Excited Local Quantum Annealing (ExcLQA)

• Excited local quantum annealing (ExcLQA) is a family of protocols that use controlled excursions into excited states from a seeded configuration for local optimization.
• The method integrates reverse annealing, pause‐and‐quench cycles, and controlled diagonal catalysts to explore nearby energy landscapes efficiently.
• ExcLQA is particularly promising for constrained optimization tasks, such as the shortest vector problem, where local search can outperform global annealing in deceptive energy landscapes.

Searching arXiv for the cited papers and closely related work on local and excited-state quantum annealing. I’m going to look up recent arXiv records relevant to ExcLQA and reverse-annealing-based local search. Excited local quantum annealing (ExcLQA) denotes a family of quantum-annealing procedures that perform a local, in-state-space search around a specified configuration while deliberately using excited-state dynamics instead of enforcing a purely adiabatic trajectory. In the 2016 local-search literature, the term itself is not used; the corresponding protocol is the local-search quantum annealing scheme that seeds the annealer in a specified classical state, briefly increases quantum fluctuations to explore a controlled neighborhood in state space, and then returns to classical readout, which in current hardware nomenclature is equivalent to reverse annealing with optional pause-and-quench (Chancellor, 2016). Later work realizes the same excite–explore–return logic through a controlled diagonal catalyst that steers diabatic passages through small-gap regions (Hattori et al., 19 Mar 2025), through reverse annealing with a hot start and explicit longitudinal-field control for excited-state preparation on D-Wave hardware (Imoto et al., 2023), and through a fully classical, physics-inspired extension of local quantum annealing that targets low-lying excited states of classical Ising Hamiltonians (Altelarrea-Ferré et al., 16 Jul 2025). This suggests that ExcLQA is best understood as a family of local or locally controlled excited-state annealing strategies rather than a single protocol.

1. Terminology, scope, and distinction from global quantum annealing

In the reverse-anneal local-search formulation, “local” refers to confining exploration to configurations within a limited Hamming distance of a seeded bitstring. In the transverse-field Ising model, the classical configurations are the vertices of a hypercube, and locality means that only a small fraction of those vertices near the seed are visited. “Excited” denotes that, during the local excursion, the system occupies and transitions among excited eigenstates of the problem-plus-driver Hamiltonian in the vicinity of the seed. Because the device is coupled to a low-temperature bath that mediates transitions in the energy eigenbasis with detailed balance, these excited-state excursions preferentially relax toward lower energies in the local neighborhood (Chancellor, 2016).

This differs structurally from traditional global quantum annealing. Global QA initializes in the ground state of the transverse field and monotonically increases the problem term so as to drive the full system toward the global ground state, without a seeded starting configuration or mid-anneal steering. ExcLQA instead starts from a chosen classical seed, makes a controlled local dip into quantum fluctuation, and returns to classical conditions for readout. The input and output are therefore classical, which is precisely why the protocol can be embedded into iterative local-search and metaheuristic loops (Chancellor, 2016).

A common misconception is to read “local” as necessarily meaning geometrically local on the hardware graph. In the reverse-annealing literature the primary notion is locality in state space, measured by Hamming distance. In the controlled-diagonal-catalyst literature, by contrast, the excitations are called local because they are driven by local $z$-fields; in the reported implementation the same $z$-field is applied to all spins, but the method remains compatible with per-qubit anneal-offset controls on realistic devices (Hattori et al., 19 Mar 2025).

