F^{(k^{\prime})}_{1}(\Delta\tau)=\sum_{k,k_{1},\cdots,k_{M}}^{\infty}\delta_{k^{\prime},k+\sum
k_{i}}\left[\prod_{j=1}^{M}\frac{(\mathrm{i}h_{j}\Delta
t)^{k_{j}}}{k_{j}!}\right]$
$\displaystyle\times\sum_{j_{1},\cdots,j_{k}}^{M}\frac{\prod^{k}_{a=1}(-\mathrm{i}h_{j_{a}}\Delta
t)}{k!}\sigma_{j_{k}}\cdots\sigma_{j_{1}}\sigma_{1}^{k_{1}}\cdots\sigma_{M}^{k_{M}}.$
(120)

To evaluate this expression, indices $k$ and $k^{\prime}_{j},k_{j}$ are
sampled from appropriate Poisson distributions for all $j$. It is this
evaluation that resembles previously discussed ideas from DiagMC most closely.
Ultimately, this sampling prescription can be used to evaluate the correlation
function
$\langle\psi_{f}|\mathrm{e}^{\mathrm{i}Ht}O\mathrm{e}^{-\mathrm{i}Ht}|\psi_{i}\rangle$
on a quantum device. Wavefunctions $|\psi_{i}\rangle$ and $|\psi_{f}\rangle$
correspond to initial and final state, $O$ is an arbitrary operator. As this
approach still suffers from the sign problem (unless the entire dynamics is
performed on the quantum computer), the applicability and scope of the method
are currently limited. Nonetheless, recently, a similar QCMC algorithm was
used to perform imaginary-time evolution by sampling random quantum circuits
with finite depth [250]. In this work, the authors demonstrated the promise of
QCMC even without a fully fault-tolerant quantum computer by applying it to
the ground state calculations.

###  V.2 Hybrid Path Integral Monte Carlo for Time Correlation Functions

Recently, Eklund and Ananth reported a hybrid Path Integral Monte Carlo
(hPIMC) algorithm to calculate real-time thermal correlation functions [251].
The classical Path Integral Monte Carlo (PIMC) algorithm was first proposed by
Barker in 1979 [252]. While the method can be used in high-dimensional
computations, there are still system size limitations in the computation of
the time-evolution operator matrix elements.

In the work from Eklund and Ananth, these time-evolution operator matrix
elements are computed on a quantum computer, and they demonstrate that this
can be performed accurately using the probabilistic imaginary-time evolution
(PITE) algorithm [253]. The symmetrized real-time time-correlation function
between two operators $A$ and $B$ is given by

$C_{AB}(t)=\frac{1}{Z}\text{Tr}\left[U^{\dagger}(t_{c})AU(t_{c})B\right]$
(121)

where $U(t_{c})=\mathrm{e}^{-\mathrm{i}Ht_{c}}$, and
$t_{c}=t-\mathrm{i}\beta/2$. In PIMC, $U(t_{c})$ is discretized into the
Trotter product of $N$ time evolution operators such that

$\tilde{C}_{AB}(t)=\frac{1}{Z}\text{Tr}\left[\left(\prod_{k=1}^{N}\tilde{U}^{\dagger}(\Delta
t_{c})\right)A\left(\prod_{k=1}^{N}\tilde{U}(\Delta t_{c})\right)B\right]$
(122)

where $\tilde{U}$ is the approximation for the unitary $U$ and $\Delta
t_{c}=t_{c}/N$. If the Hamiltonian takes the form $H=H_{1}+H_{2}$, where
$[H_{1},H_{2}]\neq 0$, $U(t_{c})$ is approximated with a 2$k$-th order
Trotter–Suzuki product.

