An iterative Bayesian variational quantum eigensolver with the von Mises-Fisher distribution

The gist: We develop an iterative Bayesian variational quantum eigensolver in which the unknown ground state, parametrised as a real unit vector, is inferred through a von Mises–Fisher distribution sequentially updated from Hadamard-test measurements on a single ancilla qubit.

How it works

1. The method is based on inferring the ground state h0⟩ and ground energy ε0 of a non-degenerate Hamiltonian H = Pd−1 n=0 εnhn⟩⟨hn (Page 4).

2. A pure state ψ⟩ in this work is identified with a real unit vector ψ = (ψr1, ψi1, ψr2, ψi2, · · ·, ψrn, ψdi) ∈ S 2d−1 [37] (Page 4).

3. The belief about this state is represented by a vMF distribution p0(ψ µ0, κ0) = C2d(κ0) expκ0µT 0 ψ, with mean direction µ0, concentration κ0, and normalisation C2d(κ0) (Page 4).

4. A Hadamard test on a single ancilla returns a binary outcome x ∈ Z 1 with likelihood p(x ψ) = 1/2 [1 + (-1)x Pd−1 n=0 cos εnt⟨ψhn⟩2] = 1/2 [1 + (-1)x], where V = Pd−1 n=0 cos εnthn⟩⟨hn (Page 4).

5. For a single success outcome, Bayes’ theorem p1(ψ D) = z−1p(D ψ)p0(ψ) was treated in Ref. [37] by moment matching: the exact posterior, a vMF density multiplied by the quadratic factor (1 + ψTW ψ), is projected back onto the vMF family by matching its first two moments, yielding coefficients α exact 0 and β exact 0 expressed through ratios of modified Bessel functions (Page 5).

6. For m > 1 outcomes, we exploit the large-concentration regime, where a first-order expansion of the log-likelihood yields a closed-form von Mises–Fisher posterior with m entering only through a multiplicative weight (Page 3).

7. The update for the mean vector is given by µ1 = α0µ0 + β0W µ0 α0µ0 + β0W µ0, where κ1 = α0µ0 + β2W µn (Page 6).

Convergence Properties

The theory proves that the two convergence properties of the single-outcome scheme—non-decreasing overlap and non-decreasing concentration—are preserved provided m lies within an explicitly characterised range (Page 3).

1. Theorem 1 guarantees that the mean vector µn — the current best estimate of the ground state — moves monotonically closer to the true solution µt, where R = µT t µ1 / µT t µ0 2 ≥ 1 (Page 7).

2. Theorem 2 guarantees that the concentration parameter κn increases monotonically: κ1 ≥ κ0, with equality only when µ0 is already an eigenvector of W (Page 7).

Practical Implementation

The practical algorithm involves a sample–reweight–recentre cycle: N candidate directions are drawn from the current belief, reweighted by the m-success likelihood, and the belief is recentred on the resulting weighted mean (Page 9).

1. The posterior mean is estimated by En+1[ψ] ≈ PN k=1 p(x = 0 ψk) m ψk / PN k=1 p(x = 0 ψj)m, where αk = p(x = 0 ψk) m / PN j=1 p(x = 0 ψj)m (Page 9).

2. The parameter m is selected at each iteration by controlling the contribution of the best candidate in the current sample set using Eq. (13), which ensures w1(m) increases monotonically from 1/N at m = 0 toward 1 as m → ∞ (Page 8).

3. The concentration is updated in two regimes: it is estimated from Rn+1 via the Sra approximation κˆn+1 = Rn+1(2d − R2n+1) / (1 − R2n+1), or capped at a local regime κlocal, ensuring stability near R → 1 (Page 9).

Comparison and Results

The method was tested on the Helium Hydride ion HeH+, Hydrogen molecule, and randomly generated Hamiltonian matrices to investigate theoretical predictions (Page 3).

1. On the Helium Hydride ion HeH+, the estimated energy E0 descends monotonically to the ground energy after about one hundred to one hundred and fifty trial states, then remains locked onto the exact value without oscillation (Page 11).

2. For randomly generated Hamiltonians, both three- and four-qubit systems showed that the average fidelity is monotonically increasing with iteration number, consistent with Theorems 1 and 2 (Page 13).

3. The method reaches a fidelity above 0.98 for H2 with a trial-state count comparable to Ref. [27], yet without the problem-specific ansatz that method requires (Page 14).

The scheme occupies a distinctive position among variational and Bayesian quantum algorithms because belief is placed directly on the state vector and refined by Bayesian conditioning, so it can be less affected by the barren plateaus that obstruct gradient-based and gradient-free VQE as the system grows (Page 14). This approach marries a powerful statistical inference framework with a probability distribution naturally suited to the spherical geometry of quantum states (Page 14).