2. Core Hamiltonians and schedule constructions

The standard transverse-field Ising QA Hamiltonian is written as

$H(s)=A(s)H_D+B(s)H_P,$

with

$$H_D=-\sum_i \sigma_i^x,\qquad H_P=\sum_i h_i \sigma_i^z+\sum_{i<j}J_{ij}\sigma_i^z\sigma_j^z.$$

In global QA, $A(s)$ decreases and $B(s)$ increases monotonically as $s$ is ramped from $0$ to $1$. In the local-search variant, the trajectory is modified to

$$s(t): 1 \rightarrow s_{\min}, \quad \text{hold at } s_{\min}\text{ for }\tau,\quad s_{\min}\rightarrow 1,$$

so that the device is initialized at $z$0 in a specified classical seed, taken to an excursion point $z$1 where transverse-field fluctuations are restored, optionally paused, and then returned to classical readout. The ratio $z$2 determines both the locality of the search and the extent to which the dynamics explores nearby excited states (Chancellor, 2016).

Seed preparation can itself be realized on the annealer. A proposed initialization Hamiltonian with a desired bitstring $z$3 as an easy ground state is

$z$4

which is a gauge transform of a strong ferromagnet. A standard forward anneal prepares $z$5 with high probability; the problem Hamiltonian is then reprogrammed to the target instance, after which the reverse-anneal, pause, and quench perform the local search. The paper emphasizes that schedule-only confinement via tuning $z$6 requires minimal or no changes to current annealer designs, whereas anneal offsets and per-qubit schedules are optional refinements rather than prerequisites (Chancellor, 2016).

A second Hamiltonian construction realizes ExcLQA through a controlled diagonal catalyst:

$z$7

where $z$8 and, in the reported study, $z$9. The control field $H(s)=A(s)H_D+B(s)H_P,$0 is time-dependent and satisfies $H(s)=A(s)H_D+B(s)H_P,$1. In that study, $H(s)=A(s)H_D+B(s)H_P,$2 and $H(s)=A(s)H_D+B(s)H_P,$3, while $H(s)=A(s)H_D+B(s)H_P,$4 is optimized variationally. Because the catalyst is diagonal and the driver is transverse, the total Hamiltonian remains stoquastic (Hattori et al., 19 Mar 2025).

A third schedule family appears in reverse annealing with a hot start for excited-state targeting on D-Wave hardware. There the Hamiltonian is

$H(s)=A(s)H_D+B(s)H_P,$5

with $H(s)=A(s)H_D+B(s)H_P,$6, $H(s)=A(s)H_D+B(s)H_P,$7, and $H(s)=A(s)H_D+B(s)H_P,$8. The schedule is divided into three segments with durations $H(s)=A(s)H_D+B(s)H_P,$9 and a cap parameter $$H_D=-\sum_i \sigma_i^x,\qquad H_P=\sum_i h_i \sigma_i^z+\sum_{i<j}J_{ij}\sigma_i^z\sigma_j^z.$$0 that limits the maximum transverse amplitude to $$H_D=-\sum_i \sigma_i^x,\qquad H_P=\sum_i h_i \sigma_i^z+\sum_{i<j}J_{ij}\sigma_i^z\sigma_j^z.$$1. The middle segment holds $$H_D=-\sum_i \sigma_i^x,\qquad H_P=\sum_i h_i \sigma_i^z+\sum_{i<j}J_{ij}\sigma_i^z\sigma_j^z.$$2 constant while the longitudinal control $$H_D=-\sum_i \sigma_i^x,\qquad H_P=\sum_i h_i \sigma_i^z+\sum_{i<j}J_{ij}\sigma_i^z\sigma_j^z.$$3 is adiabatically removed, so that an initially pinned excited eigenstate is morphed into a targeted excited eigenstate of $$H_D=-\sum_i \sigma_i^x,\qquad H_P=\sum_i h_i \sigma_i^z+\sum_{i<j}J_{ij}\sigma_i^z\sigma_j^z.$$4 (Imoto et al., 2023).

