Reverse Annealing in Quantum Optimization

Updated 7 July 2026

Reverse annealing is a quantum protocol that starts from a user-specified classical state and reverses part of the anneal to refine local solutions.
It leverages restored transverse-field fluctuations to explore the vicinity of an initial seed, effectively escaping local minima in complex landscapes.
Empirical and theoretical studies reveal that its performance, impacted by initialization quality and environmental dynamics, spans optimization and memory probing applications.

Reverse annealing is a quantum annealing protocol in which the evolution begins from a user-specified classical bitstring, reverses partway into a regime with restored transverse-field fluctuations, may include a pause, and then returns to the classical limit for readout. In contrast to conventional forward annealing, which starts from the transverse-field ground state and performs a global search from scratch, reverse annealing is ordinarily used as a seeded local refinement procedure around a candidate solution; more recently, it has also been used as a controlled probe of memory retention and erasure under quantum fluctuations (Ohkuwa et al., 2018, Jattana, 2024, Pelofske et al., 13 Sep 2025).

1. Protocol and operational meaning

On D-Wave hardware, the annealing Hamiltonian is commonly written as

$H(s)= -\frac{A(s)}{2}\Big(\sum_{i=1}^n \hat{\sigma}_x^{(i)}\Big) + \frac{B(s)}{2}\Big(\sum_{i=1}^n h_i \hat{\sigma}_z^{(i)} + \sum_{i\le j} J_{ij}\hat{\sigma}_z^{(i)}\hat{\sigma}_z^{(j)}\Big),$

where $A(s)$ controls the transverse field, $B(s)$ controls the problem Hamiltonian, and $s\in[0,1]$ is the anneal fraction (Pelofske et al., 2023). In forward annealing, $s$ moves monotonically from $0$ to $1$. In reverse annealing, the schedule instead starts at $s=1$ with a classical state encoded, descends to an intermediate value $s_{\mathrm{inv}}$ , may pause there, and then returns to $s=1$ (Pelofske et al., 2023).

This schedule makes reverse annealing operationally distinct from forward annealing. It is not designed to explore the full state space from a uniform superposition, but to search in the neighborhood of a supplied seed state. Several papers therefore describe it as a local-search or refinement mode rather than a global search method (Jattana, 2024, Haba et al., 3 Jan 2025). The key schedule control is often the reverse or reversal distance, expressed as $A(s)$ 0 in one NBMF study, or equivalently by the inversion point $A(s)$ 1 (Haba et al., 3 Jan 2025). If the reversal is too deep, the protocol can forget the initial state and behave more like forward annealing; if it is too shallow, it may not escape the initial basin effectively (Haba et al., 3 Jan 2025).

Because the protocol begins from a classical bitstring, reverse annealing is tightly coupled to prior information. That prior information may come from a classical heuristic, a forward-annealing run, a previous reverse-annealing iteration, or a related optimization instance (Jattana, 2024, Osaba et al., 27 Jan 2025). This dependence on an explicit seed is both the defining strength of reverse annealing and its principal limitation.

2. Analytical frameworks and phase-structure picture

A standard analytical formulation of adiabatic reverse annealing introduces an explicit initialization Hamiltonian:

$A(s)$ 2

with

$A(s)$ 3

Here $A(s)$ 4 is the target problem Hamiltonian, $A(s)$ 5 encodes the initial classical guess, and $A(s)$ 6 is the transverse-field driver; conventional quantum annealing is recovered at $A(s)$ 7 (Ohkuwa et al., 2018).

The fully connected ferromagnetic $A(s)$ 8-spin model became the canonical mean-field setting for reverse-annealing theory because conventional quantum annealing on that model encounters a first-order quantum phase transition for $A(s)$ 9, whereas reverse annealing can break the first-order line if the initial state is sufficiently close to the solution (Ohkuwa et al., 2018). In the zero-random-field case, closeness is quantified by

$B(s)$ 0

the fraction of spins initially aligned with the correct ferromagnetic solution (Ohkuwa et al., 2018).

