# Resource-Optimized Fermionic Local-Hamiltonian Simulation on a Quantum
Computer for Quantum Chemistry

Qingfeng Wang<EMAIL_ADDRESS>Chemical Physics Program and Institute for
Physical Science and Technology, University of Maryland, College Park, MD
20742, USA  Ming Li<EMAIL_ADDRESS>IonQ, College Park, MD 20740, USA  Christopher
Monroe<EMAIL_ADDRESS>IonQ, College Park, MD 20740, USA  Joint Quantum
Institute, Department of Physics, and Joint Center for Quantum Information and
Computer Science, University of Maryland, College Park, MD 20742, USA
Yunseong Nam<EMAIL_ADDRESS>IonQ, College Park, MD 20740, USA  Department of
Physics, University of Maryland, College Park, MD 20742, USA

###### Abstract

The ability to simulate a fermionic system on a quantum computer is expected
to revolutionize chemical engineering, materials design, nuclear physics, to
name a few. Thus, optimizing the simulation circuits is of significance in
harnessing the power of quantum computers. Here, we address this problem in
two aspects. In the fault-tolerant regime, we optimize the ${\textsc{r}}_{z}$
and t gate counts along with the ancilla qubit counts required, assuming the
use of a product-formula algorithm for implementation. We obtain a savings
ratio of two in the gate counts and a savings ratio of eleven in the number of
ancilla qubits required over the state of the art. In the pre-fault tolerant
regime, we optimize the two-qubit gate counts, assuming the use of the
variational quantum eigensolver (VQE) approach. Specific to the latter, we
present a framework that enables bootstrapping the VQE progression towards the
convergence of the ground-state energy of the fermionic system. This
framework, based on perturbation theory, is capable of improving the energy
estimate at each cycle of the VQE progression, by about a factor of three
closer to the known ground-state energy compared to the standard VQE approach
in the test-bed, classically-accessible system of the water molecule. The
improved energy estimate in turn results in a commensurate level of savings of
quantum resources, such as the number of qubits and quantum gates, required to
be within a pre-specified tolerance from the known ground-state energy. We
also explore a suite of generalized transformations of fermion to qubit
operators and show that resource-requirement savings of up to more than
$20\%$, in small instances, is possible.

##  1 Introduction

Simulating fermionic matter on a quantum computer has recently been receiving
much attention. Already available in the literature are various chemistry and
nuclear physics simulation results [19, 15, 34, 31, 8, 42], performed across
multiple quantum computing platforms, including superconducting [19, 34, 8]
and trapped-ion [15, 31, 42] based approaches. The spotlight on the simulation
of fermionic systems on a quantum computer is not accidental. Simulations of
these systems are useful for furthering fundamental science and practical
engineering [26, 13], and quantum computers are expected to enable the quantum
simulations of fermions undergoing local interactions [9, 23], a task believed
to be classically difficult to scale.

Broadly speaking, simulations of fermionic systems on a quantum computer may
be classified into two categories: a variational, quantum-classical hybrid
simulation [37], suitable for imperfect, pre-fault tolerant (pre-FT) quantum
computers, and a Hamiltonian dynamics simulation based on pure quantum
simulation algorithms [6], typically considered in fault-tolerant (FT) quantum
computers. In the context of estimating the ground-state energy of a fermionic
system, the former leverages efficient preparation of ansatz states and
evaluation of operator expectation values, both enabled by quantum computers.
The latter leverages the ability of a quantum computer to efficiently simulate
evolution of quantum systems with a local Hamiltonian, which, when combined
with quantum phase estimation [47], allows us to evaluate the ground-state
energy of the system.

In the pre-FT regime, quantum computational cost is dominated by the use of
multi-qubit gates. In the FT regime, where quantum circuits are typically
composed of gates in the Clifford+t gateset, quantum computational cost is
dominated by the use of $\textsc{t}:=\left(\begin{smallmatrix}1&0\\\
0&e^{i\pi/4}\end{smallmatrix}\right)$ gates, many of which are used in the FT
implementation of
${\textsc{r}}_{z}(\theta):=\left(\begin{smallmatrix}e^{-i\theta/2}&0\\\
0&e^{i\theta/2}\end{smallmatrix}\right)$ gates (see, e.g., [4]). In this
paper, we present approaches that optimize quantum simulations of fermionic
systems in either the pre-FT regime, reducing the number of multi-qubit gates,
or in the FT regime, reducing the number of ${\textsc{r}}_{z}$ gates or the
number of time steps that ${\textsc{r}}_{z}$ gates need to be applied, in the
simulation circuit.

Our paper is structured as follows. In Sec. 2 we briefly define the scope of
the problem in both the pre-FT and the FT regimes and lay out the
preliminaries. In Sec. 3, we present a highly-optimized circuit construction
for the FT quantum simulations, where we improve ${\textsc{r}}_{z}$ and t
counts, as well as the number of ancilla qubits required over the state of the
art. In Sec. 4, we show two complementary approaches that result in quantum
resource savings in the pre-FT regime, i.e., a second-order perturbation based
approach and a generalized fermion to qubit operator transformation. In the
specific sample case of a water molecule, our method provides nearly 25%
improvement over that obtained via the method detailed in [31] in the number
of two-qubit gates required to achieve the accuracy to within chemical
accuracy. We discuss our results in Sec. 5 and conclude in Sec. 6.

##  2 Preliminaries

We consider in this paper, as a concrete example, a fermionic system evolving
according to a time-independent, local, second-quantized Hamiltonian $H$ in
the occupation basis

$H=\sum_{p,r}h_{pr}a^{\dagger}_{p}a_{r}+\sum_{p,q,r,s}h_{pqrs}a^{\dagger}_{p}a^{\dagger}_{q}a_{r}a_{s}\;,$
(1)

where $a^{\dagger}_{p}$ and $a_{r}$ denote the fermionic creation and
annihilation operators on the $p$th and $r$th levels, respectively, and
$h_{pr}$ and $h_{pqrs}$ denote single- and double- fermion Hamiltonian
coefficients, respectively. The fermionic operators follow the canonical anti-
commutation relations

$\displaystyle\\{a_{j},a_{k}\\}=0\;,{\ \ \ \
}\\{a_{j}^{\dagger},a_{k}^{\dagger}\\}=0\;,{\ \ \ \
}\\{a_{j},a_{k}^{\dagger}\\}=\delta_{jk}\mathbf{1}\;,$ (2)

where $\\{A,B\\}$ denotes the anti-commutator $AB+BA$, $\delta_{jk}$ is the
Kronecker delta, and $\mathbf{1}$ is the identity operator. For their
implementations on a quantum computer, the fermionic operators need to be
suitably transformed [18, 5, 43]. A well-known, popular choice is the Jordan-
Wigner (JW) transformation [18], defined for an $n$-qubit system according to

$a_{j}^{\dagger}=1^{\otimes n-j-1}\otimes\sigma_{+}\otimes\sigma_{z}^{\otimes
j}\;,\;\;a_{j}=1^{\otimes n-j-1}\otimes\sigma_{-}\otimes\sigma_{z}^{\otimes
j}\;,$ (3)

where $j\in[0,n-1]$, $\otimes$ denotes the tensor product,
$\sigma_{+}:=\left(\begin{smallmatrix}0&0\\\ 1&0\end{smallmatrix}\right)$,
$\sigma_{-}:=\left(\begin{smallmatrix}0&1\\\ 0&0\end{smallmatrix}\right)$, and
$\sigma_{z}:=\left(\begin{smallmatrix}1&0\\\ 0&-1\end{smallmatrix}\right)$.

In the case where we consider a quantum dynamics simulation approach more
suitable for the FT regime, we aim to implement the evolution operator $U_{\rm
evo}=e^{-iHt}$ on a quantum computer, where $H$ is the system Hamiltonian and
$t$ is the duration by which we desire to evolve the system forward in time.
Roughly, once the initial state $|\psi_{\rm init}\rangle$, sufficiently close
to the ground state $|\psi_{\rm ground}\rangle$, is evolved to
$|\psi(t)\rangle\approx e^{-iE_{\rm ground}t}|\psi_{\rm init}\rangle$ up to
the closeness, quantum Fourier transform may be used to estimate the phase
angle $E_{\rm ground}t$, thus extracting the ground-state energy $E_{\rm
ground}$. Detailed discussion on the circuit layout that implements the
quantum phase estimation algorithm is available in [33]. Note, an efficient
implementation of the quantum Fourier transform on a FT quantum computer is
known [32]. Therefore, the problem of an efficient energy-spectra calculation
on a quantum computer in the FT regime boils down to the problem of
efficiently implementing real-time quantum dynamics simulations.

A host of algorithms that (approximately) implement $U_{\rm evo}$ have been
proposed, such as the asymptotically optimal quantum signal processing [25,
24]. However, it has been shown that for certain Hamiltonians including ones
of the form (1) a more straightforward technique of the $2k$th order product
formula (PF) can be implemented more efficiently in practice [6, 3]. Thus in
this paper we choose the PF algorithm for the FT regime quantum simulation and
present methods to reduce the quantum resources required in its quantum
circuit construction, measured in ${\textsc{r}}_{z}$ gate counts and depths,
in addition to the number of t gates and the ancilla qubits required (see Sec.
3).

In the case where we consider a quantum-classical hybrid approach more
suitable for the pre-FT regime, we consider the widely-adopted technique of
the variational quantum eigensolver (VQE) [37]. Specifically, we aim to
implement a unitary ansatz evolution operator $U_{\rm ansatz}$ on a quantum
computer, an example of which is the well-established unitary coupled cluster
(UCC) ansatz [16, 2]. By tuning the variational parameters in $U_{\rm
ansatz}$, in a typical VQE approach, $\langle\psi_{0}|U_{\rm
ansatz}^{\dagger}HU_{\rm ansatz}|\psi_{0}\rangle$ is minimized, where
$|\psi_{0}\rangle$ is an initial state that is assumed to be close to the
target ground state of $H$ and can readily be prepared on a quantum computer.
The goal of this hybrid approach is then to estimate the ground state energy
of the fermionic system with the Hamiltonian of the form (1). We show in this
paper a complementary approach, based on perturbation theory, to the hybrid
method described above in order to better and more efficiently estimate the
ground state energy (see Sec. 4.1). Our perturbative approach induces a
negligible quantum resource overhead, measured in the number of multi-qubit
gates. We also consider different fermion to qubit transformations other than
the aforementioned JW transformation (see Sec. 4.2), which help reduce the
number of multi-qubit gates without any loss in the algorithmic accuracy.

##  3 Fault-Tolerant Regime - Time evolution simulation

As discussed in Sec. 2, in this section, we detail the methods to optimize the
time-evolution operator implementation on a quantum computer. In particular,
we consider the PF approach. The cost functions we consider are the number of
${\textsc{r}}_{z}$ gates and the ${\textsc{r}}_{z}$ gate depth, which are
expected to be good proxy measures for the FT-regime quantum resource
requirements. This is so since the de-facto standard gate set for the FT
regime contains cnot, single-qubit Clifford, and t gates, where t gates are
assumed to be expensive as each implementation of the gate typically requires
expending a carefully induced, distilled quantum state that is expensive to
produce, and an ${\textsc{r}}_{z}$ gate consumes multiple t gates for its
precise implementation.

