# Mixed Quantum-Semiclassical Simulation

Javier Gonzalez-Conde  Department of Physical Chemistry, University of the
Basque Country UPV/EHU, Apartado 644, 48080 Bilbao, Spain  EHU Quantum Center,
University of the Basque Country UPV/EHU, Apartado 644, 48080 Bilbao, Spain
Andrew T. Sornborger  Information Sciences, Los Alamos National Laboratory,
Los Alamos, NM, USA.

###### Abstract

We study the quantum simulation of mixed quantum-semiclassical (MQS) systems,
of fundamental interest in many areas of physics, such as molecular scattering
and gravitational backreaction. A basic question for these systems is whether
quantum algorithms of MQS systems would be valuable at all, when one could
instead study the full quantum-quantum system. We study MQS simulations in the
context where a semiclassical system is encoded in a Koopman-von Neumann (KvN)
Hamiltonian and a standard quantum Hamiltonian describes the quantum system.
In this case, because KvN and quantum Hamiltonians are constructed with the
same operators on a Hilbert space, standard theorems guaranteeing simulation
efficiency apply. We show that, in this context, many-body MQS particle
simulations give only nominal improvements in qubit resources over quantum-
quantum simulations due to logarithmic scaling in the ratio, $S_{q}/S_{c}$, of
actions between quantum and semiclassical systems. However, field simulations
can give improvements proportional to the ratio of quantum to semiclassical
actions, $S_{q}/S_{c}$. Of particular note, due to the ratio $S_{q}/S_{c}\sim
10^{-18}$ of particle and gravitational fields, this approach could be
important for semiclassical gravity. We demonstrate our approach in a model of
gravitational interaction, where a harmonic oscillator mediates the
interaction between two spins. In particular, we demonstrate a lack of
distillable entanglement generation between spins due to classical mediators,
a distinct difference in dynamics relative to the fully quantum case.

##  I Introduction

Figure 1: Examples of MQS systems of interest. (i) In molecular scattering,
the Born-Oppenheimer approximation is used to describe the light electronic
degrees of freedom quantum mechanically, but the heavy nuclear system
semiclassically [1, 2, 3, 4]. (ii) Backreaction in gravity is studied with
various semiclassical gravitational approaches [5, 6, 5, 6, 7, 8, 7, 9, 10,
11, 12, 13, 14, 15, 16, 17, 18, 19]. (iii) The Jaynes-Cummings model serves as
a toy model for many MQS systems [20, 21, 22, 23].

Mixed quantum-semiclassical (MQS) systems are studied in many areas of physics
[1, 2, 3, 4, 5, 6, 5, 6, 7, 8, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19,
20, 21, 22, 23]. In molecular scattering, the Born-Oppenheimer approximation,
Fig. 1 (i), describes the light electronic degrees of freedom quantum
mechanically, but the heavy nuclear system semiclassically [1, 2, 3, 4].
Backreaction in gravity, Fig. 1 (ii), is studied with various semiclassical
gravitational approaches [5, 6, 5, 6, 7, 8, 7, 9, 10, 11, 12, 13, 14, 15, 16,
17, 18, 19]. And the Jaynes-Cummings model, Fig. 1 (iii), serves as a toy
model for many MQS systems [20, 21, 22, 23].

One of the main problems in this area of research concerns the study of
backreaction between classical and quantum theories, which is of particular
interest, since quantum fields can generate entanglement, whereas classical
fields do not. Typical semiclassical derivations from field theory result in a
standard coupling [24]. However, some researchers, particularly in
gravitational theory, which is non-renormalizable, have suggested that MQS
theories could be fundamental, giving rise to open questions involving how to
couple quantum with semiclassical fields and giving rise to suggestions of
non-standard couplings [25].

A question that arises naturally from the study of these systems is whether
MQS theory could benefit from quantum simulation algorithms, as a significant
component of these models is quantum, and hence costly to compute classically
[26, 27, 28, 29, 30]. In this regard, algorithms for Hamiltonian simulation on
quantum computers have reached a mature stage [26, 31, 30, 32, 33, 34, 35, 36,
37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]. Quantum
simulations are known to use logarithmically fewer resources versus their
equivalents on classical computers [38, 51]. Complexity bounds are now well-
understood and current quantum simulation algorithms are at the theoretical
limit for accuracy [36, 31, 46].

Additionally, quantum algorithms for solving differential equations and
performing classical simulations are of recent interest. A range of quantum
algorithmic approaches for classical simulations has recently been studied,
including Carleman [52, 53, 54, 55] and Koopman-von Neumann methods [56, 57,
58, 59, 25, 60, 61, 62, 4] for lifting finite dimensional, classical nonlinear
differential equations to a linear description in an infinite dimensional
Hilbert space. Recent results show an exponential advantage when simulating
classical systems of harmonic oscillators [63].

It is unclear, however, if an exponential advantage may be obtained,
generally, for simulating classical systems [60, 63]. And, in some cases, it
is known that it cannot [55]. One reason for this is that although quantum
simulations of quantum systems overcome the ‘sign problem’ in quantum Monte
Carlo caused by quantum interference [64, 65], this problem does not exist for
classical Monte Carlo and therefore classical Monte Carlo methods are already
very competitive. Therefore, it can be difficult to find quantum algorithms
that give better than a polynomial advantage for classical simulation.