3. Locality control, excited-state transport, and diabatic routing

The locality of reverse-anneal ExcLQA can be derived perturbatively around a classical state. For small $$H_D=-\sum_i \sigma_i^x,\qquad H_P=\sum_i h_i \sigma_i^z+\sum_{i<j}J_{ij}\sigma_i^z\sigma_j^z.$$5,

$$H_D=-\sum_i \sigma_i^x,\qquad H_P=\sum_i h_i \sigma_i^z+\sum_{i<j}J_{ij}\sigma_i^z\sigma_j^z.$$6

where $$H_D=-\sum_i \sigma_i^x,\qquad H_P=\sum_i h_i \sigma_i^z+\sum_{i<j}J_{ij}\sigma_i^z\sigma_j^z.$$7 depends on the spectrum of $$H_D=-\sum_i \sigma_i^x,\qquad H_P=\sum_i h_i \sigma_i^z+\sum_{i<j}J_{ij}\sigma_i^z\sigma_j^z.$$8. Under dephasing in the energy eigenbasis, the transition matrix element between locally perturbed classical states scales with Hamming distance,

$$H_D=-\sum_i \sigma_i^x,\qquad H_P=\sum_i h_i \sigma_i^z+\sum_{i<j}J_{ij}\sigma_i^z\sigma_j^z.$$9

For small $A(s)$0, long-range moves are therefore exponentially suppressed in Hamming distance, and locality is obtained without explicit Hamming-distance constraints (Chancellor, 2016).

The controlled-diagonal-catalyst formulation uses a different mechanism. Near an isolated avoided crossing between the ground and first excited states, the Landau–Zener transition probability is

$A(s)$1

where $A(s)$2 is the local gap and $A(s)$3 is the relative rate of change of the levels. By adding the time-dependent local $A(s)$4-field $A(s)$5, the anneal locally reshapes $A(s)$6, modifies both $A(s)$7 and $A(s)$8, moves population to selected excited states at favorable times, and adjusts the late-anneal return path so that the population is steered back to the ground state near $A(s)$9. In the reported dynamics, the ground-state population decreases earlier than in linear QA and then rises near the minimal-gap region, which is the intended excite-and-return signature (Hattori et al., 19 Mar 2025).

A related physical mechanism appears in boson-mediated annealing. There, “excited solutions” are states with the same spin correlations as the target ground state but larger bosonic occupancy, while “local spin errors” correspond to locally incorrect spin patterns. In the near-resonant regime, the instantaneous spectrum develops an avoided crossing between the excited-solution band and the spin-error band around $B(s)$0 for the studied $B(s)$1, $B(s)$2 case. At intermediate anneal times, population interchange across that avoided crossing can reduce final spin errors relative to the direct Ising anneal; in the fully adiabatic limit the same mechanism can instead degrade performance (Pino et al., 2018).

4. Hybrid optimization workflows and classical formulations

The local-search ExcLQA cycle is explicitly designed for hybrid use. A seed bitstring $B(s)$3 is chosen from a classical heuristic such as iterated local search, tabu search, or large-neighborhood search; $B(s)$4 is programmed and annealed to prepare the seed; the target Hamiltonian $B(s)$5 is then programmed; the annealer executes the schedule $B(s)$6, optional pause at $B(s)$7 for duration $B(s)$8, and return to $B(s)$9 for readout; the classical energy is evaluated; and the move is accepted if improved or with Metropolis probability

$s$0

Illustrative parameters are $s$1, $s$2, $s$3, and $s$4–$s$5 cycles per seed, with multiple seeds, tabu memory, variable neighborhood sizes, and gauge transforms used as optional outer-loop controls (Chancellor, 2016).

The same framework supports quantum analogues of population annealing and parallel tempering. A single-qubit proxy,

$s$6

induces an effective temperature

$s$7

or, equivalently in the detailed derivation,

$s$8

In the parallel-tempering analogue, replicas at different $s$9 exchange schedules with

$0$0

while in the population-annealing analogue the expected number of offspring is proportional to $0$1. The same papers emphasize that $0$2 is a proxy for fluctuation strength rather than an exact temperature of the final readout distribution, so strict detailed balance is not guaranteed without post-processing (Chancellor, 2016).

