Linear combination of unitaries

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• LCHS — linear combination of Hamiltonian simulation Method

  Takes A block-encoding of $A(t)$ with a normalization $\alpha_A \ge \max_t \lVert A(t)\rVert$, preparation unitaries for $u_0$ and $b$, the evolution time $T$, and an error tolerance $\varepsilon$. Returns A state proportional to $u(T)$, or a history state, together with separately stated matrix-query and initial-state-query complexity.

• LCHS with the improved kernel Method

  Takes A block-encoding of $A(t)$ with a normalization $\alpha_A \ge \max_t \lVert A(t)\rVert$, preparation unitaries for $u_0$ and $b$, the evolution time $T$, and an error tolerance $\varepsilon$. Returns A state proportional to $u(T)$, or a history state, together with separately stated matrix-query and initial-state-query complexity.

• Kernel-weighted combination of unitary propagators Method

  Takes The generator $A(t)$ with no Hermiticity assumed, the interval $[0,T]$, and an error tolerance $\varepsilon$. Returns A Hermitian generator or a family of them, the enlargement of the space that carrying them cost, and the map that reads the original solution back — with the weight that map applies stated, because that weight is where the non-unitarity was moved to rather than removed.

• Chebyshev series by linear combination of unitaries Method

  Takes A block-encoding of $A$ together with its subnormalisation $\alpha$; a target function $f$ on $[-1,1]$; an error budget $\varepsilon$. Returns A circuit implementing a block-encoding of $f(A)$ to error $\varepsilon$, together with the query count in $U$ and $U^\dagger$.

• Block-encoding from a Pauli decomposition Method

  Takes An access model for $A$ — sparse-access oracles, a Pauli or LCU decomposition, a purification, or an explicit arithmetic description — plus a target precision $\varepsilon$. Returns A unitary $U$ on $s+a$ qubits, its subnormalization $\alpha$, and its ancilla/flag count $a$. Because $\lVert U\rVert = 1$, Gilyén, Su, Low and Wiebe's Definition 43 forces $\lVert A\rVert \le \alpha + \varepsilon$.

• Truncated-Taylor LCU simulation Method

  Takes An access model for $H$ — a sum of efficiently exponentiable terms, sparse-access oracles, or a block-encoding — plus an evolution time $t$ and a target error $\varepsilon$. Returns A circuit approximating $e^{-iHt}$ to within $\varepsilon$, with a stated query or gate count, an ancilla count, and the norm parameter — sparsity times $\lVert H\rVert_{\max}$, or the LCU 1-norm — that the cost is measured against.