In the case of quantum algorithms for MQS simulations, an opportunity exists
to make use of a quantum simulation of a classical system by coupling it with
a quantum simulation of a quantum system. In this case, a computational
advantage is obtained, generally, since the quantum part of the system becomes
exponentially efficient, even if the classical part does not. Therefore, with
MQS simulations, we have the opportunity to study backreaction [5, 6, 7, 8],
among other phenomena, with exponential efficiency on a quantum computer.

In this article, we construct an MQS simulation algorithm with a semiclassical
subsystem described using a Koopman-von Neumann (KvN) framework within an
otherwise quantum simulation. KvN is most amenable to MQS simulation in the
Schrödinger picture since semiclassical state evolution is governed by a
quantum Hamiltonian on a Hilbert space. The defining difference in combined
KvN and quantum Hamiltonian simulation is that the KvN simulation involves
system operators that commute with each other. Thus, MQS simulation algorithms
based on KvN are efficient and use logarithmically fewer resources than the
equivalent classical algorithm. Additionally, for MQS systems, our algorithm
can use fewer resources than the equivalent quantum-quantum algorithm. We show
that although this reduction in resources is nominal for MQS many-body
simulations, it can be significant for MQS field simulations [66]. Finally, we
implement our approach with a proof-of-principle model of two, two-level
quantum systems with an interaction mediated by a harmonic oscillator. We
compare simulation of the model with both quantum and classical oscillators
and study and contrast their entanglement properties. In particular, we show
that the MQS system does not generate distillable entanglement, whereas the
fully quantum system does.

##  II Semiclassical Dynamics

The purpose of semiclassical mechanics is to obtain a formalism that enables
us to express the dynamics of classical (Liouville) systems with quantum
operators. In quantum mechanics, we assume that the dynamics of the quantum
density matrix of a closed system, $\hat{\rho}$, is governed by the von
Neumann equation

$\partial_{t}\hat{\rho}=-\frac{i}{\hbar}[\hat{H},\hat{\rho}]\;,$ (1)

where $\hat{H}$ is the system Hamiltonian and
$[\hat{H},\hat{\rho}]=~{}\hat{H}\hat{\rho}-\hat{\rho}\hat{H}$ the commutator.
On the other hand, in the context of classical mechanics, the time evolution
of a phase space distribution function, $f(p,q)$, is described by the
Liouville equation,

$\frac{\partial f}{\partial t}=-\\{\,f,H\,\\}$ (2)

with $\\{\,f,H\,\\}=\sum_{i=1}^{N}\left(\frac{\partial f}{\partial
q_{i}}\frac{\partial H}{\partial p_{i}}-\frac{\partial f}{\partial
p_{i}}\frac{\partial H}{\partial q_{i}}\right)$, the Poisson bracket.

We aim to put quantum and semiclassical systems on the same footing in order
to make use of quantum Hamiltonian simulation methods. In this regard, we use
phase-space quantum mechanics [67, 68] to derive a classical Liouville
equation, then construct a Hamiltonian using Koopman-von Neumann mechanics
[56, 57, 58, 69, 25].

Phase-space quantum mechanics relies on transforming the quantum density
matrix, $\hat{\rho}$, into a pseudo-distribution, $\gamma({q},{p})$, in phase
space. The standard method to achieve this mapping is via the Wigner transform
[70],

$\gamma_{\omega}({q},{p})=\frac{1}{(2\pi\hbar)^{2}}\int_{\mathbb{R}^{d}}\rho({q}-\frac{1}{2}{r},{q}+\frac{1}{2}{r})e^{i{p}\cdot{r}/\hbar}d{r}\;,$
(3)

where

$\rho({q}-\frac{1}{2}{r},{q}+\frac{1}{2}{r})\equiv\phi({q}-\frac{1}{2}{r})\phi^{*}({q}+\frac{1}{2}{r})\;.$
(4)

This transformation is equivalent to the standard Schrödinger formalism, but
the transform takes the density matrix into a quasiprobability distribution in
phase space [71, 72].

Following this definition, the quasiprobability distribution, $\gamma_{w}$,
using (2), obeys the dynamical equation

$\partial_{t}\gamma_{w}=-\frac{2}{\hbar}H\sin\left(\frac{\hbar\Lambda}{2}\right)\gamma_{w}\;,$
(5)

where $H$ is the classical Hamiltonian and the differential operator $\Lambda$
is defined

$\Lambda\equiv\overleftarrow{\partial}_{p}\overrightarrow{\partial}_{q}-\overleftarrow{\partial}_{q}\overrightarrow{\partial}_{p}\;.$
(6)

We expand (5) to zeroth order in $\hbar$, resulting in a semiclassical
approximation of the dynamics given by the classical Liouville equation [70]

$\partial_{t}\gamma_{w}({p},{q})=-\frac{2}{\hbar}H\left(\frac{\hbar\Lambda}{2}\right)\gamma_{w}+O(\hbar^{2})$
(7)

giving

$\partial_{t}\gamma_{c}({p},{q})\approx-\partial_{p}H\;\partial_{q}\gamma_{c}+\partial_{q}H\;\partial_{p}\gamma_{c}\;,$
(8)

where $\gamma_{c}$ denotes semiclassical evolution in the phase space of the
pseudo-distribution $\gamma_{w}$. It is important to realize that, although
$\gamma_{c}$ is a pseudo-distribution, for any measurement probability,
$P_{A}$, where $A=\ket{a}\bra{a}$, and $\ket{a}$ is a quantum state,
$P_{A}=\mathrm{Tr}\\{\rho A\\}=\mathrm{Tr}\\{\gamma_{w}A_{w}\\}$, and $A_{w}$
is the Wigner transform of $A$. Therefore, there is no possibility of
measuring negative probabilities.