If we expand out the operator in a basis $\\{\phi_{j}\\}$, each Monte Carlo
iteration $k$ involves computing matrix elements
$\\{\langle\phi_{j_{k+1}}|\tilde{U}(\Delta t_{c})|\phi_{j_{k}}\rangle\\}$ and
$\\{\langle\phi_{j_{k+1}}|\tilde{U}^{\dagger}(\Delta
t_{c})|\phi_{j_{k}}\rangle\\}$. In classical PIMC, these matrix elements are
computed by diagonalizing the Hamiltonian in the basis, so for basis size $M$,
we require $O(M^{3d})$ cost for a $d$-dimensional system. The overall runtime
for classical PIMC is $O(N\tau+M^{3d})$ where $\tau$ is the number of Monte-
Carlo iterations determined by the desired error, and the space requirement is
$O(M^{2d})$. In hPIMC, each matrix element is computed on the quantum
computer, eliminating the need for diagonalization. Instead, $2N\tau$ calls
are made to the oracle, which computes the matrix elements. Therefore, the
total runtime of hPIMC is $O(N\tau Q(\tilde{U}))$, where $Q$ is the cost of
executing the oracle. The space requirement is $O(dn)+\text{ Anc.}(\tilde{U})$
where $n=\log_{2}(M)$ is the number of qubits required, and
$\text{Anc.}(\tilde{U})$ is the number of ancilla qubits required to implement
the time-evolution unitary. The efficiency of the algorithm, therefore,
depends on the efficiency of the oracle, and there is some potential for
significant speedup in runtime compared to a standard PIMC implementation.
Nevertheless, this algorithm also suffers from the dynamical sign problem, so
its practical applicability and scope to large-scale problems still need to be
examined.

###  V.3 Boson Sampling for Molecular Vibronic Dynamics

Boson sampling [254] and its variant, Gaussian boson sampling (GBS) [255,
256], have demonstrated the quantum advantage in experiments for artificial
problems [257, 258, 259, 260]. However, it remains an open question whether
they can be applied to real-world problems of practical interest and
demonstrated to have a quantum advantage over classical methods.

The first theoretical attempt to apply GBS to study chemical problems was made
with the computation of molecular vibronic spectra based on the GBS device
made of squeezed states of light coupled to a boson sampling optical network
[261]. With Fermi’s golden rule and the Condon approximation, the optical
absorption and emission spectra at zero temperature can be written as

$S(\omega)=\left|\mu\right|^{2}\sum_{n}\left|\langle\phi^{i}_{0}(\mathbf{q}_{\mathrm{i}})|\phi^{f}_{n}(\mathbf{q}_{\mathrm{f}})\rangle\right|^{2}\delta\left(\omega+\omega_{0}-\omega_{n}\right),$
(123)

where the frequency-dependent prefactor is omitted, $\mu$ is the transition
dipole moment, and $|\phi^{i}_{0}\rangle$ is the initial state of the
transition, which corresponds to the ground state or the lowest excited state
with energy $\omega_{0}$ for the absorption or emission spectra, respectively.
$|\phi_{n}^{f}\rangle$ corresponds to the final states of the transition. The
normal modes of the initial state and final states’ potential of energy
surfaces (PES) are shifted by $\Delta\mathbf{q}$ after applying the Duschinsky
rotation $\mathbf{S}$ for mode mixing,

$\mathbf{q}_{\mathrm{f}}=\mathbf{S}\mathbf{q}_{\mathrm{i}}+\Delta\mathbf{q}.$
(124)

Figure 12: Illustration of Gaussian boson sampling and molecular vibronic
spectroscopy. (a) The potential energy curves with a vibronic transition
modeled with a harmonic potential and represented via a Doktorov
transformation. (b) The GBS involves photon output distribution from a quantum
optical network and the example of computing molecule’s vibronic spectra.
Figures adapted from Ref. 262, 261.

