Exponential Quantum Speedup on Structured Hard Instances of Maximum Independent Set

TL;DR

This work studies maximum independent set (MIS) on a structurally defined family of graphs called GIC graphs, which embed a planted MIS alongside numerous competing local minima. It introduces the Dic-Dac-Doa non-stoquastic adiabatic algorithm, combining Phase I TFQA-based identification of an independent-clique structure with Phase II a three-stage, XX-driver–assisted anneal that removes anti-crossings via structural steering and sign-generating interference. The main result is a polynomial-time evolution on GIC instances, achieving exponential speedup over both transverse-field quantum annealing and advanced classical solvers under supported assumptions, and providing scalable reduced models for verification on near-term quantum devices. These findings shed light on a concrete quantum mechanism—expanding the admissible subspace through non-stoquasticity and harnessing interference—that underpins potential quantum advantage in structured optimization problems, with implications for practical MIS applications and hardware validation.

Abstract

Establishing quantum speedup for computationally hard problems of practical relevance, particularly combinatorial optimization problems, remains a central challenge in quantum computation. In this work, we identify a structurally defined family of classically hard maximum independent set (MIS) instances, and design and analyze a non-stoquastic adiabatic quantum optimization algorithm that exploits this structure. The algorithm runs in polynomial time and achieves an exponential speedup over both transverse-field quantum annealing and state-of-the-art classical solvers on these instances, under assumptions supported by analytical and numerical evidence. We identify the essential quantum mechanism enabling the speedup as the use of a non-stoquastic XX-driver to access a larger sign-structured admissible subspace beyond the stoquastic regime, which allows sign-generating quantum interference to create smooth evolution paths that bypass tunneling. This identifies a distinctive quantum mechanism underlying the speedup and explains why no efficient classical analogue is likely to exist. In addition, our analysis produces scalable small-scale models, derived from our structural reduction, that capture the essential dynamics of the algorithm. These models provide a concrete opportunity for verification of the quantum advantage mechanism on currently available universal quantum computers.

Figures (8)

• Figure 1: Structure of a $\mathtt{dMIC}$ block and a $\mathtt{GIC}$ graph. (a) A single $\mathtt{dMIC}$ block consisting of three cliques of sizes $(4,3,3)$. Each maximal independent set is obtained by selecting exactly one vertex from each clique, yielding $4 \times 3 \times 3$ degenerate maximal independent sets. The highlighted vertex (blue) denotes the planted MIS vertex; all remaining vertices are non-MIS vertices. (b) Schematic structure of a $\mathtt{GIC}$ graph composed of multiple blocks. Edges within cliques and between blocks are omitted for clarity. Blocks are initially connected by complete bipartite couplings, after which cross-block edges are selectively removed to embed a unique planted global MIS spanning multiple blocks, while preserving degree-oblivious local structure.
• Figure 2: Anti-crossings and their iterative removal by Dic-Dac-Doa.(a--c) Illustration of an $(\mathtt{LM},\mathtt{GM})$-anti-crossing. (a) An anti-crossing between the lowest two levels $E_0(t)$ and $E_1(t)$. (b) Bare energies $E_0^{\mathtt{LM}}(t)$ and $E_0^{\mathtt{GM}}(t)$ cross. (c) Overlay showing that the anti-crossing originates from this bare crossing. (d--f) Conceptual illustration of the iterative removal of anti-crossings by Dic-Dac-Doa. (d) Under TFQA, the bare energy $E_0^{\mathtt{GM}}(t)$ (blue) crosses those of two local minima, $E_0^{\mathtt{LM}^{(1)}}(t)$ and $E_0^{\mathtt{LM}^{(2)}}(t)$ at $x^{(1)}=\mathtt{x}(t_1)$ and $x^{(2)}=\mathtt{x}(t_2)$. Each such bare crossing corresponds to an $(\mathtt{LM}^{(i)},\mathtt{GM})$-anti-crossing in the system spectrum. (e) After the first iteration of Dic-Dac-Doa, the first crossing is removed by lifting $E_0^{\mathtt{LM}^{(1)}}(t)$, while the second remains. (f) A second iteration applies the same procedure to lift $E_0^{\mathtt{LM}^{(2)}}(t)$, completing the removal of both anti-crossings.
• Figure 3: Schematic illustration of Phase I. (a) An anti-crossing between the lowest two energy levels under stoquastic TFQA. (b) The anti-crossing originates from competition between the energies associated with $\mathtt{LM}$ and $\mathtt{GM}$; the $\mathtt{LM}$ corresponds to a degenerate set of maximal independent sets generated by cliques forming a $\mathtt{dMIC}$.
• Figure 4: Example graphs illustrating the $G_{\mathsf{dis}}$ and $G_{\mathsf{share}}$ structures. Recall that each $\mathtt{LM}$ here is structurally a $\mathtt{dMIC}$. (a) Disjoint-structure graph $G_{\mathsf{dis}}$: The set $L$ consists of $m_l = 2$ disjoint cliques, each of size $n_1 = n_2 = 4$, whose vertices (pink) generate the local minima $\mathtt{LM}$. The set $R$ (blue) consists of $m_r = 3$ independent vertices forming the global minimum $\mathtt{GM}$. (a) Shared-structure graph $G_{\mathsf{share}}$: Relative to $G_{\mathsf{dis}}$, two vertices in $L$ are shared between the $\mathtt{LM}$ and $\mathtt{GM}$, and are therefore shown in blue instead of pink. The global minimum (in blue) has $m_g = 5$ vertices. In both cases, edges between the pink vertices in $L$ and all vertices in $R$ are complete, though not all are shown for visual clarity.
• Figure 5: Annealing parameter schedule for the system Hamiltonian $\mathrm{H}(t) = \mathtt{x}(t) \mathrm{H}_{\mathsf{X}} + \mathtt{jxx}(t) \mathrm{H}_{\mathsf{XX}} + \mathtt{p}(t)\mathrm{H}_{\mathsf{problem}}$. The transverse field $\mathtt{x}(t)$ (blue) begins at $\Gamma_0$, decreases to $\Gamma_1$ in Stage 0, then to $\Gamma_2$ in Stage 1, and reaches 0 at the end of Stage 2. The $\mathsf{XX}$-coupling strength $\mathtt{jxx}(t)$ (red) increases linearly to $J_{\mathsf{xx}}$ in Stage 0, remains constant in Stage 1, and decreases to 0 in Stage 2. Vertex weights $\mathtt{w}_i$ (orange) and $\mathsf{ZZ}$-couplings $J_{\mathsf{zz}}, J_{\mathsf{zz}}^{\mathsf{clique}}$ (green) ramp to their final values in Stage 0 and remain fixed thereafter.