Note that when we take the limit $\hbar\rightarrow 0$, we really are using a
shorthand for taking a limit $\hbar/S_{0}\rightarrow 0$, where $S_{0}$ is a
characteristic action, or a variable relevant to the action, in the theory.

##  III Koopman-von Neumann Hamiltonian Dynamics

In the Liouville equation, (8), derived above for semiclassical mechanics,
Hamiltonian dynamics governs motion in phase space. Defining the square-root
of the pseudo-distribution as $\gamma_{c}\equiv~{}\psi(x,p)^{*}\psi(x,p)$,
where $\psi(x,p)$ is a complex function, it is possible to derive an equation
for a wavefunction, $\psi$,

$i\partial_{t}\psi=\hat{L}\psi\;,$ (9)

where $\hat{L}$ is the Hermitian operator

$\hat{L}=i\left(-\partial_{p}H\;\partial_{q}\psi+\partial_{q}H\;\partial_{p}\psi\right)\;.$
(10)

Note the close resemblance to a quantum theory here, since $\hat{L}$ is
Hermitian. However, some work remains to obtain a representation that fully
obeys the axioms of Quantum Mechanics.

Our goal is to develop an operator representation of the phase space dynamics
of $\psi(q,p)$ in a Hilbert space [25, 56]. This will give us a quantum,
Schrödinger representation of classical Hamiltonian dynamics that may be
implemented with quantum registers and methods from quantum simulation on a
quantum computer, while still maintaining semiclassical dynamics. We note that
nonlinear Hamiltonian systems are readily represented with this formalism,
called Koopman-von Neumann dynamics.

We introduce axioms that are standard in quantum theory, except that for the
classical theory, we require that the commutator

$[\hat{q},\hat{p}]=0\;.$ (11)

We note that the classical commutator may formally be considered to be the
limit as $\hbar\rightarrow 0$ of the quantum commutator

$[\hat{q},\hat{p}]=i\hbar\;.$ (12)

The axioms that we will use in our framework [25] are: (1) The wavefunction,
$\Psi$, is normalized as $\langle\Psi(t)|\Psi(t)\rangle=1$; (2) The
expectation value of an observable $\hat{A}$ is given by $\mathrm{Tr}\\{\rho
A\\}=\langle\Psi(t)|\hat{A}|\Psi(t)\rangle$; (3) The probability of measuring
a value, $a$, of an observable is
$\mathrm{Pr}(a)=|\Pi_{a}|\Psi(t)\rangle|^{2}$, with $\Pi_{a}$ a projection
operator on the space defined by $\hat{A}$ with eigenvalue $a$; (4) The
composition of two systems, $A$ and $B$, is given by the tensor product of
their Hilbert spaces, $\Psi_{AB}\equiv\Psi_{A}\otimes\Psi_{B}$.

In this formalism, in order to preserve probability (i.e. axiom (1) above),
evolution in time is governed by a unitary operator, $U(t)$. However, note the
important fact that $\hat{L}$ is not the generating operator for time
evolution. It turns out that, in order for $\hat{q}$ and $\hat{p}$ to commute,
two new operators, $\hat{\lambda}_{q}$ and $\hat{\lambda}_{p}$, must be
introduced. We call these ghost momenta. In this formalism, both $\hat{q}$ and
$\hat{p}$ can be considered as configuration space operators in a quantum
theory, and, as such, they commute. The ghost momenta, $\hat{\lambda}$, do not
commute with their corresponding configuration operators:

$\displaystyle[\hat{q},\hat{\lambda}_{q}]$ $\displaystyle=$ $\displaystyle i$
(13) $\displaystyle[\hat{p},\hat{\lambda}_{p}]$ $\displaystyle=$
$\displaystyle i\;.$ (14)

So, in this theory, the size of the Hilbert space must be doubled to
accommodate the commuting configuration space operators, $\hat{q}$ and
$\hat{p}$.

With the introduction of ghost momenta, we can now write down the generator of
time evolution for this theory,

$\hat{H}_{\mathrm{KvN}}\equiv\widehat{\partial_{p}H}\hat{\lambda}_{q}-\widehat{\partial_{q}H}\hat{\lambda}_{p}\;,$
(15)

where $H$ is the classical Hamiltonian and the hat over the classical
Hamiltonian derivative terms represents the promotion of phase space
variables, $(q,p)$, to operators $(\hat{q},\hat{p})$. $\hat{H}_{\mathrm{KvN}}$
gives the time evolution of the semiclassical system on Hilbert space,

$|\Psi(t)\rangle=e^{{-i\hat{H}_{\mathrm{KvN}}t}}|\Psi(0)\rangle\;.$ (16)

This approach is depicted in Fig. 2.

##  IV Quantum-Semiclassical Dynamics

In the KvN dynamics described in Sec. III, we ignored the global phase of the
KvN wavefunction since it did not contribute to any observables and had no
impact on the classical dynamics. However, when we couple quantum and
semiclassical systems, we can no longer ignore the semiclassical phase, since
the dynamics now becomes sensitive to it [59].

