[Paper Review] Quantum Annealing: a journey through Digitalization, Control, and hybrid Quantum Variational schemes

Glen Bigan Mbeng, Rosario Fazio+1 more|arXiv (Cornell University)|Jun 21, 2019

Quantum Computing Algorithms and Architecture41 citations

TL;DR

The paper links digitized quantum annealing (dQA) with QAOA and optimal quantum control, proving a residual-energy bound for MaxCut on 2-regular graphs and showing how a regular, adiabatic-like digitized-QA schedule can be constructed within QAOA.

ABSTRACT

We establish and discuss a number of connections between a digitized version of Quantum Annealing (QA) with the Quantum Approximate Optimization Algorithm (QAOA) introduced by Farhi et al. (arXiv:1411.4028) as an alternative hybrid quantum-classical variational scheme for quantum-state preparation and optimization. We introduce a technique that allows to prove, for instance, a rigorous bound concerning the performance of QAOA for MaxCut on a $2$-regular graph, equivalent to an unfrustrated antiferromagnetic Ising chain. The bound shows that the optimal variational error of a depth-$\mathrm{P}$ quantum circuit has to satisfy $ε^\mathrm{res}_{\mathrm{P}}\ge (2\mathrm{P}+2)^{-1}$. In a separate work (Mbeng et al., arXiv:1911.12259) we have explicitly shown, exploiting a Jordan-Wigner transformation, that among the $2^{\mathrm{P}}$ degenerate variational minima which can be found for this problem, all strictly satisfying the equality $ε^\mathrm{res}_{\mathrm{P}}=(2\mathrm{P}+2)^{-1}$, one can construct a special {\em regular} optimal solution, which is computationally optimal and does not require any prior knowledge about the spectral gap. We explicitly demonstrate here that such a schedule is adiabatic, in a digitized sense, and can therefore be interpreted as an optimized digitized-QA protocol. We also discuss and compare our bound on the residual energy to well-known results on the Kibble-Zurek mechanism behind a continuous-time QA. These findings help elucidating the intimate relation between digitized-QA, QAOA, and optimal Quantum Control.

Motivation & Objective

• Establish connections between digitized QA, QAOA, and optimal quantum control in the NISQ era.
• Derive a rigorous bound on the residual energy for QAOA applied to MaxCut on 2-regular graphs.
• Demonstrate the existence of a regular, adiabatic-like digitized-QA schedule within QAOA.
• Clarify how boundary conditions and reduced spin chains yield tractable analytical bounds.

Proposed method

• Model MaxCut as an antiferromagnetic Ising problem encoded in a cost Hamiltonian H_z.
• Describe continuous-time QA and its digitized/Trotterized variants with parameters gamma_m and beta_m.
• Define QAOA as a P-depth digitized alternation of H_z and H_x and its residual energy epsilon^res_P.
• Use translational invariance and boundary-condition freedom to derive a lower bound epsilon^res_P >= 1/(2P+2) for 2P < N.
• Apply a Jordan-Wigner transformation to map the problem to independent two-level systems and obtain expressions for epsilon^res_P in terms of k-modes.

Experimental results

Research questions

• RQ1 What is the relationship between digitized-QA and QAOA in solving combinatorial optimization?
• RQ2 Can one derive a rigorous bound on the residual energy for QAOA in simple regular graphs, and under what conditions is it tight?
• RQ3 Can an adiabatic-like, regular QAOA schedule be identified within the digitized QA framework without spectral-gap information?
• RQ4 How do boundary conditions and reduced spin chains influence residual energy and optimality of QAOA schedules?

Key findings

• A variational bound on residual energy for MaxCut on 2-regular graphs: epsilon^res_P >= 1/(2P+2) for 2P < N, and epsilon^res_P = 0 for 2P >= N.
• The bound is tight when using anti-periodic boundary conditions, and the reduced-chain argument justifies the bound via translational invariance.
• Jordan-Wigner mapping shows the problem decomposes into independent two-level systems, enabling explicit expressions for residual energy contributions from each mode.
• There exists a regular, adiabatic-like QAOA schedule within digitized-QA that saturates the residual energy bound and can be constructed without spectral-gap knowledge.
• Optimal QAOA landscapes exhibit periodicity and symmetry properties, with evidence that in certain regimes the optimal parameters satisfy beta = gamma' (symmetric manifold) for moderate P.