Abstract
Molecular simulations with the variational quantum eigensolver (VQE) are a promising application for emerging noisy intermediatescale quantum computers. Constructing accurate molecular ansätze that are easy to optimize and implemented by shallow quantum circuits is crucial for the successful implementation of such simulations. Ansätze are, generally, constructed as series of fermionicexcitation evolutions. Instead, we demonstrate the usefulness of constructing ansätze with "qubitexcitation evolutions”, which, contrary to fermionic excitation evolutions, obey "qubit commutation relations”. We show that qubit excitation evolutions, despite the lack of some of the physical features of fermionic excitation evolutions, accurately construct ansätze, while requiring asymptotically fewer gates. Utilizing qubit excitation evolutions, we introduce the qubitexcitationbased adaptive (QEBADAPT)VQE protocol. The QEBADAPTVQE is a modification of the ADAPTVQE that performs molecular simulations using a problemtailored ansatz, grown iteratively by appending evolutions of qubit excitation operators. By performing classical numerical simulations for small molecules, we benchmark the QEBADAPTVQE, and compare it against the original fermionicADAPTVQE and the qubitADAPTVQE. In terms of circuit efficiency and convergence speed, we demonstrate that the QEBADAPTVQE outperforms the qubitADAPTVQE, which to our knowledge was the previous most circuitefficient scalable VQE protocol for molecular simulations.
Introduction
Quantum computers are anticipated to enable simulations of quantum systems more efficiently and accurately than classical computers^{1,2}. A promising algorithm to perform this task on emerging noisy intermediatescale quantum (NISQ)^{3,4,5} computers is the variational quantum eigensolver (VQE)^{6,7,8,9,10,11,12}. The VQE is a hybrid quantumclassical algorithm that estimates the lowest eigenvalue of a Hamiltonian H by minimizing the energy expectation value E(θ) = 〈ψ(θ)∣H∣ψ(θ)〉 with respect to a parametrized state \(\left\psi (\boldsymbol\theta )\right\rangle =U(\boldsymbol\theta )\left{\psi }_{0}\right\rangle\). Here, θ is a set of variational parameters, and the unitary U(θ) is an ansatz. Compared to other purely quantum algorithms for eigenvalue estimation, like the quantumphase estimation algorithm^{13,14}, the VQE requires shallower quantum circuits. This makes the VQE more noise resistant, at the expense of requiring a higher number of quantum measurements and additional classical postprocessing.
The VQE can solve the electronic structure problem^{6,15} by estimating the lowest eigenvalue of an electronic Hamiltonian. A major challenge for the practical realization of a molecular VQE simulation on NISQ computers is to construct a variationally flexible ansatz U(θ) that: (1) accurately approximates the ground state of H; (2) is easy to optimize; and (3) can be implemented by a shallow circuit.
These desired qualities are satisfied, to various levels, by several types of ansätze. The unitary coupledcluster (UCC) type, was the first to be used for molecular VQE simulations^{16}. The UCC is motivated by the classical coupledcluster theory^{15}, and corresponds to a series of unitary evolutions of fermionicexcitation operators, which we refer to as “fermionic excitation evolutions” (see the section “Ansatz elements”). A prominent example of a UCC ansatz is the UCC Singles and Doubles (UCCSD)^{17,18,19,20,21,22}, which corresponds to a series of single and doublefermionicexcitation evolutions. The UCCSD has been used successfully to implement the VQE for small molecules^{16,17,23}. Due to their physically motivated fermionic structure, UCC ansätze respects the symmetries of electronic wavefunctions, which makes these ansätze accurate and easy to optimize. Even a relatively simple UCC ansatz, like the UCCSD, is highly accurate for weakly correlated systems, such as molecules near their equilibrium configuration^{16,17,24,25}. However, UCC ansätze are generalpurpose built and do not take into account details of the system of interest. They contain redundant excitation terms, resulting in unnecessarily high numbers of variational parameters as well as long ansatz circuits. Moreover, to simulate strongly correlated systems, UCC ansätze requires higherorder excitations and/or multiplestep Trotterization^{24}, which creates additional overhead for the quantum hardware.