Retaining the phase, $\hat{W}$, in KvN dynamics gives us the KvN Hamiltonian

$\hat{H}_{\mathrm{KvN}}\equiv\widehat{\partial_{p}H}\hat{\lambda}_{q}-\widehat{\partial_{q}H}\hat{\lambda}_{p}+\hat{W}\;,$
(17)

where $\hat{W}$ generates a phase on the semiclassical Hilbert space.

Kostant [73], and later recapped in Bondar, et al. [59] and Joseph [61], note
that without the phase, $\hat{W}$, the KvN Lagrangian (used in a variational
formulation of KvN dynamics) is not covariant with respect to local phase
transformations, $\Psi(q,p)\rightarrow e^{i\varphi(q,p)}\Psi(q,p)$. They
further note that this may be overcome by considering a minimally coupled
gauge theoretic approach, where a $U(1)$ gauge-covariant Liouvillian is
constructed with the replacements

$i\hbar\partial_{t}\rightarrow
i\hbar\partial_{t}-\Phi\;\;\mathrm{and}\;\;i\hbar\nabla\rightarrow
i\hbar\nabla+\mathcal{A}\;,$ (18)

where $\Phi=H$ is the gauge potential, and $\mathcal{A}\cdot dz$ is the
symplectic potential (with $z=(q,p))$. In this paper, we use the harmonic
oscillator gauge

$\mathcal{A}\cdot z=\frac{1}{2}\left(p\cdot dq-q\cdot dp\right)\;,$ (19)

with which the phase of the covariant Liouvillian is eliminated for the
homogeneous Hamiltonians with quadratic coupling that we consider in our
examples [59].

A variety of approaches to coupling quantum and semiclassical systems have
been considered [59, 74]. Typically, from a fully quantum system, components
are identified with parameters that allow one or more components to be made
semiclassical in some limit.

For some systems, such as gravity coupled with quantum fields, it has even
been considered that gravity is fundamentally classical [75, 25].
Additionally, more formal approaches have been considered, where in an
initially classical theory, one system is promoted to a quantum system via
canonical quantization, and the other system is (conceptually) separately
promoted to a Liouvillian system encoded in a KvN Hamiltonian [76].

The general form of the quantum Hamiltonian that we use for these quantum-
semiclassical systems is

$\hat{H}_{\mathrm{qc}}=\hat{H}_{\mathrm{KvN}}+\hat{H}_{\mathrm{int}}+\hat{H}_{\mathrm{q}}\;.$
(20)

where the semiclassical term $\hat{H}_{\mathrm{int}}$ is derived according to
the KvN formalism, Eq. (17), from the fully quantum Hamiltonian. We illustrate
this with an example in section VII.1.

Figure 2: Scheme followed to obtain a quantum representation of semiclassical-
dynamics. First, we transform the state $\hat{\rho}$, and Hamiltonian of our
system $\hat{H}$ to phase space via the Wigner Transform. In phase space, we
obtain a dynamical equation associated with a quasiprobability distribution,
$\gamma_{w}$. In order to move to the semiclassical framework we take the
limit of the characteristic action $S_{0}$, to be much larger than $\hbar$,
which brings us to the approximate dynamics of the semiclassical
quasiprobability distribution, $\gamma_{c}$. Then, with a KvN formulation, we
rewrite the semiclassical Liouvillian dynamics in the Schrödinger formulation
in order to be able to implement the semiclassical dynamics on a quantum
computer.

For this quantum-semiclassical system, the quantum density matrix is positive
definite. However, the Liouville density, $\gamma_{c}(p,q)$, which is derived
from a Wigner transform can become negative for general quantum-semiclassical
interactions. This indicates that the semiclassical subsystem can retain some
quantum characteristics. One means of understanding this is given in [70].
Alternatively, the derivation of MQS theories may be studied from an effective
field theory approach [77].

##  V Dynamical Quantum-Semiclassical Simulations are Efficient

Quantum signal processing (QSP) methods currently provide optimal quantum
algorithms for quantum simulation [78, 79]. The following theorem outlines
bounds on the time-interval, error, and success probability for QSP methods.

######  Theorem 1 (Optimal sparse Hamiltonian simulation using quantum signal
processing [78]).

A $d$-sparse Hamiltonian, $\hat{H}$, on $n$ qubits with matrix elements
specified to $m$ bits of precision can be simulated for time-interval $t$,
error $\epsilon$, and success probability at least $1-2\epsilon$ with
$O\left(td\|\hat{H}\|_{\mathrm{max}}+\frac{\log(1/\epsilon)}{\log\log(1/\epsilon)}\right)$
queries and a factor of $O[n+~{}m\,\mathrm{polylog}(m)]$ additional quantum
gates. The quantum simulation is valid for
$td\|\hat{H}\|_{\mathrm{max}}=~{}O[\log(1/\epsilon)/\log\log(1/\epsilon)]$.

Since the combined KvN, interaction, and quantum Hamiltonians that we have
constructed for the MQS systems considered here consist of a fully quantum
operator acting on a Hilbert space that may be encoded in a quantum computer,
we have,

######  Corollary 1 (Optimal Mixed Quantum-Classical Simulation).

Simulations of d-sparse interacting quantum and KvN-embedded semiclassical
systems are optimal in the sense of Thm. 1.

