Optimal T-Count for Block Encodings of Fermionic and Spin Hamiltonians
Abstract
We determine the non-Clifford T-gate cost of constructing block encodings of structured fermionic and spin Hamiltonians in a unitary Clifford+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 n-qubit operator with a clean ancillas and at most s T gates can be compressed to use at most min{a,n+2s} ancillas, without increasing the absolute error or T-count. For general second-quantized Hamiltonians with bounded one- and two-body coefficients, at operator-norm block-encoding error ε, a volume-covering argument combined with circuit counting gives the worst-case lower bound Ω(n2√(log(n4/ε))), matching the existing upper bound at fixed precision. For the bond-dependent Kitaev honeycomb family on n spins, we obtain independent lower bounds Ω(n) from stabilizer nullity and 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)). As an application, we evaluate the T-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 T-count scales as the query count times the optimal T-count per query.
Related
- Efficient Block Encodings of Vibrational Hamiltonians
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- Explicit block encodings of rate matrices for simulating polymerization kinetics on quantum computers
BibTeX
@misc{ma2026optimal,
title = {{Optimal T-Count for Block Encodings of Fermionic and Spin Hamiltonians}},
author = {Jiaxin Ma and Kevin J. Joven and Yuan Liu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11153},
eprint = {2609.11153},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11153},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Optimal T-Count for Block Encodings of Fermionic and Spin Hamiltonians
AU - Jiaxin Ma
AU - Kevin J. Joven
AU - Yuan Liu
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.11153
UR - https://arxiv.org/abs/2609.11153
AB - We determine the non-Clifford T-gate cost of constructing block encodings of structured fermionic and spin Hamiltonians in a unitary Clifford+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 n-qubit operator with a clean ancillas and at most s T gates can be compressed to use at most min{a,n+2s} ancillas, without increasing the absolute error or T-count. For general second-quantized Hamiltonians with bounded one- and two-body coefficients, at operator-norm block-encoding error ε, a volume-covering argument combined with circuit counting gives the worst-case lower bound Ω(n^2√(log(n^4/ε))), matching the existing upper bound at fixed precision. For the bond-dependent Kitaev honeycomb family on n spins, we obtain independent lower bounds Ω(n) from stabilizer nullity and 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)). As an application, we evaluate the T-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 T-count scales as the query count times the optimal T-count per query.
ER -