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 $Ω(n^2\sqrt{\log(n^4/ε)})$, 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.