We consider MQS simulations that are governed by a $d$-sparse Hamiltonian
$\hat{H}=\hat{H}_{\mathrm{KvN}}+\hat{H}_{\mathrm{int}}+\hat{H}_{\mathrm{Q}}$.
Here, KvN and interaction Hamiltonians are built from the same operators as
the quantum theory. The only difference is the introduction of the
$\hat{\lambda}$ operators, which are standard quantum momentum operators and
may be implemented efficiently. Thus, the full Hamiltonian inherits the
optimality of quantum simulation using quantum signal processing techniques.

##  VI Resource Savings with Quantum-Semiclassical vs. Quantum-Quantum
Dynamics

Figure 3: Visualization of wavepackets induced by differing actions. This
figure gives a geometrical depiction of the difference in wavepacket shapes. A
free particle wavepacket with a mass of $10$ units (blue) is plotted with a
wavepacket with a mass of $0.1$ units (black). When the ratio between quantum
and classical actions, $S_{q}/S_{c}$, is large, the particle corresponding to
the large action ($S_{c}$, blue) can be treated semiclassically.

In a theory describing two systems with drastically different effective
actions (depending on disparate masses, momenta, or potentials), MQS
descriptions can simplify computations, both analytically and numerically,
relative to quantum-quantum descriptions. For instance, in Born-Oppenheimer
theory describing electron-nucleus interactions, the parameter used to
distinguish between quantum electronic degrees of freedom and semiclassical
nuclear degrees of freedom is $\sqrt{m_{e}/m_{n}}$, where $m_{e}$ is the
electron mass, and $m_{n}$ is an appropriate nuclear mass scale, typically
more than three orders of magnitude larger than the electron mass [1, 2, 3,
4].

In this limit, the semiclassical field is described by a high frequency
wavepacket, where the effective nuclear action is very large compared to the
electromagnetic action, which is described by a low frequency wavepacket. This
situation (see Fig. 3) will occur whenever the characteristic action of one
(semiclassical) system, $S_{c}$, is significantly larger than that of another.
This will be relevant for understanding quantum computational resource savings
that give an advantage for MQS systems.

As a specific example, a (dispersive) plane wave describing a free particle of
mass, $m$, may be described by the expression

$\psi(x,t)=\frac{1}{\sqrt{2\pi\hbar}}e^{ip(x-\frac{p}{2m}t)/\hbar}\;.$ (21)

Here, the spatial wavenumber is $p=\sqrt{2mE}$. To numerically describe this
particle wave function for a fully quantum simulation, we require a grid with
wavenumber $2\sqrt{2mE}$ to resolve the high frequency oscillations due to the
particle mass, $m$. This corresponds to the Shannon-Nyquist frequency [80],
whereas for a numerical simulation of the semiclassical equation, the high-
frequency quantum wave packet structure does not contribute since it is
coarse-grained in the semiclassical approximation. Thus, for an MQS simulation
of a quantum particle with mass $m_{q}$ and a semiclassical particle with mass
$m_{c}$, we only need to resolve the quantum particle, which is lighter and
induces lower frequency power in the wave function. In this example, we save,
of order $\sqrt{m_{q}/m_{c}}$ grid points, assuming some maximum energy scale,
$E_{\mathrm{max}}$.

More generally, we have wavefunctions of the form

$\psi(x,t)_{q,c}=A(x,t)_{q,c}e^{iS(x,t)_{q,c}/\hbar}\;.$ (22)

where $A$ is a complex amplitude. In this case, we save, of order
$S_{q}/S_{c}$ grid points with an MQS simulation.

There are two different cases that must be considered to understand the
quantum computational resources needed for such simulations. In the first
case, consider a many-body simulation. Here, each quantum and semiclassical
particle’s amplitude is defined on a semiclassical spatial grid of size,
$N^{\mathrm{semi}}_{\mathrm{grid}}$, which is binarily encoded in a quantum
register. Letting $n_{\mathrm{q,c}}$ represent the numbers of particles
simulated for quantum and semiclassical systems (i.e. particles with small or
large action, $S_{q,c}$), respectively, the number of qubits required to
encode the combined system is
$n=(2n_{\mathrm{c}}+n_{\mathrm{q}})\log{N^{\mathrm{semi}}_{\mathrm{grid}}}$,
where the factor of $2$ multiplying the classical particle number,
$n_{\mathrm{c}}$, comes from the doubling of the Hilbert space of KvN systems.
For this type of system,
$N^{\mathrm{semi}}_{\mathrm{grid}}=S_{q}/S_{c}\;N^{\mathrm{quant}}_{\mathrm{grid}}$
where $N^{\mathrm{quant}}_{\mathrm{grid}}$ represents the number of grid
points that would be necessary if no semiclassical approximation were made.
Thus, for instance, the resource savings in terms of the ratio, $R$, of qubits
used in a quantum-semiclassical simulation versus a fully quantum simulation
for the encoding is

$R=\frac{3}{2}\log_{N^{\mathrm{quant}}_{\mathrm{grid}}}\left(\frac{S_{q}}{S_{c}}N^{\mathrm{quant}}_{\mathrm{grid}}\right)\;,$
(23)

assuming equal numbers of particles, $n_{c}=n_{q}$. Here, for instance for a
mixed simulation of a proton-electron system, where $S_{q}/S_{c}\sim 10^{-3}$,
$R\approx 0.6$, for a system of size
$N^{\mathrm{quant}}_{\mathrm{grid}}=10^{6}$. Hence, for this example, the
advantage of using an MQS simulation, as opposed to a fully quantum
simulation, is nominal since it takes just a few more qubits to resolve the
full grid.