A distinct, fully classical formulation later adopted the same term. In that work, ExcLQA extends Local Quantum Annealing by constraining the state to a product-state ansatz,

$0$3

and by replacing the final objective with

$0$4

The total cost is

$0$5

and the continuous stationary point of $0$6 lies at $0$7, so $0$8 selects a target excitation scale. The method uses stochastic gradient descent, randomized shots, and binary search over $0$9; it does not use overlaps, exclusion sets, or deflation (Altelarrea-Ferré et al., 16 Jul 2025).

5. Demonstrations, benchmarks, and problem classes

In the controlled-diagonal-catalyst study, empirical evidence is reported on maximum-weight independent set instances with perturbative crossings on complete bipartite graphs $1$0. At a fixed long anneal time $1$1, the mean time-to-solution over hard instances scales exponentially with $1$2 for both methods but with distinct exponents:

$1$3

and

$1$4

The ratio $1$5 is reported as consistent with an approximate square speedup, in the sense of improving the effective gap dependence by roughly a square factor. The catalyst-enabled dynamics also succeeds on instances where linear QA fails to obtain the ground state even as $1$6 increases (Hattori et al., 19 Mar 2025).

Direct hardware demonstrations of excited-state search have been performed on a D-Wave quantum annealer. In a two-qubit Ising example with

$1$7

and schedule parameters $1$8, $1$9, $$s(t): 1 \rightarrow s_{\min}, \quad \text{hold at } s_{\min}\text{ for }\tau,\quad s_{\min}\rightarrow 1,$$0, $$s(t): 1 \rightarrow s_{\min}, \quad \text{hold at } s_{\min}\text{ for }\tau,\quad s_{\min}\rightarrow 1,$$1, the hot-start state $$s(t): 1 \rightarrow s_{\min}, \quad \text{hold at } s_{\min}\text{ for }\tau,\quad s_{\min}\rightarrow 1,$$2 is used to target the first excited manifold. For $$s(t): 1 \rightarrow s_{\min}, \quad \text{hold at } s_{\min}\text{ for }\tau,\quad s_{\min}\rightarrow 1,$$3, freezing dominates; for $$s(t): 1 \rightarrow s_{\min}, \quad \text{hold at } s_{\min}\text{ for }\tau,\quad s_{\min}\rightarrow 1,$$4, relaxation to the ground manifold dominates; near $$s(t): 1 \rightarrow s_{\min}, \quad \text{hold at } s_{\min}\text{ for }\tau,\quad s_{\min}\rightarrow 1,$$5, the sum of the target excited-state populations is maximized. In a 4-qubit shortest vector problem embedding with the same timing parameters and $$s(t): 1 \rightarrow s_{\min}, \quad \text{hold at } s_{\min}\text{ for }\tau,\quad s_{\min}\rightarrow 1,$$6, the desired first excited state is dominant near $$s(t): 1 \rightarrow s_{\min}, \quad \text{hold at } s_{\min}\text{ for }\tau,\quad s_{\min}\rightarrow 1,$$7, while the other degenerate first excited state appears with much smaller probability, consistent with local tracking of the state closer in Hamming distance to the hot-start configuration. The reported statistics use 100000-shot success maps for the two-qubit and SVP experiments, together with 1000-shot population scans for the 4-qubit case (Imoto et al., 2023).

The original modernization program also includes proof-of-principle quantum Monte Carlo experiments. On a modified 16-qubit Ising instance, 1000 traditional PIQA forward anneals did not find the correct ground state, whereas reverse local excursions of the ExcLQA type could succeed. The same experiments show three regimes as $$s(t): 1 \rightarrow s_{\min}, \quad \text{hold at } s_{\min}\text{ for }\tau,\quad s_{\min}\rightarrow 1,$$8 varies: a frozen regime with no tunneling, a local tunneling regime with improved success, and a global tunneling regime in which excessive exploration falls into a false basin (Chancellor, 2016).

