Noise operators and approximate state vectors

Prathyush Poduval, Zhuowen Zou, Mohsen Imani

Grover’s quantum search algorithm achieves a quadratic speedup over classical search methods but requires an exact oracle that perfectly distinguishes marked states. We propose a modification that replaces the exact oracle with an approximate oracle constructed from a local similarity operator. This operator applies phase shifts based on the similarity between candidate states and the marked state, enabling approximate search while improving robustness to noise. Our analysis shows that the algorithm reliably amplifies marked states even under oracle imperfections, with a tunable trade-off between optimality and error resilience. We validate the method with analytical results and simulations, demonstrating efficient performance in both single- and multi-object search scenarios. Potential applications include error-prone quantum systems and quantum machine learning tasks involving similarity-based search over feature vectors.

Philip L. Bowers

Franz Schwabl

S. T.

The temporal development of the state vectors in the interaction representation is investigated for a quantum-mechanical system that has bound states. It is shown that there is a non-unitary operator which determines the variation of the state vectors of the free states. This operator satisfies the same differential equation and initial condition as the unitary operator which transforms a state vector at the remote past to the state vector at a finite time or in the remote future. The present investigation is relevant to a remark made by H. S. Snyder.No iteration process is made use of in the general investigation. Born's method of successive approximations is discussed at the end.

Franz Schwabl

Ziqing Guo, Ziwen Pan, Alex Khan +1 more

Vectorized quantum block encoding provides a way to embed classical data into Hilbert space, offering a pathway for quantum models, such as Quantum Transformers (QT), that replace classical self-attention with quantum circuit simulations to operate more efficiently. Current QTs rely on deep-parameterized quantum circuits (PQCs), rendering them vulnerable to QPU noise, and thus hindering their practical performance. In this paper, we propose the Vectorized Quantum Transformer (VQT), a model that supports ideal masked-attention matrix computation through quantum approximation simulation and efficient training via vectorized nonlinear quantum encoder, yielding shot-efficient and gradient-free quantum circuit simulation (QCS) and reduced classical sampling overhead. In addition, we demonstrate an accuracy comparison for IBM and IonQ in quantum circuit simulation and competitive results in benchmarking natural language processing tasks on IBM’s state-of-the-art, high-fidelity Kingston QPU. Our noise intermediate-scale quantum (NISQ)-friendly VQT approach unlocks a novel architecture for end-to-end machine learning in quantum computing.