For completeness, we start by introducing the $2k$th order PF algorithm,
simulating the fermionic system described in Section 2. Assuming the JW
transformation has been performed on the Hamiltonian $H$ in (1), the time
evolution operator we aim to implement may be written as

$\exp\biggl{(}-i\sum_{j=1}^{L}\theta_{j}\hat{\sigma}^{(j)}\biggr{)}\approx[S_{2k}(\lambda)]^{r},$
(4)

where $\lambda:=1/r$,
$\hat{\sigma}^{(j)}=\bigotimes_{i}\tilde{\sigma}^{(i,j)}+\text{h.c.}$, where
$\tilde{\sigma}^{(i,j)}\in\\{\sigma_{+},\sigma_{-},\sigma_{z}\\}$ and h.c.
denotes the Hermitian conjugate operator, and

$\displaystyle S_{1}(\lambda)$
$\displaystyle:=\prod_{j=1}^{L}\exp(-i\theta_{j}\hat{\sigma}^{(j)}\lambda),$
$\displaystyle S_{2}(\lambda)$
$\displaystyle:=\prod_{j=1}^{L}\exp(-i\theta_{j}\hat{\sigma}^{(j)}\lambda/2)\prod_{j=L}^{1}\exp(-i\theta_{j}\hat{\sigma}^{(j)}\lambda/2),$
$\displaystyle S_{2k}(\lambda)$
$\displaystyle:=[S_{2k-2}(p_{k}\lambda)]^{2}S_{2k-2}((1-4p_{k})\lambda)[S_{2k-2}(p_{k}\lambda)]^{2},$
(5)

with $p_{k}:=1/(4-4^{1/(2k-1)})$ for $k>1$ [44]. Inspecting (4) and (5), we
observe that the individual exponential terms, hereafter referred to as a
Trotter term, are of the form
$\exp\left[-i\theta^{\prime}_{j}(\bigotimes_{i}\tilde{\sigma}^{(i,j)}+\text{h.c.})\right]$,
where $\theta^{\prime}_{j}$ is a suitably scaled $\theta_{j}$. A standard
circuit that implements a Trotter term is readily available in [47]. Employing
the circuit tricks in [31], in Fig. 1, we show an optimized quantum circuit
that implements an example Trotter term
$e^{-i\theta/2(\sigma_{+}\sigma_{+}\sigma_{-}\sigma_{-}+{\rm h.c.})}$.

$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$h$\textstyle{{\textsc{r}}_{z}}$
$\textstyle{{\textsc{r}}_{z}}$ $\textstyle{{\textsc{r}}_{z}^{\dagger}}$
$\textstyle{{\textsc{r}}_{z}^{\dagger}}$ $\textstyle{{\textsc{r}}_{z}}$
$\textstyle{{\textsc{r}}_{z}^{\dagger}}$
$\textstyle{{\textsc{r}}_{z}^{\dagger}}$
$\textstyle{{\textsc{r}}_{z}}$h$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$s
$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\textsc{s}^{\dagger}}$$\textstyle{\bullet}$
s
$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\textsc{s}^{\dagger}}$$\textstyle{\bullet}$
s $\textstyle{\textsc{s}^{\dagger}}$$\textstyle{\bullet}$ s Figure 1: Standard
two-body interaction circuit that implements
$\exp[-i(\theta/2)\sigma_{+}\otimes\sigma_{+}\otimes\sigma_{-}\otimes\sigma_{-}+\mathrm{h.c.}]$.
Following the steps detailed in [31] closely, expanding
$\sigma_{+}\sigma_{+}\sigma_{-}\sigma_{-}+\mathrm{h.c.}$ (we suppress
$\otimes$ hereafter whenever contextually clear) into the particular ordering
of $\sigma_{x}\sigma_{x}\sigma_{x}\sigma_{x}$,
$\sigma_{x}\sigma_{x}\sigma_{y}\sigma_{y}$,
$\sigma_{x}\sigma_{y}\sigma_{y}\sigma_{x}$,
$\sigma_{x}\sigma_{y}\sigma_{x}\sigma_{y}$,
$\sigma_{y}\sigma_{y}\sigma_{x}\sigma_{x}$,
$\sigma_{y}\sigma_{x}\sigma_{x}\sigma_{y}$,
$\sigma_{y}\sigma_{x}\sigma_{y}\sigma_{x}$, and
$\sigma_{y}\sigma_{y}\sigma_{y}\sigma_{y}$, and implementing them one after
another with the last qubit as the target qubit, we obtain the circuit shown
in this figure after applying the circuit optimization routines detailed in
[30, 31].
$\textstyle{+\;}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{{\textsc{r}}_{x}(\theta)}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{+\;}$
$\textstyle{\bullet}$ $\textstyle{-\;}$$\textstyle{\bullet}$
$\textstyle{\bullet}$ $\textstyle{\bullet}$$\textstyle{-\;}$
$\textstyle{\bullet}$ Figure 2: FT-regime optimized circuit for the two-body
term $e^{-i\theta/2(\sigma_{+}\sigma_{+}\sigma_{-}\sigma_{-}+{\rm h.c.})}$. We
marked each qubit lines with either $+$ or $-$ to denote which qubits are
associated with $\sigma_{+}$ or $\sigma_{-}$, respectively, for this
particular example.

To more efficiently implement the two-body term in terms of ${\textsc{r}}_{z}$
counts, we first directly evaluate the operator. It is straightforward to see
that the two-body term
$e^{-i\theta/2(\sigma_{+}\sigma_{+}\sigma_{-}\sigma_{-}+{\rm h.c.})}$
implements an ${\textsc{r}}_{x}$-rotation between
${\left|{0011}\right\rangle}$ state and ${\left|{1100}\right\rangle}$ state.
Note that, for an input sate of ${\left|{b_{0}b_{1}b_{2}b_{3}}\right\rangle}$,
where the $m$th qubit variable $b_{m}$ is either 0 or 1, ${\left|{b_{0}\ \
b_{1}\oplus b_{2}\ \ b_{0}\oplus b_{2}\ \ b_{0}\oplus b_{3}}\right\rangle}$
maps ${\left|{0011}\right\rangle}$ to ${\left|{0111}\right\rangle}$ and
${\left|{1100}\right\rangle}$ to ${\left|{1111}\right\rangle}$. A triply-
controlled ${\textsc{r}}_{x}$ rotation on the zeroth qubit would thus
implement the desired rotation on the mapped state, and all that needs to be
done at this point is to map the rotated state back. We show a quantum circuit
that implements the two-body term, constructed according to the aforementioned
method, in Fig. 2.

We refer to Supplementary Material (SM) Sec. A for the circuit-level
implementation details of Fig. 2. Briefly, a triply-controlled
${\textsc{r}}_{x}$ gate in Fig. 2 may be implemented with Hadamard,
${\textsc{r}}_{z}$, and triply-controlled not gates. Specifically, we choose
to use relative-phase triply-controlled not gates (see [27] and SM Sec. A for
its decomposition into single- and two-qubit gates), as it admits less
expensive circuit construction. Overall, our circuit decreases the
${\textsc{r}}_{z}$ count required per two-body Trotter term to two.

Our construction can be contrasted to [39] wherein (see Fig. 6 of the appendix
of [39]) an optimized implementation of a generic two-body term for the FT
regime is discussed. Because the construction in [39] considers
$\sigma_{x/y}$-decomposed form of a two-body term with distinct coefficients –
note our two-body term
$e^{-i\theta/2(\sigma_{+}\sigma_{+}\sigma_{-}\sigma_{-}+{\rm h.c.})}$ will
have the same coefficients upon decomposition – the circuit construction there
requires eight parallel ${\textsc{r}}_{z}$ gates with different angles of
rotation that are commensurate to the different coefficients. Considering our
specific example, these eight angles would be made the same, and the parallel
application of the same angle ${\textsc{r}}_{z}$ gates is known to admit a
more efficient implementation using the weightsum trick [10, 29]. As detailed
in SM Sec. A, employing this trick reduces the gate requirement from the
original $8\,{\textsc{r}}_{z}$ gates to $32\,\textsc{t}+4\,{\textsc{r}}_{z}$
gates. The total number of ancilla qubits required is eleven in this approach.
These may be compared with $16\,\textsc{t}+2\,{\textsc{r}}_{z}$ gates and one
ancilla qubit – note that a relative-phase triply-controlled not does not
require an ancilla, unlike a conventional triply-controlled not gate (see SM
Sec. A) – required for our method that is optimized for ${\textsc{r}}_{z}$
depth. Both of these methods admit an ${\textsc{r}}_{z}$ depth of one.

We note in passing that our method described above can be viewed as a
generalization of the single-body term implementation discussed elsewhere [17,
21]. Indeed, the generalization may be applied to an arbitrary many-body term,
should such a term be of interest to quantum simulations of a more complex
kind. See SM Sec. A for detail for the single-body term description, inserted
for completeness.

##  4 Pre-Fault Tolerant Regime - Variational Quantum Eigensolver

In the pre-FT regime, we aim to calculate the ground state energy of the
system whose Hamiltonian is given in the form (1). This is typically achieved
by the VQE approach. In this approach, by iteratively calling the quantum
computer to compute the energy of parametrized ansatz states, one aims to
minimize the energy and variationally obtain the ground state of the target
system.

We consider in this paper a type of ansatz states ${\left|{\Psi_{\rm
ansatz}}\right\rangle}$ that are transformed from an initial state
${\left|{\Psi_{0}}\right\rangle}$ using a parametrizable unitary ansatz
evolution operator $U_{\rm ansatz}$, as ${\left|{\Psi_{\rm
ansatz}}\right\rangle}=U_{\rm ansatz}{\left|{\Psi_{0}}\right\rangle}$. For
brevity, we drop the subscript in $U_{\rm ansatz}$ in the following text.
Since $U$ is a unitary operator, it can always be written in the exponential
form $e^{Z-Z^{\dagger}}$ and $Z$ is in turn parametrized. By varying the
parameters in $Z$, the ansatz state energy ${\left\langle{\Psi_{\rm
ansatz}}\right|}H{\left|{\Psi_{\rm
ansatz}}\right\rangle}={\left\langle{\Psi_{0}}\right|}U^{\dagger}HU{\left|{\Psi_{0}}\right\rangle}$
is variationally minimized. With a proper fermion to qubit basis
transformation the energy expectation value can be evaluated efficiently on a
quantum computer, where the quantum resource cost of this hybrid approach is
the implementation cost of the operator $U$. We note that the procedures
proposed in this section can be implemented with any ansatz with a unitary
evolution operator $U$. As a concrete example for discussion, we use the
widely-adopted UCC ansatz with single and double excitations (UCCSD), a
systematic method that is universally applicable to any quantum hardware
backend. Other ansatzes, for instance the $k$-UpCCGSD ansatz of Lee et al.
[22] or the UCCGSD ansatz of Grimsley et al. [12], can also be readily
applied.

The UCCSD method starts with a ground state Hartree-Fock (HF) wavefunction
$|\Psi_{0}\rangle$ that can be easily calculated on a classical computer and
implemented on a quantum computer. It then evolves $|\Psi_{0}\rangle$ with a
unitary operator $U=e^{Z_{\rm UCCSD}-Z_{\rm UCCSD}^{\dagger}}$, where $Z_{\rm
UCCSD}=\sum_{l=1}^{2}Z_{l}$ is the so-called cluster operator. $Z_{1,2}$ are
the single and double excitation operators and are written in second
quantization as

$\displaystyle\begin{split}Z_{1}=&\sum_{\begin{subarray}{c}p\in\mathrm{virt}\\\
r\in\mathrm{occ}\end{subarray}}t_{pr}a^{\dagger}_{p}a_{r},\\\
Z_{2}=&\sum_{\begin{subarray}{c}p,q\in\mathrm{virt}\\\
r,s\in\mathrm{occ}\end{subarray}}t_{pqrs}a^{\dagger}_{p}a^{\dagger}_{q}a_{r}a_{s},\end{split}$
(6)

where $t_{pr}$ and $t_{pqrs}$ are variational parameters and “virt” and “occ”
denote virtual and occupied levels respectively. Notice that for the UCCSD
ansatz, to minimize the size of the circuit that implements $U$, not all terms
in the cluster operator are needed to achieve a certain precision. In other
words, only a subset of all the parameters $t_{pq}$ and $t_{pqrs}$ need to be
included in the VQE approach, while the rest can be set to zero. In fact, it
is very much the name of the game in the pre-FT regime to find a minimal set
of variational parameters so that the final ground state energy satisfies
certain error criteria.