Another type of “hardwareefficient” ansätze^{26,27,28,29,30} corresponds to universal unitary transformations implemented as periodic sequences of parametrized one and twoqubit gates. These ansätze are implemented by shallow circuits, and can be highly variationally flexible. However, as they lack a physically motivated structure, these ansätze require a large number of variational parameters and may suffer by vanishing energy gradients along their variational parameters, making classical optimization intractable for large molecules^{31,32}. In some scenarios, this is known as the barren plateau problem^{32,33,34,35}.
Recently, a number of works^{36,37,38,39,40,41,42,43,44} suggested new “iterative” VQE protocols, which instead of using generalpurpose, fixed ansätze, construct problemtailored ansätze on the go. These algorithms can construct arbitrarily accurate ansätze that are optimized in the number of variational parameters and the ansatzcircuit depth, at the expense of requiring a larger number of quantum computer measurements. The ADAPTVQE protocols^{36,37} are perhaps the most prominent family of iterative VQE protocols. The fermionicADAPTVQE^{36}, which was the first iterative VQE protocol, constructs its ansatz by iteratively appending parametrized unitary operators, which we refer to as “ansatz elements”. The ansatz element at each iteration is sampled from a pool of spincomplement single and doublefermionicexcitation evolutions, based on an energygradient hierarchy. The fermionicADAPTVQE was demonstrated to achieve chemical accuracy (10^{−3} Hartree), using an ansatz with several times fewer variational parameters, and a correspondingly shallower circuit, than the UCCSD. In the followup work^{37}, the qubitADAPTVQE utilizes an ansatzelement pool of more variationally flexible and rudimentary Pauli string exponentials. Due to this, the qubitADAPTVQE constructs even shallower ansatz circuits than the fermionicADAPTVQE, thus being, to the best of our knowledge, the currently most circuitefficient, physically motivated VQE algorithm. However, the use of more rudimentary unitary operations comes at the expense of requiring additional variational parameters and iterations to construct an ansatz for a given accuracy.
In this work, we utilize unitary operations that, despite the lack of some of the physical features captured by fermionicexcitation evolutions, achieve the accuracy of fermionic excitations evolutions as well as the hardware efficiency of Pauli string exponentials. These operations can be used to construct circuitefficient molecular ansätze without incurring as many additional variational parameters and iterations, as the qubitADAPTVQE. We call these unitary operations “qubit excitation evolutions”. Qubitexcitation evolutions^{23,45,46,47} are unitary evolutions of “qubit excitation operators”, which satisfy “qubit commutation relations”^{46,47}. Qubitexcitation evolutions can be implemented by circuits that act on fixed numbers of qubits, as opposed to fermionicexcitation evolutions, which act on a number of qubits that scale at least as \(O({{{{{{{\mathrm{log}}}}}}}\,}_{2}{N}_{{{{{{{{\rm{MO}}}}}}}}})\) with the number of considered molecular spin orbitals N_{MO}. We show numerically that qubitexcitation evolutions can approximate an electronic wavefunction almost as accurately as fermionicexcitation evolutions can. On the other hand, qubitexcitation evolutions enjoy higher complexity than Pauli string exponentials, thus allowing for more rapid construction of the ansatz. We utilize qubitexcitation evolutions to introduce the qubitexcitationbased adaptive variational quantum eigensolver (QEBADAPTVQE) protocol. As the name suggests, the QEBADAPTVQE is an ADAPTVQE protocol for molecular simulations that grows a problemtailored ansatz from an ansatzelement pool of qubitexcitation evolutions. The QEBADAPTVQE also features a modified ansatzgrowing strategy, which allows for a more efficient ansatz construction at the expense of a constantfactor increase of quantum computer measurement. We benchmark the performance of the QEBADAPTVQE with classical numerical simulations for small molecules: LiH, H_{6}, and BeH_{2}. In the section “Energy dissociation curves”, we compare the QEBADAPTVQE to the standard UCCSDVQE by presenting energydissociation curves obtained with each of the two methods. In the section “Energy convergence”, we compare the QEBADAPTVQE to the fermionicADAPTVQE and to the qubitADAPTVQE by presenting energy convergence plots, obtained with each of the three ADAPTVQE protocols.
