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Sensor Data Fusion: Trends, Solutions, Applications (SDF)

Digitized Adiabatic Bayesian Update via Quantum Approximate Optimization Algorithm

Norman Bunk, Felix Govaers

Published Nov 24, 2025• 0 citations
Publisher

Abstract

We present a gate-based, digitized adiabatic (QAOAstyle) realization of the Bayesian filtering update on discrete grids. Our method encodes the likelihood into a diagonal cost Hamiltonian and implements the anneal via a symmetric Trotter sequence of mixer and cost blocks, avoiding state-conditioned gate constructions and any variational parameter training. We show theoretically that the measurement statistics exhibit the same effective inverse temperature as in the continuous anneal up to$O\left(T^{3} / p^{2}\right)$Trotter error, and that a single energy-scale parameter$\alpha$(applied only to the cost) restores the target posterior variance after a one-shot calibration. On Gaussian test cases, the calibrated digitized anneal closely matches the analytical posterior while maintaining shallow, structured circuits that scale naturally with the register size. These results position digitized adiabatic filtering as a practical alternative to both analog annealing and state-conditioned gate approaches in quantum data fusion.

DOI
10.1109/sdf67080.2025.11330688

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We present a gate-based, digitized adiabatic (QAOAstyle) realization of the Bayesian filtering update on discrete grids. Our method encodes the likelihood into a diagonal cost Hamiltonian and implements the anneal via a symmetric Trotter sequence of mixer and cost blocks, avoiding state-conditioned gate constructions and any variational parameter training. We show theoretically that the measurement statistics exhibit the same effective inverse temperature as in the continuous anneal up toO(T3/p2)O\left(T^{3} / p^{2}\right)O(T3/p2)Trotter error, and that a single energy-scale parameterα\alphaα(applied only to the cost) restores the target posterior variance after a one-shot calibration. On Gaussian test cases, the calibrated digitized anneal closely matches the analytical posterior while maintaining shallow, structured circuits that scale naturally with the register size. These results position digitized adiabatic filtering as a practical alternative to both analog annealing and state-conditioned gate approaches in quantum data fusion.
We propose a quantum algorithm for the filtering step in Bayesian state estimation, based on adiabatic quantum computing (AQC).The approach embeds the likelihood function into the energy landscape of an Ising Hamiltonian, enabling the posterior distribution to emerge through quantum annealing.Unlike existing gate-based methods, our formulation avoids complex conditional operations and scales more naturally with the size of the state space.Numerical simulations confirm that the resulting distributions closely match the correct posterior, even with approximate parameter settings.This demonstrates the feasibility of AQC as a tool for probabilistic inference in high-dimensional filtering problems.
Digitizing an adiabatic evolution is a strategy able to combine the good performance of gate-based quantum processors with the advantages of adiabatic algorithms, thus providing a hybrid model for efficient quantum information processing. In this work we develop validity conditions for high-fidelity digital adiabatic tasks. To this end, we assume a digitizing process based on the Suzuki-Trotter decomposition, which allows us to introduce a digitized adiabatic theorem. As a consequence of this theorem, we show that the performance of such a hybrid model is limited by the fundamental constraints on the adiabatic theorem validity, even in ideal quantum processors. We argue how our approach predicts the existence of intrinsic nonadiabatic errors reported by Barends et al. [Nature (London) 534, 222 (2016)] through an empirical study of digital annealing. In addition, our approach allows us to explain the existence of a scaling of the number of Suzuki-Trotter blocks for the optimal digital circuit with respect to the optimal adiabatic total evolution time, as reported by Mbeng et al. [Phys. Rev. B 100, 224201 (2019)] through robust numerical analysis of digital annealing. We illustrate our results through two examples of digitized adiabatic algorithms, namely, the two-qubit exact-cover problem and the three-qubit adiabatic factorization of the number 21.
