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arXiv:2510.08644

Low-gate-count block encodings for second-quantized fermionic Hamiltonians

Diyi Liu, Shuchen Zhu, Lin Lin, Guang Hao Low, Chao Yang

Published Oct 9, 2025• 1 citation
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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.

DOI
10.48550/arxiv.2510.08644

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We determine the non-Clifford TTT-gate cost of constructing block encodings of structured fermionic and spin Hamiltonians in a unitary Clifford+T+T+T model, when arbitrarily many clean ancillas and unrestricted block-encoding subnormalization are allowed, but without mid-circuit measurements or classical feed-forward. Our main technical tool is an ancilla-compression theorem: any block encoding of an nnn-qubit operator with aaa clean ancillas and at most sss TTT gates can be compressed to use at most min⁡{a,n+2s}\min\{a,n+2s\}min{a,n+2s} ancillas, without increasing the absolute error or TTT-count. For general second-quantized Hamiltonians with bounded one- and two-body coefficients, at operator-norm block-encoding error εεε, a volume-covering argument combined with circuit counting gives the worst-case lower bound Ω(n2log⁡(n4/ε))Ω(n^2\sqrt{\log(n^4/ε)})Ω(n2log(n4/ε)​), matching the existing upper bound at fixed precision. For the bond-dependent Kitaev honeycomb family on nnn spins, we obtain independent lower bounds Ω(n)Ω(n)Ω(n) from stabilizer nullity and Ω(log⁡(1/ε))Ω(\log(1/ε))Ω(log(1/ε)) from one-qubit state preparation, established using different Hamiltonian instances. Together with an explicit LCU construction, they give the tight worst-case scaling Θ(n+log⁡(1/ε))Θ(n+\log(1/ε))Θ(n+log(1/ε)). As an application, we evaluate the TTT-count of a Hamiltonian simulation circuit based on quantum singular value transformation, with each block-encoding query compiled separately. When phase synthesis and controlled queries add at most constant-factor overhead, the simulation TTT-count scales as the query count times the optimal TTT-count per query.
Simulation of molecular vibrations is emerging as a promising application of quantum computing in quantum chemistry. In this work, we develop new strategies for block-encoding the second-quantized nnn-mode representation of the vibrational Hamiltonian, whose algebraic structure closely parallels that of the second-quantized electronic Hamiltonian. We derive linear-combination-of-unitaries (LCU) representations of this Hamiltonian based on Pauli decomposition and tensor hypercontraction (THC) and construct efficient circuits for the corresponding block encodings. We also extend the Block-Invariant Symmetry Shift (BLISS) technique to vibrational Hamiltonians, yielding a lower LCU 1-norm and, consequently, a lower overall circuit cost. For the molecules considered here, BLISS-THC reduces the Toffoli cost of quantum phase estimation by more than two orders of magnitude relative to existing block-encoding approaches. In comparison with recent Trotter-based methods, it shows up to a ninefold reduction in the T-gate count, with the relative savings tending to increase with system size. Asymptotically, and up to polylogarithmic factors, both the Pauli and THC LCUs trade additional ancillas for a quadratic improvement in scaling with the number of modes, while THC provides an additional quadratic improvement in scaling with the number of modals. These results highlight the importance of Hamiltonian factorizations and Toffoli-qubit tradeoffs for reducing the computational cost of quantum simulations of molecular vibrations.
In fault-tolerant quantum computing, the cost of calculating Hamiltonian eigenvalues using the quantum phase estimation algorithm is proportional to the constant scaling the Hamiltonian matrix block-encoded in a unitary circuit. We present a method to reduce this scaling constant for the electronic Hamiltonians represented as a linear combination of unitaries. Our approach combines the double tensor-factorization method of Burg et al. with the block-invariant symmetry shift method of Loaiza and Izmaylov. By extending the electronic Hamiltonian with appropriately parametrized symmetry operators and optimizing the tensor factorization parameters, our method achieves a 25% reduction in the block-encoding scaling constant compared to previous best techniques for industrially and biologically relevant molecules, including the nitrogenase cofactor (FeMoCo) and cytochrome P450. Benchmark calculations comparing our method with the original double-factorization method show a 75% reduction in the block-encoding scaling constant. The resulting savings in the number of non-Clifford T-gates, which are an essential resource for fault-tolerant quantum computation, are expected to accelerate the journey to the feasibility of practical Hamiltonian simulations.
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Reversible logic is essential in quantum computing due to its ability to preserve information and establish a one-to-one mapping between inputs and outputs. The Clifford+T gate set, known for supporting fault-tolerant quantum circuits, offers resilience against decoherence but introduces high implementation costs, particularly due to the T gate's resource intensity. In this work, we propose a new and efficient quantum multiplexer design, the Quantum Clifford T-MUX, designed entirely within the Clifford+T framework. This multiplexer minimizes T-count and avoids the use of ancillary qubits, both critical factors for practical, fault-tolerant quantum computing. As part of this architecture, we present a newly designed fault-tolerant Peres gate using only Clifford+T gates, which serves as a fundamental building block in the T-MUX construction. The proposed T-MUX achieves reduced T-count, no garbage output, and improved efficiency compared to traditional designs. These improvements position it as a valuable component for building scalable quantum Arithmetic Logic Units (ALUs), data buses, and memory systems. Overall, this work contributes toward the development of low-overhead, modular quantum control structures suited for near-term and long-term quantum computing architectures. The proposed circuit is simulated using the Quirk online tool, and the result confirms the accuracy of the design. This design achieves an average 35% improvement in T-Count without any constant input, compared to its counterpart, respectively.