Quantum singular value transformation without block encodings: Near-optimal complexity with minimal ancilla

Understand

We develop new algorithms for Quantum Singular Value Transformation (QSVT), a unifying framework that encapsulates most known quantum algorithms and serves as the foundation for new ones.

• Existing implementations of QSVT rely on block encoding, incurring an intrinsic $O(\log L)$ ancilla overhead and circuit depth $\widetilde{O}(L d\lambda )$ for polynomial transformations of a Hamiltonian $H=\sum_{k=1}^L H_k$, where $d$ is the polynomial degree and $\lambda=\sum_{k}\|H_k\|$.
• We introduce a simple yet powerful approach that utilizes only basic Hamiltonian simulation techniques, namely, Trotter methods, to: (i) eliminate the need for block encoding, (ii) reduce the ancilla overhead to only a single qubit, and (iii) still maintain near-optimal complexity.
• Our method achieves a circuit depth of $\widetilde{O}(L(d\lambda_{\mathrm{comm}})^{1+o(1)})$, without requiring any complicated multi-qubit controlled gates.