Adaptive Basis Algorithms for Hybrid Quantum-Classical

• Basis adaptive algorithms are computational methods that iteratively refine the processing basis during runtime based on physical and statistical criteria.
• They integrate quantum sampling with classical optimization to dynamically construct and prune the basis, thereby improving simulation accuracy and reducing resource demands on NISQ devices.
• These algorithms support applications in quantum chemistry, many-body physics, and quantum machine learning, achieving high accuracy with reduced circuit depths and shot counts.

A basis adaptive algorithm for hybrid quantum-classical platforms refers to any hybrid computational scheme in which the effective basis used for quantum or classical processing is dynamically constructed, expanded, or pruned during runtime, based on physical, informational, or statistical criteria derived from quantum hardware outputs and classical post-processing. This adaptivity offers a route to resource-efficient simulation, learning, and optimization of quantum many-body systems, quantum chemistry, and quantum machine learning tasks, particularly within the current limits set by noisy intermediate-scale quantum (NISQ) hardware. Several lines of research have recently established algorithmic methodologies for quantum circuit design, basis selection, and subspace diagonalization that exemplify this paradigm (Nakagawa et al., 2023, Biswas et al., 14 Dec 2025, Feniou et al., 2023, Santini et al., 28 Oct 2025, Levi et al., 31 Aug 2025, Zhao et al., 2 Apr 2025, Volkmann et al., 2023).

1. Foundational Principles and General Framework

A basis adaptive algorithm combines quantum hardware sampling with classical optimization, with the central concept being the iterative construction (and adaptation) of a computational basis or circuit ansatz that is problem- and measurement-adaptive. This basis may refer to:

• The set of quantum states (often bitstrings or Slater determinants) used for subspace diagonalization, as in Quantum-Selected Configuration Interaction (QSCI) and its variants.
• The set of parameterized quantum gates or operators incrementally added to a variational circuit ansatz in quantum eigensolver or machine learning settings.
• Dynamically refined hypothesis or measurement bases in quantum learning problems to maximize information gain.

The generic hybrid workflow encodes a loop: quantum circuit preparation (possibly shallow or short-time), measurement (sampling over computational or adaptively chosen bases), construction or pruning of the basis, and classical diagonalization or post-processing. The loop may be stopped by convergence thresholds on energy, information gain, or another task-specific metric (Nakagawa et al., 2023, Biswas et al., 14 Dec 2025, Feniou et al., 2023).

2. Algorithmic Realizations

(a) Adaptive Subspace Methods for Ground-State and Spectral Estimation

The "ADAPT-QSCI" protocol constructs a state via a sequence of adaptive Pauli rotations acting on a reference, with the basis for subspace diagonalization grown via measurement statistics from this state. The top $R$ bitstrings (by empirical frequency) define a subspace on which the Hamiltonian is projected and classically diagonalized. Operators for further ansatz growth are ranked using commutators evaluated in the classical subspace state, with adaptive optimization of rotation angles via analytic minimization (Nakagawa et al., 2023).

The "Basis Adaptive Algorithm" for many-body spin chains proceeds by evolving a set of basis bitstrings via shallow Trotterized real-time evolution, sampling outcomes, filtering by discrete symmetries, and then projecting the Hamiltonian onto the surviving subspace for classical diagonalization. Iterative enrichment is performed by selecting states with largest subspace amplitude in the ground-state eigenvector and repeating the process (Biswas et al., 14 Dec 2025).

(b) Variational Ansatz Adaptivity

The Greedy Gradient-free Adaptive VQE (GGA-VQE) builds a variational ansatz one operator at a time, using a closed-form fit of the one-parameter energy landscape to determine both the optimal parameter and operator selection at each step. This avoids the high-dimensional and noise-sensitive global optimizations and gradient evaluations of conventional ADAPT-VQE (Feniou et al., 2023). Qubit-ADAPT with explicitly correlated bases (Volkmann et al., 2023) enhances resource-efficiency by selecting operators and optimizing ansatz in a first-quantized basis tailored to rapid convergence for chemical systems.

