Universal Resources for QAOA and Quantum Annealing

Abstract

The Quantum Approximate Optimization Algorithm (QAOA) is a variational ansatz that resembles the Trotterized dynamics of a Quantum Annealing (QA) protocol. This work formalizes this connection formally and empirically, showing the angles of a multilayer QAOA circuit converge to universal QA trajectories. Furthermore, the errors in both QAOA circuits and QA paths act as thermal excitations in pseudo-Boltzmann probability distributions whose temperature decreases with the invested resource -- i.e. integrated angles or total time -- and which in QAOA also contain a higher temperature arising from the Trotterization. This also means QAOA and QA are cooling protocols and simulators of partition functions whose target temperature can be tuned by rescaling the universal trajectory. The average cooling power of both methods exhibits favorable algebraic scalings with respect to the target temperature and problem size, whereby in QAOA the coldest temperature is inversely proportional to the number of layers, $T\sim 1/p$, and to the integrated angles -- or integrated interactions in QA.

Figures (9)

• Figure 1: (Color online) Overview of the QAOA algorithm with $p$-layers, depicted schematically. In panel (a), we present a $p$-layer QAOA algorithm. The red and green modules represent the cost and mixer layers, respectively. The cost layer is defined by a quantum spin network Ising Hamiltonian, as described in Eq. \ref{['H_QSNet']}. Each experimental sample, denoted as the $m$-th sample, corresponds to a quantum spin network with topology $\mathcal{G}_m$. Panel (b) illustrates the rescaling of the energies in Eq. \ref{['eq_energyrescaling']}, which maintains the same energy levels structure.
• Figure 2: (Color online) Optimized QAOA state for one problem with $N=16$ qubits and a circuit with $p=1$ (left panel) and $p=15$ layers (right panel). We plot the numerical probability amplitude of the QAOA state as dots, together with a solid line representing the best fit to a pseudo-Boltzmann distribution \ref{['eq_bimodalBoltzmann']}, both as a function of the normalized energy $E_z$. The upper histograms show the density of states. The right-side histograms depict the contributions to the probability of the hot and cold components, with a Gaussian fits revealing signatures of the bimodal distribution. The bottom panels show the cumulative probability $\mathscr{C}(E < E_z)$, to illustrate the dominant contribution of the $\beta_\text{high}$ component.
• Figure 3: (Color online) Evolution of binomial pseudo-Boltzmann states described by Eq. \ref{['eq_bimodalBoltzmann']} with the number of QAOA layers $p$, for 800 random QUBO instances (500 instances for $N=20$). (top) Average effective hot and cold temperatures, together with standard deviation as error bars. (bottom) Distribution of overlaps with the ground state, showing the first to the third quartile (box), the median (solid line) and the 90% percentile interval (error bars).
• Figure 4: (Color online) Annealing trajectories defined by the rescaled optimal QAOA angles $\left(\overline{\Theta}_n/\overline{\Theta}_{\textnormal{max}}, \overline{\Gamma}_n/\overline{\Gamma}_{\textnormal{max}}\right)$. (a) Average trajectories fitted to Eq. \ref{['eq_AQCcirclefit']} for $N=$ 8, 10, 12, 14, 16, 18, and 20 qubits, computed from $p=30$ QAOA angles on 800 QUBO instances (500 for $N=$ 20). We also plot the average angles for $N=20$ qubits as circles. (a) Average rescaled optimal angles for QAOA circuits with $N=18$ spins and increasing number of layers $p=$ 5, 10, 15, 20, 25, and 30 layers, computed over 800 QUBO instances, together with a numerical fit to Eq. \ref{['eq_AQCcirclefit']} for $p=30$ layers (solid line).
• Figure 5: (Color online) Scaling of the average total resources $\Gamma_{\textnormal{max}}\sqrt{N}$ (triangles) and $\Theta_{\textnormal{max}}$ (circles) with increasing number of QAOA layers $p$ for different problem sizes. We plot the average of 800 random QUBO instances (500 instances for $N=20$) using a 95% confidence interval that results indistinguishable from the plot marker.