Optimal Interpolation Coordinate Descent

• OICD is a hybrid quantum-classical optimization method for parameterized quantum circuits, leveraging optimal interpolation of trigonometric cost slices.
• It utilizes the finite Fourier structure of cost functions to perform statistically robust coordinate descent, outperforming SGD and RCD in efficiency.
• The method minimizes quantum measurement overhead by reconstructing full cost slices using optimal interpolation node selection for precise parameter updates.

Optimal Interpolation-based Coordinate Descent (OICD) is a hybrid quantum-classical optimization method designed to address the parameter optimization problem in parameterized quantum circuits (PQCs). OICD exploits the trigonometric structure of the cost function slices with respect to individual parameters, leveraging optimal interpolation strategies to achieve efficient, statistically robust coordinate descent steps without increasing quantum measurement overhead compared to established methods such as stochastic gradient descent (SGD) and random coordinate descent (RCD) (Lai et al., 6 Mar 2025).

1. Parameterized Quantum Circuit Optimization

In the variational quantum algorithm framework, the goal is to select real-valued parameters $\boldsymbol\theta = (\theta_1, \dots, \theta_m)$ that minimize the expected value of a Hermitian observable $M$, encoded as:

$$\min_{\boldsymbol\theta \in \mathbb{R}^m}\;C(\boldsymbol\theta) = \langle 0 | U(\boldsymbol\theta)^\dagger M U(\boldsymbol\theta) | 0\rangle$$

with the parameterized quantum circuit $U(\boldsymbol\theta)$ defined as a sequence of fixed unitaries $V_j$ and single-parameter gates $e^{iH_j\theta_j}$. Fixing all parameters except $\theta_j$, the restricted cost function $f_j(x)$ exhibits a finite Fourier (trigonometric) series structure:

$$f_j(x) = \frac{1}{\sqrt{2}}a_0 + \sum_{k=1}^{r_j}\left[a_k \cos(\Omega^j_k x) + b_k \sin(\Omega^j_k x)\right]$$

where the $\Omega^j_k$ are the distinct positive eigenvalue gaps of $M$0. This structure underpins the OICD approach.

2. Interpolation and Slice Recovery

Quantum devices yield noisy estimates of $M$1 due to finite measurement (shot) sampling. OICD uses interpolation to reconstruct the trigonometric slice from $M$2 quantum measurements at distinct nodes $M$3:

$M$4

where $M$5 is the matrix of trigonometric basis functions evaluated at the nodes $M$6, and $M$7 estimates the true Fourier coefficients. Once $M$8 is computed, the surrogate function $M$9 and its derivatives can be evaluated classically without further quantum queries, supporting efficient subsequent optimization.

3. Optimal Interpolation Node Selection

The precision in reconstructing $$\min_{\boldsymbol\theta \in \mathbb{R}^m}\;C(\boldsymbol\theta) = \langle 0 | U(\boldsymbol\theta)^\dagger M U(\boldsymbol\theta) | 0\rangle$$0 depends on the choice of interpolation nodes. Assuming Gaussian measurement noise, the covariance of the estimated coefficients $$\min_{\boldsymbol\theta \in \mathbb{R}^m}\;C(\boldsymbol\theta) = \langle 0 | U(\boldsymbol\theta)^\dagger M U(\boldsymbol\theta) | 0\rangle$$1 is governed by:

$$\min_{\boldsymbol\theta \in \mathbb{R}^m}\;C(\boldsymbol\theta) = \langle 0 | U(\boldsymbol\theta)^\dagger M U(\boldsymbol\theta) | 0\rangle$$2

and the mean-squared error is minimized by minimizing $$\min_{\boldsymbol\theta \in \mathbb{R}^m}\;C(\boldsymbol\theta) = \langle 0 | U(\boldsymbol\theta)^\dagger M U(\boldsymbol\theta) | 0\rangle$$3. Additional criteria include minimizing the condition number $$\min_{\boldsymbol\theta \in \mathbb{R}^m}\;C(\boldsymbol\theta) = \langle 0 | U(\boldsymbol\theta)^\dagger M U(\boldsymbol\theta) | 0\rangle$$4 and the average variance of higher derivatives. Notably, when the frequencies $$\min_{\boldsymbol\theta \in \mathbb{R}^m}\;C(\boldsymbol\theta) = \langle 0 | U(\boldsymbol\theta)^\dagger M U(\boldsymbol\theta) | 0\rangle$$5 are equidistant (as with Pauli generators), equidistant nodes $$\min_{\boldsymbol\theta \in \mathbb{R}^m}\;C(\boldsymbol\theta) = \langle 0 | U(\boldsymbol\theta)^\dagger M U(\boldsymbol\theta) | 0\rangle$$6 (for arbitrary shift $$\min_{\boldsymbol\theta \in \mathbb{R}^m}\;C(\boldsymbol\theta) = \langle 0 | U(\boldsymbol\theta)^\dagger M U(\boldsymbol\theta) | 0\rangle$$7) are globally optimal—simultaneously minimizing the estimator MSE, achieving optimal condition number (unity), and minimizing average derivative variances.

