TARE: Block Encoding Linear Combinations of Pauli Strings Without Ancilla State Preparation

TL;DR

TARE is a novel block-encoding method for linear combinations of Pauli strings that reduces T-gate count and circuit depth, enhancing resource efficiency in quantum algorithms based on QSP.

Contribution

Introduces TARE, a new block-encoding technique that absorbs coefficients into anti-commuting Pauli strings, reducing ancilla qubits and improving efficiency over standard methods.

Findings

TARE reduces T-gate count compared to LCU.

TARE improves circuit depth in numerical simulations.

TARE scales logarithmically with the number of Pauli strings.

Abstract

Quantum algorithms based on Quantum Signal Processing (QSP) offer the potential for speedups across a broad range of applications, with block encodings serving as the central input model. In this framework, non-unitary matrices are embedded into larger unitary operators, and the cost of constructing these encodings often dominates the overall gate complexity. In this work, we introduce Tag-and-Restore Encoding (TARE), a block-encoding method for linear combinations of Pauli strings. In this method coefficient magnitudes are absorbed into a unitary built from a set of mutually anti-commuting Pauli strings acting on the system register. These Pauli strings are then mapped to the target Pauli strings through appropriate transformations, yielding a block encoding of the target operator. The ancilla register size scales logarithmically with the number of Pauli strings and can be extended to larger registers providing a width/depth tradeoff. We evaluate TARE through numerical simulations of the transverse-field Ising model, the Jordan-Wigner image of a fermionic star Hamiltonian, and random Pauli-string operators. Compared with standard Linear Combination of Unitaries (LCU), TARE substantially reduces the T-gate count while improving circuit depth in several cases. These results suggest that TARE can provide resource-efficient block encodings for a wide range of relevant systems.