Results
Theoretical background and notation
We begin with a theoretical introduction (required for the selfcompleteness of the paper) and by defining our notation. Finding the groundstate electron wavefunction \(\left{E}_{0}\right\rangle\) and corresponding energy E_{0} of a molecule (or an atom) is known as the “electronic structure problem”^{15}. This problem can be solved by solving the timeindependent Schrödinger equation \(H\left{{{\Phi }}}_{0}\right\rangle ={E}_{0}\left{{{\Phi }}}_{0}\right\rangle\), where H is the electronic Hamiltonian of the molecule. Within the Born–Oppenheimer approximation, where the nuclei of the molecule are treated as motionless, H can be secondly quantized as
As already mentioned, N_{MO} is the number of molecular spin orbitals, \({a}_{i}^{{{{\dagger}}} }\) and a_{i} are fermionic creation and annihilation operators, corresponding to the ith molecular spin orbital, and the factors h_{ij} and h_{ijkl} are one and twoelectron integrals, written in a spinorbital basis^{15}. The Hamiltonian expression in Eq. (1) can be mapped to quantumgate operators using an encoding method, e.g., the Jordan–Wigner^{48} or the Bravyi–Kitaev^{49} methods. Throughout this work, we assume the more straightforward Jordan–Wigner encoding, where the occupancy of the ith molecular spin orbital is represented by the state of the ith qubit.
The fermionic operators \({a}_{i}^{{{{\dagger}}} }\) and a_{i} satisfy anticommutation relations
Within the Jordan–Wigner encoding, \({a}_{i}^{{{{\dagger}}} }\) and a_{i} can be written in terms of quantumgate operators as
where
We refer to \({Q}_{i}^{{{{\dagger}}} }\) and Q_{i} as qubit creation and annihilation operators, respectively. They act to change the occupancy of spinorbital i. The Pauliz strings, in Eqs. (3) and (4), compute the parity of the state and act as exchange phase factors that account for the fermionic anticommutation of a^{†} and a. Substituting Eqs. (3) and (4) into Eq. (1), H can be written as
where σ_{s} is a Pauli operator (X_{s}, Y_{s}, Z_{s}, or I_{s}) acting on qubit s, and h_{r} (not to be confused with h_{ik} and h_{ijkl}) is a real scalar coefficient. The number of terms in EQ. (7) scales as \(O({N}_{{{{{{{{\rm{MO}}}}}}}}}^{4})\).
Once H is mapped to a Pauli operator representation, the VQE can be used to minimize the expectation value E(θ) = 〈ψ(θ)∣H∣ψ(θ)〉. The VQE relies upon the Rayleigh–Ritz variational principle
to find an estimate for E_{0}. The VQE is a hybrid quantumclassical algorithm that uses a quantum computer to prepare the trial state \(\left\psi (\boldsymbol\theta )\right\rangle\) and evaluate E(θ), and a classical computer to process the measurement data and update θ at each iteration. The trial state \(\left\psi (\boldsymbol\theta )\right\rangle =U(\boldsymbol\theta )\left{\psi }_{0}\right\rangle\) is generated by an ansatz, U(θ), applied to an initial reference state \(\left{\psi }_{0}\right\rangle\).
The ADAPTVQE protocols
The ADAPTVQE protocols iteratively construct problemtailored ansätze on the go. At the mth iteration one or several unitary operators, \(\{{U}_{r}^{(m)}({\theta }_{r}^{(m)})\}\), which we refer to as ansatz elements, are appended to the left of the already existing ansatz, U(θ^{(m−1)}):
The ansatz elements, \(\left\{{U}_{r}^{(m)}({\theta }_{r}^{(m)})\right\}\), at each iteration, are chosen from a finite ansatzelement pool \({\mathbb{P}}\), based on an ansatzgrowing strategy that aims to achieve the lowest estimate of E(θ^{(m)}). After a new ansatz U(θ^{(m)}) is constructed, the new set of variational parameters \({\boldsymbol\theta }^{(m)}={\boldsymbol\theta }^{(m1)}\cup \left\{{\theta }_{r}^{(m)}\right\}\) is optimized by the VQE, and a new estimate for E(θ^{(m)}) is obtained. This iterative greedy strategy results in an ansatz that is tuned specifically to the system being simulated, and can approximate the ground eigenstate of the system with considerably fewer variational parameters and a shallower ansatz circuit, than generalpurpose fixed ansätze, like the UCCSD.
