[Paper Review] Polynomial scaling of QAOA for ground-state preparation of the fully-connected p-spin ferromagnet

Matteo M. Wauters, Glen Bigan Mbeng+1 more|arXiv (Cornell University)|Mar 16, 2020

Quantum Computing Algorithms and Architecture9 citations

TL;DR

This paper demonstrates that the Quantum Approximate Optimization Algorithm (QAOA) can prepare the ground state of the fully-connected p-spin ferromagnet with polynomial resources, overcoming the exponentially small gaps that hinder Quantum Annealing for p ≥ 3. When the number of QAOA parameters P exceeds a critical threshold P* ∝ N, the parameter space simplifies, enabling perfect ground-state fidelity with extensive parameter scaling and polynomial resource cost.

ABSTRACT

We show that the quantum approximate optimization algorithm (QAOA) can construct with polynomially scaling resources the ground state of the fully-connected p-spin Ising ferromagnet, a problem that notoriously poses severe difficulties to a Quantum Annealing (QA) approach, due to the exponentially small gaps encountered at first-order phase transition for ${\ m p} \\ge 3$. For a target ground state at arbitrary transverse field, we find that an appropriate QAOA parameter initialization is necessary to achieve a good performance of the algorithm when the number of variational parameters $2{\ m P}$ is much smaller than the system size ${\ m N}$, because of the large number of sub-optimal local minima. Instead, when ${\ m P}$ exceeds a critical value ${\ m P}^*_{\ m N} \\propto {\ m N}$, the structure of the parameter space simplifies, as all minima become degenerate. This allows to achieve the ground state with perfect fidelity with a number of parameters scaling extensively with ${\ m N}$, and with resources scaling polynomially with ${\ m N}$.

Motivation & Objective

• To address the challenge of ground-state preparation for the fully-connected p-spin ferromagnet, a model where Quantum Annealing fails due to exponentially small gaps at first-order phase transitions for p ≥ 3.
• To investigate whether QAOA can efficiently prepare the ground state of this model, given its known difficulties in adiabatic evolution.
• To analyze the structure of the QAOA parameter space and identify conditions under which high-fidelity ground-state preparation becomes feasible.
• To determine the scaling of resources and parameters required for successful ground-state preparation, especially in the regime where P ≪ N and P ≳ N.
• To establish the necessity of proper parameter initialization for achieving good performance when the number of variational parameters is sub-extensive.

Proposed method

• Employ the QAOA framework, using a product state with all spins aligned in the x-direction as the initial state, and applying alternating unitaries generated by the transverse field (Hx) and the p-spin Hamiltonian (Hz).
• Use variational parameters γ and β to parameterize the QAOA evolution, with the goal of minimizing the energy expectation value via classical optimization.
• Analyze the symmetry properties of the QAOA energy landscape, including invariance under (γ, β) → (−γ, −β), β → β + π, and γ → γ + π or γ + π/2^p−1 depending on N and p.
• Identify critical thresholds for P, the number of QAOA layers, with P* ∝ N marking a transition in the parameter space structure.
• Leverage the simplification of the energy landscape when P > P*, where all local minima become degenerate, enabling perfect ground-state fidelity.
• Use analytical and numerical methods to study the energy landscape and fidelity scaling, particularly focusing on the role of initial parameter choice for P ≪ N.

Experimental results

Research questions

• RQ1 Can QAOA efficiently prepare the ground state of the fully-connected p-spin ferromagnet, a system where Quantum Annealing fails due to exponentially small gaps?
• RQ2 What is the role of parameter initialization in QAOA performance when the number of variational parameters P is much smaller than the system size N?
• RQ3 How does the structure of the QAOA parameter space change as P increases, particularly when P exceeds a critical value P* ∝ N?
• RQ4 Under what conditions does the QAOA energy landscape simplify, leading to degenerate minima and perfect ground-state fidelity?
• RQ5 What is the scaling of resources (quantum circuits and classical optimization steps) required for QAOA to achieve high-fidelity ground-state preparation in this model?

Key findings

• QAOA can prepare the ground state of the fully-connected p-spin ferromagnet with polynomially scaling resources, even for p ≥ 3, where Quantum Annealing fails due to exponentially small gaps.
• When P ≪ N, the QAOA performance is highly sensitive to initialization due to a large number of sub-optimal local minima in the parameter space.
• When P > P* ∝ N, the parameter space structure simplifies such that all local minima become degenerate, enabling perfect ground-state fidelity.
• For P > P*, the number of variational parameters scales extensively with system size N, and the total resources scale polynomially with N.
• The energy landscape exhibits symmetries including invariance under (γ, β) → (−γ, −β), β → β + π (for p odd), and γ → γ + π or γ + π/2^p−1 depending on the parity of N and p.
• The critical threshold P* scales linearly with N, marking a transition from a complex, rugged landscape to a simplified, degenerate one that supports exact ground-state preparation.