In the second case, consider a quantum simulation of two fields [66] (or, in
general, a second-quantized system), where each individual grid point is
represented by one quantum register per field (representing the occupation
number of the quantum or classical field ($n_{\mathrm{q,c\;field}}$) at a
given spatial location). Here, the resources for a fully quantum simulation
would be

$2\log(n_{\mathrm{q\;field}})\;N^{\mathrm{quant}}_{\mathrm{grid}}\;.$ (24)

And those for an MQS simulation would be

$(\log(n_{\mathrm{q\;field}})+2\log(n_{\mathrm{c\;field}}))\;\frac{S_{q}}{S_{c}}N^{\mathrm{quant}}_{\mathrm{grid}}\;.$
(25)

From these expressions, we find that a quantum-semiclassical simulation will
use

$R=\left(\frac{3}{2}\right)\;\frac{S_{q}}{S_{c}}$ (26)

fewer qubits for field simulations, where we have set
$n_{\mathrm{q\;field}}=n_{\mathrm{c\;field}}$. In contradistinction with the
situation for MQS particle simulations, MQS field simulations exhibit a
reduction in qubit resources that is linear in the ratio $S_{q}/S_{c}$.

One use-case where this is of particular value is semiclassical gravity.
Semiclassical gravity (an MQS system) corresponds to summing all Feynman
diagrams with no graviton loops (i.e. the semiclassical system), but an
arbitrary number of matter loops (the quantum system) [81, 82]. For the
example of MQS simulation of gravitational and scalar fields, we have the
Einstein-Hilbert action

$S_{\mathrm{EH}}=-\frac{1}{16\pi G}\int d^{4}x\sqrt{-g}\;R$ (27)

and the action of a scalar field in a classical gravitational background

$S_{\mathrm{scalar}}=\frac{1}{2}\int
d^{4}x\sqrt{-g}\;\left(g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi+m^{2}\phi^{2}+\xi\phi^{2}R\right)$
(28)

Here,

$G=1/M_{P}=1/1.22\times 10^{19}\mathrm{GeV}^{-1}\;,$ (29)

with $M_{P}$ the Planck mass, giving $S_{q}/S_{c}\approx 4\times 10^{-18}$. It
is clear that for this case, MQS simulations can give a significant advantage
relative to quantum-quantum simulations. We note that for other MQS scenarios,
such as Born-Oppenheimer, significant advantages can also exist [77] with
field-based MQS simulation.

##  VII Applications

###  VII.1 Theoretical Model

Figure 4: Gravitational massive mediator between two quantum particles [19].
This model presents a system of two particles that interact through a massive
mediator, which in the fully quantum case results in an effective non-
classical interaction between the two test-systems that grows with the mass of
the mediator. On the other hand, when the mediator is a classical harmonic
oscillator the effective non-classical interaction (measured by the
logarithmic negativity) vanishes. See quantum simulation results in text.
Figure 5: Evolution of entanglement in quantum and MQS systems (Eqs. (30) and
(34), resp.). We numerically simulated the dynamics of both quantum and
classical oscillators with $n=7$ qubits for each coordinate, plus $1$ qubit
for each spin subsystem, giving a total of $2+7=9$ qubits for the fully
quantum case and $2+2\times 7=16$ qubits for the KvN model. a) The purity of
spin and oscillator subsystems as a function of time for the fully quantum
model (Eq. (30)). b) The purity of spin and oscillator subsystems as a
function of time for the MQS model (Eq. (34)). c) The logarithmic negativity
between the two spin subsystems in both quantum and MQS models. Note that in
the MQS model, the interaction generates no distillable entanglement between
the spin subsystems, whereas in the fully quantum model, the interaction
generates significant entanglement.

Below, we apply our quantum algorithm to study a model from both fully quantum
and MQS perspectives. We choose a toy model of special relevance to the study
of gravity, a system comprised of a massive harmonic oscillator mediating the
interaction between two quantum particles [19]. When the oscillator is quantum
the dynamics results in an effective non-classical interaction between the two
test-systems (spins) that grows with the mass of the mediator and is
independent of its initial state and, therefore, its temperature. Under the
assumptions given in the original proposal [19], the Hamiltonian of this model
reads:

$\hat{H}=\hbar\omega_{a}\hat{\sigma}^{z}_{a}+\hbar\omega_{b}\hat{\sigma}^{z}_{b}+\hbar\omega\hat{a}^{\dagger}\hat{a}\\\
+\hbar(g_{a}\hat{\sigma}^{z}_{a}+g_{b}\hat{\sigma}^{z}_{b})(\hat{a}+\hat{a}^{\dagger})$
(30)

with $\omega_{a}$, $\omega_{b}$ the frequencies of the spin subspaces,
$\omega$ the frequency of the quantum harmonic oscillator, and $g_{a}$,
$g_{b}$ the couplings between each spin and the mediator. For our quantum
simulation, we replace the ladder operators $a$, $\hat{a}^{\dagger}$ with
their associated position and momentum operators, $\hat{q}$ and $\hat{p}$,
obtaining

$\hat{H}=\hbar\omega_{a}\hat{\sigma}^{z}_{a}+\hbar\omega_{b}\hat{\sigma}^{z}_{b}+\frac{m\omega^{2}}{2}\hat{q}^{2}+\frac{1}{2m}\hat{p}^{2}\\\
+g_{a}\sqrt{2m\hbar\omega}\hat{\sigma}^{z}_{a}\hat{q}+g_{b}\sqrt{2m\hbar\omega}\hat{\sigma}^{z}_{b}\hat{q}$
(31)

with $m$ the mass of the oscillator.