In this section, we propose two general procedures to reduce circuit
complexity of the approach detailed above, represented by the total number of
multi-qubit gates. To be concrete with our example, we use the first-order PF
algorithm to implement the UCCSD ansatz, although extensions to other ansatzes
or higher order PF algorithms are straightforward. In Sec. 4.1, we detail a
hybrid framework that is based on perturbation theory, wherein we apply a
perturbation correction to each instance of the VQE circuit. As will be
discussed in Sec. 4.1 in detail, our perturbation approach naturally provides
a means to determine the variational parameters to further include in the
ansatz for a given VQE circuit. This enables us to systematically build, in an
iterative fashion, a larger and larger VQE circuit, effectively bootstrapping
the VQE progression towards the ground-state energy estimate of the simulated
system. In Sec. 4.2, we explore a suite of generalized fermion to qubit
operator transformations to reduce the cost of quantum computation without
sacrificing algorithmic accuracy.

###  4.1 Perturbation Assisted Quantum Simulation

In this section, we describe a general framework that leverages the power of
perturbation theory to optimize VQE-based quantum simulations by predicting
the subset of variational parameters to include in the ansatz and correcting
the VQE result via post processing. The framework optimizes both the total
number of VQE executions as well as the size of the ansatz state preparation
circuits used to reach convergence in the ground-state energy estimate. In
Sec. 4.1.1 we outline the framework. The derivation of a simple perturbation
scheme that can be straightforwardly implemented in the framework is shown in
Sec. 4.1.2, which we hereafter refer to as a hybrid second order Møller-
Plesset perturbation (HMP2) method. In Sec. 4.1.3 we present a classically
simulated comparison between the HMP2 scheme and a more conventional VQE
approach [31], by computing the ground state energy of a water molecule using
the UCCSD ansatz.

####  4.1.1 Perturbative predictor and corrector

Figure 3: Flow diagram for the proposed framework incorporating perturbation
methods to VQE simulations. See main text for detailed description.

We detail, in this section, a general, systematic framework of using
perturbation methods to improve quantum resources required for the VQE
approach. Specifically, we aim to rapidly converge to the ground state energy
with small quantum circuits.

Any general (not specific to any system) unitary ansatz operator can be
written as

$U=e^{Z-Z^{\dagger}}\;,\;Z=\sum_{\\{\alpha\\}}f_{\alpha}(\\{t_{\beta}\\})D_{\alpha},$
(7)

where the summation is over a set of orbital sequences $\\{\alpha\\}$,
$f_{\alpha}$ are real functions of the set of variational parameters
$\\{t_{\beta}\\}$, and $D_{\alpha}$ is a generic orbital substitution operator
that substitutes the orbitals in a wavefunction according to the orbital
sequence $\alpha$. For instance, a single orbital substitution operator
$D_{p^{\dagger}q}=a_{p}^{\dagger}a_{q}$ substitutes orbital $q$ with $p$. We
can then define the orbital substituted wavefunction
${\left|{\Psi_{D_{\alpha}}}\right\rangle}=D_{\alpha}{\left|{\Psi_{0}}\right\rangle}$.
For a specific general ansatz approach there is a full set of variational
parameters $T_{\rm full}=\\{t_{\beta}\\}$ that can be used to parametrize the
ansatz evolution operator. In the case of UCCSD, by comparing (6) with (7), we
note that $T_{\rm full}=\\{t_{pr},t_{pqrs}\leavevmode\nobreak\
|\leavevmode\nobreak\ p,q\in{\rm virt};\leavevmode\nobreak\ r,s\in{\rm
occ}\\}$,
$\\{\alpha\\}=\\{p^{\dagger}r,p^{\dagger}q^{\dagger}rs\leavevmode\nobreak\
|\leavevmode\nobreak\ p,q\in{\rm virt};\leavevmode\nobreak\ r,s\in{\rm
occ}\\}$, and $f_{\alpha}=t_{\alpha}$.

Note further that we can generate a set of ansatz evolution operators
$\\{U_{T_{i}}\\}$, each element of which is the unitary operator that induces
an ansatz state with a subset $T_{i}$ of the full variational parameters
$T_{\rm full}$ ($T_{i}\subset T_{\rm full}$). The rest of the parameters in
the complement set $T_{\rm full}\setminus T_{i}$ are fixed to an ordered set
of default values $T_{\rm full}^{(0)}$. Without loss of generality, we assume
the default values are absorbed into $f_{\alpha}$ in (7) and the elements in
the complement set $T_{\rm full}\setminus T_{i}$ are simply set to zero. For
every $U_{T_{i}}$ in $\\{U_{T_{i}}\\}$, the ansatz state energy can be
variationally minimized as $E_{T_{i}}=\mathop{\bm{\min}}_{{\rm
elements}(T_{i})\in\mathbb{R}}{\left\langle{\Psi_{0}}\right|}U_{T_{i}}^{\dagger}HU_{T_{i}}{\left|{\Psi_{0}}\right\rangle}$,
where ${\rm elements}(T_{i})$ denotes the elements of the set $T_{i}$. For an
application of a particular general ansatz through the VQE approach in the
pre-FT regime, one aims to find a $U_{T_{i}}$ that has the least circuit
complexity while satisfying $|E_{T_{i}}-E_{T_{\rm full}}|<\varepsilon$ with
$\varepsilon$ a pre-determined error criterion.

A concrete strategy to reach convergence with minimal circuit complexity is to
start with an operator with a small set of variational parameters that is
known to be necessary (but not necessarily sufficient) to achieve the error
threshold and iteratively grow it. Going from the $m$th iteration to the
$(m+1)$th one, the set of variational parameters in the operators grows and
satisfies $T_{m}\subset T_{m+1}$. A critical part in ensuring a rapid
convergence is the selection of the additional variational parameter(s) to
include between iterations. In the case of UCCSD, they directly correspond to
additional individual excitation terms in (6) used to prepare the ansatz
state. Previous works have, e.g., used the full configuration interaction
(FCI) results [31] in the case the system is sufficiently small to be
simulatable on a classical computer, and have demonstrated the significance of
the excitation term selection for the resource requirement. Here, we propose
to iteratively select the next variational parameters to include in the ansatz
state preparation based on the size of the perturbatively predicted
wavefunction correction amplitude for a given ansatz state whose variational
parameters are already optimized via the conventional means of VQE. Our
strategy also evaluates a perturbative energy correction in addition to the
conventional VQE approach, directly contributing to the fast convergence,
while incurring no overhead in the pre-FT regime resource requirement.

Figure 3 shows the flow diagram of the proposed framework. As in the
conventional VQE approach for fermionic simulations, we start with the ground
state Hartree-Fock wavefunction of the single particle Hamiltonian as
${\left|{\Psi_{0}}\right\rangle}$. The first step is to determine the initial
evolution operator $U_{T_{1}}$ to use for the first iteration of VQE. As an
example, using the two-particle Hamiltonian as perturbation, energy and
wavefunction corrections can be calculated using classical algorithms with
relative ease. From the perturbed wavefunction, we can extract the amplitudes
of individual excited states in the single-particle Hamiltonian basis. These
amplitudes will serve as the initial guesses of the variational parameters for
the first round of VQE simulation, which could significantly reduce the number
of evaluations of the quantum circuit comparing to all-zeros or random initial
guesses [40]. If we demand the energy convergence criteria of $\delta_{E}$ –
The entire simulation is considered to be converged when the magnitude of
energy change associated with an addition of any extra ansatz term is smaller
than $\delta_{E}$ – we include in the initial ansatz set the ansatz terms
whose contribution to the correlation energy is greater than $f(\delta_{E})$,
where a standard choice of function $f(\delta_{E})$ may simply be
$\delta_{E}$.

With the initial set of ansatz terms and their initial variational parameter
values, we run the first round of VQE simulation to minimize the energy
$E_{T_{1}}={\left\langle{\Psi_{0}}\right|}U_{T_{1}}^{\dagger}HU_{T_{1}}{\left|{\Psi_{0}}\right\rangle}$.
Once the energy is minimized and the ansatz state converges, we proceed to
compute the expectation values of a set of operators that were chosen in
advance to inform us about the perturbation around the converged ansatz state.
We refer to this method as a hybrid perturbation since unlike the conventional
perturbation where we know the unperturbed wavefunction in advance, we start
the perturbation from the converged VQE ansatz state.

Based on the additional measurements performed on a quantum computer, we may
now use the hybrid perturbation method to calculate corrections to the
correlation energy and the wavefunction. The resulting total energy of this
cycle is the summation of the energy correction and the VQE energy. Before
starting the next cycle, we first identify a pool of operators that can be the
immediate successor of the current operator. Then we evaluate how much each
operator in the pool can account for towards the perturbative wavefunction
correction. The operator that can recover the wavefunction correction the most
gets the nod for the next cycle and the way it recovers the correction informs
the initial guesses of the newly added variational parameters in the next
cycle. A more detailed illustration of such a procedure is given in the next
subsection. The initial variational parameter values of the ansatz terms that
continue to be a part of the next VQE simulation may simply be imported from
the cycle before.

The cycle iterates the outlined procedure of running the VQE simulation and
the hybrid perturbation calculation, until convergence of the total energy is
achieved. We next detail the steps of a simple implementation of our hybrid
perturbation framework that is inspired by the classical MP2 method.

####  4.1.2 HMP2 method

We derive in this section our hybrid perturbation method, inspired by the
Møller-Plesset perturbation theory, applied to the VQE simulation. Our hybrid
MP2-based perturbation method, HMP2, aims to help each VQE cycle in terms of
both improving the resulting energy via a perturbative energy correction and
optimizing the ansatz operator to use in the next cycle through a perturbative
wavefunction correction.

In the $m$th VQE cycle, we can write the energy of the ansatz state
$E_{T_{m}}$ in the context of first-order perturbation theory, i.e.,

$\displaystyle\begin{split}E_{T_{m}}&={\left\langle{\Psi_{0}}\right|}U_{T_{m}}^{\dagger}HU_{T_{m}}{\left|{\Psi_{0}}\right\rangle}\\\
&={\left\langle{\Psi_{0}}\right|}F+(U_{T_{m}}^{\dagger}HU_{T_{m}}-F){\left|{\Psi_{0}}\right\rangle}\\\
&={\left\langle{\Psi_{0}}\right|}F{\left|{\Psi_{0}}\right\rangle}+{\left\langle{\Psi_{0}}\right|}U_{T_{m}}^{\dagger}HU_{T_{m}}-F{\left|{\Psi_{0}}\right\rangle}\\\
&=E_{0}+(E_{T_{m}}-E_{0})\\\ &=E^{(0)}+E^{(1)}_{T_{m}},\end{split}$ (8)

where $F$ is the Fock operator, $E_{0}$ is the sum of orbital energies,
$E^{(0)}=E_{0}$ is the zeroth-order energy, and
$E^{(1)}_{T_{m}}=E_{T_{m}}-E_{0}$ is the first-order correction energy. Based
on perturbation theory, a second-order correction to the energy can now be
written as

$E^{(2)}_{T_{m}}=\sum_{\\{\alpha^{\prime}\\}}\frac{\left|{\left\langle{\Psi_{D_{\alpha^{\prime}}}}\right|}V_{T_{m}}{\left|{\Psi_{0}\vphantom{\Psi_{D_{\\{}}}}\right\rangle}\right|^{2}}{\Delta
E_{D_{\alpha^{\prime}}}},$ (9)

where $V_{T_{m}}=U_{T_{m}}^{\dagger}HU_{T_{m}}-F$. The set of orbital
sequences $\\{\alpha^{\prime}\\}$ does not necessarily need to be the same as
the set $\\{\alpha\\}$ used in (7) and should be chosen according to accuracy
requirement and resource constraints. See Sec. 5 for more detailed discussion.
In our case study using the UCCSD ansatz, we choose
$\\{\alpha^{\prime}\\}=\\{\alpha\\}$. The energy $\Delta
E_{D_{\alpha^{\prime}}}$ is defined as the orbital energy difference of the
orbital substitutions. For instance, $\Delta
E_{D_{p^{\dagger}q}}=E_{q}-E_{p}$, where $E_{p}$ and $E_{q}$ are the orbital
energies of the $p$th and the $q$th orbitals, respectively. Inserting
$V_{T_{m}}=U_{T_{m}}^{\dagger}HU_{T_{m}}-F$ in (9), the numerator becomes

$\displaystyle\begin{split}\left|{\left\langle{\Psi_{D_{\alpha^{\prime}}}}\right|}V_{T_{m}}{\left|{\Psi_{0}\vphantom{\Psi_{D_{\\{}}}}\right\rangle}\right|^{2}&=\left|{\left\langle{\Psi_{D_{\alpha^{\prime}}}}\right|}U_{T_{m}}^{\dagger}HU_{T_{m}}-F{\left|{\Psi_{0}\vphantom{\Psi_{D_{\\{}}}}\right\rangle}\right|^{2}\\\
&=\left|{\left\langle{\Psi_{D_{\alpha^{\prime}}}}\right|}U_{T_{m}}^{\dagger}HU_{T_{m}}{\left|{\Psi_{0}\vphantom{\Psi_{D_{\\{}}}}\right\rangle}-{\left\langle{\Psi_{D_{\alpha^{\prime}}}}\right|}F{\left|{\Psi_{0}\vphantom{\Psi_{D_{\\{}}}}\right\rangle}\right|^{2}\\\
&=\left|{\left\langle{\Psi_{D_{\alpha^{\prime}}}}\right|}U_{T_{m}}^{\dagger}HU_{T_{m}}{\left|{\Psi_{0}\vphantom{\Psi_{D_{\\{}}}}\right\rangle}\right|^{2},\end{split}$
(10)

where we used
${\left\langle{\Psi_{D_{\alpha^{\prime}}}}\right|}F{\left|{\Psi_{0}\vphantom{\Psi_{D_{\\{}}}}\right\rangle}=0$.