The key quantity to solve Eq. 123 is the Franck–Condon factor (FCF)

$\textrm{FCF}=\left|\langle\phi_{0}^{i}(\mathbf{q}_{\mathrm{i}})|\phi_{n}^{f}(\mathbf{q}_{\mathrm{f}})\rangle\right|^{2},$
(125)

which corresponds to a vibronic transition between states of two harmonic
oscillators. Solving in Eq. 123 involves $d^{N}$ summations if the maximal
boson occupation for each mode is truncated at $d$ for $N$ vibrational modes
in total, which is naively exponentially hard for classical computers. Since
FCFs with larger amplitudes will contribute more to the spectra intensity, one
only needs to sample that $n$ having large FCFs. However, it is hard for
classical algorithms to determine which $n$ contributes the most importance
unless computing their FCF directly [263].

Huh et al. [261] proposed to directly measure the FCF on a photonic quantum
device, and naturally, the more important $n$ has a larger probability of
being sampled. The quantum measurement of FCF is performed with the Doktorov
rotation circuit and the photon Fock states,

$\textrm{FCF}=|\langle\bm{m}|\hat{U}_{\text{Dok }}|\bm{n}\rangle|^{2},$ (126)

where $|\bm{n}\rangle=\left|n_{1},n_{2},\ldots,n_{N}\right\rangle$ describes
the ground state with $n_{i}$ excitations for the $i-$th mode (for zero
temperature, $|\bm{n}\rangle$ is the ground state (mathematically a vacuum
state), and $|\bm{m}\rangle$ denotes the excited state). After this original
theoretical work, a series of experimental works using GBS with a special
focus on molecular vibronic spectra [264, 265, 266] were implemented.
Additional theoretical studies addressed more complex models, such as the non-
Condon effect [267], anharmonic PESs [268], and vibrational dynamics [262].

Demonstrating that the GBS algorithm offers a quantum computational advantage
for calculating molecular vibronic spectra requires solving the challenges in
computing spectra (Eq. 123), a task that is exponentially challenging for
classical computers using a brute-force sum-of-states approach. However, an
exact classical method exists for efficiently computing the time correlation
function in the time domain, applicable when there is no anharmonicity. This
method, known as the thermal vibrational correlation function (TVCF) [269,
*Peng2007Mar, 271], challenges the potential advantages of boson sampling.

##  VI Summary

In this review, we have explored an interface between quantum computing and
computational chemistry, focusing particularly on the complex sampling tasks
essential for studying chemical systems’ ground state, equilibrium, and non-
equilibrium properties. Due to their common occurrences in computational
chemistry, tackling these tasks is a good target for quantum algorithms.
Current quantum hardware still grapples with noise, coherence times, and qubit
scalability limitations. Nonetheless, these challenges are driving innovative
developments in both fully quantum and quantum-classical hybrid algorithms as
we summarized in this Review.

We have reviewed various sampling approaches, from hybrid quantum-classical
algorithms to fully quantum algorithms, with a special emphasis on the Monte
Carlo methods and other relevant sampling techniques. We have summarized the
theoretical frameworks of quantum and classical algorithms and discussed the
practical challenges of these algorithms. The rapid advancement of quantum
simulation presents a moving target, and this Review captures only a transient
moment in the ongoing progress.

We hope our Review offers new perspectives and useful background to quantum
information scientists and computational chemists developing quantum and
classical algorithms for complex sampling problems in chemistry, materials
science, and physics. While we have not yet identified a good use case for
these quantum algorithms with a definitive quantum advantage, this continued
effort in developing and assessing new algorithms will bring us one step
closer to a practical quantum advantage.

###### Acknowledgements.

This work was supported by Harvard University’s startup funds. Additionally,
the work of T.J., H.D., and J.L. was supported by the DOE Office of Fusion
Energy Sciences “Foundations for Quantum Simulation of Warm dense Matter”
project and Google’s research gift. The support from Wellcome Leap as part of
the Quantum for Bio Program was partially used for the work done by T.J. and
J.L. N.M. acknowledges support by the Department of Energy Computational
Science Graduate Fellowship under award number DE-SC0021110.

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