$B(s)$ 1	Threshold $B(s)$ 2 beyond which the first-order line breaks
3	$B(s)$ 3
5	$B(s)$ 4
7	$B(s)$ 5
11	$B(s)$ 6

Even when a first-order transition cannot be eliminated, the magnetization jump $B(s)$ 7 is reduced as $B(s)$ 8 increases, which the mean-field analysis associates with a larger tunneling rate (Ohkuwa et al., 2018). Closed-system dynamical simulations of the same model later corroborated the equilibrium picture for adiabatic reverse annealing: under favorable parameters, conventional quantum annealing retained exponential scaling while adiabatic reverse annealing exhibited polynomial scaling, whereas iterated reverse annealing of the D-Wave type was ineffective on that model (Yamashiro et al., 2019).

Subsequent theory broadened this picture in two directions. First, Counterdiabatic Reverse Annealing combined reverse annealing with approximate counterdiabatic driving based on low-order nested commutators, with performance gains in ground-state fidelity and time to solution in the short-time regime, including with local counterdiabatic potentials (Passarelli et al., 2022). Second, the mean-field interpretation itself was challenged by the introduction of simulated reverse annealing, a classical analogue based on thermal fluctuations; in the infinite-range non-disordered $B(s)$ 9-spin model, simulated reverse annealing succeeds in every case where adiabatic reverse annealing does and in a narrow range where adiabatic reverse annealing fails (Baldwin, 31 Oct 2025). Reverse annealing in mean-field models is therefore understood not only as a quantum protocol but also as a protocol for reshaping a free-energy landscape around a marked initial state.

3. Initialization strategies, hybridization, and knowledge transfer

Because reverse annealing requires an initial classical state, seed generation has become a central research topic. A graph-coloring study described the core difficulty directly: an inappropriate initial state generally yields no improvement, while finding a good initial state is problem dependent and often difficult (Jattana, 2024). That paper proposed a generic workaround by feeding reverse annealing with low-quality solutions obtained from forward annealing. On $s\in[0,1]$ 0 random Erdős–Rényi graphs with $s\in[0,1]$ 1 vertices, forward annealing found a valid coloring for only $s\in[0,1]$ 2 instances; among the remaining $s\in[0,1]$ 3 cases, reverse annealing seeded with the lowest-energy invalid forward output found at least one valid solution for $s\in[0,1]$ 4 problems, about $s\in[0,1]$ 5 (Jattana, 2024). The same study also found that random initial states perform much worse than best-bitstring seeds, and that the number of unique valid solutions peaks near reverse distance $s\in[0,1]$ 6 (Jattana, 2024).

A parallel line of work uses classical relaxations to generate higher-quality seeds. In nonnegative/binary matrix factorization, the $s\in[0,1]$ 7-update subproblem is relaxed from $s\in[0,1]$ 8 to $s\in[0,1]$ 9, solved classically by projected gradient descent, rounded by the threshold rule $s$ 0 if $s$ 1 and $s$ 2 otherwise, and then supplied to reverse annealing as the initial configuration (Haba et al., 3 Jan 2025). On facial image datasets, this relaxation-assisted initialization produced better convergence than previously studied reverse-annealing initializations and performance close to exact optimization methods (Haba et al., 3 Jan 2025).

Another strand of research broadens seeded annealing beyond native reverse annealing itself. The h-gain feature applies a time-dependent gain to linear biases and can encode a target initial state through a large early-time bias that is later removed; this produces a forward-only analogue of initial-state encoding and can be combined with reverse annealing in hybrid RA+HG schedules (Pelofske et al., 2023, Pelofske et al., 2020). On weighted Maximum Cut and weighted Maximum Clique, these studies found that h-gain can be a viable alternative to reverse annealing and that the relative ranking of RA, HG, and RA+HG depends on problem class and graph density (Pelofske et al., 2023, Pelofske et al., 2020).

Reverse annealing has also been interpreted as a vehicle for transfer of knowledge between related instances. In a preliminary knapsack study on $s$ 3 instances derived from two parent instances, using the best solution of a related instance as the reverse-annealing input often improved robustness and, for the larger parent instance, also improved best results (Osaba et al., 27 Jan 2025). The salient empirical observation was that Hamming-distance closeness showed a clear trend, whereas energy closeness showed no clear general correlation (Osaba et al., 27 Jan 2025). In that setting, structural overlap in bitstring space mattered more than energetic proximity under the target Hamiltonian.