In order to apply the VQE results, especially the ansatz wavefunction
${\left|{\Psi_{\rm ansatz}^{T_{m}}}\right\rangle}$ directly to the
perturbation calculation, we proceed as follows. For brevity, we introduce
$\tilde{Z}_{T_{m}}=Z_{T_{m}}-Z_{T_{m}}^{\dagger}$ with
$U_{T_{m}}=e^{\tilde{Z}_{T_{m}}}$ and
$\tilde{D}_{\alpha^{\prime}}=D_{\alpha^{\prime}}-D_{\alpha^{\prime}}^{\dagger}$.
Inserting these into (10), we obtain

$\displaystyle\begin{split}|{\left\langle{\Psi_{D_{\alpha^{\prime}}}}\right|}U_{T_{m}}^{\dagger}HU_{T_{m}}{\left|{\Psi_{0}}\right\rangle}|^{2}&=|{\left\langle{\Psi_{0}}\right|}\tilde{D}_{\alpha^{\prime}}^{\dagger}U_{T_{m}}^{\dagger}HU_{T_{m}}{\left|{\Psi_{0}}\right\rangle}|^{2}\\\
&=|{\left\langle{\Psi_{\rm
ansatz}^{T_{m}}}\right|}U_{T_{m}}\tilde{D}_{\alpha^{\prime}}^{\dagger}U_{T_{m}}^{\dagger}H{\left|{\Psi_{\rm
ansatz}^{T_{m}}}\right\rangle}|^{2}\;.\end{split}$ (11)

Next we Taylor expand the $e^{\pm\tilde{Z}_{T_{m}}}$ in
$e^{\tilde{Z}_{T_{m}}}\tilde{D}_{\alpha^{\prime}}^{\dagger}e^{-\tilde{Z}_{T_{m}}}H$
up to first order in $\tilde{Z}_{T_{m}}$. This is consistent with our choice
of the first-order PF algorithm for the implementation of the ansatz. We
obtain

$\displaystyle\begin{split}&|{\left\langle{\Psi_{\rm
ansatz}^{T_{m}}}\right|}e^{\tilde{Z}_{T_{m}}}\tilde{D}_{\alpha^{\prime}}^{\dagger}e^{-\tilde{Z}_{T_{m}}}H{\left|{\Psi_{\rm
ansatz}^{T_{m}}}\right\rangle}|^{2}\\\ =&|{\left\langle{\Psi_{\rm
ansatz}^{T_{m}}}\right|}(1+\tilde{Z}_{T_{m}}+\cdots)\tilde{D}_{\alpha^{\prime}}^{\dagger}(1-\tilde{Z}_{T_{m}}+\cdots)H{\left|{\Psi_{\rm
ansatz}^{T_{m}}}\right\rangle}|^{2}\\\ \approx&|{\left\langle{\Psi_{\rm
ansatz}^{T_{m}}}\right|}\tilde{D}_{\alpha^{\prime}}^{\dagger}H-\tilde{D}_{\alpha^{\prime}}^{\dagger}\tilde{Z}_{T_{m}}H+\tilde{Z}_{T_{m}}\tilde{D}_{\alpha^{\prime}}^{\dagger}H{\left|{\Psi_{\rm
ansatz}^{T_{m}}}\right\rangle}|^{2}.\end{split}$ (12)

With (9) and (12) we can obtain the energy correction to the VQE result.

We note that
$\tilde{D}_{\alpha^{\prime}}^{\dagger}H-\tilde{D}_{\alpha^{\prime}}^{\dagger}\tilde{Z}_{T_{m}}H+\tilde{Z}_{T_{m}}\tilde{D}_{\alpha^{\prime}}^{\dagger}H$
is a sum of products of Pauli operators
$\sum_{j}\epsilon_{j}\hat{\sigma}^{(j)}$ in the qubit basis, after applying a
suitable fermion to qubit basis transformation. Since $\hat{\sigma}^{(j)}$
have eigenvalues $+1$ or $-1$, denoting
${\left|{\Psi_{j}^{(p)}}\right\rangle}$ and
${\left|{\Psi_{j}^{(m)}}\right\rangle}$ as the eigenvectors with the
respective eigenvalues, we may then write

$\displaystyle\begin{split}&\left|\left\langle\Psi_{\rm
ansatz}^{T_{m}}\left|\sum_{j}\epsilon_{j}\hat{\sigma^{(j)}}\right|\Psi_{\rm
ansatz}^{T_{m}}\right\rangle\right|^{2}\\\ =&\left|\left\langle\Psi_{\rm
ansatz}^{T_{m}}\left|\sum_{j}\epsilon_{j}\left(\sum_{p}{\left|{\Psi_{j}^{(p)}}\right\rangle}{\left\langle{\Psi_{j}^{(p)}}\right|}-\sum_{m}{\left|{\Psi_{j}^{(m)}}\right\rangle}{\left\langle{\Psi_{j}^{(m)}}\right|}\right)\right|\Psi_{\rm
ansatz}^{T_{m}}\right\rangle\right|^{2}\\\
=&\left|\sum_{j}\epsilon_{j}\left(\sum_{p}\left|\left\langle\Psi_{\rm
ansatz}^{T_{m}}\left.\right|\Psi_{j}^{(p)}\right\rangle\right|^{2}-\sum_{m}\left|\left\langle\Psi_{\rm
ansatz}^{T_{m}}\left.\right|\Psi_{j}^{(m)}\right\rangle\right|^{2}\right)\right|^{2}\;.\end{split}$
(13)

Note that (13) requires only a simple projection of ${\left|{\Psi_{\rm
ansatz}}\right\rangle}$ onto ${\left|{\Psi_{j}^{(p)}}\right\rangle}$ or
${\left|{\Psi_{j}^{(m)}}\right\rangle}$. Thus the second order correction
energy may be obtained without any quantum resource overhead in the circuit
size.

To optimize the choice of the ansatz evolution operator to use in the next
cycle, we turn to the wavefunction correction. The ansatz state wavefunction
can be considered as the zeroth-order wavefunction
${\left|{\Psi^{(0)}}\right\rangle}$ and a first order correction is given by

${\left|{\Psi^{(1)}_{T_{m}}}\right\rangle}=\sum_{\\{\alpha\\}}\frac{{\left\langle{\Psi_{D_{\alpha}}}\right|}V_{T_{m}}{\left|{\Psi_{0}\vphantom{\Psi_{D_{\\{}}}}\right\rangle}}{\Delta
E_{D_{\alpha}}}{\left|{\Psi_{D_{\alpha}}}\right\rangle}.$ (14)

We then use the following procedure to determine which ansatz evolution
operator to use next in the $(m+1)$th round. In the $m$th round we have
variationally determined a set of values for the parameters in the set
$T_{m}$, denoted as $S_{m}$. We then look for a set $R_{m}=\\{U_{T_{i}}\\}$ of
all the evolution operators $U_{T_{i}}$ that satisfies $T_{m}\subset T_{i}$
and $n(T_{i}\setminus T_{m})=\min_{j\neq m}[n(T_{j}\setminus T_{m})]$, where
$n(A)$ denotes the number of elements in set $A$. $R_{m}$ is then the pool of
operators from which we are going to choose the next ansatz evolution
operator. For each $U_{T_{i}}$ in $R_{m}$, we can then compute the overlap
$F_{m}^{i}=1/w_{m}^{i}\sum_{t_{\beta}\in T_{i}\setminus
T_{m}}\left|\left\langle\Psi_{t_{\beta}}^{m,i}|\Psi^{(1)}_{T_{m}}\right\rangle\right|$,
where

${\left|{\Psi_{t_{\beta}}^{m,i}}\right\rangle}=\left(\left.\frac{\partial
U_{T_{i}}}{\partial
t_{\beta}}\right|_{S_{m}}\right){\left|{\Psi_{0}}\right\rangle}\;.$ (15)

Here we use $w_{m}^{i}$ to capture the additional cost associated with
implementing $U_{T_{i}}$ instead of $U_{T_{m}}$. For example, it can be the
number of additional two-qubit gates needed to compute
${\left\langle{\Psi_{0}}\right|}U^{\dagger}HU{\left|{\Psi_{0}}\right\rangle}$
for $U=U_{T_{i}}$ instead of $U=U_{T_{m}}$. For our concrete example, we
simply take $w_{m}^{i}$ to be the number of additional fermionic operators in
$T_{i}$ comparing to $T_{m}$. Using the identity
$\left\langle\Psi_{D_{\alpha}}|\Psi_{D_{\alpha}^{\prime}}\right\rangle=\delta_{\alpha\alpha^{\prime}}$,
it is straightforward to evaluate all of the $F_{m}^{i}$ using (9), (12), and
(15). Then we simply pick the $U_{T_{i}}$ out of $R_{m}$ with the largest
$F_{m}^{i}$ to be the ansatz evolution operator to use in the next cycle. We
can also use the wavefunction correction to guess the initial values for an
additional variational parameter $t_{\beta}\in T_{m+1}\setminus T_{m}$ as
$\pm\left\langle\Psi_{t_{\beta}}^{m,m+1}|\Psi^{(1)}_{T_{m}}\right\rangle$.

In the case of UCCSD, each $U_{T_{i}}$ in $R_{m}$ only has one more
$D_{\alpha}$ term in $Z_{T_{i}}$ than $Z_{T_{m}}$ of $U_{T_{m}}$, which
corresponds to an incremental change in the terms included in the ansatz
operator $Z$. Using $e^{\tilde{Z}_{T_{i}}}\approx 1+\tilde{Z}_{T_{i}}$,
$F_{m}^{i}$ is approximately the perturbative amplitude
$|{\left\langle{\Psi_{D_{\alpha}}}\right|}V_{T_{m}}{\left|{\Psi_{0}}\right\rangle}/\Delta
E_{D_{\alpha}}|$ of the corresponding additional $D_{\alpha}$.

All the procedures introduced above for the $m$th round of a VQE cycle can
also be used before the first cycle by simply taking $m=0$. In this case, all
the calculations are classical and the energy correction is simply the
classical MP2 energy correction. The wavefunction correction however can
inform the first operator to be tried and suggest the initial values for its
variational parameters.

