Digital-Analog Quantum Computing Framework

Updated 7 January 2026

Digital-analog quantum computing (DAQC) is a universal paradigm that interleaves programmable single-qubit operations with native analog multi-qubit interactions to generate entanglement.
DAQC employs stepwise (sDAQC) and banged (bDAQC) protocols, reducing circuit depth and enhancing resilience on NISQ devices by leveraging continuous analog dynamics.
The framework enables scalable simulation of arbitrary two-body Hamiltonians with robust error mitigation, adaptable to both qubit and qudit architectures.

Digital-analog quantum computing (DAQC) is a universal quantum computational paradigm that constructs quantum algorithms by interleaving programmable single-qubit gates (“digital” layers) with native multi-qubit evolutions under an analog Hamiltonian (“analog blocks”). DAQC leverages the entanglement-generating capacity and physical robustness of continuous-time analog dynamics, while retaining the algorithmic flexibility of digital quantum logic. The approach supports both stepwise (sDAQC, with analog Hamiltonian switched off during single-qubit gates) and banged (bDAQC, with the analog resource left always-on) protocols. Its universality, resource scaling, and algorithmic flexibility have been rigorously established for both qubit and qudit systems, with particular attention to NISQ devices and beyond (Garcia-de-Andoin et al., 2023, Garcia-de-Andoin et al., 14 Nov 2025, Alvarez-Ahedo et al., 19 Dec 2025).

1. Core Principles and Universal Construction

The essential principle of DAQC is to exploit a native two-body (or more general) many-qubit Hamiltonian

$H_S = \sum_{i<j} \sum_{\mu,\nu \in \{x, y, z\}} h_{ij}^{\mu\nu} \sigma_i^\mu \sigma_j^\nu$

as a persistent entangling resource. This is alternated with fast, high-fidelity, fully programmable single-qubit gates (SQGs). The universality of DAQC arises from the Lie-algebraic result that any entangling two-qubit Hamiltonian, together with arbitrary single-qubit gates, suffices to generate the full unitary group on n qubits (Garcia-de-Andoin et al., 2023, Parra-Rodriguez et al., 2018).

Two main protocols are used:

Stepwise DAQC (sDAQC): Analog evolution is strictly separated from the application of SQGs. Each analog block is followed by a layer of local rotations.
Banged DAQC (bDAQC): The analog Hamiltonian remains continuously engaged, and digital gates are “banged” on top, reducing hardware complexity at the cost of a small coherent error due to non-commutation.

This structure allows the DAQC framework to transform a large number of discrete two-qubit digital gates into fewer analog evolutions, thereby reducing total circuit depth and enhancing resilience to certain noise sources (García-Molina et al., 2021, Martin, 2024).

2. Simulation of Arbitrary Two-Body Hamiltonians

A defining technical challenge for DAQC is the digital-analog compilation of an arbitrary two-body target Hamiltonian

$H_T = \sum_{i<j} \sum_{\mu,\nu} g_{ij}^{\mu\nu} \sigma_i^\mu \sigma_j^\nu$

using a fixed native source Hamiltonian $H_S$ and SQGs. The canonical compilation strategy proceeds as follows (Garcia-de-Andoin et al., 2023, Garcia-de-Andoin et al., 14 Nov 2025):

Block Decomposition: For each pair (i, j), apply all nine Pauli-pair “sandwich” conjugations $U_{ij}^{\mu\nu} = \sigma_i^\mu \otimes \sigma_j^\nu$ before and after a period of analog evolution. Each block produces a conjugated Hamiltonian $H_S^{(k)} = U_k H_S U_k^\dagger$ with selected sign flips.
Linear System Construction: The total analog time $T$ is partitioned into $M = 9 \cdot \frac{n(n-1)}{2} = O(n^2)$ blocks, relating via a sign matrix $M$ the vector of block times $t$ to the scaled target couplings $(g/h)_k$ . In matrix form:

$M t = T (g/h)$

The full unitary is

$U_T(T) \approx \prod_{k=1}^M \exp( -i t_k H_S^{(k)} )$

Guaranteed Solution Properties: The block-sign matrix $M$ is proven non-singular for arbitrary n, ensuring an invertible construction for arbitrary two-body targets (Garcia-de-Andoin et al., 2023). To enforce $t_k \geq 0$ (for physical implementability), the problem is cast as a non-negative least-squares (NNLS) instance, for which polynomial-time algorithms apply (Garcia-de-Andoin et al., 14 Nov 2025).

For native Ising sources, an explicit matrix eigen-decomposition yields an exact protocol using no more than $O(n^2)$ analog blocks, with only polynomial classical compile time, sidestepping the exponential gate-sequence search of fully digital approaches (Garcia-de-Andoin et al., 14 Nov 2025).

