---
title: 'LCHS: Linear Combination of Hamiltonian Simulation'
url: https://www.emergentmind.com/topics/linear-combination-of-hamiltonian-simulation-lchs
type: topic
---

# LCHS: Linear Combination of Hamiltonian Simulation

Linear Combination of Hamiltonian Simulation (LCHS) is a quantum-algorithmic framework for simulating linear non-unitary dynamics by rewriting the target propagator as a linear combination of unitary evolutions, each of which is a Hamiltonian simulation problem. In its standard form, LCHS addresses linear ordinary differential equations of the form $\partial_t u(t)=-A(t)u(t)$ or $\partial_t u(t)=-A(t)u(t)+b(t)$, with the generator decomposed as $A(t)=L(t)+iH(t)$, where $L(t)$ and $H(t)$ are Hermitian and $L(t)\succeq 0$ after a possible identity shift. The central idea is that the non-unitary propagator $\mathcal T e^{-\int_0^t A(s)\,ds}$ can be expressed as an integral over unitaries of the form $\mathcal T e^{-i\int_0^t(kL(s)+H(s))\,ds}$, after which truncation, quadrature, block-encoding, linear-combination-of-unitaries, and amplitude amplification yield a quantum implementation [2303.01029, 2312.03916].

## 1. Foundational identity and problem setting

The modern LCHS formalism emerged from a theorem showing that a general dissipative generator can be represented as a continuous linear combination of Hamiltonian evolutions. For the time-dependent homogeneous ODE
\[
\partial_t u(t)=-A(t)u(t),\qquad A(t)=L(t)+iH(t),
\]
with $L(t)\succeq 0$, one has
\[
\mathcal T e^{-\int_0^t A(s)\,ds}
=
\int_{-\infty}^{\infty}\frac{f(k)}{1-ik}\;
\mathcal T e^{-i\int_0^t(H(s)+kL(s))\,ds}\,dk,
\]
for a suitable kernel $f$ satisfying normalization and decay conditions [2303.01029]. The same framework extends to the inhomogeneous case via Duhamel’s principle, and in the time-independent case it reduces to a direct integral formula for $e^{-At}$ [2312.03916].

A widely used one-parameter family of kernels is
\[
f_\beta(z)=\frac{1}{C_\beta\,\exp((1+iz)^\beta)},\qquad 0<\beta<1,\qquad C_\beta=2\pi e^{-2^\beta},
\]
which leads to the identity
\[
M(t)=\int_{k\in\mathbb R}\frac{f(k)}{1-ik}\cdot
\mathcal T\exp\!\left[-i\int_0^t(kL(s)+H(s))\,ds\right]dk
\]
for the propagator $M(t)=\mathcal T\exp[-\int_0^tA(s)\,ds]$ [2312.03916]. In the time-independent specialization summarized in later constant-factor analyses,
\[
e^{-At}=\int_{k=-\infty}^{\infty} g(k)\,U(k)\,dk,
\qquad
U(k)\equiv \exp[-it(kL+H)],
\qquad
g(k)=\frac{f(k)}{1-ik},
\]
with $f(k)=C_\beta^{-1}\exp[-(1+ik)^\beta]$ [2506.20760].

This identity places LCHS conceptually between dissipative dynamics and standard Hamiltonian simulation. A plausible implication is that the method transfers advances in unitary simulation—such as qubitization, generalized quantum signal processing, and improved block-encoding analyses—into the non-unitary setting with comparatively little change in the outer algorithmic structure.

## 2. Kernel design, truncation, and quadrature

An LCHS algorithm becomes finite only after truncating the infinite $k$-integral and discretizing it. Early formulations used a cutoff $K=O(1/\epsilon)$ and trapezoidal discretization for error $O(\epsilon)$ [2303.01029]. The subsequent near-optimal formulation replaced this with a sharper kernel family and composite Gaussian quadrature, obtaining
\[
K=O((\log 1/\epsilon)^{1/\beta}),\qquad
h_1=\frac{1}{e\,T\,\|L\|_\infty},\qquad
Q=O(\log(1/\epsilon)),
\]
and hence
\[
r=O\!\left(T\|L\|(\log 1/\epsilon)^{1+1/\beta}\right)
\]
LCU terms after discretization [2312.03916]. This is the source of the widely cited logarithmic-to-sub-polynomial error dependence of LCHS.

