In this paper, we introduce a major advancement in Quantum Singular Value Transformation (QSVT) through the development of Feedforward QSVT (FQSVT), a framework that significantly enhances the efficiency and robustness of quantum algorithm design. By leveraging intermediate measurements and feedforward operations, FQSVTs reclaim quantum information typically discarded in conventional QSVT, enabling more efficient transformations. Our results show that FQSVTs can exponentially accelerate the projection of quantum states onto energy subspaces, outperforming probabilistic projection and adiabatic algorithms with superior efficiency and a drastic reduction in query complexity. In the context of superconducting qubits, FQSVTs offer a powerful tool for managing energy subspaces, improving efficiency for state preparation and leakage detection.
In this paper, we introduce a major advancement in Quantum Singular Value Transformation (QSVT) through the development of Feedforward QSVT (FQSVT), a framework that significantly enhances the efficiency and robustness of quantum algorithm design. By leveraging intermediate measurements and feedforward operations, FQSVTs reclaim quantum information typically discarded in conventional QSVT, enabling more efficient transformations. Our results show that FQSVTs can exponentially accelerate the projection of quantum states onto energy subspaces, outperforming probabilistic projection and adiabatic algorithms with superior efficiency and a drastic reduction in query complexity. In the context of superconducting qubits, FQSVTs offer a powerful tool for managing energy subspaces, improving efficiency for state preparation and leakage detection.
Quantum singular value transformation (QSVT) is a framework that has been shown to unify many primitives in quantum algorithms. In this work, we leverage the QSVT framework in two directions. We first show that the QSVT framework can accelerate one recently introduced quantum power method, which substantially improves its running time. Additionally, we incorporate several elementary numerical integration techniques, such as the rectangular method, Monte Carlo method, and quadrature method, into the QSVT framework, which results in polynomial speedup with respect to the size or the number of points of the grid. Our results thus provide further examples to demonstrate the potential of the QSVT and how it may enhance quantum algorithmic tasks.
Quantum singular value transformation (QSVT) is a powerful framework that unifies many quantum algorithms, including Grover’s search, by enabling polynomial transformations of the singular values of linear operators embedded in unitary matrices. We present an improved quantum search algorithm within this framework. Instead of amplifying the target state amplitude, our method suppresses the non-target amplitude using a modified Chebyshev polynomial. This approach achieves the optimal O(Nlog(1/δ)) query complexity while requiring lower-degree polynomials. For N = 512, our method reduces the polynomial degree from 71 to 43 compared to prior QSVT-based fixed-point search. We also demonstrate how QSVT enables exact search construction.
We introduce shadow quantum singular value transformation (Shadow QSVT): given an initial state ∣ψ⟩, a Hermitian matrix H, a polynomial f, and a set of observables {O1,…,Om}, the goal is to estimate ⟨ψ∣f(H)†Ojf(H)∣ψ⟩ for all j∈{1,…,m}. Shadow QSVT provides a systematic route to reduce the quantum resources required by standard QSVT, which constructs a unitary block-encoding of f(H). It uses structure in the input state and observables, together with the fact that many applications require only observable estimates rather than synthesizing the full unitary.
We present three algorithms that exploit structure in the initial state and observables to reduce quantum circuit depth. First, we develop a state-aware QSVT algorithm that prepares the target state with low circuit depth when the Krylov subspace associated with H and ∣ψ⟩ is low-dimensional or admits an accurate low-dimensional approximation. Second, we introduce an observable-aware Shadow QSVT algorithm that combines a new observable-aware Krylov subspace with history states to further reduce circuit depth and gate complexity. Finally, we develop Classical Shadow QSVT, which constructs a classical representation from H, f, and ∣ψ⟩ without prior knowledge of the observables or explicit preparation of the target state proportional to f(H)∣ψ⟩. This representation enables estimation of the target quantities for observables specified after the quantum computation. Together, these three algorithms provide tools for reducing the circuit depth of QSVT-based computations across a range of settings.
Quantum singular-value transformation (QSVT) is a framework that has been shown to unify many primitives in quantum algorithms. In this work we leverage the QSVT framework in two directions. We first show that the QSVT framework can accelerate one recently introduced quantum power method, which substantially improves its running time. Additionally, we incorporate several elementary numerical integration techniques, such as the rectangular method, Monte Carlo method, and quadrature method, into the QSVT framework, which results in polynomial speedup with respect to the size or the number of points of the grid. Our results thus provide further examples to demonstrate the potential of the QSVT and how it may enhance quantum algorithmic tasks.
An n-qubit quantum circuit performs a unitary operation on an exponentially large, 2n-dimensional, Hilbert space, which is a major source of quantum speed-ups. We develop a new “Quantum singular value transformation” algorithm that can directly harness the advantages of exponential dimensionality by applying polynomial transformations to the singular values of a block of a unitary operator. The transformations are realized by quantum circuits with a very simple structure - typically using only a constant number of ancilla qubits - leading to optimal algorithms with appealing constant factors. We show that our framework allows describing many quantum algorithms on a high level, and enables remarkably concise proofs for many prominent quantum algorithms, ranging from optimal Hamiltonian simulation to various quantum machine learning applications. We also devise a new singular vector transformation algorithm, describe how to exponentially improve the complexity of implementing fractional queries to unitaries with a gapped spectrum, and show how to efficiently implement principal component regression. Finally, we also prove a quantum lower bound on spectral transformations.
Block encoding is a key ingredient in the recently developed quantum singular value transformation (QSVT) framework, which provides a unifying description for many quantum algorithms. Initially introduced to simplify and optimize resource utilization in various problems, such as searching, amplitude estimation, and Hamiltonian simulation, it is reasonable to expect that the capabilities of QSVT extend beyond these applications and offer untapped potential for designing new quantum algorithms. In this article, we affirm this perspective by leveraging block encoding to substantially enhance two previously proposed quantum algorithms: largest eigenvalue estimation and quantum gradient descent. Unlike previous works that rely on sophisticated approaches, our findings demonstrate that even just elementary operations within the unitary block encoding framework can eliminate major scaling factors present in their original counterparts. This results in significantly more efficient quantum algorithms capable of tackling target computational problems with remarkable efficiency. Furthermore, we illustrate how our proposed method can be extended to other contexts, including matrix inversion and multiple eigenvalue estimation.