Rotosolve: Gradient-Free Quantum Optimizer

• Rotosolve is a gradient-free optimizer that uses Fourier reconstruction to transform parameter updates into exact one-dimensional minimizations.
• It exploits the single-frequency structure from self-inverse Hermitian generators, enabling analytic recovery of cost function minima with minimal evaluations.
• Extensions like multi-frequency generalizations and hybrids (e.g., Rotosolve-Haar, ExcitationSolve) broaden its applicability in variational quantum algorithms.

Rotosolve is a gradient-free coordinate-descent optimizer for parametrized quantum circuits that exploits an exact structural property of many variational quantum objectives: when a single circuit parameter controls a gate with a self-inverse Hermitian generator satisfying $G^2=I$, and all other parameters are held fixed, the one-dimensional cost slice is a single-frequency trigonometric function. In that setting, each coordinate update is an analytic one-dimensional minimization rather than a stepsize-based descent step. In the variational quantum eigensolver literature, the same method also appears as the Nakanishi–Fujii–Todo (NFT) algorithm or as Sequential Minimal Optimization for VQE (SMO-VQE), and subsequent work has extended the basic idea to multi-frequency generators, excitation-operator ansätze, hybrid optimizer stacks, and finite-shot convergence analysis (Pramanik et al., 28 Apr 2026).

1. Mathematical setting and single-parameter structure

A standard formulation considers a parametrized quantum circuit

$$U(\theta)=\prod_{j=1}^d V_j U_j(\theta_j), \qquad U_j(\theta_j)=\exp(-i\theta_j \sigma_j/2),$$

with fixed unitaries $V_j$, a reference state $|\iota\rangle$, and objective

$$f(\theta)=\langle \psi(\theta)|H|\psi(\theta)\rangle,\qquad |\psi(\theta)\rangle=U(\theta)|\iota\rangle,$$

where $H$ is Hermitian and, in the convergence analysis, has spectrum in $[-\Lambda,\Lambda]$. Fixing all coordinates except $\theta_j$ yields a univariate function $f_j(\phi)$ that, for Pauli-generated gates and more generally for two-eigenvalue generators, is a single-frequency sinusoid. Equivalent forms used in the literature are

$f_j(\phi)=A_j\sin(\phi+B_j)+C_j,$

and

$$U(\theta)=\prod_{j=1}^d V_j U_j(\theta_j), \qquad U_j(\theta_j)=\exp(-i\theta_j \sigma_j/2),$$0

with $$U(\theta)=\prod_{j=1}^d V_j U_j(\theta_j), \qquad U_j(\theta_j)=\exp(-i\theta_j \sigma_j/2),$$1 and $$U(\theta)=\prod_{j=1}^d V_j U_j(\theta_j), \qquad U_j(\theta_j)=\exp(-i\theta_j \sigma_j/2),$$2 (Pramanik et al., 28 Apr 2026).

The same structure can be derived directly from the gate form

$$U(\theta)=\prod_{j=1}^d V_j U_j(\theta_j), \qquad U_j(\theta_j)=\exp(-i\theta_j \sigma_j/2),$$3

under the assumption $$U(\theta)=\prod_{j=1}^d V_j U_j(\theta_j), \qquad U_j(\theta_j)=\exp(-i\theta_j \sigma_j/2),$$4. After absorbing all preceding and following subcircuits into an effective observable-state pair, the single-parameter dependence of the expectation value reduces to

$$U(\theta)=\prod_{j=1}^d V_j U_j(\theta_j), \qquad U_j(\theta_j)=\exp(-i\theta_j \sigma_j/2),$$5

or equivalently to polar form

$$U(\theta)=\prod_{j=1}^d V_j U_j(\theta_j), \qquad U_j(\theta_j)=\exp(-i\theta_j \sigma_j/2),$$6

with $$U(\theta)=\prod_{j=1}^d V_j U_j(\theta_j), \qquad U_j(\theta_j)=\exp(-i\theta_j \sigma_j/2),$$7 and $$U(\theta)=\prod_{j=1}^d V_j U_j(\theta_j), \qquad U_j(\theta_j)=\exp(-i\theta_j \sigma_j/2),$$8. This exact sinusoidal dependence is the core structural assumption behind Rotosolve (Pankkonen et al., 9 Oct 2025).

The same viewpoint extends beyond Pauli rotations. If the generator has $$U(\theta)=\prod_{j=1}^d V_j U_j(\theta_j), \qquad U_j(\theta_j)=\exp(-i\theta_j \sigma_j/2),$$9 distinct frequency components, then along a single coordinate the objective becomes a trigonometric polynomial with $V_j$0 harmonics,

$V_j$1

where $V_j$2 is the spectral gap in the two-eigenvalue case. This already places Rotosolve within a broader Fourier-reconstruction framework rather than a purely Pauli-specific construction (Pramanik et al., 28 Apr 2026).