(c) Adaptive Subspace Discovery for Spectroscopy and Dynamical Correlations

In the adaptive algorithm for molecular spectra, a perturbed ground state is classically prepared, computational-basis samples are evolved via short, shallow quantum trajectories, and all basis states encountered are collected to define a dynamically relevant subspace. The Hamiltonian is projected onto this subspace for high-resolution spectral reconstruction or long-time dynamics, entirely classically (Santini et al., 28 Oct 2025).

(d) Adaptive Measurement and Inference in Quantum Learning

Quantum Likelihood Estimation (QLE) employs an information-theoretic criterion to adaptively select the initial state, measurement basis, and evolution time at each iteration, maximizing mutual information (minimizing conditional entropy) between quantum measurement outcomes and hypotheses about the system Hamiltonian. This achieves exponential reductions in iterations to convergence in Hamiltonian learning, extending to broader quantum learning tasks (Levi et al., 31 Aug 2025).

(e) Adaptive Quantum Machine Learning Structures

Hybrid quantum-classical classifiers (HQCC) leverage an LSTM-driven controller to adaptively select gate sequences and parameters for parameterized quantum circuits (PQC), optimizing architectural plasticity for accuracy and noise-robustness on real-world tasks, such as MNIST (Zhao et al., 2 Apr 2025).

3. Mathematical Formalism and Circuit Structures

The basis-adaptive class of algorithms is characterized by mathematically precise selection or update rules:

• In QSCI-based methods, the adaptive subspace $S$ is iteratively constructed with $|S| = R$ chosen from the highest-probability basis states in quantum measurements, and the reduced Hamiltonian $H^S_{kl} = \langle r_k|H|r_l\rangle$ is diagonalized classically (Nakagawa et al., 2023).
• In variational ansatz schemes, the circuit is grown via $$U(\boldsymbol\theta) = \prod_{j=1}^M e^{-i\theta_jP_j}$$, with $P_j$ drawn based on maximum commutator norm or energy-drop criteria (Feniou et al., 2023, Volkmann et al., 2023).
• Molecular spectra algorithms build an orthonormal basis for the span of states sampled in quantum short-time evolution, with error and scaling controlled by subspace fidelity bounds and empirical sampling efficiency (Santini et al., 28 Oct 2025).

Circuit costs per iteration are driven by state preparation and short Trotter evolution, with measurement performed natively in the computational basis. The principal classical cost arises from subspace Hamiltonian assembly and diagonalization, typically cubic in the subspace dimension but amenable to sparse methods for large dimensions (Biswas et al., 14 Dec 2025, Nakagawa et al., 2023).

4. Benchmarking, Resource Estimates, and Performance

Numerical benchmarks consistently demonstrate that basis adaptive algorithms achieve ground-state energy errors and spectral estimation below 1% (even sub-milli-Hartree accuracy in molecular cases) with orders-of-magnitude reductions in shot count, quantum circuit depth, and classical optimization overhead compared to non-adaptive or fixed-basis variational methods:

• In hydrogen chains and N₂ dissociation, ADAPT-QSCI yields $|E-E_\text{exact}|<10^{-3}$ Ha, converging in a small number of iterations with shot counts and CNOT counts far lower than qubit-ADAPT VQE (Nakagawa et al., 2023).
• For XXZ spin chains up to 24 qubits, the basis-adaptive algorithm achieves $\Delta E_\text{gs} < 1\%$ with subspace dimensions $<5\times10^5$, outperforming Sampling Krylov Quantum Diagonalization (SKQD) by an order of magnitude in energy error for comparable subspace sizes (Biswas et al., 14 Dec 2025).
• GGA-VQE achieves >98% overlap with the ground state of a 25-qubit Ising model in error-mitigated QPU experiments, exhibiting higher resilience to shot noise than gradient-based adaptive algorithms (Feniou et al., 2023).

