Quantum Approximate Walk Algorithm

TL;DR

The paper introduces the Quantum Approximate Walk Algorithm (QAWA), a near-term quantum-classical framework that combines a QAOA-inspired ansatz with mid-circuit measurements, activation-encoded weights, and cascaded weighted-sum blocks to learn multivariate correlations in optimization tasks. By encoding classical data via an $R_y$ encoder and employing a coin-controlled mid-circuit measurement, QAWA extracts correlation structure with a shallow circuit, enabling scalable copula learning and Bayesian model averaging without full quantum state tomography. Theoretical analysis shows exact convex interpolation of marginals within the learned correlation space and a distributed learning scheme across multiple QPUs, while experiments on IBM's Pittsburgh hardware and copula benchmarks demonstrate exponential convergence to true dependencies and practical hardware advantages. The approach yields a resource-efficient path to quantum-classical optimization for industrial problems, bridging near-term quantum capabilities with interpretable, scalable correlation learning and partial tomography-free insight into solution quality.

Abstract

The encoding of classical to quantum data mapping through trigonometric functions within arithmetic-based quantum computation algorithms leads to the exploitation of multivariate distributions. The studied variational quantum gate learning mechanism, which relies on agnostic gradient optimization, does not offer algorithmic guarantees for the correlation of results beyond the measured bitstring outputs. Consequently, existing methodologies are inapplicable to this problem. In this study, we present a classical data-traceable quantum oracle characterized by a circuit depth that increases linearly with the number of qubits. This configuration facilitates the learning of approximate result patterns through a shallow quantum circuit (SQC) layout. Moreover, our approach demonstrates that the classical preprocessing of mid-quantum measurement data enhances the interpretability of quantum approximate optimization algorithm (QAOA) outputs without requiring full quantum state tomography. By establishing an inferable mapping between the classical input and quantum circuit outcomes, we obtained experimental results on the state-of-the-art IBM Pittsburgh hardware, which yielded polynomial-time verification of the solution quality. This hybrid framework bridges the gap between near-term quantum capabilities and practical optimization requirements, offering a pathway toward reliable quantum-classical algorithms for industrial applications.

Figures (8)

• Figure 1: Quantum Approximate Walk Algorithm (QAWA) architecture: mid-circuit measurements applied to the QAOA ansatz yields outcomes that feed an $\mathrm{R}_y$ encoding layer with sign negation; the unitary block $C(\theta)$ generates a walkable state that, via one ancilla qubit, controls the weighted-summation learning layer conditioned on the measurement results. The top register $\ket{\psi_a}$ functions as an activation multiplier that globally modulates the learning procedure, and the training loop employs a classical optimizer.
• Figure 2: The architecture of quantum circuit for activation encoding and walkable weighted sum learning.
• Figure 3: The standard quantum-walk oracle uses Hadamard gates to drive the red- and green-marked transition unitaries; a reset gate then clears the qubit, preserving an unbiased coin state. Here, 'INC' and 'DEC' excite and de-excite the state.
• Figure 4: The QAWA oracle is trained by sequentially applying local inversion unitaries $U'_{\textsc{qawa}}$ to individual qubits. Subsequent application of their conjugate transposes reverses the learned transformations, enabling a final SWAP gate to extract the correlations encoded in each qubit.
• Figure 5: (a) The ground truth Gaussian copula density, characterized by a correlation coefficient of $\rho = 0.7$ between asset pairs, establishes the target multivariate dependency structure for subsequent portfolio optimization. (b) The empirical copula density, learned via QAWA after 150 training iterations utilizing mid-circuit measurements. (c) The convergence of the Kullback-Leibler (KL) divergence between the learned copula $\hat{C}_{\text{QAWA}}$ and the true copula $C_{\text{true}}$ is presented as a function of training iterations. This convergence exhibits an exponential decay, modeled by $\text{KL}(t) = 0.082e^{-0.031t} + 0.004$, reaching a threshold below $\varepsilon = 0.01$ (indicated by the red dashed line) after approximately 75 iterations. (d) A comparative analysis of pairwise correlation coefficients for all $\binom{4}{2} = 6$ asset pairs confirms the accurate recovery of both strong intra-sector correlations (specifically for pairs $(0,1)$ and $(2,3)$) and weaker cross-sector correlations. Error bars denote one standard deviation across 10 independent experimental runs.

Theorems & Definitions (9)

• Theorem 1
• Proposition 1
• Claim 1
• Corollary 1
• Corollary 2
• proof
• Lemma 1
• proof
• Definition 1