2. Analytic coordinate minimization

For the single-frequency case, the global minimizer along one coordinate is available in closed form. In the representation

$V_j$3

the minimum occurs at

$V_j$4

Equivalently, in the form $V_j$5, the minimizer can be written as

$V_j$6

and in the $V_j$7 convention as $V_j$8 (Jäger et al., 2024).

A standard reconstruction uses three cost evaluations. One algebraically convenient probe set is $V_j$9, with

$|\iota\rangle$0

followed by the exact update

$|\iota\rangle$1

An equivalent reconstruction can be written in terms of

$|\iota\rangle$2

together with amplitude and phase recovery from the three probes (Pramanik et al., 28 Apr 2026).

Other probe schedules are also used. For $|\iota\rangle$3, three distinct angles determine the coefficients $|\iota\rangle$4; one standard choice is three equally spaced points such as $|\iota\rangle$5. In iterative coordinate descent, the energy at the current unshifted angle can be reused from the previous step, so only two new evaluations are needed per parameter update. This is why the single-frequency Rotosolve update is often described as requiring “three evaluations” in full reconstruction but “effective 2 with reuse” in sweeping implementations (Jäger et al., 2024).

The method can also be interpreted as implicitly using both first- and second-order local information. For Pauli-generated gates,

$|\iota\rangle$6

and

$|\iota\rangle$7

From the same three probes, Rotosolve reconstructs the exact sinusoid and therefore performs exact coordinate minimization without any explicit stepsize parameter (Pramanik et al., 28 Apr 2026).

3. Resource model and Fourier generalizations

The basic resource comparison is between analytic function reconstruction and parameter-shift gradients. For single-frequency rotations with $|\iota\rangle$8, full reconstruction requires $|\iota\rangle$9 evaluations and effective $$f(\theta)=\langle \psi(\theta)|H|\psi(\theta)\rangle,\qquad |\psi(\theta)\rangle=U(\theta)|\iota\rangle,$$0 with reuse, while a parameter-shift gradient requires $$f(\theta)=\langle \psi(\theta)|H|\psi(\theta)\rangle,\qquad |\psi(\theta)\rangle=U(\theta)|\iota\rangle,$$1 evaluations. For excitation operators with $$f(\theta)=\langle \psi(\theta)|H|\psi(\theta)\rangle,\qquad |\psi(\theta)\rangle=U(\theta)|\iota\rangle,$$2, the analogous line-search method is ExcitationSolve: the one-dimensional objective is an exact second-order Fourier series, reconstruction requires $$f(\theta)=\langle \psi(\theta)|H|\psi(\theta)\rangle,\qquad |\psi(\theta)\rangle=U(\theta)|\iota\rangle,$$3 evaluations and effective $$f(\theta)=\langle \psi(\theta)|H|\psi(\theta)\rangle,\qquad |\psi(\theta)\rangle=U(\theta)|\iota\rangle,$$4 with reuse, and the standard four-term parameter-shift rule likewise requires $$f(\theta)=\langle \psi(\theta)|H|\psi(\theta)\rangle,\qquad |\psi(\theta)\rangle=U(\theta)|\iota\rangle,$$5 evaluations per partial derivative (Jäger et al., 2024).

Regime	Reconstruction evaluations	Gradient evaluations
Rotations, $$f(\theta)=\langle \psi(\theta)|H|\psi(\theta)\rangle,\qquad |\psi(\theta)\rangle=U(\theta)|\iota\rangle,$$6	$$f(\theta)=\langle \psi(\theta)|H|\psi(\theta)\rangle,\qquad |\psi(\theta)\rangle=U(\theta)|\iota\rangle,$$7 (effective $$f(\theta)=\langle \psi(\theta)|H|\psi(\theta)\rangle,\qquad |\psi(\theta)\rangle=U(\theta)|\iota\rangle,$$8)	$$f(\theta)=\langle \psi(\theta)|H|\psi(\theta)\rangle,\qquad |\psi(\theta)\rangle=U(\theta)|\iota\rangle,$$9
Excitations, $H$0	$H$1 (effective $H$2)	$H$3

The deeper general principle is spectral. For a single-parameter gate $H$4, the expectation value is a finite Fourier series whose frequencies are determined by pairwise differences of eigenvalues of $H$5. When the frequencies are equidistant and of order $H$6, exact reconstruction requires $H$7 evaluations. This yields a generalized Rotosolve procedure: reconstruct the trigonometric polynomial exactly, then minimize it analytically when possible or numerically on the exact reconstructed surface when higher harmonics are present (Wierichs et al., 2021).