The fully classical ExcLQA algorithm was benchmarked on shortest vector problem instances mapped to Ising Hamiltonians. For local dimension $$s(t): 1 \rightarrow s_{\min}, \quad \text{hold at } s_{\min}\text{ for }\tau,\quad s_{\min}\rightarrow 1,$$9 and ranks $z$00–$z$01, the solved ratio is reported as consistently above $z$02, averaging approximately $z$03; the average number of shots remains below $z$04; and the approximation factor stays below $z$05. With local dimension $z$06, the method solves some instances up to rank $z$07, specifically $z$08, $z$09, $z$10, and $z$11. On the tested instances it outperforms a Metropolis–Hastings baseline in solved ratio, number of shots, and approximation factor (Altelarrea-Ferré et al., 16 Jul 2025).

6. Applications, limitations, and open issues

A principal application domain is the shortest vector problem. In the excited-state QA formulation of SVP, the ground state of the problem Hamiltonian corresponds to the zero vector, so the shortest nonzero lattice vector lies in the first excited manifold. Excited-state search with nonuniform transverse fields and excited-state initialization yields higher success probability than ground-state search in numerical experiments, and the D-Wave reverse-annealing implementation demonstrates explicit first-excited-state targeting on a small SVP instance (Ura et al., 2022). A plausible implication is that ExcLQA is especially natural for constrained optimization tasks in which the unconstrained ground state is trivial and the desired feasible solution is the lowest state in a restricted excited manifold.

The 2016 local-search papers argue that ExcLQA should outperform global QA when high-quality seeds are available, when the landscape contains deceptive basins that challenge monotonic schedules, when global energies are corrupted by mis-specification or hardware noise, and when improvements depend on multiqubit transitions within a neighborhood. The same papers emphasize mis-specification robustness: if the explored neighborhood is restricted to a typical Hamming radius $z$12, then the relevant calibration error scales as $z$13 rather than $z$14, because only local energy differences among visited states matter (Chancellor, 2016).

Several limitations recur across formulations. In reverse annealing with hot starts, too small $z$15 leads to freeze-out, while too large $z$16 increases relaxation to the ground state; successful operation therefore requires an intermediate regime and co-optimization of dwell time. In the controlled-diagonal-catalyst approach, too-short total runtime can make the method underperform linear QA because the excite-and-return pathway does not complete, and schedules may fail to transfer between instances with similar spectra but different Hamming-distance structure between low-lying states. If the bottleneck gap is highly nonlocal and local $z$17-fields do not significantly reshape the crossing, the method may offer limited benefit (Imoto et al., 2023, Hattori et al., 19 Mar 2025).

Practical implementation remains conditioned by embedding, calibration, and control fidelity. Problems must still be embedded on the hardware graph $z$18; local search does not remove sparse-connectivity constraints. Performance depends on accurate knowledge of $z$19 and $z$20, stable timing of $z$21, and mitigation of bias through gauge averaging or spin-reversal transforms. The catalyst amplitudes in the controlled-diagonal formulation are small relative to $z$22 and must vanish at the boundaries. In the classical product-state ExcLQA, the method is heuristic and hyperparameter-sensitive, and cryptographically relevant lattices of size around $z$23 remain out of reach (Chancellor, 2016, Hattori et al., 19 Mar 2025, Altelarrea-Ferré et al., 16 Jul 2025).

Across these variants, the central principle is stable: ExcLQA treats quantum annealing not as a one-shot attempt to remain in the instantaneous ground state, but as a controlled local or locally controlled excursion through excited manifolds, followed by a return to a useful classical or low-energy output. The different realizations—reverse annealing around seeded states, controlled diabatic routing with diagonal catalysts, hot-start excited-state tracking, and product-state classical surrogates—differ in implementation details, but all reinterpret annealing as an excite–route–return process directed toward low-lying structure that global adiabatic search may miss.