A later constant-factor study derived explicit non-asymptotic formulas for both truncation and Gaussian quadrature. For truncation error $\epsilon_{\rm trunc}$, the cutoff may be chosen as
\[
K =\Bigl(\frac{2\beta}{\cos(\beta\pi/2)}\,W_0\!\bigl((B_\beta/\epsilon_{\rm trunc})^{1/\beta}\frac{\cos(\beta\pi/2)}{2\beta}\bigr)\Bigr)^{1/\beta},
\]
where $W_0$ is the principal Lambert $W$-function. For discretization error $\epsilon_{\rm disc}$ on $[-K,K]$, the order-$Q$ Gauss–Legendre rule can be chosen as
\[
Q = -(\ln e/4)\;W_{-1}\!\bigl[-(3\,C_\beta)/(2\pi\,e^{1/3}\,\ln e)\; (\epsilon_{\rm disc}/K)\bigr],
\]
with total LCU size
\[
M=(2K/h)\,Q=t\,\|L\|\,e\,K\,Q
=
O\!\bigl(t\,\|L\|[\log(1/\epsilon)]^{1+1/\beta}\bigr)
\]
[2506.20760]. These expressions recover the asymptotic estimates but, more importantly for resource estimation, fix the constants that dominate realistic fault-tolerant costs.

Kernel construction itself has also diversified. A Fourier-transform-based reformulation shows that one may choose any $F\in C^2(\mathbb R)\cap W^{2,1}(\mathbb R)$ satisfying $F(x)=e^{-x}$ for $x\ge 0$, and then define
\[
\frac{f(k)}{1-ik}=\frac1{2\pi}\int_{-\infty}^{+\infty}F(x)e^{ikx}\,dx.
\]
This removes the technical requirements beyond the real axis present in the original half-plane formulation and yields a new kernel family. For truncation error $\epsilon\le 10^{-8}$, that formalism reports a $1.81$ times reduction in the quantum differential equation algorithms based on LCHS and an $8.27$ times reduction in its quantum circuit depth [2508.19596].

## 3. Coherent implementation: block-encoding, SELECT, and amplification

The coherent realization of LCHS follows the linear-combination-of-unitaries pattern. In the time-dependent oracle model, one assumes access to HAM-T oracles that block-encode time samples of $L$ and $H$ on short time slices, prepares an ancilla superposition over quadrature nodes, and implements
\[
\mathrm{SEL}=\sum_j |j\rangle\langle j|\otimes U_j,
\qquad
U_j=\mathcal T e^{-i\int_0^t(k_jL(s)+H(s))\,ds},
\]
after which
\[
W=(O_{c,l}^\dagger\otimes I)\,\mathrm{SEL}\,(O_{c,r}\otimes I)
\]
block-encodes the discrete approximation to the non-unitary propagator [2312.03916].

For the time-independent block-encoding model, later work makes the SELECT step substantially more explicit. If $U_A$ is an $(\alpha_A,m_A,\epsilon_A)$ block encoding of $A$, define the effective Hamiltonian
\[
\mathcal S=\sum_{j=1}^M |j\rangle\langle j|\otimes (k_jL+H).
\]
This $\mathcal S$ can be $(\sqrt{1+K^2}\,\alpha_A,m_A+2,\epsilon_S)$-encoded using just one call to $U_A$ and one to $U_A^\dagger$ plus $2M$ controlled-$R_z$ gates. Qubitization or generalized quantum signal processing then implements
\[
Q(t)=e^{-it\mathcal S}
=
\sum_{j=1}^M|j\rangle\langle j|\otimes e^{-it(k_jL+H)}
\]
as a $(1,m_A+4,\epsilon_Q)$ block encoding using
\[
N_{\rm qb}=e\sqrt{1+K^2}\,\alpha_A\,t+2\ln(2\eta/\epsilon_{\exp})
\]
calls to controlled-$U_A$ or its inverse, with $\eta\approx1.4776$ [2506.20760].

The outer LCU circuit is then
\[
U_{\rm LCHS}:=(O_{c,l}^\dagger\otimes I)\,Q(t)\,(O_{c,r}\otimes I),
\]
which realizes the approximate propagator $\mathcal L=\sum_j c_j e^{-it(k_jL+H)}$ as a block encoding. Applying $U_{\rm LCHS}$ to an initial-state block encoding $U_0$ yields a block encoding of $v(t)=\mathcal L u_0$ with
\[
\epsilon_v=\|v(t)-u(t)\|\le \|u_0\|(\epsilon_{\rm trunc}+\epsilon_{\rm disc})
\]
[2506.20760].

