Noise-Directed Adaptive Remapping (NDAR)

• NDAR is a heuristic optimization technique that exploits structured hardware noise by remapping the cost-function so that noise attractors yield high-quality solutions.
• It iteratively applies gauge transformations in shallow QAOA circuits, alternating between exploration and exploitation to align the noise bias with promising solution candidates.
• Empirical studies on superconducting hardware reveal that NDAR significantly boosts approximation ratios in binary, integer, and qudit optimization problems.

Noise-Directed Adaptive Remapping (NDAR) is a heuristic optimization method that uses structured hardware noise as a computational resource rather than treating it solely as an error source. In its original formulation, NDAR approximately solves binary optimization problems on noisy quantum processors whose dynamics feature a global attractor state by iteratively gauge-transforming the cost-function Hamiltonian so that the attractor corresponds to progressively better candidate solutions. The method was introduced in the context of shallow noisy QAOA on superconducting hardware, then extended to exploration/exploitation studies with controlled amplitude damping, multilevel QUBO pipelines, and integer optimization over qudits and encoded qubits (Maciejewski et al., 2024, Tam et al., 17 Apr 2025, Maciejewski et al., 2024, Hadfield et al., 26 Jun 2026).

1. Origin, motivation, and conceptual premise

NDAR was introduced to address a central limitation of near-term quantum optimization: shallow QAOA on noisy hardware often fails because noise both limits circuit depth and distorts the output distribution. The original setting is a noisy quantum processor with a global attractor, such as amplitude-damping-like noise, where repeated noisy evolution tends to favor a specific computational-basis state. In the conceptual model, that attractor is the canonical state $|0\dots 0\rangle$, although on Rigetti hardware the measured output distribution was observed to concentrate strongly near a physical state resembling $|1\dots 1\rangle$, attributed to hardware effects such as overheating (Maciejewski et al., 2024).

The key premise is that if the device is already biased toward producing a particular bitstring, then the optimization problem can be re-encoded so that the noise-favored bitstring represents a higher-quality solution. NDAR is therefore a symmetry-aware strategy for exploiting dissipative bias. This is especially natural for Ising and QUBO objectives, where gauge-like bitflip transformations alter the interpretation of logical variables without changing the underlying optimization problem (Maciejewski et al., 2024).

Later work made the same logic explicit in heuristic-design language by describing NDAR as alternating between exploration and exploitation. Exploration consists of sampling candidate bitstrings from a circuit, while exploitation consists of remapping the Hamiltonian so that the best sampled bitstring is aligned with the noise attractor, typically the all-zero state under amplitude damping (Tam et al., 17 Apr 2025). Across these formulations, NDAR is not presented as a new unitary ansatz with formal guarantees; it is a meta-algorithm that leverages a hardware-dependent asymmetry created by noise.

2. Core algorithm and mathematical structure

The original NDAR formulation begins from an Ising-type objective

$$\min \sum_i h_i s_i + \sum_{i<j} J_{ij} s_i s_j, \qquad s_i \in \{\pm 1\},$$

with qubit Hamiltonian

$$H = \sum_i h_i Z_i + \sum_{i<j} J_{ij} Z_i Z_j + \dots$$

and, in the main experiments, the Sherrington–Kirkpatrick Hamiltonian

$H = \sum_{i<j} J_{ij} Z_i Z_j,$

where $J_{ij}\in\{\pm 1\}$ are drawn from a bimodal random distribution on a fully connected graph (Maciejewski et al., 2024).

The remapping is implemented with the bitflip gauge operator

$$P_{\mathbf y} = \bigotimes_{i=0}^{n-1} X_i^{y_i}, \qquad \mathbf y \in \{0,1\}^n,$$

which acts by conjugation,

$$H^{\mathbf y} = P_{\mathbf y} H P_{\mathbf y} = \sum_i (-1)^{y_i} h_i Z_i + \sum_{i<j} (-1)^{y_i+y_j} J_{ij} Z_i Z_j + \dots$$

This preserves the spectrum while permuting eigenvectors, and in particular maps $|0\dots 0\rangle$ to $|y_0\dots y_{n-1}\rangle$ (Maciejewski et al., 2024).