We note in passing that, in principle, the number of individual Pauli-string
operators in (13) whose expectation values are to be evaluated using a quantum
computer scales as $O(n^{4})$ per UCCSD ansatz where $n$ is the number of
qubits. This is so since $H$ consists of $O(n^{4})$ terms, while, for a
particular ansatz, $Z$ is fixed and provides a constant prefactor. Thus the
predictive feature using (14) requires $O(n^{8})$ Pauli-string measurements
because there are $O(n^{4})$ elements in $\\{\alpha\\}$. The perturbative
correction using (9) can reuse all the measurement results from the predictive
feature because we choose $\\{\alpha^{\prime}\\}=\\{\alpha\\}$. While this may
appear challenging, in practice, a series of techniques can be applied to
significantly reduce the number of evaluations. First, we can take advantage
of the fact that only the qubits that are coupled via a chosen ansatz are
entangled. By carefully choosing the qubits to use to create the ansatz state,
we can treat the qubits that have not been operated on classically. A
straightforward extension to small, disjoint sets of qubits with set sizes
that admit inexpensive classical postprocessing may be considered as well,
although we do not consider such an aggressive optimization in this paper. In
addition, if we have two qubits that represent the same spatial orbital with
two opposite spins, in the case where they are not distinguishable in their
energy due to a particular choice of the ansatz (see Sec. 4.2 and [31] for
more details), the two qubits would encode a redundant piece of information,
and this allows us to encode the information using just one qubit. We call
this optimization technique qubit space reduction (QSR) (see SM Sec. H for the
implementation details). Secondly, it is possible to measure multiple Pauli
strings simultaneously to reduce the total number of measurements provided
that these Pauli strings commute with each other. For instance, two methods,
the general commuting (GC) partition and the qubit-wise commuting (QWC)
partition, have been proposed to enable such simultaneous measurements [11].
Generally speaking, GC can reduce the number of measurements significantly but
could incur additional cost in terms of additional two-qubit gates while QWC
can reduce the number of measurements, though to a lesser degree than GC,
without increasing the number of two-qubit gates. If using GC, it is
beneficial to use QSR beforehand to reduce the size of the qubit space needed
to be partitioned, which in turn reduces the number of extra two-qubit gates
incurred from using the GC partition.

####  4.1.3 Comparison to prior state-of-the-art

Ansatz | $E_{\mathrm{UCCSD}}$ | $E_{\mathrm{P_{D}}}^{\text{corr}}$ | $E_{\mathrm{UCCSD+HMP2}}$
---|---|---|---
Ansatz terms in HMP2 order | HF+1 | -74.9749 | -0.0249 | -74.9998
HF+2 | -74.9781 | -0.0220 | -75.0001
HF+4 | -74.9854 | -0.0170 | -75.0024
HF+5 | -74.9881 | -0.0155 | -75.0036
HF+6 | -74.9909 | -0.0139 | -75.0048
HF+8 | -74.9966 | -0.0091 | -75.0057
HF+9 | -74.9996 | -0.0068 | -75.0063
HF+11 | -75.0038 | -0.0039 | -75.0077
HF+12 | -75.0047 | -0.0034 | -75.0081
| HF+14 | -75.0074 | -0.0023 | -75.0098
| HF+16 | -75.0091 | -0.0015 | -75.0106
| HF+17 | -75.0100 | -0.0009 | -75.0109
| HF+19 | -75.0102 | -0.0007 | -75.0109
| HF+21 | -75.0107 | -0.0004 | -75.0111
| HF+23 | -75.0111 | -0.0002 | -75.0113
| HF+24 | -75.0113 | -0.0001 | -75.0114
| HF+26 | -75.0113 | -0.0001 | -75.0114
| HF+28 | -75.0113 | -0.0001 | -75.0114
$E_{\text{HF}}$ |  |  |  | -74.9624
$E_{\text{MP2}}$ |  |  |  | -74.9977
$E_{\text{CCSD}}$ |  |  |  | -75.0114
$E_{\text{FCI}}$ |  |  |  | -75.0116
Table 1: Ground state energy calculations for a water molecule using STO-3G
basis. HF+$N$ represents a VQE cycle with the UCCSD ansatz with $N$
$D_{\alpha}$ terms, as described in Sec. 4.1.1.
$E_{\mathrm{P_{D}}}^{\text{corr}}$ is the HMP2 correction based on (12).
$E_{\mathrm{UCCSD+HMP2}}=E_{\mathrm{UCCSD}}+E_{\mathrm{P_{D}}}^{\text{corr}}$
is the total energy obtained in one VQE cycle. For the classical optimization
step in VQE, we used L-BFGS-B optimizer [48], and we assumed perfect
measurement with an infinite number of circuit repetitions to suppress the
effect of finite measurement. The classically computed energies
$E_{\text{HF}}$, $E_{\text{MP2}}$, $E_{\text{CCSD}}$, and $E_{\text{FCI}}$ for
the water molecule with the same geometry and basis set are also listed for
comparison. All energies are in units of Hartree.

To demonstrate our framework of perturbation assisted quantum simulation using
the HMP2 method, we performed classically emulated VQE calculations with the
UCCSD ansatz of the ground state energy of a water molecule at its equilibrium
geometry. Using the STO-3G basis, the calculation contains 14 qubits in total.
Table 1 shows the incremental changes of the UCCSD correlation energy as well
as the HMP2 correction as more ansatz terms are included according to our
framework. Note that our choice of $\\{\alpha^{\prime}\\}$ in (9) for this
example guarantees that the final energy reaches the UCCSD energy when all the
ansatz terms are included in (6). To compare, we also listed the classically
calculated HF, MP2, CCSD, and FCI energies since it is difficult to obtain the
UCCSD energy via classical algorithms. With the HMP2 correction, the total
energy $E_{\text{UCCSD+HMP2}}$ descends quickly towards FCI energy.

Figure 4: Comparison of the ground state energies of a water molecule at its
equilibrium geometry using the STO-3G basis set, calculated by various methods
as a function of $N$, the number of $D_{\alpha}$ terms included in the HF$+N$
ansatz. The orange dashed line is the FCI energy calculated using the PSI4
package [35], which serves as the benchmark. The black diamonds connected by
the dotted lines are the UCCSD energies $E_{\mathrm{UCCSD}}$, calculated using
different numbers of ansatz terms ordered by the contribution of corresponding
determinants to the FCI energy. The red squares and blue circles connected by
the dotted lines are the ground-state energies computed according to the
proposed framework with the HMP2 ordering, with the circles containing the
additional HMP2 energy correction at each VQE cycle. The inset shows in semi-
log the differences between the energies obtained by the aforementioned
methods and the FCI energy as a function of $N$. The purple dashed line shows
the chemical accuracy given by $10^{-3}$ hartree. As stated in the caption of
Table 1 as well, for the classical optimization step in VQE, we used L-BFGS-B
optimizer [48], and we assumed perfect measurement with an infinite number of
circuit repetitions to suppress the effect of finite measurement.  Figure 5:
Demonstration of the reduction of the number of measurements as a secondary
cost for the water molecule calculation using various optimization schemes.
The calculation is carried out using the HMP2 framework with the UCCSD ansatz.
The total number of expectation values to be evaluated on a quantum computer
as a function of the number $N$ of $D_{\alpha}$ terms in the UCCSD ansatz
operator for a VQE cycle is shown in (a) for three different optimization
schemes. The black solid line represents the reduction using only QSR. The
blue dotted line shows the reduction using the QWC partition and QSR at the
same time. The red dashed line illustrates the most reduction, using the GC
partition along with QSR. Note the plateau-like behaviors that make the lines
to appear like staircases are due to our QSR technique (orange, dot-dashed
line with the $y$ axis on the right), i.e., the jumps in the number of
expectation values to be measured per VQE iteration occur when the QSR
technique makes a transition from one space requirement to another. The
average number of additional two-qubit gates incurred by using the GC
partition along with QSR, $n_{\mathrm{GC+QSR}}$, is shown in (b) as the the
blue dotted line. This may be compared with the black solid line, representing
the total number of two-qubit gates $n_{\mathrm{JW}}$ required to induce UCCSD
ansatz states, implemented with the JW transformation. The distribution of
$n_{\mathrm{GC+QSR}}$ is given in the SM Sec. I. The red dashed line with an
alternative $y$-axis label on the right shows the ratio
$R=n_{\mathrm{GC+QSR}}/(n_{\mathrm{JW}}+n_{\mathrm{GC+QSR}})$.

Figure 4 shows the convergence of the ground state energies using different
approaches as the number $N$ of $D_{\alpha}$ terms included in the UCCSD
ansatz operator increases. For the conventional approach [31], the insertion
order of the ansatz terms is obtained from the prior knowledge of the order of
contribution of determinants in a classical FCI calculation, which closely
mimics the ideal case but is rarely realistic to obtain. For the UCCSD
energies obtained using the proposed HMP2, we bootstrapped the ordering of the
ansatz terms as detailed in Sec. 4.1.2. Comparing the convergence of the UCCSD
energies, we find that the HMP2-bootstrapped ordering effectively captures the
major ansatz terms. This is confirmed by the good agreement between the FCI
ordering and the HMP2 ordering shown in Fig. 4. We also observe that the HMP2
correction to the ground-state energy helps accelerate the energy convergence
towards the FCI energy significantly. Provided that implementation of each
additional ansatz term leads to a substantial accumulation of noise in the
NISQ hardware, we find the rapid energy convergence enabled by the HMP2 method
to be particularly useful for the near-term quantum computers.

In addition to focusing on reducing the circuit complexity, which is the most
important cost for NISQ era applications, we examine the total number of
measurements as a secondary cost using the water molecule calculation as an
example. Figure 5 (a) shows the number of measurements using various
optimization schemes discussed in Sec.4.1.2. Comparing to the number of
measurements optimized using only QSR, the number optimized using the GC
partition method and QSR is more than two orders of magnitude smaller while
the number is about one order of magnitude smaller using the QWC partition
method along with QSR. Figure 5 (b) shows the additional cost in terms of the
number of two-qubit gates associated with using the GC partition method along
with QSR comparing to the total number of two-qubit gates using the JW
transformation. The ratio between the two quickly reduces as the number of
ansatz terms increases. Thus one should decide whether to use GC or QWC along
with QSR after weighing their benefits against their cost on a case-by-case
basis.

###  4.2 Generalized transformations for Fermion to Qubit operator

In this section, we investigate how a variety of fermion to qubit
transformations may be used to reduce the quantum resource requirements for
the pre-FT fermion simulations. It is important to note that all of the
transformations are equivalent and thus the resulting quantum circuits for
each transformation implements exactly the same fermion simulation. Therefore,
the resource savings we obtain in this section are independent of the accuracy
of the simulations, in general.

Well-known fermion to qubit transformations, such as Jordan-Wigner (JW) [18]
or Bravyi-Kitaev (BK) [5] transformations, map fermionic creation
(annihilation) operators to Pauli strings. However, there exist numerous other
transformations available for use. Below, we introduce a generalized
transformation (GT) method (see also [43]), of which the JW and BK
transformations are a part. We show that, when used with the PF algorithm as a
concrete example for the implementation of the UCCSD ansatz, significant
quantum resource savings may be achieved by a suitable choice of the mapping
for a given cluster operator input, together with a carefully chosen sequence
of heuristic optimization methods.

All transformations in the GT method must respect relations specified in (2).
This may be achieved by considering the following invertible, upper-triangular
basis-transformation matrix $\beta$, which transforms the occupation number
basis to a GT basis according to

$\displaystyle\beta=\left(\begin{matrix}&\beta_{n-1,n-1}\quad&\beta_{n-1,n-2}&\;...&\beta_{n-1,0}\\\
&0\quad&\beta_{n-2,n-2}&\;...&\beta_{n-2,0}\\\ &\vdots&\vdots&&\vdots\\\
&0\quad&0&\;...&\beta_{0,0}\end{matrix}\right),$ (16)

where $\beta_{i,j}\in\\{0,1\\}$ for $i>j$, $\beta_{i,i}=1$, and $n$ is the
number of qubits involved in the transformation. Following closely the
notations used in [41], we define the following sets of indices for
convenience, which we detail below. We note that all matrix operations are
performed in modulo-2 space and the main diagonal elements are excluded when
generating these sets.