In the fermionicADAPTVQE, the ansatzelement pool \({\mathbb{P}}\) is a set of spincomplement pairs of single and doublefermionicexcitation evolutions. In the qubitADAPTVQE, \({\mathbb{P}}\) is a set of parametrized exponentials of XYPauli strings. The growth strategy of the fermionicADAPTVQE and the qubitADAPTVQE is to add, at each iteration, the ansatz element with the largest energygradient magnitude
where \(\left{\psi }^{(m1)}\right\rangle\) is the trial state at the end of the (m−1)th iteration. For detailed descriptions of the fermionicADAPTVQE and the qubitADAPTVQE, we refer the reader to refs. ^{36} and ^{37}, respectively.
Ansatz elements
Single and doublefermionicexcitation evolutions can construct an ansatz that approximates an electronic wavefuction to an arbitrary accuracy^{50,51}. Single and doublefermionicexcitation operators are defined, respectively, by the skewHermitian operators
Single and doublefermionicexcitation evolutions are thus given, respectively, by the unitaries
Using Eqs. (3) and (4), for i < j < k < l, A_{ik} and A_{ijkl} can be expressed in terms of quantumgate operators as
As seen from Eqs. (14) and (15), fermionicexcitation evolutions act on a number of qubits that scales as O(N_{MO}). Therefore, they are implemented by circuits whose size (in terms of number of CNOTs) also scales as O(N_{MO}). We derived a CNOTefficient method to construct circuits for fermionic excitations evolutions in ref. ^{47}. The circuits for a single and doublefermionicexcitation evolution have CNOT counts of 2(k − i) + 1 and 2(l + j − i − k) + 9, respectively.
Qubitexcitation operators are defined by the qubit annihilation and creation operators, Q_{i} and \({Q}_{i}^{{{{\dagger}}} }\) (Eqs. (5) and (6)), which satisfy the qubitcommutation relations
Some authors have referred to these commutation relations as parafermionic^{46}. Single and doublequbitexcitation operators are given, respectively, by the skewHermitian operators
Thus, single and doublequbitexcitation evolutions are given, respectively, by the unitary operators
Using Eqs. (5) and (6), \({\tilde{A}}_{ik}\) and \({\tilde{A}}_{ijkl}\) can be reexpressed in terms of quantumgate operators as
As seen from Eqs. (21) and (22), unlike fermionicexcitation evolutions, qubitexcitation evolutions act on a fixed number of qubits, and can be implemented by circuits that have a fixed number of CNOTs. Singlequbitexcitation evolutions can be performed by the circuit in Fig. 1, with a CNOT count of 2. Doublequbitexcitation evolutions can be performed by the circuit in Fig. 2, which was introduced in ref. ^{47}, with a CNOT count of 13.
For larger systems, qubitexcitation evolutions are increasingly more CNOTefficient compared to fermionicexcitation evolutions, whose CNOT count scales as O(N_{MO}) in the Jordan–Wigner encoding and as \(O({{{{{{\mathrm{log}}}}}}}\,{N}_{{{{{{{{\rm{MO}}}}}}}}})\) in the Bravyi–Kitaev encoding. On the other hand, single and doublequbitexcitation evolutions, as seen from Eqs. (21) and (22), correspond to combinations of 2 and 8, mutually commuting Pauli string exponentials, respectively. Hence, by constructing ansätze with qubitexcitation evolutions instead of Pauli string exponentials, we decrease the number of variational parameters. A further advantage of qubitexcitation evolutions is that they allow for the local circuit optimizations of ref. ^{47}, which Pauli string exponentials do not.
When comparing the QEBADAPTVQE with the fermionicADAPTVQE (see the section “Energy convergence”), we assume the use of the qubit and fermionicexcitation evolutions circuits derived in ref. ^{47}. To our knowledge, these are the most CNOTefficient circuits for these two types of unitary operations. For the qubitADAPTVQE, we assume that an exponential of a Pauli string of length l is implemented by a standard CNOT staircase construction^{6,47,52}, with a CNOT count of 2(l − 1). Global circuit optimization is beyond the scope of this paper.
The QEBADAPTVQE protocol
In the previous section, we formally introduced qubitexcitation evolutions and presented the circuits that implement such unitary evolutions. Here, we describe the three preparation components, and the fourth iterative component, of the QEBADAPTVQE protocol.
First, we transform the molecular Hamiltonian H to a quantumgateoperator representation as described earlier. This transformation is a standard step in every VQE algorithm. It involves the calculation of the one and twoelectron integrals h_{ik} and h_{ijkl} (Eq. (1)), which can be done efficiently (in time polynomial in N_{MO}) on a classical computer^{6}.