4. Open-system behavior, relaxation, and memory

A recurring result across reverse-annealing studies is that hardware behavior is often governed by open-system relaxation rather than by closed-system adiabatic dynamics alone. For the $s$ 4 ferromagnetic $s$ 5-spin model, unitary reverse annealing was found to be ineffective unless the system already started in the solution, but adding dephasing in the instantaneous energy eigenbasis allowed thermal relaxation to repopulate the ground state, and pausing near the avoided crossing improved performance further (Passarelli et al., 2019). In that work, collective dephasing gave somewhat better performance than independent dephasing (Passarelli et al., 2019).

The structure of the environment matters. Reverse-annealing experiments on the D-Wave 2000Q for the $s$ 6 $s$ 7-spin model revealed strong asymmetry between the two degenerate ground states, depending on the initial state, in the partial success probabilities for all-up and all-down outcomes (Bando et al., 2021). Weak-coupling adiabatic master equations, which predict symmetry between those two ground states, failed to reproduce the experiment, whereas the polaron transformed Redfield equation agreed closely (Bando et al., 2021). A different experimental study on Advantage_5.4 interpreted long reverse-anneal times as approaching equilibrium-like sampling; from one-spin and ferromagnetic-chain data it extracted fitted effective temperatures of approximately $s$ 8 and $s$ 9, respectively, and showed that a classical Markovian master equation can reproduce the observed long-time behavior for larger systems (Mehta et al., 12 Feb 2025).

Open-system effects need not be purely detrimental. In the weak-coupling adiabatic limit, low-temperature decoherence can preserve the usefulness of adiabatic reverse annealing provided the path in the $0$0 plane avoids discontinuous finite-temperature transitions and the final equilibrium state remains ferromagnetic (Le et al., 20 Nov 2025). That analysis also identified two distinct high-temperature failure mechanisms: the disappearance of transition-avoiding paths and the disordering of the final equilibrium state (Le et al., 20 Nov 2025). Remarkably, for a range of initial-guess quality around $0$1, nonzero temperature can create transition-avoiding paths that do not exist at zero temperature (Le et al., 20 Nov 2025).

Reverse annealing is now also used to study memory directly. A 2025 experimental paper reframed reverse annealing as a memory-erasure protocol on odd-numbered antiferromagnetic rings containing a single pinned domain wall (Pelofske et al., 13 Sep 2025). After initialization in a specific $0$2-basis state, the protocol turns on transverse fluctuations, holds at a chosen pause value, and then returns for readout; the central observable is the Shannon entropy of the domain-wall distribution,

$0$3

where $0$4 is the number of edges (Pelofske et al., 13 Sep 2025). In this diagnostic, $0$5 corresponds to perfect memory retention and $0$6 to complete memory loss. The experiments showed three regimes: pinned domain walls and near-zero entropy at low transverse field, a window of partial memory at intermediate field, and nearly uniform domain-wall positions with $0$7 at high transverse field (Pelofske et al., 13 Sep 2025). The window of partial memory broadened with pause time—from about $0$8 orders of magnitude for $0$9 to $1$0 for $1$1 and $1$2 for $1$3—and the entropy diagnostic was sensitive enough to reveal hardware faults, including a case where two malfunctioning qubits produced a baseline entropy of about $1$4 at low $1$5 (Pelofske et al., 13 Sep 2025).

5. Application domains and empirical benchmarks

Reverse annealing has been applied to a diverse set of optimization and learning tasks. In portfolio optimization, mean-variance selection was mapped to a QUBO and solved on the D-Wave 2000Q with seeds from a greedy local search; the optimized reverse-annealing protocol was reported to be more than $1$6 times faster on average than forward quantum annealing for the hardest instances in expected time-to-solution, although that advantage depended on excluding programming, thermalization, readout, and seed-generation overheads (Venturelli et al., 2018).

In multi-AGV routing, reverse annealing was coupled to a fast greedy heuristic that first builds a feasible route assignment and then lets the annealer search nearby improvements (Haba et al., 2022). In a virtual plant with $1$7 AGVs, reverse annealing achieved $1$8 completed tasks and a working rate of $1$9, nearly matching Gurobi’s $s=1$ 0 completed tasks and $s=1$ 1 working rate, while forward annealing reached $s=1$ 2 completed tasks and $s=1$ 3 (Haba et al., 2022). For small problem sizes, reverse annealing was almost $s=1$ 4 faster than Gurobi in time-to-solution, but beyond about $s=1$ 5 variables reverse annealing often failed to find the optimal solution frequently enough for TTS benchmarking and classical performance reasserted itself (Haba et al., 2022).

