DIC-DAC-DOA: Overcoming Quantum Annealing Gaps

• DIC-DAC-DOA is a decomposition method that partitions the Hamiltonian into LM and GM subspaces, enabling precise isolation of anti-crossing bottlenecks in quantum annealing.
• It employs a three-step process—DIC, DAC, and DOA—to transform complex tunneling scenarios into a tractable generalized eigenvalue problem by segregating same-sign and opposite-sign sectors.
• By facilitating the incorporation of non-stoquastic drivers, the algorithm mitigates exponentially small energy gaps, potentially shifting quantum annealing complexity from exponential to polynomial scaling.

The DIC-DAC-DOA algorithm is a structural decomposition and analysis technique designed to elucidate and overcome the exponentially small energy gap (anti-crossing) that arises in stoquastic transverse-field quantum annealing (TFQA) for certain structured combinatorial optimization problems, particularly the Maximum Independent Set (MIS) problem. By systematically decomposing the problem Hamiltonian into distinct substructures and reformulating the spectral problem in a non-orthogonal basis, the algorithm enables precise isolation of the bottleneck responsible for quantum tunneling limitations and provides the foundation for incorporating non-stoquastic driving terms to mitigate these bottlenecks.

1. Structural Decomposition Framework

The DIC-DAC-DOA algorithm implements a three-step structural factorization of the Hamiltonian. Each component targets a specific aspect of the low-energy spectrum relevant to the tunneling process in stoquastic TFQA applied to structured MIS instances.

• DIC (Disjoint Inner decomposition of Cliques): The MIS instance is constructed on a graph $G$ partitioned into cliques (L) associated with degenerate local minima (LM) and an independent set (R) associated with the global minimum (GM) in the Ising mapping. In the case of strictly disjoint LM and GM subgraphs, the decomposition focuses the analysis on relevant subspaces for the LM and GM configurations.

Each $k$-clique can be decomposed in an angular momentum or total spin basis. The subspace of single excitations in the empty and singleton configurations separates naturally into a two-dimensional "same-sign" sector (spin-1/2) and several "opposite-sign" (spin-0) blocks, reflecting symmetries in the local Hilbert space.

• DAC (Decomposition and Analysis of Crossings): After constructing the tensor product of local clique subspaces, the algorithm further separates the full Hilbert space into same-sign and opposite-sign sectors, confining the global ground state to the same-sign block (denoted $H_C$), due to the Perron-Frobenius property of stoquastic Hamiltonians.

An "L-inner" decomposition further separates the LM and GM bare subspaces. At the critical value of the transverse field, the bare energies of LM ($H_L^{(0)}$) and GM ($H_R^{(0)}$) cross, but the shared empty set links these two subspaces, converting the level crossing to an avoided crossing (anti-crossing).

• DOA (Decomposition via Orthogonalization/Diagonalization of the Overlap – Analysis): The restriction of $H_C$ to the subspace spanned by the bare LM and GM ground states leads to consideration of a non-orthogonal set, as the shared empty state appears in both. The eigenvalue problem is thus naturally reformulated as a generalized eigenvalue problem:

$\tilde{H}|\psi\rangle = E\,\tilde{S}|\psi\rangle,$

where $\tilde{H}$ is the projected Hamiltonian and $\tilde{S}$ is the overlap matrix in the non-orthogonal bare-state basis.

2. Tunneling Bottleneck and Stoquastic TFQA Limitations

In stoquastic TFQA, the evolution encounters a tunneling bottleneck governed by an avoided crossing, where the system must transition from the LM sector to the GM sector. The effective coupling responsible for this transition can be quantified as the overlap $g_0 = \langle L_0|R_0 \rangle$ between the ground states of the LM and GM subsystems. This overlap typically exhibits exponential suppression with system size, $g_0 \sim 2^{-(m_L + m_R)/2}$ for LM and GM sectors of size $m_L$ and $m_R$.

The effective two-level Hamiltonian (in the "bare" basis) takes the form:

$$B(w, x) = \begin{bmatrix} -w & -\frac{1}{2}x \ -\frac{1}{2}x & 0 \end{bmatrix},$$

with eigenvalues

$$\beta_k = -\frac{1}{2}\left(w + (-1)^k \sqrt{w^2 + x^2}\right), \quad k=0,1.$$

The energy gap at the anti-crossing is determined by the off-diagonal element $g_0$ and the bare energy $e_0$, yielding

$\Delta(H_{\text{core}}) \approx 2|e_0|g_0.$

This exponentially small gap sharply limits the efficiency of stoquastic TFQA, leading to exponential runtime scaling for problem classes exhibiting this structure.