Second, we define an ansatzelement pool \({\mathbb{P}}(\tilde{A},{N}_{{{{{{{{\rm{MO}}}}}}}}})\) of all unique single and doublequbitexcitation evolutions, \({\tilde{A}}_{ik}(\theta )\) and \({\tilde{A}}_{ijkl}(\theta )\), respectively, for i, j, k, l ∈ {0, N_{MO} − 1}. The size of this pool is \(  {\mathbb{P}}(\tilde{A},{N}_{{{{{{{{\rm{MO}}}}}}}}})  =\left({{N}_{{{{{{{{\rm{MO}}}}}}}}}} \atop {2}\right)+3\left({{N}_{{{{{{{{\rm{MO}}}}}}}}}} \atop {4}\right)\). Here, ∣∣ ⋅ ∣∣ denotes a set’s cardinality.
Third, we choose an initial reference state \(\left{\psi }_{0}\right\rangle\). For faster convergence, \(\left{\psi }_{0}\right\rangle\) should have a significant overlap with the unknown ground state, \(\left{E}_{0}\right\rangle\). In the classical numerical simulations presented in this paper, we use the conventional choice of the Hartree–Fock state^{53}.
Fourth, we initialize the iteration number to m = 1, and the ansatz to the identity U → U^{(0)} = I. Then, we initiate the QEBADAPTVQE iterative loop. We start by describing the six steps of the mth iteration of the QEBADAPTVQE. We then comment on these steps.

1.
Prepare state \(\left{\psi }^{(m1)}\right\rangle =U({\boldsymbol\theta }^{(m1)})\left{\psi }_{0}\right\rangle\), with θ^{(m−1)} as determined in the previous iteration.

2.
For each qubitexcitation evolution \({\tilde{A}}_{p}({\theta }_{p})={e}^{{\theta }_{p}{\tilde{T}}_{p}}\in {\mathbb{P}}(\tilde{A},{N}_{{{{{{{{\rm{MO}}}}}}}}})\), calculate the energy gradient:
$$\, {\left.\frac{\partial }{\partial {\theta }_{p}}{E}^{(m1)}({\theta }_{p})\right}_{{\theta }_{p} = 0}={\left.\frac{\partial }{\partial {\theta }_{p}}\left\langle {\psi }^{(m1)}\right{\tilde{A}}_{p}^{{{{\dagger}}} }({\theta }_{p})H{\tilde{A}}_{p}({\theta }_{p})\left{\psi }^{(m1)}\right\rangle \right}_{{\theta }_{p} = 0}\\ \, \qquad\qquad\quad\qquad\;\; =\left\langle {\psi }^{(m1)}\right[H,\tilde{{T}_{p}}]\left{\psi }^{(m1)}\right\rangle .$$(23) 
3.
Identify the set of n qubitexcitation evolutions, \({\tilde{{\mathbb{A}}}}^{(m)}(n)\), with largest energy gradient magnitudes. For \({\tilde{A}}_{p}({\theta }_{p})\in {\tilde{{\mathbb{A}}}}^{(m)}(n)\):

(a) Run the VQE to find \(\mathop{\min }\nolimits_{{\boldsymbol\theta }^{(m1)},{\theta }_{p}}E({\boldsymbol\theta }^{(m1)},{\theta }_{p})=\mathop{\min }\nolimits_{{\boldsymbol\theta }^{(m1)},{\theta }_{p}}\left\langle {\psi }_{0}\right{U}^{{{{\dagger}}} }({\boldsymbol\theta }^{(m1)}){\tilde{A}}_{p}^{{{{\dagger}}} }({\theta }_{p})H{\tilde{A}}_{p}({\theta }_{p})U({\boldsymbol\theta }^{(m1)})\left{\psi }_{0}\right\rangle .\)

(b) Calculate the energy reduction \({{\Delta }}{E}_{p}^{(m)}={E}^{(m1)}\mathop{\min }\nolimits_{{\boldsymbol\theta }^{(m1)},{\theta }_{p}}E({\boldsymbol\theta }^{(m1)},{\theta }_{p})\) for each p.

(c) Save the (re)optimized values of θ^{(m−1)} ∪ {θ_{p}} as \({\boldsymbol\theta }_{p}^{(m)}\) for each p.