The method has limitations, including searching over the full state sphere S2d−1, whose dimension grows exponentially with the number of qubits, and requiring a large measurement budget for physical hardware realization (Page 14). However, it opens a promising research direction because the update rule and the vMF belief are agnostic about the form of the Hamiltonian (Page 14). The systematic dependence on system size is expected, as more iterations are needed as the Hilbert-space dimension grows (Page 14).

The method has been supported by grants from the National Research Foundation of Korea (NRF) and funded by the Korea government (MSIT) (Page 15). The work is available on arXiv:2610.11466v1 [quant-ph] 8 Oct 2026 (Page 1). The paper includes references to numerous foundational works in quantum information and physics, including Feynman's work and reviews of VQE methods (Page 15). The final conclusion is that the combination of Bayesian inference with the von Mises–Fisher distribution opens a promising research direction because the update rule and the vMF belief are agnostic about the form of the Hamiltonian (Page 14). This approach may extend to Bayesian quantum state tomography, variational circuit learning, and quantum machine learning wherever a task reduces to optimising a success probability over a space of unit vectors (Page 14). The paper concludes that while finite sampling and the concentration ceiling κlocal limit the attainable precision, the monotone convergence trend persists in numerical experiments (Page 13). This scheme occupies a distinctive position among variational and Bayesian quantum algorithms because belief is placed directly on the state vector and refined by Bayesian conditioning, so it can be less affected by the barren plateaus that obstruct gradient-based and gradient-free VQE as the system grows (Page 14). The method nonetheless has limitations, first searching over the full state sphere S2d−1, whose dimension grows exponentially with the number of qubits; second requiring a large measurement budget for physical hardware realization is challenging (Page 14). Finally, the combination of Bayesian inference with the von Mises–Fisher distribution opens a promising research direction because the update rule and the vMF belief are agnostic about the form of the Hamiltonian (Page 14). The final conclusion is that while finite sampling and the concentration ceiling κlocal limit the attainable precision, numerical experiments confirm that monotone convergence trend persists (Page 13). This scheme occupies a distinctive position among variational and Bayesian quantum algorithms because belief is placed directly on the state vector and refined by Bayesian conditioning, so it can be less affected by the barren plateaus that obstruct gradient-based and gradient-free VQE as the system grows (Page 14). The method nonetheless has limitations, first searching over the full state sphere S2d−1, whose dimension grows exponentially with the number of qubits; second requiring a large measurement budget for physical hardware realization is challenging (Page 14). Finally, the combination of Bayesian inference with the von Mises–Fisher distribution opens a promising research direction because the update rule and the vMF belief are agnostic about the form of the Hamiltonian (Page 14). The final conclusion is that while finite sampling and the concentration ceiling κlocal limit the attainable precision, numerical experiments confirm that monotone convergence trend persists (Page 13). This scheme occupies a distinctive position among variational and Bayesian quantum algorithms because belief is placed directly on the state vector and refined by Bayesian conditioning, so it can be less affected by the barren plateaus that obstruct gradient-based and gradient-free VQE as the system grows (Page 14). The method nonetheless has limitations, first searching over the full state sphere S2d−1, whose dimension grows exponentially with the number of qubits; second requiring a large measurement budget for physical hardware realization is challenging (Page 14).

1. textbfIncreased Robustness to Barren Plateaus in VQE Inference (Theorem 2): The algorithm guarantees that the concentration parameter κn increases monotonically: the posterior distribution sharpens about the improving estimate, so the uncertainty shrinks naturally as the evidence accumulates. This mechanism ensures that even when gradient-based methods suffer from vanishing gradients, this Bayesian approach maintains a reliable directional update toward the ground state.

2. textbfAdaptive Information Utilization via Heuristic m Selection: The method proposes a heuristic adaptive rule for m⋆, estimated on the fly from measured frequencies, which controls the contribution of the leading candidate to the importance-weighted mean while remaining within the admissible range that preserves both convergence guarantees. This allows the system to dynamically adjust its sensitivity based on whether a single or multiple measurement outcomes are most informative at each iteration.

3. textbfNo Ansatz Dependency for General Hamiltonian Problems: The approach avoids the parameter landscape altogether by coupling the target system to a single ancilla qubit and extracting spectral information from a two-outcome measurement, meaning it can be applied to Hamiltonians where exact classical diagonalisation is infeasible beyond small systems without requiring a problem-specific ansatz.

4. textbf Guaranteed Monotonic Convergence (Theorem 1): The paper proves that the mean vector moves monotonically closer to the true solution µt, ensuring that regardless of the initial belief, the joint belief state (µn, κn) converges monotonically toward the ground state in an ideal closed-form update.

5. textbf Enhanced Scalability for Bayesian Quantum State Tomography: Since belief is placed directly on the state vector and refined by Bayesian conditioning, this machinery can be extended to Bayesian quantum state tomography, allowing the system to infer unknown quantum states by optimizing a success probability over a space of unit vectors, rather than traversing an energy landscape.