3. Non-Orthogonal Basis and Generalized Eigenvalue Problem

The most critical technical step in DIC-DAC-DOA is the projection onto a non-orthogonal basis formed by the bare ground states $|L_0\rangle$ and $|R_0\rangle$. The overlap matrix is

$$\tilde{S} = \begin{bmatrix} 1 & g_0 \ g_0 & 1 \end{bmatrix},$$

and the block-diagonal bare Hamiltonian is

$$\tilde{H}_{PP} = \begin{bmatrix} e_{l0} & 0 \ 0 & e_{r0} \end{bmatrix}.$$

Through this formulation, the key off-diagonal coupling between $|L_0\rangle$ and $|R_0\rangle$ is isolated, and the 2×2 effective Hamiltonian $H_\text{core}$ fully captures the anti-crossing structure governing the low-energy tunneling dynamics. The full spectrum of the original problem is mirrored in this reduced generalized eigenvalue structure, as proved via the generalized eigenpair equivalence theorem.

Away from the crossing point, the eigenvalues are well approximated by the unperturbed energies of the LM and GM blocks, while near the crossing, the off-diagonal $g_0$ dominates and determines the gap.

4. Non-Stoquastic Drivers and Bottleneck Mitigation

The principal motivation for the DIC-DAC-DOA algorithm is to facilitate the design and analysis of non-stoquastic driver terms that can mitigate the bottleneck arising from the exponentially small gap. In the stoquastic setting, the system's trajectory in Hilbert space is dominated by same-sign states, leading to destructive interference and confinement within the LM well.

Insertion of an appropriately designed non-stoquastic driver modifies the dynamics of the crossing region. For moderate, problem-sized non-stoquastic terms, the interference can be steered so that the system explores regions avoiding deep localization in the LM sector, potentially opening a larger gap or entirely bypassing the avoided crossing mechanism. If such a driver is too strong, the adiabatic pathway may be spoiled—thereby, an optimal regime for non-stoquasticity exists. This mechanism is rigorously supported by the explicit structural reduction provided by DIC-DAC-DOA.

A plausible implication is that, by controlling the interference structure at the crossing, non-stoquastic drivers may transform certain problem instances from exhibiting exponential to polynomial quantum annealing complexity.

5. Comparison with Related DOA Algorithms and Contextual Significance

While the DIC-DAC-DOA acronym bears similarity to “DOA” (direction-of-arrival) estimation algorithms found in array signal processing, in this context it explicitly refers to the decomposition steps: Disjoint Inner decomposition of Cliques, Decomposition and Analysis of Crossings, and Decomposition via Orthogonalization/Diagonalization of the Overlap – Analysis.

No claims are made regarding the direct relation to direction-of-arrival estimation frameworks, although the algorithm's structural decomposition and overlap-based eigenvalue problem have methodological parallels to subspace-based analysis and perturbative treatments in other fields.

The DIC-DAC-DOA approach is specific to Hamiltonian complexity and quantum adiabatic optimization. Its explicit disentangling of the bottleneck and openness to generalization for incorporating designed (e.g., non-stoquastic) terms sets it apart from prior heuristic or qualitative treatments of tunneling gaps in quantum annealing.

6. Summary Table: Core Features

Step	Description	Mathematical Construct
DIC	Partition problem into LM/GM (cliques/independent)	Subsystem angular momentum
DAC	Separate same/opposite sign, perform L-inner decomposition	Block-diagonal Hamiltonian
DOA	Reformulate as generalized eigenvalue problem	Non-orthog. overlap, $H_\text{core}$

The significance of the DIC-DAC-DOA algorithm is its ability to reduce complex problem Hamiltonians to analytically tractable low-dimensional structures that capture the essential dynamics of tunneling-induced bottlenecks and to guide the design of non-stoquastic drivers capable of bypassing these limitations (Choi, 18 Sep 2025). Its rigorous formalization provides both diagnostic and prescriptive value for quantum annealing and related quantum optimization algorithms.