In order to simplify the simulations of this model, we nondimensionalize the
model with the change of variables,

$\hat{\Pi}=\hat{q}\frac{\sqrt{m\omega}}{\sqrt{\hbar}}\ \ \ \ \
\hat{\xi}=\hat{p}\frac{1}{\sqrt{\hbar m\omega}}$ (32)

obtaining the Hamiltonian

$\hat{H}=\hbar\omega_{a}\hat{\sigma}^{z}_{a}+\hbar\omega_{b}\hat{\sigma}^{z}_{b}+\hbar\frac{\omega}{2}\left(\hat{\Pi}^{2}+\hat{\xi}^{2}\right)\\\
+\hbar\sqrt{2}\left(g_{a}\hat{\sigma}^{z}_{a}+g_{b}\hat{\sigma}^{z}_{b}\right)\hat{\Pi}.$
(33)

We will refer to the dynamics of this system as the fully quantum system and
its analytical solution is given in [19].

In the semiclassical regime, the massive mediator is taken to be classical. In
the KvN formalism, Eq. (15), the Hamiltonian of the system reads

$\hat{H}=\hbar\omega_{a}\hat{\sigma}^{z}_{a}+\hbar\omega_{b}\hat{\sigma}^{z}_{b}+\hbar\frac{\omega}{2}\left(\hat{\xi}\hat{\lambda}_{\Pi}-\hat{\Pi}\hat{\lambda}_{\xi}\right)\\\
-\hbar\sqrt{2}\left(g_{a}\hat{\sigma}^{z}_{a}+g_{b}\hat{\sigma}^{z}_{b}\right)\hat{\lambda}_{\xi}\;.$
(34)

This system is an MQS system. Note that this Hamiltonian is composed entirely
of quantum operators, $\hat{\sigma}_{a}^{z}$, $\hat{\sigma}_{b}^{z}$,
$\hat{\xi}$, $\hat{\Pi}$, $\hat{\lambda}_{\Pi}$, and $\hat{\lambda}_{\xi}$.
The primary difference is that position, $\hat{\xi}$, and momentum,
$\hat{\Pi}$, operators commute for the semiclassical subsystem. Hence, Cor. 1,
holds, and the system may be implemented with exponentially fewer resources on
a quantum computer than an equivalent algorithm implemented on a classical
computer.

In order to implement our simulations, for both quantum, Eq. (33), and
classical, Eq. (34), models, we chose the initial state of the mediator to be
a coherent state (for KvN, the distribution in phase space matching the Wigner
transform of a coherent state) $\ket{\alpha}$, whose wavefunction reads

$\small\psi^{(\alpha)}(x,0)=C\exp\Bigg(-\frac{1}{2}\left(\hat{\Pi}-\real[\alpha]\right)^{2}+i\imaginary[\alpha]\hat{\Pi}\Bigg{missing})\normalsize$
(35)

where $C$ is a normalization factor. Its phase space distribution, given by
the Wigner function, results in

$W(\hat{\Pi},\hat{\xi})=\frac{2}{\pi}e^{-[(\hat{\Pi}-\Pi_{0})^{2}+(\hat{\xi}-\xi_{0})^{2}]}\;,$
(36)

with $\Pi_{0}=\sqrt{2}\text{Re}(\alpha)$ and
$\xi_{0}=\sqrt{2}\text{Im}(\alpha)$. For simplicity we chose
$\alpha\in\mathbb{R}$ . Note that the operators $\hat{\Pi}$ and $\hat{\xi}$ of
these distributions are independent and therefore the distribution can be
factored as a product of its marginal distributions, which can be encoded in
disjoint Hilbert spaces.

###  VII.2 Numerical Simulations

We numerically simulated the dynamics of both quantum and classical
oscillators with $n=7$ qubits for each coordinate, plus $1$ qubit for each
spin subsystem, giving a total of $2+7=9$ qubits for the fully quantum case
and $2+2\times 7=16$ qubits for the KvN model. We adopted a Trotter method,
[35, 41, 42], where we leverage the quantum Fourier transform to construct the
momentum operators. During the system’s evolution we tracked two physical
quantities, the purity of each subsystem, and the logarithmic negativity
between the spins where the mediator was traced out. The physical values of
the frequencies and coupling used in the simulation are depicted in Tab. 1.

Purity and logarithmic negativity are both used to understand correlations
between quantum systems. Of the two, purity is easier to compute on a quantum
computer with just two copies of the system. However, the logarithmic
negativity is an entanglement monotone and serves as an upper bound on the
distillable entanglement, and therefore of interest for our results here, in
that it can increase under mixing, when classical information is lost.

In the case of fully quantum systems, it is expected that entanglement between
quantum spins, upon tracing over any of the subsystems, and the mediator will
generate decoherence and hence an initially pure state will lose its purity.
Less expected, but seen and understood in the literature [69], MQS systems can
also exhibit a similar loss of purity due to quantum-classical correlations.

