1. Introduction
One of the important motivations for developing quantum computers is their potential to simulate strongly correlated many-body systems efficiently [
1,
2]. Algorithms that exactly diagonalize the electronic Hamiltonian, known as the full configuration interaction approach, scale exponentially with the size of the Hilbert space, making it applicable to very few cases [
3] on classical computers. The configuration interaction (CI) method offers an approximate solution by truncating the Hilbert space to only include the most important basis states. However, the energy calculated by the CI method does not scale properly with the size of the system when used on molecules with varying sizes, nor does it predict the dissociation energy correctly because it cannot produce factorized atomic states. The coupled-cluster (CC) method addresses these issues by being both size consistent and size extensive. Size consistency means that the method would yield the same energy of two particles separated by an infinite distance as the sum of the energies calculated individually. Size extensivity means the energy scales linearly with the number of particles for a homogeneous system [
4]. The CC method is also memory efficient because it does not explicitly construct the energy eigenstate. Instead, the set of amplitudes for the CC ansatz is calculated iteratively by the so-called amplitude equations [
4,
5,
6], which correspond to zeroing out the row (or column) of the similarity-transformed Hamiltonian matrix that corresponds to the initial single-reference state. The CC method with single, double and (perturbative) triple excitations is regarded as the “gold standard” for computational chemistry [
7].
Quantum computers have been proposed as being capable of solving a set of quantum chemistry problems that are otherwise difficult or very challenging on classical machines: namely, molecules that contain both weakly and strongly correlated electrons. One of the most promising algorithms for the noisy intermediate-scale quantum (NISQ) era is the variational quantum eigensolver (VQE), where the trial wave function is prepared on the quantum hardware and the expectation value of the Hamiltonian is measured there as well; the parameters in the eigenstate are optimized variationally on classical machines [
8,
9]. The conventional coupled-cluster ansatz is given as
, where
is a trial wave function (often chosen to be the single-reference Hartree–Fock state), and
is the cluster operator consisting of up to rank-
n excitations (
n electrons are removed from the Hartree–Fock state and replaced by
n electrons in virtual orbitals). The excitation operator is given as
and
, where
is the creation operator acting on virtual orbital
a and
is the annihilation operator acting on occupied orbital
i. Traditionally, the CC method employs a similarity-transformed Hamiltonian to obtain a set of equations to determine the amplitudes
t:
where
, and
is a set of states that covers the entire space generated by
acting on the reference state [
5,
10]. In practice, this set of amplitude equations is solved iteratively, which yields the energy without needing to construct the energy eigenstate. The total number of amplitude equations is given by the number of amplitudes in the expansion of the
operator, which is much smaller than the total number of Slater determinants in the
(which is typically exponentially larger). The properties of size consistency and size extensivity for the CC ansatz stem from the facts that the similarity-transformed Hamiltonian
is additively separable and the term
is multiplicatively separable. Notice that the electronic Hamiltonian for the molecule (in second quantization) is given by
where
are the one-electron integrals and
are the two-electron integrals:
Here,
M is the number of atoms in the system,
are their atomic numbers,
,
, and
are the single-particle optimized orbitals from the HF solution [
11,
12]. In order to solve the amplitude Equations (
2) and (
3), we need to explicitly compute the similarity-transformed Hamiltonian. Using the Hadamard lemma (also called the Baker–Campbell–Hausdorff expansion or the Baker–Campbell formula), we can rewrite the transformed Hamiltonian as
Conveniently, the series truncates at the fourth order due to the Hamiltonian having only one- and two-body interaction terms [
5,
13] and the excitations always being from real to virtual orbitals. Traditionally, this projective method to determine the CC amplitudes is preferred over variational methods due to the non-unitarity of the
operator [
5,
7].