4.
Identify the largest energy reduction \({{\Delta }}{E}^{(m)}\equiv {{\Delta }}{E}_{p^{\prime} }^{(m)}=\mathop{\max }\nolimits_{p}\{{{\Delta }}{E}_{p}^{(m)}\}\), and the corresponding qubitexcitation evolution \({\tilde{A}}^{(m)}({\theta }^{(m)})\equiv {\tilde{A}}_{p^{\prime} }({\theta }_{p^{\prime} })\).
If ΔE^{(m)} < ϵ, where ϵ > 0 is an energy threshold:

(a)
Exit
Else:

(a)
Append \({\tilde{A}}^{(m)}({\theta }^{(m)})\) to the ansatz: \(U({\boldsymbol\theta }^{(m)})={\tilde{A}}^{(m)}({\theta }^{(m)})U({\boldsymbol\theta }^{(m1)})\)

(b)
Set \({E}^{(m)}={E}^{(m1)}{{\Delta }}{E}_{p^{\prime} }^{(m)}\)

(c)
Set the values of the new set of variational parameters, \({\boldsymbol\theta }^{(m)}={\boldsymbol\theta }^{(m1)}\cup \{{\theta }_{p^{\prime} }\}\), to \({\boldsymbol\theta }_{p^{\prime} }^{(m)}\)

(a)

5.
(Optional) If the ground state of the system of interest is known, a priori, to have the same spin as \(\left{\psi }_{0}\right\rangle\), append to the ansatz the spincomplementary of \({\tilde{A}}^{(m)}({\theta }^{(m)})\), \({\tilde{A}}^{^{\prime} (m)}({\theta }^{^{\prime} (m)})\), unless \({\tilde{A}}^{(m)}({\theta }^{(m)})\equiv {\tilde{A}}^{^{\prime} (m)}\big({\theta }^{^{\prime} (m)}\big)\):
$$U({\boldsymbol\theta }^{(m)})={\tilde{A}}^{^{\prime} (m)}({\theta }^{^{\prime} (m)}){\tilde{A}}^{(m)}({\theta }^{(m)})U({\boldsymbol\theta }^{(m1)}).$$(24) 
6.
Enter the m + 1 iteration by returning to step 1
We now provide some more information about the steps of the protocol. The QEBADAPTVQE loop starts by preparing the trial state \(\left{\psi }^{(m1)}\right\rangle\) obtained in the (m − 1)th iteration. To identify a suitable qubitexcitation evolution to append to the ansatz, first we calculate (step 2) the gradient of the energy expectation value, with respect to the variational parameter of each qubitexcitation evolution in \({\mathbb{P}}(\tilde{A},{N}_{{{{{{{{\rm{MO}}}}}}}}})\).The gradients are evaluated at θ_{p} = 0 because of the presumption that \(\left{\psi }_{0}\right\rangle\) is close to the ground state, which suggests that the optimized value of θ_{p} is close to 0. The gradients (Eq. (23)) are calculated by measuring, on a quantum computer, the expectation value of the commutator of H and the corresponding qubitexcitation operator \({\tilde{T}}_{p}\), with respect to \(\left{\psi }^{(m1)}\right\rangle\). The expression for the gradient in Eq. (23) is derived explicitly in Supplementary Note 1. Note that, steps 1 and 2 are identical to those of the original fermionicADAPTVQE.
The gradients calculated in step 2, indicate how much each qubit excitation can decrease E^{(m − 1)}. However, the largest gradient does not necessarily correspond to the largest energy reduction, optimized over all variational parameters. In step 3, we identify the set of n qubitexcitation evolutions with the largest energygradient magnitudes: \({\tilde{{\mathbb{A}}}}^{(m)}(n)\in {\mathbb{P}}(\tilde{A},{N}_{{{{{{{{\rm{MO}}}}}}}}})\). We assume that \({\tilde{{\mathbb{A}}}}^{(m)}(n)\) likely contains the qubitexcitation evolution that reduces E^{(m − 1)} the most. For each of the n qubitexcitation evolutions in \({\tilde{{\mathbb{A}}}}^{(m)}(n)\), we run the VQE with the ansatz from the previous iteration to calculate how much it contributes to the energy reduction. Step 3 is not present in the original fermionicADAPTVQE, which directly grows its ansatz by the ansatz element with largest energygradient magnitude (equivalent to n = 1). Performing step 3 for n > 1 further reduces the ansatz circuit at the expense of more quantum computer measurements. A study of the performance of the QEBADAPTVQE for different values of n is presented in Supplementary Note 5. The study shows that for the three molecules considered in this paper, LiH, H_{6}, and BeH_{2}, a CNOT reduction between 15 and 25% is achieved for n = 10.