  * •

Update set $U(j)$: elements of this set are the row indices with non-zero
entries in column $j$ of the basis-transformation matrix $\beta$.

  * •

Parity set $P(j)$: elements of this set are the column indices with non-zero
entries in row $j$ of the matrix $(\pi\beta^{-1}-\beta^{-1})$, where
$\pi_{i,j}=1$ if $i\leq j$, otherwise 0.

  * •

Remainder set $R(j)$: elements of this set are the column indices with non-
zero entries in row $j$ of the matrix $\pi\beta^{-1}$.

The GT-based creation and annihilation operators are then

$\displaystyle a_{j}^{\dagger}$
$\displaystyle\equiv\left[\sigma_{x}^{U(j)}\otimes\sigma_{x}^{j}\otimes\sigma_{z}^{P(j)}-i\sigma_{x}^{U(j)}\otimes\sigma_{y}^{j}\otimes\sigma_{z}^{R(j)}\right]/2,$
$\displaystyle a_{j}$
$\displaystyle\equiv\left[\sigma_{x}^{U(j)}\otimes\sigma_{x}^{j}\otimes\sigma_{z}^{P(j)}+i\sigma_{x}^{U(j)}\otimes\sigma_{y}^{j}\otimes\sigma_{z}^{R(j)}\right]/2,$
(17)

which can straightforwardly be shown to satisfy (2). We note that, e.g., the
JW transformation is a special case of $\beta=\mathbf{1}$.

The state of the art implementation of the UCCSD ansatz [31] relies on the JW
transformation and it considers a series of heuristics that were chosen
carefully to optimize the resulting quantum circuits. To enable a proper
comparison of those with our results then, it is necessary for us to consider
a similar series of heuristics as well. Details of the heuristics we consider
are provided in the SM Secs. B–G. Below, we briefly outline the steps in the
order of applications. The cost function we consider for our concrete example
is the number of two-qubit gates.

The outer-most loop of our approach considers different transformation
matrices $\beta$. For a given mapping matrix $\beta$, we execute the following
routines to construct and optimize our circuit that implements the UCCSD
ansatz. The routines are designed to repeatedly call a suite of dedicated,
automated circuit optimization tools, whose technical details may be found in
[31, 30]. The efficiency of these tools allows us to quickly evaluate the cost
function for different cases we consider in each of the subroutines.

  1. Routine 1

_Fermionic level labeling:_ Unlike in the JW transformation where $U(j)$ is an
empty set and $P(j)=R(j)$, in the GT approach, myriads of combinations of sets
$U(j)$, $P(j)$, and $R(j)$ are possible. To best take advantage of this, it is
critical to carefully select which fermion level is mapped onto which qubit
index. Exploring all possible mappings is however computationally
prohibitively expensive. We thus resort to a simple greedy approach, whereby
we explore one permutation at a time from a given fermion-level to qubit-index
mapping. Specifically, from the given mapping, we apply the permutation that
results in the most reduction in the quantum resource requirements. We iterate
this process until no single permutation results in the reduction of quantum
resource requirements. See SM Sec. B for detail. See Subroutine 1 below for
the cost function evaluation for each permutation.

  2. Routine 2

_Inter-Trotter term ordering:_ Demonstrated in [38, 14, 46, 1, 45, 31] was
that ordering the Trotter terms appropriately can lead to large savings in the
two-qubit gate counts due to gate cancellations between the neighboring
Trotter term circuits. While a similar approach is indeed possible – and we
base our method on the work reported in [31] – in our GT approach, a non-
trivial modification to a method reported in [31] needs to be made.
Specifically, we need to preprocess each Trotter term to determine its
eligibility for being classified under the same equivalence class, the
elements of which have the opportunities for resource savings when placed next
to one another on a quantum circuit via the aforementioned gate cancellations.
The eligibility criteria are straightforward for the JW transformation,
considered in [31]. See SM Sec. D for details for the GT method. Once the
equivalence classes according to the eligibility of each Trotter term is
determined, we use a simple greedy approach to order the Trotter-term elements
of each equivalence class to reduce the two-qubit gate counts.

  1. Subroutine 1

_Intra-Trotter term ordering:_ For a given fermion-label to qubit-index
mapping, an efficient method to implement a single- or a double-fermion
excitation Trotter term is known [47] in the case where the JW transformation
is used, and an aggressively optimized circuit construction that leverages
full qubit-to-qubit connectivity is available in [31]. The method in [31]
relies on a careful ordering of the intra-Trotter operator implementation,
where, for instance, an example double-fermion Trotter term
$e^{t_{pqrs}(a_{p}^{\dagger}a_{q}^{\dagger}a_{r}a_{s}-h.c)}$ is expanded to
$\sigma_{x,y,z}$-based intra-Trotter terms. To enable an efficient
implementation in other transformations used in our GT approach, we compute
the cost function for every possible permutation of the intra-Trotter terms
(see SM Sec. C for detail). We choose an ordering with the least cost.

The resulting optimized circuit implements the UCCSD ansatz in the chosen
transformation basis defined by $\beta$. However, with the exception of the JW
transformation where $\beta={\bf 1}$, the GT $\beta$ matrix requires us to
implement the initial mapping of the basis at the beginning of the circuit.
This incurs an overhead $O(n^{2}/\log(n))$ in the two-qubit gate counts [36].
To obtain the final quantum resource requirement, we call an automated
optimizer with the input quantum circuit that consists of the prefix
subcircuit that implements $\beta$ and the postfix subcircuit that implements
the $\beta$-basis UCCSD ansatz.

We note that in [31] the concept of “bosonic” excitations is discussed.
Effectively, in the JW transformation, whenever a pair of neighboring qubits,
whichever appropriate fermion levels they correspond to, are excited to yet
another pair of neighboring qubits that denote another set of fermion levels,
the circuit that implements such an excitation term can be dramatically
simplified to require only two two-qubit gates, while requiring only half the
number of qubits that would otherwise be required (see SM Sec. E for the cases
where the pairs do not neighbor). To take advantage of this, we use a
juxtaposition of the bosonic circuit written according to the JW
transformation and non-bosonic circuit written according to our GT approach.
We note in passing that, to return from the half-qubit space of the bosonic
circuit to the full-qubit space of the non-bosonic circuit, at most $n/2$ cnot
gates are expended. All our circuit metrics appropriately reflect this.

Table 2 shows circuit metrics, measured according to the number of two-qubit
gates used to implement the UCCSD ansatz circuit, for different molecules of
our choice. We show the results for the JW, BK, and the best GT
transformations that our heuristic toolchain specified above found for
comparison. To find the best GT transformations, we used a particle swarm
optimization, as detailed in SM Secs. F and G. The advantages offered by the
GT transformations vary in the suite of molecules we consider, ranging from
$1.44\%$ to $21.43\%$. This demonstrates the capability of our heuristics that
it is indeed possible to further optimize the quantum circuits over the
previous state of the art obtained via the JW transformation by considering GT
transformations, custom selected for different input cases.

We make an open-ended note here on the savings ratio obtained. The savings
ratio for the GT method is obtained by comparison to the JW transformation,
which often admits better circuit optimization over the BK transformation in
the instances we considered, especially around the interesting regimes where
the ansatz circuit is large enough to recover, e.g., chemical accuracy, but is
sufficiently small to merit VQE implementations on a near-term, noisy device.
In the asymptotic limit, the BK transformation, one of the GTs, is known to
incur $O(\log(n))$ overhead, an exponential advantage over the JW
transformation that incurs $O(n)$ overhead [48]. Provided that, in the small
instances, the savings ratio can be larger than $20\%$, a question arises as
to how the ratio will behave in the intermediate-sized, aforementioned
interesting regime. We mention here a couple possibilities. It may be that the
improved circuit efficiency by the GT method for a particular excitation term
could be offset by the increased cost of implementing other excitation terms.
In this case, the ratio is expected to drop. It could also be that the JW
transformation, due to its lack of nimbleness in the definition of
transformation to help reduce potentially large cost of adding the excitation
terms to the ansatz circuit, incurs a large overhead, thus resulting in the
more significant savings ratio. Further research is required to provide better
predictions. The low reduction rate observed for larger sized circuits in
Table 2 may in part be due to the ineffectiveness of the classical
optimization strategy and the limited classical computational resources used,
which is further discussed in SM Sec. G.

Molecule | $n$ | NE | JW | BK | GT | Improve(%)
---|---|---|---|---|---|---
HF | 6 | 3 | 30 | 29 | 25 | 16.67
LiH | 6 | 3 | 30 | 29 | 25 | 16.67
BeH2 | 8 | 9 | 70 | 71 | 60 | 14.29
NH3 | 14 | 52 | 485 | 607 | 478 | 1.44
|  |  |  |  |  |
H2O(4) | 8 | 4 | 42 | 50 | 33 | 21.43
H2O(5) | 8 | 5 | 44 | 52 | 35 | 20.45
H2O(6) | 8 | 6 | 46 | 47 | 37 | 19.56
H2O(8) | 10 | 8 | 68 | 88 | 63 | 7.35
H2O(9) | 10 | 9 | 71 | 89 | 66 | 7.04
H2O(11) | 10 | 11 | 93 | 110 | 87 | 6.45
H2O(12) | 10 | 12 | 95 | 112 | 89 | 6.32
H2O(14) | 10 | 14 | 114 | 140 | 111 | 2.63
H2O(16) | 10 | 16 | 135 | 166 | 131 | 2.96
H2O(17) | 10 | 17 | 137 | 168 | 133 | 2.91
Table 2: Number of two-qubit gates required for the VQE simulation of
different molecules with different fermion to qubit transformations. $n$ is
the dimension of the $\beta$ matrix. NE is the number of excitation terms
considered in the UCCSD ansatz. JW/BK are the number of two-qubit gates with
JW or BK transformations. GT is the number of two-qubit gates given by the
best $\beta$ other than JW or BK. All molecules use the STO-3G basis, and
ansatz terms are determined according to the HMP2 ordering. (Top): Here, we
report the results for the cases where the HMP2 method was used to reach
chemical accuracy. (Bottom): Here, we report the results for the HMP2
progression for a water molecule. The number in the parentheses next to H2O
indicates the total number of excitation terms $(D_{\alpha})$ considered for
the UCCSD ansatz. The cutoff of H2O(17) is chosen based on the recovery of
chemical accuracy (see Fig. 4).

##  5 Discussion

So far in this paper, in the optimization of the FT-regime circuits, we have
considered efficient implementations of each Trotter term. It should however
be noted that a parallel implementation of multiple Trotter terms should also
be considered in the ${\textsc{r}}_{z}$ gate depth reduction of the quantum
simulation circuits. Based on our circuit construction detailed in Sec. 3, we
propose the following methodology for optimization over the parallel
implementation.

As discussed in Sec. 3, the circuit that implements the Trotter term (see Fig.
2) consists largely of three parts: an initial cnot gate network that computes
a linear function of Boolean input variables, a triply-controlled
${\textsc{r}}_{x}$ gate, and the inverse of the initial cnot gate network.
Denoting the Boolean variables at the input as, in the order from top to
bottom, $a$, $b$, $c$, and $d$, the cnot gate network outputs $a$, $b\oplus
c$, $a\oplus c$, and $a\oplus d$. The bottom three outputs remain invariant
over the action of the triply-controlled ${\textsc{r}}_{x}$ gate. This means
that any linear functions of $b\oplus c$, $a\oplus c$, and $a\oplus d$ are
accessible for use in the implementation of cnot gates that corresponds to the
JW $\sigma_{z}$ strings for other Trotter terms. This is so, because the
invariance is required to implement the appropriate inverse of the cnot gates
that were used to take the JW $\sigma_{z}$ strings into consideration in the
first place, as per our circuit construction shown in SM Fig. 10. Heuristic
methods that collect those Trotter terms and simultaneously implement them can
then be used to optimize the depth of the quantum circuit.