Parameter | $\omega/(2\pi)$ | $g_{a}/(2\pi)$ | $g_{b}/(2\pi)$ | $\omega_{a}/(2\pi)$ | $\omega_{b}/(2\pi)$
---|---|---|---|---|---
Value | 1.59 Hz | 1.59 Hz | 2.23 Hz | 7.0 e-6 Hz | 3.34e-4 Hz
Table 1: Physical values of the frequencies and coupling used in the
simulation of Eqs. (30) and (34)

We present the results from simulating Eqs. (30) and (34) in Fig. 5. As we can
see, during the evolution of both models, purity (a,b) and logarithmic
negativity (c) evolve. In the fully quantum case, Fig. 5 a), once the first
oscillation is completed, the oscillator is decoupled from the spins, and the
spins become entangled, as we can see in the logarithmic negativity of the
fully quantum system, Fig. 5 c). On the other hand, in the KvN model, Fig. 5
b), the classical harmonic oscillator purity evolves, indicating quantum-
classical correlations. However, as may be seen via the logarithmic
negativity, Fig. 5 c), these correlations do not generate distillable
entanglement between the spins as expected of a classical mediator.

In both quantum and MQS systems, purities and logarithmic negativities evolved
differently. This indicates that quantum-quantum and quantum-classical
backreactions can usefully be studied with MQS simulation algorithms on
quantum hardware.

##  VIII Conclusions

This article presents a novel quantum algorithmic approach for simulating MQS
systems by combining Koopman-von Neumann (KvN) and standard quantum simulation
methods. By combining semiclassical and quantum simulation techniques, MQS
methods take advantage of resource reductions made possible with quantum
algorithms for semiclassical simulation, and additionally provide a
superpolynomial speedup relative to classical simulation methods.

The KvN approach is suitable for MQS simulations in the Schrödinger picture,
benefiting from the uniform encoding of both KvN and quantum Hamiltonians. The
key distinction between KvN and fully quantum Hamiltonian simulation lies in
the system variables’ commutativity in the KvN (semiclassical) part of the
simulation.

We have demonstrated that MQS simulation algorithms based on the KvN formalism
can significantly reduce resource requirements when compared to equivalent
classical algorithms. Additionally, MQS algorithms can outperform their
equivalent quantum-quantum algorithms in terms of resource usage as long as
the ratio of actions, $S_{q}/S_{c}$, of quantum and semiclassical systems is
large, and can give a particular advantage in the case of simulations of
quantum and semiclassical fields.

To validate the effectiveness of our approach, we implemented a proof-of-
principle gravitational interaction model involving two two-level quantum
systems with interaction mediated by a harmonic oscillator. By comparing
measures of entanglement dynamics between quantum and classical oscillators,
this model provided valuable insights into the two systems’ entanglement
properties, confirming the viability and usefulness of our approach.

Our work highlights the potential of the mixed KvN/quantum simulation approach
for MQS simulation and paves the way for further advances in MQS simulations,
offering a promising direction for optimizing resource utilization in quantum
computational studies.

Our proposal establishes a general framework for simulating MQS systems with a
focus on achieving more efficient MQS simulations relative to fully quantum
simulations and establishing differences in the resulting behavior of
entanglement due to backreaction in these systems.

Finally, we note that recent research studying Born-Oppenheimer systems [4]
takes a complementary approach, showing that KvN methods with phase-kickback
from an electronic simulation may be used for computing thermodynamical
properties in the Born-Oppenheimer approximation. This work outlines a
semiclassical KvN algorithm and detailed computational complexity analysis for
the computation of thermodynamical quantities from micro-canonical and
canonical ensembles. In this work, the authors read out the thermodynamic
state of the KvN system with an efficient quantum algorithm providing a direct
(efficient) estimation of the free energy. Of interest in this study, is that
the semiclassical KvN algorithm makes use of a many-particle encoding. Given
the results shown here, this could potentially be further improved by
converting the algorithm to a coarse-grained, second-quantized formalism [83].

###### Acknowledgements.

We thank Julen Padernales for extensive help with the spin-mediator model that
we used for proof-of-principle. We thank Prof. Iñigo Egusquiza for discussions
regarding Koopman-Von Neumann theory, Rolando Somma for discussions regarding
resource differences in quantum-quantum and quantum-semiclassical systems,
Michael Ragone and Joe Gibbs for helpful discussions. ATS acknowledges initial
support by the U.S. Department of Energy through the Los Alamos National
Laboratory, under the ASC Beyond Moore’s Law project when this project was
conceived. Los Alamos National Laboratory is operated by Triad National
Security, LLC, for the National Nuclear Security Administration of U.S.
Department of Energy (Contract No. 89233218CNA000001). During the project’s
research phase, ATS was subsequently supported by the U.S. Department of
Energy (DOE), Office of Science, Office of High Energy Physics, QuantISED
program. JGC acknowledges financial support from OpenSuperQ+100 (Grant No.
101113946) of the EU Flagship on Quantum Technologies, as well as from the EU
FET-Open project EPIQUS (Grant No. 899368), also from Project Grant No.
PID2021-125823NA-I00 595 funded by MCIN/AEI/10.13039/501100011033 and by “ERDF
A way of making Europe” and “ERDF Invest in your Future,” Basque Government
through Grant No. IT1470-22 and the IKUR Strategy under the collaboration
agreement between Ikerbasque Foundation and BCAM on behalf of the Department
of Education of the Basque Government, as well as from and UPV/EHU Ph.D. Grant
No. PIF20/276.

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