The NDAR loop wraps a stochastic optimizer, usually noisy $|1\dots 1\rangle$0 QAOA with classical parameter search. Its basic procedure is:

1. start from an original Hamiltonian $|1\dots 1\rangle$1;
2. run the optimizer on the current Hamiltonian $|1\dots 1\rangle$2;
3. collect sampled bitstrings $|1\dots 1\rangle$3;
4. evaluate each sample and choose the best one;
5. define a gauge transform from the best bitstring $|1\dots 1\rangle$4;
6. update the Hamiltonian by

$|1\dots 1\rangle$5

1. repeat until a heuristic termination rule is met, such as no improvement in the best-found energy or the mean cost (Maciejewski et al., 2024).

The QAOA circuit used in the original formulation is

$|1\dots 1\rangle$6

with

$|1\dots 1\rangle$7

and initial state $|1\dots 1\rangle$8. In the noiseless case, gauge-transformed Hamiltonians are equivalent under QAOA: $|1\dots 1\rangle$9 so measurement probabilities and expectation values are invariant up to relabeling. NDAR is useful precisely because noise breaks that symmetry (Maciejewski et al., 2024).

3. Noise attractors, exploration, and exploitation

Amplitude damping is the paradigmatic noise model in NDAR. In the qubit case, amplitude damping biases the computation toward $$\min \sum_i h_i s_i + \sum_{i<j} J_{ij} s_i s_j, \qquad s_i \in \{\pm 1\},$$0, so low-Hamming-weight states become noise attractors. NDAR exploits this by remapping the current best solution so that, in the new Hamiltonian, the attractor acquires the favorable energy of that solution. The 2025 study summarized this with

$$\min \sum_i h_i s_i + \sum_{i<j} J_{ij} s_i s_j, \qquad s_i \in \{\pm 1\},$$1

which makes explicit how the logical meaning of the attractor changes under remapping (Tam et al., 17 Apr 2025).

That study also recast NDAR in classical heuristic terms. Exploration is the circuit-based sampling step; exploitation is the adaptive Hamiltonian remapping step. Two exploration circuits were compared on IBM’s Heron processor: single-layer QAOA and depth-2 random circuits. The principal control parameter for exploitation was a delay gate duration $$\min \sum_i h_i s_i + \sum_{i<j} J_{ij} s_i s_j, \qquad s_i \in \{\pm 1\},$$2, inserted before measurement to vary the degree of amplitude damping. Increasing delay time strengthened the bias toward low-Hamming-weight states and often improved the best objective value found in each NDAR iteration, but the same work emphasized a tradeoff: additional damping can also produce information loss (Tam et al., 17 Apr 2025).

The same paper introduced a classical NDAR analogue in which each bit is sampled independently with probability $$\min \sum_i h_i s_i + \sum_{i<j} J_{ij} s_i s_j, \qquad s_i \in \{\pm 1\},$$3 of being 0 and $$\min \sum_i h_i s_i + \sum_{i<j} J_{ij} s_i s_j, \qquad s_i \in \{\pm 1\},$$4 of being 1. In that setting, $$\min \sum_i h_i s_i + \sum_{i<j} J_{ij} s_i s_j, \qquad s_i \in \{\pm 1\},$$5 controls how concentrated the samples are around the attractor. The reported conclusion was that controlling the Hamming-weight distribution of sampled bitstrings yields higher quality solutions, and that the closer the exploration samples are concentrated around a well-chosen attractor, the better the solution quality (Tam et al., 17 Apr 2025). Within the NDAR framework, this makes the sampling distribution itself a design variable.

4. Empirical results on superconducting hardware and multilevel optimization

The original experimental demonstration used Rigetti’s Ankaa-2 superconducting QPU on 10 random fully connected 82-qubit Sherrington–Kirkpatrick instances with $$\min \sum_i h_i s_i + \sum_{i<j} J_{ij} s_i s_j, \qquad s_i \in \{\pm 1\},$$6, using $$\min \sum_i h_i s_i + \sum_{i<j} J_{ij} s_i s_j, \qquad s_i \in \{\pm 1\},$$7 QAOA compiled with an efficient SWAP network. For $$\min \sum_i h_i s_i + \sum_{i<j} J_{ij} s_i s_j, \qquad s_i \in \{\pm 1\},$$8, one $$\min \sum_i h_i s_i + \sum_{i<j} J_{ij} s_i s_j, \qquad s_i \in \{\pm 1\},$$9 circuit used roughly $$H = \sum_i h_i Z_i + \sum_{i<j} J_{ij} Z_i Z_j + \dots$$0 iSWAP gates and $$H = \sum_i h_i Z_i + \sum_{i<j} J_{ij} Z_i Z_j + \dots$$1 single-qubit rotations. Parameter optimization used Tree-Structured Parzen Estimators via Optuna, with $$H = \sum_i h_i Z_i + \sum_{i<j} J_{ij} Z_i Z_j + \dots$$2 trials and $$H = \sum_i h_i Z_i + \sum_{i<j} J_{ij} Z_i Z_j + \dots$$3 samples per trial for NDAR, and a fair comparison gave standard QAOA the same total number of function calls and the same effective number of gate-ordering choices (Maciejewski et al., 2024).

