Modern quantum algorithms are increasingly formulated as coherent procedures for implementing polynomial transformations of operators and singular values. This perspective provides a powerful and unifying language for quantum algorithm design, connecting a wide range of distinct problems through five closely related key tools: block-encoding, qubitization, QSP, QSVT and GQSP. Block-encoding embeds non-unitary matrices into larger unitaries; qubitization converts block-encodings into structured operators; QSP, QSVT and GQSP enable polynomial transformations with near-optimal query complexity. Together, these techniques form a general toolkit for transforming matrix functions into implementable quantum circuits. This paper develops these techniques from first principles as a unified framework for constructing quantum algorithms. We apply this framework to representative applications to highlight design principles and demonstrate how distinct algorithms can be constructed from a unified sequence of operator transformations. A central contribution is a systematic decision workflow for selecting the appropriate approach according to the operator structure and the desired transformation polynomial. This perspective clarifies when direct GQSP or through qubitization, or Laurent expansion, or QSVT is most appropriate. We organize algorithmic design into an end-to-end pipeline: identifying the target matrix function, constructing an appropriate block-encoding, determining the relevant spectral domain, designing a polynomial or Laurent-polynomial approximation, synthesizing the phase factors, and translating the transformation into an executable quantum circuit. By applying this unified framework to example applications, we showcase a practical methodology for reasoning, designing, and implementing quantum algorithms based on polynomial transformations.
Tal Gurfinkel, Kaushika De Silva, A. Mahasinghe et al.· 0 citations
Preparation and verification of specific quantum states is an important capability for quantum devices to realise advantages over classical computations and algorithms. In this work, we have demonstrated an end-to-end framework that combines resource-efficient quantum state preparation with rapid, robust fidelity verification on near-term quantum hardware. By experimentally preparing and validating a structured complex quantum state encoding a digitized acoustic signal on the Quantinuum H2-1 trapped-ion platform, we achieved a high hardware fidelity of $F_{\mathrm{hw}} = 0.929$. Crucially, this milestone was realized without relying on idealized assumptions or deep fault-tolerant overhead, but rather through resource-minimal circuits optimized for NISQ-era and early fault-tolerant devices. Furthermore, we addressed a key limitation in current quantum state certification. While validation methods like shadow overlap work well for random states, their sample complexity can become prohibitively high for the structured states used in practical algorithms. We mitigate this by introducing a pre-measurement basis-change technique that reduces the verification parameter, $\tau$, by over 10 orders of magnitude for structured targets. This approach tightens the theoretical certification guarantees of the shadow overlap method and integrates tensor-network preparation and shadow validation into a unified workflow. These results shift the paradigm of how structured classical data can be mapped to and verified on quantum hardware under realistic noise and measurement budgets. By compressing a robust verification procedure to just 1,000 measurement shots, this framework offers an immediate, scalable benchmarking standard.
Archie Butterworth, Josh Green, Yu-Sen Wu et al.· 0 citations
Learning the generator of an open many-body system is more challenging than Hamiltonian learning: local responses, which can directly reveal coherent interaction terms in closed-system dynamics, may also contain dissipative contributions in open-system dynamics. In this paper, we address this challenge by developing an efficient Lindbladian learning framework for a known local candidate generator dictionary with bounded dissipative support and either bounded dual-interaction-graph degree or bounded unweighted local strength. The framework resolves the coherent-dissipative ambiguity by treating local Pauli responses as a linear system over both types of generator terms. Inverting this response system separates their contributions and makes the individual Lindbladian coefficients accessible from local response data in a fixed short-time window. Within this framework, we develop two efficient learning algorithms: Chebyshev--Lobatto response interpolation, which uses logarithmically many short evolution times and has a post-mean cost linear in $M$, with the stated dependence on $\epsilon$, and Single-time projected response contraction, which uses a single fixed evolution time and globally inverts a truncated response function. Both procedures estimate $M$ candidate coefficients to entrywise accuracy $\epsilon$ using $\widetilde{\mathcal{O}}(M/\epsilon^2)$ sample and classical post-processing complexity. Our theoretical results establish local response inversion as a scalable paradigm for learning, calibrating, and diagnosing complex quantum systems from experimentally accessible short-time data.
Jiaxing Song, Yukun Zhang, Xiao Yuan et al.· 0 citations
Quantum advantage is widely expected to require sufficiently deep circuits, where correlations and global computational structure can grow beyond the reach of efficient classical simulation. This expectation is especially stark for constant-depth circuits with local readout: the expectation value of any fixed local observable lies within a bounded backward lightcone and is therefore classically tractable. Here we show that measurement feedback changes this picture. We establish a strict hierarchy of computational power: at fixed coherent depth, increasing the number of feedback outcomes strictly enlarges the class of functions accessible through a local expectation value. The two ends of this hierarchy exhibit distinct computational regimes. With logarithmic feedback, local expectation values for product-state inputs are efficiently classically simulable. Polynomial feedback, by contrast, enables an explicit family of adaptive shallow circuits to encode prime-field discrete logarithm problem~(DLP) into a fixed single-qubit expectation. Assuming the standard worst-case classical hardness of DLP, estimating this expectation value is classically hard. These results reveal a feedback-driven complexity transition, with further implications for resource lower bounds on DLP and the complexity of local-observable estimation under area-law entanglement. Our results open a new route to quantum advantage with shallow quantum circuits.
Yusen Wu, Yukun Zhang, Xiao-Ming Zhang et al.· 0 citations
This work proposes a non-Hermitian QRC in which non-Hermitian dynamics are employed as a tunable resource to significantly enhance the QRC performance, and demonstrates that the non-Hermitian reservoir can be tuned toward the edge of chaos by varying a single parameter that controls the non-Hermitian strength.
Lu-Fan Zhang, Yu-Sen Wu, Yong-Pan Gao et al.· Quantum Science and Technolo...· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