4. Coordinate Descent Update Mechanism

Each OICD iteration selects a coordinate $$\min_{\boldsymbol\theta \in \mathbb{R}^m}\;C(\boldsymbol\theta) = \langle 0 | U(\boldsymbol\theta)^\dagger M U(\boldsymbol\theta) | 0\rangle$$8, reconstructs $$\min_{\boldsymbol\theta \in \mathbb{R}^m}\;C(\boldsymbol\theta) = \langle 0 | U(\boldsymbol\theta)^\dagger M U(\boldsymbol\theta) | 0\rangle$$9 via interpolation, and updates the parameter by performing an exact argmin over the reconstructed univariate function:

$U(\boldsymbol\theta)$0

The argmin problem reduces to finding the roots of a trigonometric polynomial derivative, which, for equidistant frequencies, is solved efficiently by the eigenvalue method of Boyd–Murray (unit-circle eigenproblem for a $U(\boldsymbol\theta)$1 companion matrix). This procedure yields the global minimizer within the restricted slice.

5. Algorithmic Workflow and Quantum Resource Cost

The OICD algorithm comprises an initial preparation phase and an iterative update loop:

• Preparation: For each parameter, determine $U(\boldsymbol\theta)$2, solve for optimal node locations (analytically or numerically), and precompute $U(\boldsymbol\theta)$3.
• Iteration:

1. Select parameter $U(\boldsymbol\theta)$4 (cyclically or randomly).
2. Query the quantum device at the $U(\boldsymbol\theta)$5 optimal nodes for $U(\boldsymbol\theta)$6 to collect noisy $U(\boldsymbol\theta)$7 samples.
3. Solve $U(\boldsymbol\theta)$8 to recover $U(\boldsymbol\theta)$9.
4. Compute $V_j$0.
5. Update only the chosen coordinate, keeping others fixed.

The quantum cost of each update is $V_j$1 circuit evaluations. This aligns with the shot count per coordinate of RCD and is substantially more efficient than full-gradient methods (which require $V_j$2 shots per iteration). OICD does not necessitate hyperparameter tuning.

6. Theoretical Properties and Error Analysis

OICD guarantees that $V_j$3 is an unbiased estimator of $V_j$4. With optimal node choice, the variance $V_j$5 is independent of $V_j$6, and the derivative variance is given by $V_j$7. The interpolation noise propagates exclusively via the coefficient estimator with covariance as above. Comparison with parameter-shift rule (PSR) approaches highlights that OICD amortizes quantum shots more efficiently: PSR requires $V_j$8 shots per derivative evaluation, while OICD reconstructs the full slice for the same quantum cost, enabling repeated classical evaluation. Per-iteration, OICD achieves a larger descent by computing the slice minimizer directly rather than taking a gradient or stochastic step.

7. Numerical Experiments and Comparative Performance

Benchmarks using Qiskit Aer (1000 shots per measurement) evaluated OICD on MaxCut (4-qubit), the transverse field Ising model (TFIM, 6-qubit, HVA ansatz, $V_j$9), and the XXZ model (6-qubit). In each case, OICD achieved target fidelities or near-ground energies substantially faster than SGD or RCD:

Model	OICD Calls to Solution	SGD/RCD Calls to Solution	Frequency Set
MaxCut (4-node)	$e^{iH_j\theta_j}$0	$e^{iH_j\theta_j}$1	$e^{iH_j\theta_j}$2
TFIM ($e^{iH_j\theta_j}$3, $e^{iH_j\theta_j}$4)	$e^{iH_j\theta_j}$5	$e^{iH_j\theta_j}$6	$e^{iH_j\theta_j}$7
XXZ ($e^{iH_j\theta_j}$8, $e^{iH_j\theta_j}$9)	Noted as superior	Noted as inferior	$\theta_j$0

All RCD and SGD settings used fixed learning rates (0.02 and 0.01, respectively), while OICD required none.

8. Implications and Prospective Developments

OICD exemplifies a statistically and computationally efficient optimization strategy for PQCs, balancing measurement cost against robust descent by leveraging the analytic Fourier structure of cost functions, optimal node selection to minimize statistical error, and efficient classical minimization techniques. Prospective directions include convergence rate analysis under realistic noise, exploiting sparsity in the trigonometric expansion, multi-parameter block updates, and adaptive nodes to mitigate shot-noise drift. OICD's approach is applicable to a wide range of variational quantum algorithms where the underlying circuit generators and cost observables endow the cost landscape with tractable Fourier structure (Lai et al., 6 Mar 2025).