Hiroshi Ueda

While extracting the entanglement spectrum is essential for probing exotic quantum many-body phases, standard tomographic methods are limited by exponential measurement overhead. To overcome this scalability barrier, we propose a hybrid quantum-classical algorithm for the partial singular value decomposition (SVD) of bipartite states, grounded in the canonical form of matrix product states. Our framework extracts the dominant and subdominant Schmidt components via sequential deflation-based optimization. Because finite circuit depths and hardware noise degrade the mutual orthogonality between these sequentially extracted vectors, we introduce an explicit classical orthogonality correction using pseudo-inverses. Acting as an error-filtering mechanism, this post-processing enforces orthogonality to high numerical precision. Consequently, it relaxes the expressivity requirements on the quantum processor, allowing the use of shallow and suboptimal ansatzes. This tolerance for shallow circuits also enables a concurrent, synergistic architecture. The classically tractable evaluation of overlap matrices is offloaded to tensor-network contractions. Concurrently, the quantum processor is dedicated solely to computing cross-terms with the complex target state, facilitated by an auxiliary shallow reference state. This quantum evaluation design bypasses the need for controlled target-state preparations, thereby suppressing the error accumulation from massive gate sequences while maintaining linear signal sensitivity. We benchmarked our deflation-based algorithm on the ground states of one- and two-dimensional Heisenberg models, where it demonstrates improved precision over global single-circuit optimization methods that target the entire spectrum. By structurally decoupling numerical accuracy from the quantum circuit optimization, our framework provides a robust solution for large-scale entanglement spectrum estimation on advanced near-term quantum devices and early fault-tolerant platforms.