Practical resource requirements for near-term quantum computers are summarized as follows:

Algorithm	Quantum Depth/Iter	Classical Diagonalization	Shot Count/Iter	Max Subspace Dimension
ADAPT-QSCI	O(n)	O(R³)	$N_s$	$R\lesssim10^3$
Basis Adaptive BA	$3N$ (Trotter)	O($D^2$)	$m_i M_s$	$D \sim 5\times10^5$
GGA-VQE	$\sim1$	$n/a$ (VQE)	$2M$–$4M$	$m < 100$ (ansatz)

(Nakagawa et al., 2023, Biswas et al., 14 Dec 2025, Feniou et al., 2023)

5. Error Mitigation, Noise Robustness, and Symmetry Exploitation

Basis adaptive algorithms exhibit enhanced noise robustness for several reasons:

• Quantum sampling is restricted to basis states with high physical relevance, reducing the impact of measurement noise.
• Classical post-processing can include measurement error mitigation (e.g., readout debiasing) and circuit-level noise suppression via zero-noise extrapolation (Nakagawa et al., 2023).
• Imposing symmetry filtering (e.g., $U(1)$ total $S^z$ conservation and spatial reflection) further truncates unphysical or noisy outcomes and concentrates resources on the relevant sector (Biswas et al., 14 Dec 2025).
• Variational pruning and plasticity mechanisms discard inessential parameters or gates based on gradient sensitivity or architectural cost functions, improving generalization and limiting noise accumulation (Zhao et al., 2 Apr 2025).
• Information-theoretic adaptivity (as in QLE) maximizes information extraction per shot, reducing the total number of quantum queries required for high-confidence quantum learning (Levi et al., 31 Aug 2025).

6. Extensions, Applications, and Limitations

Basis adaptive algorithms apply across a spectrum of quantum information processing tasks:

• Quantum chemistry (ground- and excited-state calculations, molecular spectra) (Nakagawa et al., 2023, Santini et al., 28 Oct 2025, Volkmann et al., 2023).
• Quantum many-body physics (spin chains, strongly correlated materials) (Biswas et al., 14 Dec 2025, Feniou et al., 2023).
• Quantum machine learning (adaptive circuits for classical–quantum classifiers and feature extraction) (Zhao et al., 2 Apr 2025).
• Hamiltonian learning and quantum property estimation, enabled by information-efficient adaptive querying and Bayesian updates (Levi et al., 31 Aug 2025).

Principal limitations arise from the scaling of classical subspace diagonalization (typically $O(R^3)$, prohibitive for large $R$), potential stalling if commutator-based operator selection loses signal before true convergence, and the inherent shot noise or gate infidelities of current NISQ hardware.

Future directions include blockwise ansatz updates, parallel adaptive basis discovery, integration of orbital rotations in operator pools, use of adaptively constructed bases as high-fidelity initial states for quantum phase estimation, and extensions to open quantum system dynamics and non-Hermitian settings (Nakagawa et al., 2023, Santini et al., 28 Oct 2025, Feniou et al., 2023).

7. Comparison with Non-Adaptive and Traditional Approaches

Basis adaptive algorithms consistently outperform fixed-basis or non-adaptive variants in terms of sample efficiency, ansatz compactness, and resilience to quantum noise:

• ADAPT-QSCI and related methods surpass fixed-basis QSCI and conventional ADAPT-VQE in both accuracy for a given quantum/classical resource budget and robustness to gate and measurement errors (Nakagawa et al., 2023).
• Symmetry-adaptive basis construction dramatically reduces subspace dimension and energy error versus non-filtered sampling schemes (e.g., SKQD) for quantum lattice models (Biswas et al., 14 Dec 2025).
• Machine learning circuits with adaptive structure yield higher task accuracy and lower noise sensitivity than fixed-structure PQC models (Zhao et al., 2 Apr 2025).

These findings collectively validate basis adaptation as a unifying strategy for the broad class of hybrid quantum-classical algorithms aimed at making near-term quantum hardware operationally effective for scientific computing and data-driven applications.