For the pre-FT regime VQE simulations, we have proposed a general framework
that leverages the predictive and corrective power of perturbation theory. The
predictive feature of our framework resembles that of the ADAPT-VQE method
[12], which aims to iteratively add terms to a generalized Trotterized
exponential ansatz by following the steepest descent based on the calculations
of numerical gradients. Despite trying to solve the same problem, our
framework adopts a different philosophy and computes an approximate
(perturbative) correction to the wavefunction and chooses the following
operator based on how much of a correction an operator could potentially
provide. Due to the two completely different approaches to the problem, the
two methods could have vastly different use cases for which they are suitable
respectively. To identify the difference in use cases, further thorough
investigation into different areas of applications are needed which is beyond
the scope of this paper.

Additionally, our framework is exceedingly flexible and provides means to
balance between resource constraints and accuracy requirement. First of all,
our framework not only works with the UCCSD ansatz, but any unitary ansatz,
including the generalized Trotterized exponential ansatz used in the ADAPT-VQE
method. Second, the predictive feature and the corrective feature can be used
independently from each other. For instance, the perturbative corrections can
be used along with the ADAPT-VQE method, which provides the ansatz term
prediction. Note that doing so, according to our framework, does not incur
additional cost in terms of circuit complexity; it incurs overhead only in
terms of the number of measurements. Our specific examples described in this
manuscript results in smaller number of measurements required, since both the
prediction and the correction are obtained by the same set of measurements, as
opposed to two different set of measurements that would otherwise be required,
had we combined ADAPT-VQE with our perturbative correction. Third, the choice
of the set of orbital sequence $\\{\alpha^{\prime}\\}$ in (9) can be leveraged
to balance between accuracy requirement and resource constraints. On one hand,
the size of the set can be decreased to reduce the number of measurements
needed. On the other hand, it is also possible to include orbital sequences
outside of the set $\\{\alpha\\}$ for the ansatz operator in order to achieve
better accuracy without sacrificing circuit complexity. For instance, one
could include triple excitation sequences in $\\{\alpha^{\prime}\\}$ along
with the UCCSD ansatz to provide a perturbative “T” correction to the final
ground state energy. In this case the increased accuracy does not correspond
to increased circuit complexity but only incurs overhead in terms of the
number of measurements.

We note that, extending and generalizing the framework used for considering
the bosonic terms in the JW transformation and the non-bosonic terms in the GT
transformation, the use of multiple fermion to qubit transformations $\beta$
for a given set of excitation terms could be of value in reducing the overall
resource requirement. Drawing from the fact that different $\beta$
transformations result in different resource requirements in implementing the
excitation terms of a target UCCSD ansatz circuit, it is reasonable to expect
that a certain subset of the excitation terms may be more efficiently
implemented by one $\beta$ transformation and some other excitation terms may
be implemented more efficiently by yet another $\beta$ transformation. Thus,
dividing the set of excitation terms required for preparing a UCCSD ansatz
state into subsets of excitation terms that may be more efficiently
implemented by respective, appropriate choices of $\beta$ transformations for
each of the subsets may prove to be more advantageous in the quantum resource
requirement. Optimizing over the tug of war expected between the overhead cost
incurred due to the switching of the transformations and the savings obtained
via the tailor-made choices of the transformations remains as future work.

##  6 Conclusion

Quantum simulations performed by quantum computers have long been thought to
be one of the most promising quantum applications that will prove advantageous
over classical computers. Despite the recent technological advancements made
by the community, there still remains a gap between what a quantum device can
realistically achieve and what is required to demonstrate a practical
advantage in running a quantum simulation on a quantum device. In an attempt
to bridge this gap, in this paper, we focused on optimizing the quantum
resource requirements for both the FT and pre-FT regime quantum simulations of
fermionic systems. Our FT-regime optimized approach yields a
${\textsc{r}}_{z}$-depth of one quantum circuit for each Trotterized evolution
operator with only two ${\textsc{r}}_{z}$ gates to implement in total, while
requiring just a single ancilla qubit. Our pre-FT regime optimized approach
leverages perturbation theory, capable of determining the best ansatz terms to
consider to enable a rapid convergence to the sought-after ground-state energy
of a simulated molecule that in turn results in the reduction of gate counts
required to reach a certain accuracy threshold. We also considered GT methods
to obtain significant quantum resource savings, upwards of 20% in some
instances, over the conventional JW or BK transformations in practice. While
these results were applied to a specific set of fermionic systems in this
paper as a concrete example, we expect similarly-spirited works could leverage
our methodologies. We believe the savings in the quantum resource requirement
demonstrated by our approaches help bring the day of solving practical
problems using a quantum computer closer.

## Acknowledgement

This work was supported, in part, by the DOE BES QIS Program (de-sc0019449).
The authors thank Diego Acosta and Germán Mauricio Muñoz at IonQ, Inc., for
their efforts in the implementation of computing systems used to generate the
data presented in this manuscript. The authors further thank Prof. Reinhold
Blümel for helpful discussion.