The reported performance difference was large. Standard $$H = \sum_i h_i Z_i + \sum_{i<j} J_{ij} Z_i Z_j + \dots$$4 QAOA achieved best-found approximation ratios in the range $$H = \sum_i h_i Z_i + \sum_{i<j} J_{ij} Z_i Z_j + \dots$$5 to $$H = \sum_i h_i Z_i + \sum_{i<j} J_{ij} Z_i Z_j + \dots$$6, whereas NDAR-enhanced $$H = \sum_i h_i Z_i + \sum_{i<j} J_{ij} Z_i Z_j + \dots$$7 QAOA reached $$H = \sum_i h_i Z_i + \sum_{i<j} J_{ij} Z_i Z_j + \dots$$8 to $$H = \sum_i h_i Z_i + \sum_{i<j} J_{ij} Z_i Z_j + \dots$$9 on the same 82-qubit hardware. The same study also reported that standard QAOA could underperform uniform random sampling under the best-sample metric, while NDAR surpassed both standard QAOA and the random baseline after only about 3 iterations and converged in at most about 20 optimization runs (Maciejewski et al., 2024).

The empirical mechanism was investigated directly. A correlation study on the 82-qubit hardware data found a clear positive correlation between the approximation ratio of the attractor state and the approximation ratio achieved by QAOA under that gauge. Distributionally, the raw measured bitstrings were sharply concentrated near the physical attractor state, while the effective Hamming-weight distribution after remapping evolved toward the middle Hamming sector, which for random SK instances is where good solutions are expected to lie. In $H = \sum_{i<j} J_{ij} Z_i Z_j,$0 simulations, gauge choice had no effect without noise, positive correlation appeared under amplitude damping, and NDAR found the ground state in all tested cases after 3 iterations under those simulation settings (Maciejewski et al., 2024).

Subsequent work embedded improved NDAR into a broader hybrid pipeline. A multilevel study combined extended NDAR with Quantum Relax-and-Round (QRR), weighted-QRR, Hamming Distance Quadratic Local Search, random-search preprocessing, and Time-Block QAOA on Rigetti Ankaa-2 (Maciejewski et al., 2024). For 10 random 82-qubit fully connected SK instances with integer coefficients, the extended method obtained normalized approximation ratios in the range $H = \sum_{i<j} J_{ij} Z_i Z_j,$1; for real-valued coefficients, the range was $H = \sum_{i<j} J_{ij} Z_i Z_j,$2. The paper reported convergence in at most 4 NDAR iterations for the integer case and at most 10 for the real-valued case (Maciejewski et al., 2024).

The same work used improved NDAR and QRR as subsolvers in a multilevel algorithm for 6 large-scale graphs with at most $H = \sum_{i<j} J_{ij} Z_i Z_j,$3 variables. The QPU solved many subproblems of size at most 82 qubits, which were then used to construct the global solution. The reported conclusion was that the quantum optimization results were competitive in solution quality relative to classical heuristics used as subsolvers within the multilevel approach (Maciejewski et al., 2024).

5. Generalization from binary optimization to integer and qudit settings

A 2026 generalization extended NDAR from unconstrained binary optimization to optimization over discrete integer domains, especially $H = \sum_{i<j} J_{ij} Z_i Z_j,$4, including both native qudits and several qubit encodings (Hadfield et al., 26 Jun 2026). In this generalized setting, the remapping is no longer unique, because there are additional logical gauge degrees of freedom. The paper identified three requirements for NDAR beyond binary domains: feasibility of the noise attractor, existence of compatible gauge transformations that preserve an efficiently implementable circuit family, and a systematic rule for selecting the transform applied at each step (Hadfield et al., 26 Jun 2026).