Because postselection amplitude depends on the norm of the evolved state, fixed-point oblivious amplitude amplification is used to boost success probability to near unity. If $\Delta\le 2\|v(t)\|/(\|c\|_1\|u_0\|)$ is a known lower bound on the overlap, the amplification cost is
\[
C_{\rm LCHS}(\Delta,\epsilon_{\rm AA})+1,
\]
with the explicit expression for $C_{\rm LCHS}$ given in the constant-factor analysis, and the total number of calls to $U_A$ becomes
\[
C_A=C_{\rm LCHS}(\Delta,\epsilon_{\rm AA})
\left[e\sqrt{1+K^2}\,\alpha_A\,t+2\ln(2\eta/\epsilon_{\exp})\right],
\]
subject to the sub-error budget
\[
\epsilon_{\rm trunc}+\epsilon_{\rm disc}+\epsilon_A+\epsilon_{\exp}+\epsilon_c+\epsilon_0+\epsilon_{\rm AA}\le \epsilon
\]
[2506.20760].

## 4. Complexity, optimality, and constant factors

A defining property of LCHS is its near-optimal parameter dependence. In the 2023 near-optimal analysis, letting $q=\|u_0\|/\|u(t)\|$, the total number of Hamiltonian-simulation queries is
\[
\widetilde O\!\left(
q\,\alpha_A\,t\,(\log 1/\epsilon)^{1+1/\beta}
\right),
\]
while the number of initial-state-oracle queries is $O(q)$, which matches the $\Omega(q)$ lower bound for state preparation cost [2312.03916]. The earlier formulation stated the same optimality more explicitly: any quantum ODE solver must use $\Omega(q)$ queries to the initial-state oracle, whereas LCHS uses $O(q)$ [2303.01029].

In the time-independent special case, replacing truncated Dyson simulation by QSP or QSVT yields overall matrix-query complexity
\[
\widetilde O\!\left(
(\|u_0\|/\|u(t)\|)\,\alpha_A\,t\,(\log 1/\epsilon)^{1/\beta}
\right),
\]
which was described as fully recovering the “fast-forwarded” performance asymptotically up to the $(\log 1/\epsilon)^{1/\beta}$ factor [2312.03916]. Later work sharpened the error dependence further. For bounded, time-independent $A$, a generalized LCHS construction gives a block-encoding of $e^{-At}$ with
\[
\mathcal O\!\left(t\log\frac1\epsilon\right)
\]
queries to the block-encoding oracle for $A$, and the normalized evolved state can be prepared with
\[
\mathcal O\!\left(1/\|e^{-At}|{\vec u}_0\rangle\|\right)
\]
queries to the initial-state oracle; both complexities are stated to be optimal in all parameters [2508.19238]. The same work also states that any improvement exceeding a constant factor of approximately $3$ is infeasible [2508.19238].

Constant factors, however, are decisive in practice. The detailed 2025 resource analysis identifies several improvements: tight truncation via a Lambert $W$ bound, tighter Gaussian quadrature bounds, a diagonal block-encoding of $\mathcal S$ that avoids $O(M)$ separate controlled calls to $U_A$, generalized quantum signal processing “doubling,” and constant-factor robust-block-encoding analyses for qubitization and fixed-point amplitude amplification [2506.20760]. Under the specialization
\[
\|u_0\|=\|u(t)\|=\alpha_A=1,\qquad
\epsilon_A=\epsilon_0=\epsilon_R=0,\qquad
\Delta\approx 2/\|c\|_1,\qquad
\epsilon=10^{-10},
\]
the resulting query counts are reported as two orders of magnitude smaller than those of the “randomized linear solver” method, with approximately $110\times$ fewer block-encoding calls under identical assumptions [2506.20760]. That same analysis summarizes the resulting behavior as linear scaling in $t$, logarithmic-to-sub-polynomial scaling in $\epsilon$, and markedly reduced block-encoding query complexity in the “no-fast-forwarding” regime [2506.20760].

## 5. Variants, generalizations, and adjacent formalisms

LCHS has rapidly expanded beyond its original role as an ODE solver. One direction generalizes from matrix exponentials to broader eigenvalue transformations. A Laplace-transform-based extension represents
\[
h(A)=\int_0^\infty g(t)e^{-At}\,dt
\]
and then substitutes the LCHS representation for $e^{-At}$, enabling transformations such as $A^{-k}$ and $e^{-A^{-1}}$ without explicitly inverting $A$ [2411.04010]. A later Weyl-calculus-based theory goes further, constructing LCHS formulas for general matrix functions $f(A)$ analytic on the numerical range of $A$, including non-normal $A$, and reports optimal $\mathcal O(\log\frac1\epsilon)$ query complexity scaling for quantum eigenvalue transformation [2606.29848].

A second direction extends the domain of operators. Inf-LCHS generalizes the finite-dimensional theorem to infinite-dimensional Hilbert spaces and possibly unbounded operators, under domain, kernel, and common-domain assumptions. It introduces Inf-LCHS-Gaussian and Inf-LCHS-MC as quadrature-based and Monte-Carlo-based discretizations, respectively, and applies the method to linear parabolic PDEs, birth–death processes, Schrödinger equations with complex potentials, Lindblad equations, and black hole thermal field equations [2502.19688].