## References

  * Babbush et al. [2015] Ryan Babbush, Jarrod McClean, Dave Wecker, Alán Aspuru-Guzik, and Nathan Wiebe.  Chemical basis of trotter-suzuki errors in quantum chemistry simulation.  _Physical Review A_ , 91(2), Feb 2015.  ISSN 1094-1622.  doi: 10.1103/physreva.91.022311.
  * Bartlett et al. [1989] Rodney J. Bartlett, Stanislaw A. Kucharski, and Jozef Noga.  Alternative coupled-cluster ansätze ii. the unitary coupled-cluster method.  _Chemical Physics Letters_ , 155(1):133 – 140, 1989.  ISSN 0009-2614.  doi: 10.1016/S0009-2614(89)87372-5.
  * Berry et al. [2007] Dominic W. Berry, Graeme Ahokas, Richard Cleve, and Barry C. Sanders.  Efficient quantum algorithms for simulating sparse hamiltonians.  _Communications in Mathematical Physics_ , 270(2):359–371, 2007.  ISSN 1432-0916.  doi: 10.1007/s00220-006-0150-x.
  * Bocharov et al. [2015] Alex Bocharov, Martin Roetteler, and Krysta M. Svore.  Efficient synthesis of universal repeat-until-success quantum circuits.  _Physical Review Letters_ , 114:080502, Feb 2015.  doi: 10.1103/PhysRevLett.114.080502.
  * Bravyi and Kitaev [2002] Sergey B. Bravyi and Alexei Yu. Kitaev.  Fermionic quantum computation.  _Annals of Physics_ , 298(1):210 – 226, 2002\.  ISSN 0003-4916.  doi: 10.1006/aphy.2002.6254.
  * Childs et al. [2018] Andrew M. Childs, Dmitri Maslov, Yunseong Nam, Neil J. Ross, and Yuan Su.  Toward the first quantum simulation with quantum speedup.  _Proceedings of the National Academy of Sciences_ , 115(38):9456–9461, 2018.  ISSN 0027-8424.  doi: 10.1073/pnas.1801723115.
  * Cirac and Zoller [1995] J. I. Cirac and P. Zoller.  Quantum computations with cold trapped ions.  _Physical Review Letters_ , 74:4091–4094, May 1995.  doi: 10.1103/PhysRevLett.74.4091.
  * Dumitrescu et al. [2018] E. F. Dumitrescu, A. J. McCaskey, G. Hagen, G. R. Jansen, T. D. Morris, T. Papenbrock, R. C. Pooser, D. J. Dean, and P. Lougovski.  Cloud quantum computing of an atomic nucleus.  _Physical Review Letters_ , 120:210501, May 2018.  doi: 10.1103/PhysRevLett.120.210501.
  * Feynman [1982] Richard P. Feynman.  Simulating physics with computers.  _International Journal of Theoretical Physics_ , 21(6):467–488, 1982.  ISSN 1572-9575.  doi: 10.1007/BF02650179.
  * Gidney [2018] Craig Gidney.  Halving the cost of quantum addition.  _Quantum_ , 2:74, Jun 2018.  ISSN 2521-327X.  doi: 10.22331/q-2018-06-18-74.
  * Gokhale et al. [2019] Pranav Gokhale, Olivia Angiuli, Yongshan Ding, Kaiwen Gui, Teague Tomesh, Martin Suchara, Margaret Martonosi, and Frederic T Chong.  Minimizing state preparations in variational quantum eigensolver by partitioning into commuting families.  _arXiv preprint arXiv:1907.13623_ , 2019.
  * Grimsley et al. [2019] Harper R. Grimsley, Sophia E. Economou, Edwin Barnes, and Nicholas J. Mayhall.  An adaptive variational algorithm for exact molecular simulations on a quantum computer.  _Nature Communications_ , 10(1):3007, Jul 2019\.  ISSN 2041-1723.  doi: 10.1038/s41467-019-10988-2.
  * Grumbling and Horowitz [2019] Emily Grumbling and Mark Horowitz, editors.  _Quantum Computing: Progress and Prospects_.  The National Academies Press, Washington, DC, 2019.  ISBN 978-0-309-47969-1.  doi: 10.17226/25196.
  * Hastings et al. [2015] Matthew B. Hastings, Dave Wecker, Bela Bauer, and Matthias Troyer.  Improving quantum algorithms for quantum chemistry.  _Quantum Info. Comput._ , 15(1–2):1–21, January 2015.  ISSN 1533-7146.  doi: 10.5555/2685188.2685189.
  * Hempel et al. [2018] Cornelius Hempel, Christine Maier, Jonathan Romero, Jarrod McClean, Thomas Monz, Heng Shen, Petar Jurcevic, Ben P. Lanyon, Peter Love, Ryan Babbush, Alán Aspuru-Guzik, Rainer Blatt, and Christian F. Roos.  Quantum chemistry calculations on a trapped-ion quantum simulator.  _Physical Review X_ , 8:031022, Jul 2018.  doi: 10.1103/PhysRevX.8.031022.
  * Hoffmann and Simons [1988] Mark R. Hoffmann and Jack Simons.  A unitary multiconfigurational coupled-cluster method: Theory and applications.  _The Journal of Chemical Physics_ , 88(2):993–1002, 1988.  doi: 10.1063/1.454125.
  * Jiang et al. [2018] Zhang Jiang, Kevin J. Sung, Kostyantyn Kechedzhi, Vadim N. Smelyanskiy, and Sergio Boixo.  Quantum algorithms to simulate many-body physics of correlated fermions.  _Physical Review Applied_ , 9(4), Apr 2018.  ISSN 2331-7019.  doi: 10.1103/physrevapplied.9.044036.
  * Jordan and Wigner [1928] P. Jordan and E. Wigner.  Über das paulische äquivalenzverbot.  _Zeitschrift für Physik_ , 47(9):631–651, 1928.  ISSN 0044-3328.  doi: 10.1007/BF01331938.
  * Kandala et al. [2017] Abhinav Kandala, Antonio Mezzacapo, Kristan Temme, Maika Takita, Markus Brink, Jerry M. Chow, and Jay M. Gambetta.  Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets.  _Nature_ , 549(7671):242–246, 2017.  ISSN 1476-4687.  doi: 10.1038/nature23879.
  * Kennedy and Eberhart [1997] James Kennedy and Russell C Eberhart.  A discrete binary version of the particle swarm algorithm.  In _1997 IEEE International conference on systems, man, and cybernetics. Computational cybernetics and simulation_ , volume 5, pages 4104–4108. IEEE, 1997.  doi: 10.1109/ICSMC.1997.637339.
  * Kivlichan et al. [2020] Ian D. Kivlichan, Craig Gidney, Dominic W. Berry, Nathan Wiebe, Jarrod McClean, Wei Sun, Zhang Jiang, Nicholas Rubin, Austin Fowler, Alán Aspuru-Guzik, and et al.  Improved fault-tolerant quantum simulation of condensed-phase correlated electrons via trotterization.  _Quantum_ , 4:296, Jul 2020.  ISSN 2521-327X.  doi: 10.22331/q-2020-07-16-296.
  * Lee et al. [2019] Joonho Lee, William J. Huggins, Martin Head-Gordon, and K. Birgitta Whaley.  Generalized unitary coupled cluster wave functions for quantum computation.  _Journal of Chemical Theory and Computation_ , 15(1):311–324, Jan 2019.  ISSN 1549-9618.  doi: 10.1021/acs.jctc.8b01004.
  * Lloyd [1996] Seth Lloyd.  Universal quantum simulators.  _Science_ , 273(5278):1073–1078, 1996.  ISSN 0036-8075.  doi: 10.1126/science.273.5278.1073.
  * Low and Chuang [2017] Guang Hao Low and Isaac L. Chuang.  Optimal hamiltonian simulation by quantum signal processing.  _Physical Review Letters_ , 118:010501, Jan 2017.  doi: 10.1103/PhysRevLett.118.010501.
  * Low and Chuang [2019] Guang Hao Low and Isaac L. Chuang.  Hamiltonian Simulation by Qubitization.  _Quantum_ , 3:163, July 2019.  ISSN 2521-327X.  doi: 10.22331/q-2019-07-12-163.
  * Maslov et al. [2019] D. Maslov, Y. Nam, and J. Kim.  An outlook for quantum computing [point of view].  _Proceedings of the IEEE_ , 107(1):5–10, Jan 2019\.  ISSN 1558-2256.  doi: 10.1109/JPROC.2018.2884353.
  * Maslov [2016] Dmitri Maslov.  Advantages of using relative-phase toffoli gates with an application to multiple control toffoli optimization.  _Physical Review A_ , 93:022311, Feb 2016.  doi: 10.1103/PhysRevA.93.022311.
  * Maslov and Nam [2018] Dmitri Maslov and Yunseong Nam.  Use of global interactions in efficient quantum circuit constructions.  _New Journal of Physics_ , 20(3):033018, mar 2018\.  doi: 10.1088/1367-2630/aaa398.
  * Nam and Maslov [2019] Yunseong Nam and Dmitri Maslov.  Low-cost quantum circuits for classically intractable instances of the hamiltonian dynamics simulation problem.  _npj Quantum Information_ , 5(1):44, 2019.  ISSN 2056-6387.  doi: 10.1038/s41534-019-0152-0.
  * Nam et al. [2018] Yunseong Nam, Neil J. Ross, Yuan Su, Andrew M. Childs, and Dmitri Maslov.  Automated optimization of large quantum circuits with continuous parameters.  _npj Quantum Information_ , 4(1):23, 2018.  ISSN 2056-6387.  doi: 10.1038/s41534-018-0072-4.
  * Nam et al. [2020a] Yunseong Nam, Jwo-Sy Chen, Neal C Pisenti, Kenneth Wright, Conor Delaney, Dmitri Maslov, Kenneth R Brown, Stewart Allen, Jason M Amini, Joel Apisdorf, Aleksey Beck, Kristin M. Blinov, Vandiver Chaplin, Mika Chmielewski, Coleman Collins, Shantanu Debnath, Andrew M. Ducore, Kai M. Hudek, Matthew Keesan, Sarah M. Kreikemeier, Jonathan Mizrahi, Phil Solomon, Mike Williams, Jaime David Wong-Campos, Christopher Monroe, and Jungsang Kim.  Ground-state energy estimation of the water molecule on a trapped-ion quantum computer.  _npj Quantum Information_ , 6(1):1–6, 2020a.  doi: 10.1038/s41534-020-0259-3.
  * Nam et al. [2020b] Yunseong Nam, Yuan Su, and Dmitri Maslov.  Approximate quantum fourier transform with o(n log(n)) t gates.  _npj Quantum Information_ , 6(1), Mar 2020b.  ISSN 2056-6387.  doi: 10.1038/s41534-020-0257-5.
  * Nielsen and Chuang [2000] Michael A. Nielsen and Isaac L. Chuang.  _Quantum Computation and Quantum Information_.  Cambridge University Press, 2000.  doi: 10.1017/CBO9780511976667.
  * O’Malley et al. [2016] P. J. J. O’Malley, R. Babbush, I. D. Kivlichan, J. Romero, J. R. McClean, R. Barends, J. Kelly, P. Roushan, A. Tranter, N. Ding, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, A. G. Fowler, E. Jeffrey, E. Lucero, A. Megrant, J. Y. Mutus, M. Neeley, C. Neill, C. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White, P. V. Coveney, P. J. Love, H. Neven, A. Aspuru-Guzik, and J. M. Martinis.  Scalable quantum simulation of molecular energies.  _Physical Review X_ , 6:031007, Jul 2016.  doi: 10.1103/PhysRevX.6.031007.
  * Parrish et al. [2017] Robert M. Parrish, Lori A. Burns, Daniel G. A. Smith, Andrew C. Simmonett, A. Eugene DePrince, Edward G. Hohenstein, Uğur Bozkaya, Alexander Yu. Sokolov, Roberto Di Remigio, Ryan M. Richard, Jérôme F. Gonthier, Andrew M. James, Harley R. McAlexander, Ashutosh Kumar, Masaaki Saitow, Xiao Wang, Benjamin P. Pritchard, Prakash Verma, Henry F. Schaefer, Konrad Patkowski, Rollin A. King, Edward F. Valeev, Francesco A. Evangelista, Justin M. Turney, T. Daniel Crawford, and C. David Sherrill.  Psi4 1.1: An open-source electronic structure program emphasizing automation, advanced libraries, and interoperability.  _Journal of Chemical Theory and Computation_ , 13(7):3185–3197, 2017.  doi: 10.1021/acs.jctc.7b00174.
  * Patel et al. [2008] Ketan N. Patel, Igor L. Markov, and John P. Hayes.  Optimal synthesis of linear reversible circuits.  _Quantum Info. Comput._ , 8(3):282–294, March 2008.  ISSN 1533-7146.  doi: 10.5555/2011763.2011767.
  * Peruzzo et al. [2014] Alberto Peruzzo, Jarrod McClean, Peter Shadbolt, Man-Hong Yung, Xiao-Qi Zhou, Peter J. Love, Alán Aspuru-Guzik, and Jeremy L. O’Brien.  A variational eigenvalue solver on a photonic quantum processor.  _Nature Communications_ , 5(1):4213, 2014.  ISSN 2041-1723.  doi: 10.1038/ncomms5213.
  * Poulin et al. [2015] David Poulin, Matthew B. Hastings, Dave Wecker, Nathan Wiebe, Andrew C. Doberty, and Matthias Troyer.  The trotter step size required for accurate quantum simulation of quantum chemistry.  _Quantum Info. Comput._ , 15(5–6):361–384, April 2015.  ISSN 1533-7146.  doi: 10.5555/2871401.2871402.
  * Reiher et al. [2017] Markus Reiher, Nathan Wiebe, Krysta M. Svore, Dave Wecker, and Matthias Troyer.  Elucidating reaction mechanisms on quantum computers.  _Proceedings of the National Academy of Sciences_ , 2017.  ISSN 0027-8424.  doi: 10.1073/pnas.1619152114.
  * Romero et al. [2018] Jonathan Romero, Ryan Babbush, Jarrod R McClean, Cornelius Hempel, Peter J Love, and Alán Aspuru-Guzik.  Strategies for quantum computing molecular energies using the unitary coupled cluster ansatz.  _Quantum Science and Technology_ , 4(1):014008, oct 2018.  doi: 10.1088/2058-9565/aad3e4.
  * Seeley et al. [2012] Jacob T. Seeley, Martin J. Richard, and Peter J. Love.  The bravyi-kitaev transformation for quantum computation of electronic structure.  _The Journal of Chemical Physics_ , 137(22):224109, 2012.  doi: 10.1063/1.4768229.
  * Shehab et al. [2019] O. Shehab, K. Landsman, Y. Nam, D. Zhu, N. M. Linke, M. Keesan, R. C. Pooser, and C. Monroe.  Toward convergence of effective-field-theory simulations on digital quantum computers.  _Physical Review A_ , 100:062319, Dec 2019.  doi: 10.1103/PhysRevA.100.062319.
  * Steudtner and Wehner [2018] Mark Steudtner and Stephanie Wehner.  Fermion-to-qubit mappings with varying resource requirements for quantum simulation.  _New Journal of Physics_ , 20(6):063010, jun 2018\.  doi: 10.1088/1367-2630/aac54f.
  * Suzuki [1991] Masuo Suzuki.  General theory of fractal path integrals with applications to many-body theories and statistical physics.  _Journal of Mathematical Physics_ , 32(2):400–407, 1991.  doi: 10.1063/1.529425.
  * Tranter et al. [2019] Andrew Tranter, Peter J. Love, Florian Mintert, Nathan Wiebe, and Peter V. Coveney.  Ordering of trotterization: Impact on errors in quantum simulation of electronic structure.  _Entropy_ , 21(12):1218, Dec 2019.  ISSN 1099-4300.  doi: 10.3390/e21121218.
  * Wecker et al. [2014] Dave Wecker, Bela Bauer, Bryan K. Clark, Matthew B. Hastings, and Matthias Troyer.  Gate-count estimates for performing quantum chemistry on small quantum computers.  _Physical Review A_ , 90(2), Aug 2014.  ISSN 1094-1622.  doi: 10.1103/physreva.90.022305.
  * Whitfield et al. [2011] James D. Whitfield, Jacob Biamonte, and Alán Aspuru-Guzik.  Simulation of electronic structure hamiltonians using quantum computers.  _Molecular Physics_ , 109(5):735–750, 2011.  doi: 10.1080/00268976.2011.552441.
  * Zhu et al. [1997] Ciyou Zhu, Richard H Byrd, Peihuang Lu, and Jorge Nocedal.  Algorithm 778: L-bfgs-b: Fortran subroutines for large-scale bound-constrained optimization.  _ACM Transactions on mathematical software (TOMS)_ , 23(4):550–560, 1997.  doi: 10.1145/279232.279236.

## SUPPLEMENTARY MATERIAL

##  Appendix A Circuits for the fault-tolerant regime

In this section, we show in detail the circuit implementation of the Trotter
terms, typically encountered in quantum chemistry simulations, optimized for
the FT regime. First, we show how to implement a triply-controlled
${\textsc{r}}_{x}$ gate, needed for our two-body term circuit shown in Fig. 2
of the main text. Then, we compare the resulting quantum resource requirements
of our method against those of the method shown in [39], which can be
optimized by combining with the work discussed in [10, 29], in terms of
circuit size, width, and depth.

Fig. 6(a) shows an ${\textsc{r}}_{z}$\- and t-gate optimized implementation of
a triply-controlled ${\textsc{r}}_{x}$ gate. We use the so-called relative
phase Toffoli gate investigated in [27] instead of a conventional Toffoli gate
to reduce the t gate count, excluding that required for the ${\textsc{r}}_{z}$
gates, to 16. An additional advantage is that a relative phase Toffoli gate
with three controls does not require an ancilla qubit, whereas a conventional
triply-controlled Toffoli gate, decomposed into the standard gate set of cnot,
single-qubit Clifford, and t gates, requires at least one ancilla. For
completeness, we show in Fig. 7 the explicit construction of the triply-
controlled, relative phase Toffoli gate, imported verbatim from [27], although
other methods, sometimes with a different gate set, have been reported
elsewhere [7, 28].

Inspecting Fig. 6(a), we notice that we need to apply the two
${\textsc{r}}_{z}$ gates in series. To reduce the ${\textsc{r}}_{z}$ depth
then, we can introduce an ancilla qubit prepared to
${\left|{0}\right\rangle}$. An ${\textsc{r}}_{z}$-depth one construction of
the triply-controlled ${\textsc{r}}_{x}$ gate is shown in Fig. 6(b). Should
the same angle rotations happen in multiple qubits at the same time, a
possibility if a molecule of interest that is being simulated has symmetries,
a weight-sum trick reported and used in [10, 29] can be employed to
exponentially reduce the number of ${\textsc{r}}_{z}$ gates at a modest cost
in the number of qubits and t gates. Two ${\textsc{r}}_{z}$ gates to be
applied in parallel however do not benefit from this trick.

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