Gaussian-LCHS: Quantum Hamiltonian Simulation

• Gaussian-LCHS is a quantum algorithm that approximates Gaussian filters for ground-state and eigenvalue targeting using a linear combination of unitary evolutions.
• It discretizes the Fourier integral of the non-unitary Gaussian filter and leverages LCU methods, supporting both hybrid quantum–classical and full continuous-variable implementations.
• The method offers favorable scaling in resource and accuracy requirements compared to cosine filter and standard LCU approaches, making it NISQ-friendly for Hamiltonian simulations.

Gaussian Linear Combination of Hamiltonian Simulations (Gaussian-LCHS) is a quantum algorithmic paradigm for simulating the action of non-unitary, smooth functions of a Hamiltonian—typically Gaussian filters—by leveraging a finite linear combination of unitary Hamiltonian evolutions. Gaussian-LCHS provides a systematic method for ground-state filtering and eigenvalue targeting within the constraints of near-term quantum devices, exhibiting favorable scaling with respect to accuracy and resource requirements compared to other filter-based or product-formula Hamiltonian simulation algorithms. The approach extends naturally to infinite-dimensional and non-Hermitian settings and supports both hybrid quantum–classical and fully quantum continuous-variable (CV) ancilla implementations (He et al., 2021, Lu et al., 27 Feb 2025, Childs et al., 2012).

1. Gaussian Filter Function and Fundamental Principle

The central objective is to amplify the ground-state (or more generally, a target-eigenstate) component of an input state $\lvert\psi_0\rangle$ via the application of a Gaussian filter operator,

$$F(H) = \exp\left[ -\frac{(H - E_0)^2}{2 \sigma^2} \right],$$

where $H$ is the system Hamiltonian, $E_0$ is an energy parameter targeting the desired state (often $E_0 < \lambda_0$, the ground-state energy), and $\sigma$ is the filter width controlling energy selectivity (He et al., 2021). In the eigenbasis $\{\lvert\lambda_j\rangle\}$,

$$F(H)\lvert\lambda_j\rangle = \exp\left[ -\frac{(\lambda_j - E_0)^2}{2 \sigma^2} \right]\lvert\lambda_j\rangle,$$

so that states energetically distant from $E_0$ are exponentially suppressed.

The essential challenge is that $F(H)$ is non-unitary and thus not directly implementable on a quantum processor. However, $F(H)$, as an analytic function of $H$, admits a Fourier integral representation,

$$F(H) = \frac{1}{\sqrt{2\pi}\sigma} \int_{-\infty}^\infty e^{-y^2/(2\sigma^2)}\, e^{-i(H-E_0)y} dy,$$

effectively reducing the filter to a weighted continuous sum of unitary time-evolution operators (He et al., 2021).

2. Discretization and Linear Combination Construction

Quantum devices can implement only a finite sum of such exponentials. Discretizing the Gaussian–Fourier integral by a simple Riemann sum, one obtains

$$F(H) \approx G_{E_0,\sigma}(H) = \sum_{j=-M}^M c_j e^{-iH t_j},$$

with

$$t_j = j\Delta y,\quad c_j = \frac{\Delta y}{\sqrt{2\pi}\sigma} e^{-t_j^2/(2\sigma^2)} e^{iE_0 t_j}.$$

Parameters $M$ and $\Delta y$ are chosen according to spectral width $\Lambda$ and the desired operator-norm error $\epsilon$, achieving bounds such as $$M\Delta y = O\left(\frac{\Lambda + |E_0|}{\sigma^2} + \sqrt{\log(1/\epsilon)}/\sigma\right)$$ and $\Delta y = O(1/\Lambda)$. The number of filter terms $2M+1 = O(\sigma^{-1} \Lambda \log(1/\epsilon))$, and the maximal simulation time $t_{\max} = O(\sigma^{-2} \log(1/\epsilon))$ (He et al., 2021). This linear combination of time-evolution unitaries forms the core of Gaussian-LCHS.

3. Quantum Algorithmic Implementation

Standard LCU (Linear Combination of Unitaries) primitives, as established in (Childs et al., 2012), construct a linear combination $V=\sum_j c_j V_j$ by using an ancilla register, quantum state preparation, controlled application of the unitaries, and amplitude amplification to ensure high-probability success. For Gaussian-LCHS, an alternative NISQ-compatible approach leverages a hybrid quantum–classical workflow (He et al., 2021):

• Quantum subsystem: Circuit-based overlap measurements use a single ancilla qubit and Hadamard tests to evaluate all necessary matrix elements of the form $\langle\psi_0|U_k^\dagger U_j |\psi_0\rangle$ and $\langle\psi_0|U_k^\dagger H U_j |\psi_0\rangle$ for $U_j = e^{-iH t_j}$.
• Classical post-processing: The measured overlaps populate Hermitian matrices $S_{jk}$ and $H_{jk}$. The ground-state energy estimate is given by

$$\lambda_{E_0,\sigma} \approx \frac{c^\dagger H c}{c^\dagger S c}$$

where $c$ is the vector of coefficients. Hyperparameters $(E_0,\sigma)$ can either be swept over a grid or classically optimized to minimize $\lambda$ (He et al., 2021). This enables post-processing flexibility and adaptation to quantum resource constraints.

Alternatively, the LCU circuit can be used to directly construct the filtered state $\propto F(H)|\psi_0\rangle$ via block-encoding and ancilla preparation, following the standard LCU method (Childs et al., 2012).