In step 4, we pick the qubit excitation, \({\tilde{A}}^{(m)}({\theta }^{(m)})\), with the largest contribution to the energy reduction, ΔE^{(m)}. If ΔE^{(m)} is below some threshold ϵ > 0, we exit the iterative loop. If instead the ∣ΔE^{(m)}∣ > ϵ, we add \({\tilde{A}}^{(m)}({\theta }^{(m)})\) to the ansatz.
If it is known, a priori, that the ground state of the simulated system has spin zero as the Hartree–Fock state does, we assume that qubitexcitation evolutions come in spincomplement pairs. Hence, we append the spincomplement of \({\tilde{A}}^{(m)}({\theta }^{(m)})\), \(\tilde{A}{^{\prime} }^{(m)}(\theta {^{\prime} }^{(m)})\) (step 5) to the ansatz. However, unlike the fermionicADAPTVQE, the QEBADAPTVQE assigns independent variational parameters to the two spincomplement excitation evolutions. The reason for this is that qubitexcitation evolutions do not account for the parity of the state. Hence, additional variational flexibility is required to obtain the correct relative sign between the two spincomplement qubitexcitation evolutions. Performing step 5 roughly halves the number of iterations required to construct an ansatz for a particular accuracy.
In Supplementary Note 4, we discuss the computational complexity of the QEBADAPTVQE. As a worstcase estimate, the QEBADAPTVQE might require as many as \(O(n{{N}_{MO}}^{16})\) quantum computer measurements.
Classical numerical simulations
We perform classical numerical VQE simulations for LiH, H_{6}, and BeH_{2} to compare the use of qubit and fermionic excitations in the construction of molecular ansätze and to benchmark the performance of the QEBADAPTVQE. LiH and BeH_{2} have been simulated with VQEbased protocols on real quantum computers and are often used in the field of quantumcomputational chemistry to classically benchmark various VQE protocols^{17,18,36,39,40}. Similar to refs. ^{36,37}, we use H_{6} as a prototype of a molecule with a strongly correlated ground state. Our numerical results are based on a custom code, designed to implement ADAPTVQE protocols for arbitrary ansatzelement pools and ansatzgrowing strategies. The code is optimized to analytically calculate excitationbased state vectors (see Supplementary Note 2). The code uses the openfermionpsi4^{54} package to secondquantize the Hamiltonian, and subsequently to transform it to quantumgateoperator representation. For all simulations presented in this paper, we represent the molecular Hamiltonians in the Slater type orbital3 Gaussians (STO3G) spinorbital basis set^{55,56}, without assuming frozen orbitals. In this basis set, LiH, H_{6}, and BeH_{2}, have 12, 12, and 14 spin orbitals, respectively, which are represented by 12, 12, and 14 qubits. For the optimization of variational parameters, we use the gradientdescend Broyden Fletcher Goldfarb Shannon (BFGS) minimization method^{57} from Scipy^{58}. We also supply to the BFGS an analytically calculated energygradient vector (see Supplementary Note 3), for faster optimization. We note that in the presence of high noise levels, gradientdescend minimizers are likely to struggle to find the global energy minimum^{59,60}, while direct search minimizers, like the Nelder–Mead^{61}, are likely to perform better^{62,63}.
Qubit versus fermionic excitations
In this section, we compare qubit and fermionicexcitation evolutions in their ability to construct ansätze to approximate electronic wavefunctions. Directly comparing the QEBADAPTVQE and the fermionicADAPTVQE (as we do in the section “Energy convergence”) does not constitute a fair comparison of the two types of excitation evolutions: the QEBADAPTVQE assigns one variational parameter per qubitexcitation evolution in its ansatz, whereas the fermionicADAPTVQE assigns one variational parameter per spincomplement pair of fermionicexcitation evolutions. Consequently, here we compare the QEBADAPTVQE for n = 1 and step 5 not implemented, to the fermionicADAPTVQE when it grows its ansatz by appending individual fermionicexcitation evolutions (instead of spincomplement pairs of fermionicexcitation evolutions). In this way, the two protocols differ only in using a pool of qubitexcitation evolutions, and a pool of fermionicexcitation evolutions, respectively.