We present a quantum algorithm for adiabatic state preparation on a gate-based quantum computer, with complexity polylogarithmic in the inverse error. Our algorithm digitally simulates the adiabatic evolution between two self-adjoint operators H0H_0H0​ and H1H_1H1​, exponentially suppressing the diabatic error by harnessing the theoretical concept of quasi-adiabatic continuation as an algorithmic tool. Given an upper bound ααα on ∥H0∥\|H_0\|∥H0​∥ and ∥H1∥\|H_1\|∥H1​∥ along with the promise that the kkkth eigenstate ∣ψk(s)⟩|ψ_k(s)\rangle∣ψk​(s)⟩ of H(s)≡(1−s)H0+sH1H(s) \equiv (1-s)H_0 + sH_1H(s)≡(1−s)H0​+sH1​ is separated from the rest of the spectrum by a gap of at least γ>0γ> 0γ>0 for all s∈[0,1]s \in [0,1]s∈[0,1], this algorithm implements an operator U~\widetilde{U}U such that ∥∣ψk(1)⟩−U~∣ψk(s)⟩∥≤ε\||ψ_k(1)\rangle - \widetilde{U}|ψ_k(s)\rangle\| \leq ε∥∣ψk​(1)⟩−U∣ψk​(s)⟩∥≤ε using O(α2/γ2)polylog(α/γε)O(α^2/γ^2)\text{polylog}(α/γε)O(α2/γ2)polylog(α/γε) queries to block-encodings of H0H_0H0​ and H1H_1H1​. In addition, we develop an algorithm that is applicable only to ground states and requires multiple queries to an oracle that prepares ∣ψ0(0)⟩|ψ_0(0)\rangle∣ψ0​(0)⟩, but has slightly better scaling in all parameters. We also show that the costs of both algorithms can be further reduced under certain reasonable conditions, such as when ∥H1−H0∥\|H_1 - H_0\|∥H1​−H0​∥ is small compared to ααα, or when more information about the gap of H(s)H(s)H(s) is available. For certain problems, the scaling can even be improved to linear in ∥H1−H0∥/γ\|H_1 - H_0\|/γ∥H1​−H0​∥/γ up to polylogarithmic factors.
Parametric quantum circuits are usually trained by classical, local optimizers, which use the quantum computer only to evaluate the cost function, or its gradient, one point at a time. Quantum Hamiltonian Descent (QHD) takes a different route: in this quantum algorithm for continuous optimization, the parameters become the position of a quantum particle, with the cost function as its potential, and two time schedules control kinetic and potential energy so that the wavefunction explores the whole landscape and tunnels towards the global minimum. QHD, however, assumes a classically computable cost function, and its digital implementation is poorly understood: standard error bounds grow exponentially with the number of qubits per parameter, and the schedules are borrowed from classical optimization. Here we propose to train parametric circuits with QHD entirely on the quantum computer, through an online step in which the qubits encoding the parameters coherently control the circuit on a second register. We prove that, for cost functions with a smooth periodic extension, including those of parametric circuits, the simulation error admits a bound independent of the grid resolution, and that for circuits the bound grows at most linearly with the number of parameters. We also show that the ideal dynamics depends on the two schedules only through a single curve, which controls the adiabaticity of the search, and that the remaining freedom can be used to minimize the gate count. Simulations on benchmark functions and parametric circuits show that, at equal cost, schedules designed in this way markedly increase the probability of finding the global minimum.
We propose a simple quantum algorithm for implementing the diffusion step of grid-based Bayesian filters. The method encodes the advected state density and the process noise density into quantum registers and realizes diffusion using a quantum Fourier transform--based adder. This avoids the explicit convolution required in classical implementations and the repeated coin-flip operations used in quantum random walk approaches. Numerical simulations using a gate-based quantum computing simulator confirm that the proposed approach reproduces the desired probability densities while requiring significantly fewer quantum gates and much shallower circuit depth.
Closed-loop control algorithms for real-time calibration of quantum processors require efficient filters that can estimate physical error parameters based on streams of measured quantum circuit outcomes. Development of such filters is complicated by the highly nonlinear relationship relationship between observed circuit outcomes and the magnitudes of elementary errors. In this work, we apply the extended Kalman filter to data from quantum gate set tomography to provide a streaming estimator of the both the system error model and its uncertainties. Our numerical examples indicate extended Kalman filtering can achieve similar performance to maximum likelihood estimation, but with dramatically lower computational cost. With our method, a standard laptop can process one- and two-qubit circuit outcomes and update gate set error model at rates comparable with current experimental execution.