Block encoding with low gate count for second-quantized Hamiltonians

TL;DR

The paper delivers explicit block-encoding schemes for second-quantized Hamiltonians that achieve sublinear T-gate scaling, $ ilde{O}(\,\sqrt{L}\,)$, by coupling a SWAP-based sparsity oracle with a direct-sampling amplitude oracle. It introduces η-particle subspace block encodings that reduce the subnormalization factor to $O(\,\sqrt{L}\,)$ (with $L$ the number of terms) and provides detailed circuit constructions, resource estimates, and optimizations for structured Hamiltonians including translation-invariant and nearest-neighbor systems. The approach leverages data-lookup via SELECT-SWAP and a direct-sampling method to load coefficients efficiently, enabling practical early fault-tolerant quantum simulations of molecular and condensed-matter Hamiltonians. Overall, the work offers a practical path to lower fault-tolerant overheads in quantum chemistry and many-body physics simulations by rethinking input models through sparsity and amplitude oracles and exploiting system structure.

Abstract

Efficient block encoding of many-body Hamiltonians is a central requirement for quantum algorithms in scientific computing, particularly in the early fault-tolerant era. In this work, we introduce new explicit constructions for block encoding second-quantized Hamiltonians that substantially reduce Clifford+T gate complexity and ancilla overhead. By utilizing a data lookup strategy based on the SWAP architecture for the sparsity oracle $O_C$, and a direct sampling method for the amplitude oracle $O_A$ with SELECT-SWAP architecture, we achieve a T count that scales as $\mathcal{\tilde{O}}(\sqrt{L})$ with respect to the number of interaction terms $L$ in general second-quantized Hamiltonians. We also achieve an improved constant factor in the Clifford gate count of our oracle. Furthermore, we design a block encoding that directly targets the $η$-particle subspace, thereby reducing the subnormalization factor from $\mathcal{O}(L)$ to $\mathcal{O}(\sqrt{L})$, and improving fault-tolerant efficiency when simulating systems with fixed particle numbers. Building on the block encoding framework developed for general many-body Hamiltonians, we extend our approach to electronic Hamiltonians whose coefficient tensors exhibit translation invariance or possess decaying structures. Our results provide a practical path toward early fault-tolerant quantum simulation of many-body systems, substantially lowering resource overheads compared to previous methods.

Figures (23)

• Figure 1: Illustration of the circuit convention.
• Figure 2: A schematic illustration of a quantum circuit for the block encoding of an $s$-sparse matrix.
• Figure 3: Illustration: the schematic circuit design of quantum singular value transformation (for an odd $d$; for an even $d$ the last $U_A$ is replaced by $U_A^{\dag}$). The additional Hadamard gate selects only the real part of the polynomial $p$.
• Figure 4: Illustration of a circuit identity for the construction of $O_{\phi}$. Note that in the circuit, the two Pauli $Z$ gates are placed on the $p$-th and $q$-th gate where $p<q$.
• Figure 5: Illustration of quantum circuit for $O_{\phi}$ for one-body interaction ${a}^{\dagger}_{p}{a}_{q}, {a}_{p}{a}_{q}, {a}^{\dagger}_{p}{a}^{\dagger}_{q}$ when $p\neq q$.

Theorems & Definitions (4)

• Lemma 2.1: LCU
• Definition 3.1: Block encoding
• Theorem 3.2
• Theorem 3.3: Quantum signal processing