This multi-frequency extension is necessary when the coordinate dependence is not purely single-harmonic. If the generator has $H$8 distinct frequency components, then $H$9 probes suffice to fit the corresponding trigonometric polynomial and minimize along that coordinate. Likewise, if the same parameter appears $[-\Lambda,\Lambda]$0 times in a circuit with $[-\Lambda,\Lambda]$1, the one-dimensional line shape becomes a Fourier series of order $[-\Lambda,\Lambda]$2 rather than a simple sinusoid; for excitation generators with $[-\Lambda,\Lambda]$3, the order becomes $[-\Lambda,\Lambda]$4, and reconstruction requires $[-\Lambda,\Lambda]$5 samples (Pramanik et al., 28 Apr 2026).

ExcitationSolve is a particularly explicit generalization. For excitation-operator ansätze such as unitary coupled cluster, qubit excitation-based circuits, and Givens rotations, the generator satisfies $[-\Lambda,\Lambda]$6 and has spectrum contained in $[-\Lambda,\Lambda]$7. In that case the single-parameter energy takes the form

$[-\Lambda,\Lambda]$8

and the global minimizer is obtained exactly by solving $[-\Lambda,\Lambda]$9 via a degree-$\theta_j$0 palindromic polynomial and a $\theta_j$1 companion matrix. This preserves the Rotosolve principle—exact analytic line minimization from minimal Fourier reconstruction—while enlarging the admissible ansatz class beyond self-inverse generators (Jäger et al., 2024).

4. Convergence theory, finite-shot behavior, and bias

For several years, interpolation-based coordinate-descent methods such as Rotosolve were mainly treated as heuristics. A rigorous analysis now shows that randomized-coordinate Rotosolve converges to $\theta_j$2-stationary points for smooth non-convex objectives and to $\theta_j$3-suboptimal points when the objective additionally satisfies a Polyak–Łojasiewicz condition. Under a finite-shot oracle with unbiased function estimates $\theta_j$4 satisfying $\theta_j$5, one theorem states

$\theta_j$6

yielding worst-case iteration complexity $\theta_j$7 for smooth non-convex problems. Under the PL condition,

$\theta_j$8

which gives a global linear convergence rate (Pramanik et al., 28 Apr 2026).

The same analysis contrasts Rotosolve with Randomized Coordinate Descent. In worst-case order their rates are comparable, but the comparison is not purely asymptotic: Rotosolve is hyperparameter-free, does not require a stepsize such as $\theta_j$9, and its three-point reconstruction implicitly encodes both gradient and curvature information. Empirically, the paper reports faster descent per iteration and often per circuit evaluation, together with greater variance across runs under noise (Pramanik et al., 28 Apr 2026).

Finite-shot sequential reuse introduces an additional issue in SMO-VQE. In the NFT formulation, the previously estimated minimum energy is reused as one of the three points for the next one-dimensional fit. Under shot noise, this reused minimum becomes systematically biased downward. For the single-frequency equidistant scheme, the bias in the estimated minimum energy is approximated as

$f_j(\phi)$0

where $f_j(\phi)$1 is the true amplitude, equivalently the curvature magnitude at the minimum. The bias is therefore strongest in flat directions and at low shot counts. The same analysis finds that direct bias correction can destabilize optimization when curvature is small, whereas the original biased estimator acts as an implicit regularizer by effectively increasing the fitted curvature. A regularized variant therefore removes the spontaneous bias analytically, introduces a controlled negative offset only during fitting, and then restores unbiased reported energies (Pedrielli et al., 15 May 2026).

This combination of convergence guarantees and bias analysis clarifies two distinct aspects of the method. One concerns idealized optimization dynamics under smoothness and finite-shot assumptions; the other concerns estimator behavior inside the concrete SMO update rule. A plausible implication is that Rotosolve’s practical behavior cannot be assessed solely from its exact analytic line-search property, because its finite-shot reuse mechanism modifies the effective optimization geometry in a curvature-dependent way.

5. Variants, hybridizations, and optimizer ecosystems

Several later works retain the exact single-parameter Rotosolve update while modifying either the gate parameterization or the sweep policy. One variant, Rotosolve-Haar, replaces a fixed Pauli axis by a Haar-random conjugated generator

$f_j(\phi)$2

preserving Hermiticity, unitarity, and the condition $f_j(\phi)$3. The angle update remains the same three-point analytic minimization, but the accessible rotation axes are randomized on the Bloch sphere. This was introduced to improve early-stage convergence and expressivity in shallow circuits (Pankkonen et al., 26 Mar 2025).