3. Resource Scaling, Error Bounds, and Compilation

DAQC protocols achieve favorable scaling for Hamiltonian simulation, especially relative to digital approaches:

Analog Blocks: Number of required analog segments per Trotter step is $O(n^2)$ for arbitrary two-body target Hamiltonians, compared to $O(n^3)$ or worse for leading digital decompositions (Garcia-de-Andoin et al., 14 Nov 2025, Garcia-de-Andoin et al., 2023).
Total Analog Time: The sum of block times per Trotter step, $t_A$ , is lower bounded by $T \|H_T\| / \|H_S\|$ , supporting time-optimal schedules when $H_S \propto H_T$ (Garcia-de-Andoin et al., 2023).
Trotter Error: First-order bound on simulation error (measured in the Frobenius norm) obeys

$\epsilon \leq \frac{2}{n_T t_A^2 \|H_S\|^2 \exp( (n_T + 2) / (n_T t_A \|H_S\|) )}$

Reaching a target error $\epsilon$ requires $n_T = O(1/\epsilon)$ repeats, retaining polynomial efficiency (Garcia-de-Andoin et al., 2023).

For small K-block circuits, a compiled hybrid strategy combines Bayesian optimization (exploration via a Gaussian process surrogate and expected improvement) with local gradient-based refinement (e.g., quasi-Newton descent), yielding up to 55% reduction in simulation error relative to straightforward Trotterization, as demonstrated empirically for 6-qubit XY chains (Garcia-de-Andoin et al., 2023).

4. Hardware Realization and Practical Implementations

DAQC has been implemented and benchmarked on several hardware platforms and simulators, including superconducting circuit architectures, Rydberg atom arrays, and generic devices supporting always-on two-body interactions.

Superconducting Circuits: Natural Ising-type or “cross-resonance” Hamiltonians serve as analog resources, with single-qubit control delivered via microwave (Yu et al., 2021, Martin, 2024).
Neutral Atom Arrays: The Rydberg Hamiltonian $H = \sum_i H_i^d(t) + \sum_{j<i} H_{ij}^{\text{int}}$ , with tunable global Rabi frequencies and detunings, enables digital-analog circuits in variational and genetic optimization contexts, e.g., for quantum chemistry (Llenas et al., 2024).
General Platforms: Any device supporting a universal two-body Hamiltonian and fast single-qubit control is suitable, including trapped ions and coupled qudit arrays (Alvarez-Ahedo et al., 19 Dec 2025).

A notably practical outcome is the resource efficiency in the simulation of interacting models: DAQC protocols have demonstrated larger than 0.98 time-dependent state fidelity in fermion-boson simulations for small system sizes, while the reduction in analog and digital gate requirements compared to conventional digital quantum circuits is explicit and substantial (Kumar et al., 2023).

5. Error Mitigation, Stability, and Scalability

DAQC protocols are robust against various error sources prevalent in NISQ hardware:

Hamiltonian Calibration Errors: Explicit analytic bounds quantify the operator-norm deviation between target and implemented Hamiltonians due to source Hamiltonian miscalibration or unaccounted couplings. The error can be kept polynomial in the system size for bounded-degree graphs and local observables (Garcia-de-Andoin et al., 6 May 2025).
Dynamical-Decoupling Error Suppression: An extended “toggling frame” protocol augments the compilation to symmetrize over unknown or uncontrolled couplings, eliminating linear sensitivity in the error to unaccounted cross-terms, in analogy to dynamical decoupling (Garcia-de-Andoin et al., 6 May 2025).
Noise Mitigation in NISQ Devices: Systematic experimental and simulation studies show that bDAQC often outperforms both sDAQC and purely digital circuits in the presence of decoherence, crosstalk, and gate errors. Zero-noise extrapolation protocols further improve fidelities, achieving above 0.95 (for 8 qubit QFT under realistic noise) and demonstrating resilience to scaling (García-Molina et al., 2021, Martin, 2024).

6. Algorithmic Applications and Performance

DAQC algorithms have been applied to a wide array of computational problems:

Quantum Simulation: Efficient digital-analog circuits for quantum Fourier transform (QFT), quantum phase estimation (QPE), variational eigensolvers (VQE), and simulation of strongly-correlated fermion-boson models have been constructed and benchmarked, typically achieving higher fidelity and resource efficiency at moderate n than digital counterparts (Martin et al., 2019, Kumar et al., 2023, Martin, 2024).
Optimization and Quantum Approximate Optimization Algorithm (QAOA): The DAQC ansatz is naturally compatible with QAOA, providing variational resilience against coherent errors and yielding mean approximation ratios near ideal for sufficiently fast single-qubit operation regimes (Headley et al., 2020).
Variational Quantum Algorithms: DAQC circuits combined with classical co-optimization algorithms (e.g., genetic or Bayesian-gradient optimization) achieve high accuracy in ground-state molecular energy estimation and other variational tasks (Llenas et al., 2024).
Counterdiabatic and Shortcut-to-Adiabaticity Protocols: Nested commutator expansions required for high-order counterdiabatic driving can be compiled into DAQC circuits of fixed depth (independent of system size), by mapping nested commutators to a constant number of suitably dressed analog blocks, demonstrating constant-depth resource scaling for these algorithms (Bhargava et al., 3 Jan 2026).

7. Extensions: Qudit DAQC and Universality

DAQC has been generalized to $d$ -level systems (qudits). The extension replaces single-qubit digital blocks by single-qudit rotations from the Weyl–Heisenberg basis, and adapts conjugation and timing strategies to simulate arbitrary two-body Hamiltonians within $O(d^4 n^2)$ analog blocks (Alvarez-Ahedo et al., 19 Dec 2025). This establishes the universality of DAQC for both qubit and qudit architectures, underlining the breadth of the formalism.