bound with Grover-style mixing unitaries and (ii) show that this type of QAOA requires at least a polynomial number of rounds to guarantee any constant approximation ratios for most problems. We also (iii) show that the bound depends only on the statistical values of the objective functions, and when the problem can be modeled as a $k$-local Hamiltonian, can be easily estimated from the coefficients of the Hamiltonians. For the conventional transverse-field mixer, (iv) our framework gives a trivial lower bound to all bounded-occurrence local cost problems and for all strictly $k$-local cost Hamiltonians matching known results that constant approximation ratio is obtainable with a constant-round QAOA for a few optimization problems from these classes. Using our proof framework, (v) we recover the Grover lower bound for unstructured search and, with small modification, show that our bound applies to any QAOA-style search protocol that starts in the ground state of the mixing unitaries.

Motivation & Objective

• To establish rigorous lower bounds on the number of QAOA rounds needed to achieve a guaranteed constant approximation ratio for optimization problems.
• To connect QAOA performance to quantum annealing dynamics by leveraging the relationship between annealing times and QAOA angles.
• To analyze how the structure of the problem Hamiltonian and mixer Hamiltonian jointly determine the minimum number of rounds required.
• To show that for Grover-style mixing, QAOA requires polynomially many rounds to achieve any constant approximation ratio on most problems.
• To recover the Grover lower bound for unstructured search and extend the framework to general QAOA-style search protocols starting from the ground state of the mixer.

Proposed method

• Derives a general lower bound on QAOA rounds using the spectral norm of the commutator $[P_0, H_1]$, where $P_0$ is the initial projection and $H_1$ is the problem Hamiltonian.
• Applies the bound from [31] to QAOA by modeling the evolution as a sequence of unitary operations driven by non-commuting Hamiltonians $H_0$ and $H_1$.
• Uses the energy change per round as a proxy for state evolution, bounding the number of rounds via $ p riangleq \frac{\left|\braket{P_0}_p - \braket{P_0}_0\right|}{2\pi\|[P_0, H_1]\|} $.
• Applies the framework to Grover-style mixing, where the mixer Hamiltonian is transverse field, and computes the commutator norm explicitly.
• Demonstrates that for $k$-local or strictly $k$-local Hamiltonians, the bound depends only on the statistical properties of the coefficients.
• Extends the bound to general QAOA-style search protocols that begin in the ground state of the mixer Hamiltonian, including continuous-time quantum walks.

Experimental results

Research questions

• RQ1 What is the minimum number of QAOA rounds required to guarantee a constant approximation ratio for a given optimization problem?
• RQ2 How does the spectral norm of the commutator between the problem and mixer Hamiltonians influence the lower bound on QAOA rounds?
• RQ3 Can the framework recover known lower bounds such as the Grover bound for unstructured search?
• RQ4 Why do standard transverse field mixers yield trivial lower bounds for bounded $k$-local problems despite known constant-round success?
• RQ5 To what extent can the lower bound be tightened by incorporating angle assumptions or effective energy changes per round?

Key findings

• For Grover-style mixing, QAOA requires at least a polynomial number of rounds to guarantee any constant approximation ratio for most problems.
• The derived lower bound depends only on the statistical properties of the objective function coefficients when the problem is modeled as a $k$-local Hamiltonian.
• The framework recovers the Grover lower bound for unstructured search when the problem is cast as a QAOA-style search protocol.
• For conventional transverse field mixers, the bound becomes trivial (i.e., $p \geq 1$) for bounded $k$-local and strictly $k$-local problems due to large commutator norms.
• The bound is applicable to any QAOA-style protocol that starts in the ground state of the mixer Hamiltonian, including amplitude amplification and continuous-time quantum walks.
• The framework reveals that the choice of mixer significantly affects the theoretical round complexity, suggesting that mixer design is critical for efficient QAOA performance.