Despite its success, the lack of unitarity prevents the CC operators from being implemented on quantum computers. This suggests using the unitary coupled-cluster ansatz (UCC), whose cluster operator now includes the excitation minus the de-excitation operator
[
14,
15]. Similar to the CC approximation, only the low-rank cluster operators such as singles and doubles are usually selected for the variational eigenstate ansatz, but for more strongly correlated systems, one expects that higher-rank factors will also be needed. In practice, a projective method like the one used in the CC calculation does not work with the UCC ansatz because the similarity-transformed Hamiltonian no longer truncates after the fourth term. Common strategies for carrying it out on classical computers include truncating the Hadamard lemma series at a fixed order [
14], expanding the exponential operator in a power series and then truncating it when the higher-rank terms no longer change the eigenfunction [
16], and using an exact operator identity of the factorized form of the UCC to allow the wavefunction to be constructed in a tree structure [
17]. But, there exists no simple method to work directly with the UCC ansatz in its original form. Since we are working with non-commuting fermionic operators
in the exponent, one common way to decompose such a function is to adopt a Trotter product formula:
Another useful method is to express the ansatz in a factorized form, given by
which corresponds to the first-order approximation of the Trotter product formula in Equation (
8). The benefit of only using the
extreme case is two-fold: the quantum resources required to prepare the factorized UCC ansatz are much smaller than higher-order approximations and the Trotter errors of the first-order approximation can be ameliorated by the fact that the calculation is variational [
18,
19]. Within the classical computational chemistry framework, work by Chen et al. [
17] created an algorithm using the factorized form of the UCC that produces significantly better results for strongly correlated systems and comparable results in terms of accuracy for weakly correlated systems. In this work, we show how one can create efficient implementation of these UCC factors using the linear combination of unitaries approach. For high-rank factors, this approach is preferable.
To implement the factorized UCC ansatz on quantum computers, one needs to transform the cluster operators
expressed in the fermionic language into a spin language (via the Jordan–Wigner transformation, or other fermionic encodings). A common realization of this approach is to exactly simulate the individual exponentials of Pauli strings found after the JW transformation of
[
18,
19]. For example, a Jordan-Wigner transformed rank-2 UCC factor is given as
Such a UCC factor can be rewritten as a product of exponentials of Pauli strings because the Pauli strings in the exponentials all commute [
18,
20]. A common strategy for creating circuits of the form
is to use basis transformations where one starts with the circuit for evaluating
and then apply basis transformations to evaluate the exponential of any Pauli string [
21]. In order to evaluate a generic Pauli string, a basis transformation can be applied before the CNOT cascades such that the effective Pauli string is that of only
Z’s. If the
ith gate in the Pauli string is an
X, a Hadamard gate is sandwiched around the CNOT cascade on the
ith qubit. This leads to the effective exponential containing a
Z since
.
Figure 1 shows an example circuit to apply
. Similarly, if an exponentiated
Y gate is applied, a
gate is sandwiched around the CNOT cascade.
Figure 2 shows an example circuit to apply Equation (
10). In applying the UCC ansatz, circuits such as
Figure 2 must be re-run
times after applying all of the
different basis transformations [
18]. A general factorized doubles UCC operator can be rewritten as Equation (
10) and implemented exactly by the circuit shown in
Figure 2.
Figure 1.
Example of a circuit implementing for four qubits. To apply the X on a different qubit, Hadamard gates can be sandwiched around the respective qubits.
Figure 1.
Example of a circuit implementing for four qubits. To apply the X on a different qubit, Hadamard gates can be sandwiched around the respective qubits.
Figure 2.
Doubles UCC circuit as discussed in Refs. [
18,
19]. For a general doubles operator, the circuit must be applied eight times, with different combinations of
U gates each time. The
U-gate choices are summarized in
Table 1. The dashed CNOT gates are part of a CNOT cascade.
Figure 2.
Doubles UCC circuit as discussed in Refs. [
18,
19]. For a general doubles operator, the circuit must be applied eight times, with different combinations of
U gates each time. The
U-gate choices are summarized in
Table 1. The dashed CNOT gates are part of a CNOT cascade.
This is possible because the different
Pauli strings (for a rank-
n UCC factor) commute with each other. In our previous work, we found a way of reducing the number of control-NOT (
Cnot) gates in quantum circuits for the factorized UCC ansatz by introducing extra ancilla qubits [
20], with the largest reductions for the higher-rank factors. Alternatively, a factorization method introduced in Ref. [
22] uses a two-step low-rank factorization to approximate the UCC operator. Circuits that implement the
Select(
) subroutine for more general Jordan–Wigner strings with linear scaling have also been developed [
23]. Another framework developed in Ref. [
24] optimizes the two-qubit gates by bootstrapping the VQE iterations towards the convergence of the systems ground state energy. A quantum software developed in Ref. [
25] optimizes the number of two-qubit gates systematically by compressing adjacent
and Pauli operators using a set of rules from the
-calculus. Other works have proposed different efficient methods to simulate the UCC factors [
26], whose main idea is to directly implement the SU(2) identity of the UCC factors presented in Equation (
22) by exchanging coefficients between the two active states. In this work, we introduce a method to directly simulate the sum of terms obtained from a hidden SU(2) symmetry of the first-order Trotter product that greatly reduces the number of multi-qubit entanglement gates of factorized UCC circuits.