Other works combine Rotosolve with more expressive single-qubit optimizers. In the hybrid algorithms built around Free Quaternion Selection (FQS), Rotosolve is used for rapid low-cost initial progress and the optimizer later switches to FQS, which fully optimizes a single-qubit gate as a unit quaternion by minimizing a quadratic form and requires $f_j(\phi)$4 circuit evaluations per parameter. Cost-based switching rules were proposed in two forms: an early-stopping rule that switches when successive cost improvements remain below a threshold for a specified patience, and a cost-average rule that switches when the current cost is sufficiently close to a moving average. In the same comparison framework, the reported per-gate evaluation counts are Rotosolve $f_j(\phi)$5, Fraxis $f_j(\phi)$6, and FQS $f_j(\phi)$7 (Pankkonen et al., 9 Oct 2025).

A different modification is gate freezing. Here the analytic Rotosolve update itself is left unchanged, but gates whose parameters change only weakly between iterations are temporarily skipped so that circuit evaluations are reallocated to parameters that still move significantly. The proposed freezing criterion is based on the wrapped angular distance

$f_j(\phi)$8

with either a fixed freeze length $f_j(\phi)$9 or an incremental per-gate freeze counter. This is a scheduling modification rather than a change in the one-dimensional minimizer (Pankkonen et al., 10 Jul 2025).

Generalized parameter-shift theory provides a broader conceptual framework for these developments. In that framework, Rotosolve is one instance of exact univariate Fourier reconstruction followed by coordinate-wise minimization. ExcitationSolve, generalized Rotosolve for multi-frequency generators, and extensions to quantum analytic descent all fit within this same reconstruction-and-minimization paradigm (Wierichs et al., 2021).

6. Applications, empirical behavior, and limitations

Rotosolve has been studied across VQE, quantum machine learning, and specialized photonic variational settings. In the 2026 convergence paper’s quantum machine learning experiments, Rotosolve typically reaches lower training loss faster than Randomized Coordinate Descent, Stochastic Gradient Descent, Randomized Stochastic Gradient-Free methods, and SPSA, both per iteration and per circuit evaluation, but with higher variance across runs, consistent with sensitivity to noise in fitting the sinusoid (Pramanik et al., 28 Apr 2026).

In hybrid-optimizer studies for layered ansätze with single-qubit $f_j(\phi)=A_j\sin(\phi+B_j)+C_j,$0 rotations followed by entangling layers, cost-based Rotosolve–FQS hybrids are reported to outperform standalone Rotosolve, Fraxis, and FQS across Heisenberg and Fermi–Hubbard benchmarks on both ideal and noisy devices, while using switching criteria based on the observed cost sequence. Related studies report that Rotosolve-Haar and Rotosolve–FQS hybrids improve average performance over standalone methods across Heisenberg models, molecular Hamiltonians, and random state preparation, reflecting the recurring trade-off between the fast low-cost line search of Rotosolve and the higher expressivity of more expensive gate optimizers (Pankkonen et al., 9 Oct 2025).

A further extension moves the Rotosolve principle into linear-optical variational algorithms. Standard bosonic phase shifters generate multi-harmonic cost slices because the number operator has eigenvalues $f_j(\phi)=A_j\sin(\phi+B_j)+C_j,$1, but when phase shifters are constrained to have only two distinct eigenvalues the cost reduces to a single harmonic

$f_j(\phi)=A_j\sin(\phi+B_j)+C_j,$2

so that the same three-point Rotosolve update becomes applicable. In that setting the paper reports faster convergence and fewer evaluations than bosonic gradient-descent baselines on constrained and unconstrained QUBO tests, together with higher gradient variance and slower decay than in the bosonic case, which the authors interpret as reduced susceptibility to barren plateaus (Horner, 2 Oct 2025).

The principal limitations are structural. Exact single-frequency Rotosolve requires that the free gate’s generator have two eigenvalues, typically through $f_j(\phi)=A_j\sin(\phi+B_j)+C_j,$3, and that parameter sharing not introduce additional harmonics. If parameters are shared across several gates, or if the generator has multiple distinct eigenvalue gaps, the objective is no longer a single sinusoid and generalized Fourier reconstruction is required. Noise is a second limitation: in low-shot regimes the estimated coefficients can be noisy, and exact line minimization may overreact. A third limitation is global rather than local: each coordinate update is globally optimal along that coordinate, but high-dimensional non-convex coordinate descent can still become trapped away from the global minimum. This is why later work emphasizes randomized coordinate selection, hybridization with more expressive optimizers, generalized multi-frequency reconstructions, and, for excitation ansätze, higher-order line-search methods such as ExcitationSolve (Pramanik et al., 28 Apr 2026).