The generalized loop preserves the original greedy logic. Given a current Hamiltonian $H = \sum_{i<j} J_{ij} Z_i Z_j,$5, the optimizer produces samples $H = \sum_{i<j} J_{ij} Z_i Z_j,$6, infeasible samples may be removed or repaired, energies are evaluated, the best sample $H = \sum_{i<j} J_{ij} Z_i Z_j,$7 is selected, and the update is

$H = \sum_{i<j} J_{ij} Z_i Z_j,$8

with constraints transformed similarly. The defining property becomes

$H = \sum_{i<j} J_{ij} Z_i Z_j,$9

so the attractor still corresponds to the best solution found so far (Hadfield et al., 26 Jun 2026).

For qudits, the paper analyzed relaxation-type noise with downward population flow

$J_{ij}\in\{\pm 1\}$0

and introduced qudit shift gauges using generalized Pauli operators $J_{ij}\in\{\pm 1\}$1 and $J_{ij}\in\{\pm 1\}$2. In the Max-$J_{ij}\in\{\pm 1\}$3-colorable subgraph example, the qudit gauge family multiplies coefficients by roots of unity rather than merely by $J_{ij}\in\{\pm 1\}$4, and the paper emphasized that the family of admissible gauges is much larger than in the qubit case (Hadfield et al., 26 Jun 2026).

Encoding comparisons were organized around NDAR compatibility. Native qudits were described as the cleanest match because the attractor $J_{ij}\in\{\pm 1\}$5 is always feasible, noise naturally favors low excitation numbers, gauge transforms are shift operations, and the cost Hamiltonian and mixer have favorable locality. Binary or Gray encodings were described as NDAR-compatible when $J_{ij}\in\{\pm 1\}$6 is a power of 2, with simple bit-flip gauges but more local Hamiltonians. Standard one-hot encoding was described as poorly suited because the natural damping attractor $J_{ij}\in\{\pm 1\}$7 is invalid. The augmented 01-hot encoding repairs feasibility but worsens locality. Domain-wall encoding was identified as the strongest unary qubit option because the attractor is valid and remapping preserves structure well (Hadfield et al., 26 Jun 2026).

6. Related concepts, limitations, and interpretive boundaries

NDAR belongs to a broader family of remapping ideas, but adjacent work does not use the term in a uniform way. The 2026 paper "Remapping and navigation of an embedding space via error minimization" does not introduce Noise-Directed Adaptive Remapping as a named framework. Instead, it proposes a substrate-independent view of cognition based on remapping embedding spaces and navigating them by iterative error correction. That framework includes denoising, adaptive remapping, and prediction-error reduction, and it is explicitly described as conceptually supportive but not an explicit NDAR theory (Hartl et al., 20 Jan 2026).

An earlier numerical-analysis precursor is the 2012 paper on adaptive phase-space remapping for Vlasov–Poisson particle-in-cell simulation. That work used periodic reconstruction of the distribution function on a hierarchy of phase-space grids, high-order interpolation, positivity-preserving local redistribution, and adaptive refinement to control particle noise and error accumulation. It is accurately described as a precursor or related approach in a broad remapping sense, but not as a paper that introduces NDAR as a named framework (Wang et al., 2012).

Within quantum optimization itself, several misconceptions recur. NDAR is not a proof that noise is generically beneficial; the original work is explicit that the method is greedy and heuristic, that it depends on noise having a sufficiently structured attractor, and that it may fail when the attractor is unstructured or poorly understood. It can also get stuck in suboptimal gauge sequences, and it relies on encoding symmetries that not all cost functions or ansätze possess (Maciejewski et al., 2024). Conversely, NDAR is not restricted to QAOA alone. The original formulation notes that it can be used with other stochastic optimizers, or even classical algorithms, so long as the problem encoding can be remapped and the hardware noise has a meaningful attractor (Maciejewski et al., 2024).

The significance of NDAR therefore lies in a specific reframing: exact gauge symmetries of optimization encodings become practically relevant once noise breaks their equivalence. In that setting, remapping is not merely a relabeling trick. It is the mechanism by which a device’s dissipative bias is repeatedly redirected toward better regions of the objective landscape.