Low-ancilla block encodings via Hamiltonian simulation

Abstract: Block encodings are a central primitive in quantum algorithms, but standard constructions typically require logarithmic ancilla overhead and complicated controlled operations. Recent lower bounds further show that such ancilla overhead is unavoidable for exact constructions in broad circuit models. We show that this barrier can be bypassed in the approximate setting. Specifically, we present a simple single-ancilla construction that converts Hamiltonian evolution into a block encoding of the underlying Hamiltonian, via generalized quantum signal processing. For operators given by Hermitian decompositions $A=\sum_{j=1}^L α_j H_j$, we instantiate this block-encoding construction in two ways, which differ in how the required Hamiltonian evolution is implemented. Using higher-order Trotterization, we obtain an $\varepsilon$-approximate block encoding of $A$ with only one ancilla qubit and circuit depth $\widetilde O\big(L(α/\varepsilon)^{o(1)}\big),$ where $α=\sum_j α_j$. Using multiproduct formulas, we obtain circuit depth $\widetilde O(L)$, at the cost of $O(\log\log(1/\varepsilon))$ ancilla qubits. Our constructions provide alternatives to the standard LCU framework, with a focus on reducing the number of ancilla qubits while maintaining (near-)optimal circuit depth.