A third direction concerns implementation architecture. Hybrid oscillator–qubit LCHS replaces the discrete quadrature register by a continuous-variable ancilla mode, using
\[
(\langle\phi|_{\rm osc}\otimes I_q)\;e^{-it(\hat x\otimes L+I\otimes H)}\;(|\psi\rangle_{\rm osc}\otimes I_q)
\]
to encode the kernel directly. This removes the explicit $O(\log M_a)$ ancilla-qubit overhead of qubit-only LCHS. In heat-equation benchmarks, the Law–Eberly protocol achieves end-to-end solution fidelity at least $99.90\%$, and a comparison with a matrix-product-state-based discrete-variable implementation reports a drastically reduced CNOT count, stated as $400$ versus $4\,700$ in a Dirichlet benchmark [2605.10708].

A fourth direction targets circuit structure rather than asymptotics. Random-LCHS randomizes both the outer LCU layer and the inner Hamiltonian-simulation layer, with observable-driven and symmetry-aware variants for expectation estimation and physically symmetric models [2509.08030]. Another recent refinement shows that implementing the Hamiltonian-simulation subroutines with Multi-Product Formulas yields commutator-sensitive complexity bounds, and that the chosen quadrature rule affects not only discretization error but also commutator structure and query complexity [2606.11475].

## 6. Applications, implementations, and interpretive cautions

LCHS has been applied most visibly to linear PDEs. One line of work maps second-order PDEs with spatially varying parameters to first-order ODE systems by spatial discretization, then uses logic minimization and matrix-product-state compression to reduce the number of Pauli strings and compile the coefficient oracle. In a room-acoustics example on a $32\times 32$ grid, a naive $128+1$ terms for one coefficient collapsed to just $6$ terms, and an MPS of bond $2$ for the LCU coefficient state achieved $99.8\%$ fidelity for $2^8$ quadrature points [2407.05019]. Related work gives explicit near-optimal circuits for the advection-diffusion equation, with $k_{\max}=40$, $K=2^{11}=2048$, total qubits about $36$, and success probability $\gtrsim 0.8$ after amplitude amplification in one representative run [2501.11146]. Subsequent papers provide boundary-aware and structure-preserving PDE frameworks, including mixed Dirichlet, Neumann, Robin, and periodic boundary conditions, explicit block-encodings, and end-to-end complexity balances [2606.03407, 2605.17542].

LCHS has also been explored in hardware-oriented settings. A variational algorithm on the superconducting processor Wukong uses LCHS only at the loss-function level, replacing coherent long-depth LCU by a fixed-depth parameterized ansatz and simplified Hadamard tests; the reported circuit depth is independent of total simulation time, and experiments on dissipative transverse Ising and interacting Hatano–Nelson models tracked theoretical curves accurately [2510.20118]. At the fault-tolerant end, an end-to-end algorithm for rapidly distorted turbulence formulates the non-unitary generator through LCHS and estimates that a minimal non-trivial instance on a $256\times 64\times 64$ grid can be encoded in about $28$ qubits with circuit depth approximately $2100$ controlled-unitary layers [2511.18802].

Two interpretive cautions recur in the literature. First, LCHS should not be conflated with the broader linear-combination-of-unitaries literature for unitary Hamiltonian simulation. Earlier work on linear combinations of unitary operations and multi-product formulas established the ancilla-based block-encoding and amplitude-amplification toolkit on which LCHS relies [1202.5822], and photonic experiments demonstrated such linear-combination methods for unitary dynamics [2211.06723]; the later LCHS literature is distinguished by its specific integral representation of non-unitary evolution in terms of Hamiltonian simulations [2303.01029]. Second, the condition $L\succeq 0$ is not a cosmetic assumption. It encodes dissipativity or non-positive logarithmic norm, guarantees $\|\mathcal T e^{-\int_0^tA(s)\,ds}\|\le 1$ in the standard setting, and underlies both the correctness of the kernel identity and the controlled success probability of postselected implementations [2312.03916].

Within quantum algorithms for non-unitary dynamics, LCHS has therefore become both a specific algorithmic family and a broader design pattern: represent the target contraction as a superposition of unitary evolutions, discretize that superposition carefully, and spend algorithmic effort on the constants hidden inside truncation, quadrature, block-encoding, and amplification. The progression from the original identity to constant-factor analyses, optimal-scaling variants, continuous-variable encodings, and PDE-specific compilers suggests that the central research questions have shifted from mere asymptotic existence to resource-tight realizations and domain-specific structure exploitation.

Source: https://www.emergentmind.com/topics/linear-combination-of-hamiltonian-simulation-lchs