4. Complexity Analysis and Error Bounds

The operator-norm approximation error of the finite sum satisfies

$$\left\|F(H) - \sum_{j=-M}^M c_j e^{-iH t_j}\right\| \leq \epsilon$$

with the corresponding filter cost scaling:

Resource	Scaling with $\sigma, \epsilon$
Number of terms $M$	$O(\sigma^{-1} \log(1/\epsilon))$
Max sim. time $t_{max}$	$O(\sigma^{-2} \log(1/\epsilon))$
Gate count (Trotter)	$$O(L^3\sigma^{-3}\Lambda[\log(1/\epsilon)]^3/\epsilon_T)$$
Sample count	$$O(\sigma^{-2} \Lambda^2 [\log(1/\epsilon)]^2/\epsilon^2)$$

(He et al., 2021)

The bias in the energy estimate compared to the ground-state energy satisfies

$$\lambda(c) - \lambda_0 = O\left(e^{-\Delta^2/(2\sigma^2)}\right),$$

with $\Delta$ the gap to the first excited state. Choosing $\sigma \lesssim \Delta/\sqrt{2\log(1/\delta)}$ bounds the bias by $\delta$. The overall cost grows polynomially in $\sigma^{-1}$ (filter sharpness) and polylogarithmically in $1/\epsilon$.

5. Generalizations to Infinite Dimensions and Non-Hermitian Extensions

The Inf-LCHS–Gaussian generalization (Lu et al., 27 Feb 2025) extends Gaussian-LCHS to infinite-dimensional Hilbert spaces and unbounded, non-Hermitian generators $A = L + iH$. The corresponding evolution is implemented as

$$e^{-A t} = \int_{-\infty}^\infty \frac{f(k)}{1-ik} \exp\left[ -i(kL + H)t \right]\, dk$$

for an analytic kernel $f$ obeying normalization and decay conditions.

The integral is truncated to $[-K, K]$ and discretized via composite Gaussian quadrature, resulting in an approximation

$$e^{-A t} \approx \sum_{j=1}^N c_j\, \exp\left[-i(x_j L + H)t \right]$$

where the coefficients $c_j$ and nodes $x_j$ arise from the quadrature construction. Resource scaling is $$N = O\left( t\|L\| [\log(1/\epsilon)]^{1+1/\beta} \right)$$ for kernel exponent $\beta$ and operator norm $\|L\|$, with optimal dependence on $\epsilon$. Each unitary is synthesized via standard Hamiltonian simulation primitives, and the LCU for the full sum utilizes “PREPARE” and “SELECT” unitaries, with post-selection success probability dictated by the norm of the output state (Lu et al., 27 Feb 2025).

6. Fully Quantum Continuous-Variable Ancilla Realization

Gaussian-LCHS admits a fully quantum, continuous-variable (CV) ancilla implementation for the exact Gaussian filter (He et al., 2021). The protocol is as follows:

• Prepare the joint state $|\psi_0\rangle \otimes |\phi\rangle$, with $|\phi\rangle$ a squeezed-momentum wavepacket,

$$|\phi\rangle \propto \int dp\, e^{-p^2/(2s^2)} |p\rangle,$$

where $s=1/\sigma$.

• Apply the joint evolution $U = e^{-i H \otimes p}$.
• Project the ancilla mode back onto $|\phi\rangle$ (e.g., by homodyne measurement), producing

$$\langle\phi| e^{-i H \otimes p} |\psi_0\rangle \otimes |\phi\rangle \propto e^{-s^2 H^2/2}|\psi_0\rangle,$$

realizing $F(H)$ exactly.

Finite squeezing and physical noise bound achievable $\sigma$, and the post-selection success probability decays for sharper filters (large $s$). CV implementation circumvents the need for classical optimization or Hadamard measurements at the expense of requiring ancillary CV hardware, compatible with superconducting-cavity or ion-trap platforms.

7. Relation to, and Trade-offs with, Other Filter and LCU-Based Methods

Gaussian-LCHS exhibits distinct advantages over other quantum filter algorithms:

• Cosine filter methods expand powers of $[\cos((H-E)/L)]^K \approx e^{-(H-E)^2/(2\delta^2)}$ in LCU form, incurring a quadratic overhead in $K$; Gaussian-LCHS achieves similar resource scaling with simpler controlled-unitary structure and classical hyperparameter scan (He et al., 2021).
• Inverse-iteration filters simulate $H^{-k}$ by LCU of $e^{-i H t}$ with significantly worse scaling in $k$ for a given spectral gap.
• Standard LCU Hamiltonian simulation (Childs–Wiebe, (Childs et al., 2012)) uses multi-product formulas with arbitrary weights; Gaussian weighting can further optimize $\sqrt{\log(1/\epsilon)}$ prefactors in segment cost (Childs et al., 2012).

The table below summarizes scaling features:

Method	Term count scaling	Success probability/amplification	Ancilla requirement
Gaussian-LCHS	$O(\sigma^{-1}\log(1/\epsilon))$	Classical postproc. or OAA	1 qubit (hybrid), or CV mode (full quantum)
Cosine Filter LCU	$O(\delta^{-1}\log(1/\epsilon))$	OAA	Ancilla-index register
Multi-product LCU (Childs et al., 2012)	$O(\exp[\sqrt{\ln(1/\epsilon)}])$	OAA	Register

Gaussian-LCHS thus achieves NISQ-friendly scaling for ground-state projection, balancing filter sharpness, error, and quantum resource expenditure, and with mechanisms for both hybrid and full-quantum execution (He et al., 2021, Lu et al., 27 Feb 2025, Childs et al., 2012).