Matrix factorization has been an especially active application area. In one NBMF study, the best workflow was an initial global search with forward annealing followed by local refinement with reverse annealing using the previous iterate as the seed (Golden et al., 2020). On the facial-image dataset $s=1$ 6 with rank $s=1$ 7, reverse annealing overtook forward annealing once total QPU access time exceeded about $s=1$ 8 s and eventually leveled off at about $s=1$ 9 improvement over forward annealing (Golden et al., 2020). The later relaxation-assisted NBMF work strengthened this picture by showing that projected-gradient relaxed solutions provide better reverse-annealing seeds than alternating-least-squares or forward-annealing seeds (Haba et al., 3 Jan 2025).

In learning applications, reverse annealing has been used to realize data-seeded local sampling for Boltzmann machines. A reverse-annealing RBM study on the D-Wave 2000Q employed the training example itself as an initial boundary condition, with a $s_{\mathrm{inv}}$ 0 reverse step, an $s_{\mathrm{inv}}$ 1 pause at $s_{\mathrm{inv}}$ 2, and a $s_{\mathrm{inv}}$ 3 forward step (Rocutto et al., 2020). The reverse-annealing procedure quickly raised the sampling probability of a meaningful subset of configurations and achieved better reconstruction scores during learning than the forward-annealing alternative (Rocutto et al., 2020).

A broad 2026 benchmarking study synthesized many of these application-level observations across Max-Cut, Number Partitioning, and sparse clustering on a D-Wave Advantage system (Menger et al., 2 Jul 2026). It found that combining forward and reverse annealing consistently improves solution quality and efficiency over forward annealing alone, that the benefit is stronger than simply extending forward-annealing time, and that gains increase with problem complexity (Menger et al., 2 Jul 2026). The best performance occurred in a narrow regime, typically at reverse distance around $s_{\mathrm{inv}}$ 4 to $s_{\mathrm{inv}}$ 5, linking practical tuning to freeze-out points and energy-level crossings in the annealing schedule (Menger et al., 2 Jul 2026).

6. Limitations, controversies, and current interpretation

The empirical literature converges on several constraints. Reverse annealing is highly sensitive to the initial state, reversal depth, pause duration, annealing time, and device-specific noise (Jattana, 2024, Mehta et al., 12 Feb 2025, Menger et al., 2 Jul 2026). The best parameters are problem dependent, and even two D-Wave machines operated under similar protocols need not respond similarly: in graph coloring, Advantage System 6.2 reproduced some behavior but did not outperform the original device under the same settings (Jattana, 2024). In the domain-wall memory study, Advantage_system6.4 and Advantage_system4.1 behaved similarly, whereas Advantage2_system1.3 differed more, plausibly because of higher temperature and/or hardware-specific noise; at $s_{\mathrm{inv}}$ 6 the entropy curves became less smooth and even non-monotonic (Pelofske et al., 13 Sep 2025).

A second controversy concerns what reverse annealing is exploiting physically. One line of work emphasizes phase-transition avoidance, tunneling, and counterdiabatic control in analytically tractable quantum models (Ohkuwa et al., 2018, Passarelli et al., 2022). Another emphasizes that current hardware often behaves as a thermally assisted, open-system local-search process (Passarelli et al., 2019, Mehta et al., 12 Feb 2025). The weak-coupling description itself has been questioned by the experimental failure of adiabatic master equations to reproduce asymmetric partial success probabilities in reverse annealing on the D-Wave 2000Q (Bando et al., 2021). In parallel, the mean-field comparison with simulated reverse annealing makes any claim of uniquely quantum advantage more difficult, since the classical analogue matches or exceeds adiabatic reverse annealing in the solvable infinite-range setting studied there (Baldwin, 31 Oct 2025).

Across these results, reverse annealing is most consistently characterized not as a universal replacement for forward annealing, but as a seeded refinement engine whose value rises when a reasonably good classical state is already available and when forward annealing alone is close but insufficient. The same protocol can preserve memory, partially erase it, or erase it almost completely, depending on how far the schedule reverses into the fluctuation-dominated regime and how long the system is allowed to evolve there (Pelofske et al., 13 Sep 2025). That dual role—as both optimization heuristic and controlled probe of fluctuation-driven dynamics—has made reverse annealing a distinct subfield within quantum annealing research.