Topological Obstructions in Quantum Adiabatic Algorithms

Abstract

We point out that, when an optimization problem has more than one solution, the quantum adiabatic algorithms (QAA) encounter topological obstructions leading to adiabatic spectral flows where spectral branches unavoidably traverse the spectral gap above the ground states of the quantum Hamiltonians. This raises serious doubts about the validity of the algorithms in such situations. However, using the Max-Cut problem as an example, we explain and demonstrate here that QAAs correctly detect all existing solutions in one single run. This newly discovered capacity of QAAs to simultaneously detect multiple solutions to an optimization problem can have an important impact on future developments of quantum variational algorithms

Figures (7)

• Figure 1: Schematic of the adiabatic spectral flow for the interpolation between an elementary Hamiltonian $H(0)$ and the Max-Cut Hamiltonian $H(1)$, in the case of multiple solutions. Overlaid are two pairs of spectral intervals $(\Gamma_1, \Gamma_2)$ and $(\Gamma_3,\Gamma_4)$, which can be deformed into each other as required by the Adiabatic Theorem (see the dotted lines).
• Figure 2: Examples of graphs together with one solution of Max-Cut problem for each. At the top of each example, we display the number $|V|$ of vertices, $|E|$ of edges, and $C$ of connecting edges for the optimal solution.
• Figure 3: Spectral flow of $\Lambda(s) e^{\imath t H(s)}$, $s\in [0,1]$, $\Lambda(s)=20s$, generated by the Qiskit script from Listing \ref{['Script:2']} on the graphs from Fig. \ref{['GraphEx']}. The lowest six eigenvalues have been highlighted to facilitate their tracking, and the spectral branches landing in the ground state manifold are identified in the legends. At the top of each panel, we specify the degeneracy $D$ of the ground-levels.
• Figure 4: Outputs of the QAA algorithm for increasing number of time steps ${\rm N_t} \in \{10, 100, 10000\}$ at fixed $\Delta t = 0.1$. Each (a-d) column corresponds to the matching (a-d) graph from Fig. \ref{['GraphEx']}. The histograms were generated with $40,960$ shots and only the 8 most frequent bitstrings are displayed. The bitstrings corresponding to the solutions identified by a brute-force search are highlighted. Bitstrings are shown in MSB-first order as $z = z_{n-1} z_{n-2} \cdots z_1 z_0$, where $z_0$ denotes qubit 0.
• Figure 5: The complete set of solutions of the Max-Cut problem obtained via classical brute-force search for the graph from Fig. \ref{['GraphEx']}(d). Each classical solution can be paired with a bitstring produced by the quantum algorithm (see Fig. \ref{['Convergence_All']}(d)), demonstrating perfect agreement and validation of the QAA output.