Figure 3 shows energyconvergence plots, obtained with the two protocols as explained above, for the ground states of LiH (Fig. 3a), H_{6} (Fig. 3b), and BeH_{2} (Fig. 3c) at bond distances of r_{LiH} = 1.546 Å, r_{HH} = 1.5 Å, and r_{BeH} = 1.316 Å, respectively. All plots are terminated for ϵ = 10^{−12} Hartree. The two protocols converge similarly, with the fermionicADAPTVQE converging slightly faster for more than ~50 ansatz elements. This difference is most evident for the more strongly correlated H_{6} (Fig. 3b), where the fermionicADAPTVQE requires up to 20% fewer excitation evolutions than the QEBADAPTVQE to achieve a given accuracy. These observations suggest that fermionicexcitationbased ansätze might be able to approximate strongly correlated states a bit better than qubitexcitationbased ansätze. To further investigate this observation, in Fig. 4 we include energyconvergence plots, similar to those in Fig. 3, but for bond distances of r_{LiH} = 3 Å (Fig. 4a), r_{HH} = 3 Å (Fig. 4b), and r_{BeH} = 3 Å (Fig. 4c). At larger bond distances the ground states of the LiH, and BeH_{2} are more strongly correlated, so we expect to see a larger difference in the convergence rates of the two protocols.
In Fig. 4a, c, we see that for LiH and BeH_{2}, at r_{Li–H} = 3 Å and r_{Be–H} = 3 Å, respectively, indeed there is a larger difference in the convergence rates of the two protocols, in favor of the fermionicADAPTVQE. This is more evident for BeH_{2} where the fermionicADAPTVQE requires ~20% fewer ansatz elements, on average, than the QEBADAPTVQE, to achieve a given accuracy. These results further indicate that fermionicexcitationbased ansätze can approximate strongly correlated states better than qubitexcitationbased ansätze.
Energydissociation curves
Figure 5 shows energydissociation curves for LiH, H_{6}, and BeH_{2}, obtained with the QEBADAPTVQE for n = 10 and energyreduction thresholds ϵ_{4} = 10^{−4} Hartree, ϵ_{6} = 10^{−6} Hartree and ϵ_{8} = 10^{−8} Hartree. Dissociation curves obtained with the Hartree–Fock (HF) method, the full configuration interaction (FCI) method, and the VQE, using an untrotterized UCCSD ansatz (UCCSDVQE) are also included for comparison. The UCCSD includes spinconserving single and doublefermionic evolutions only, for a fairer comparison to the QEBADAPTVQE.
Figure 5a–c shows the absolute values for the groundstate energy estimates. All methods except the HF, produce close energy estimates that cannot be clearly distinguished. In Fig. 5d–f, the exact FCI energy is subtracted in order to differentiate better the different methods and their corresponding errors.
The UCCSDVQE achieves chemical accuracy over all bond distances for LiH (Fig. 5d) and over bond distances close to equilibrium configuration for H_{6} (Fig. 5e) and BeH_{2} (Fig. 5f). However, the UCCSDVQE fails to achieve chemical accuracy for bond distances away from equilibrium configuration for H_{6} and BeH_{2}, where the ground states become more strongly correlated.
The QEBADAPTVQE for ϵ_{4}, similarly to the UCCSDVQE, struggles to achieve chemical accuracy for strongly correlated ground states. However, for ϵ_{6} and ϵ_{8} the QEBADAPTVQE achieves chemical accuracy over all investigated bond distances, for all three molecules. This indicates that the QEBADAPTVQE can successfully construct ansätze to accurately approximate strongly correlated states.
However, the real strength of the QEBADAPTVQE, similarly to other ADAPTVQE protocols, is not just in constructing accurate ansätze, but in constructing accurate problemtailored ansätze with few variational parameters, and corresponding shallow ansatz circuits. Figure 5g–i shows plots of the number of variational parameters used by the ansatz of each method as a function of bond distance. In the cases of LiH (Fig. 5g) and BeH_{2} (Fig. 5i), the ansätze constructed by the QEBADAPTVQE for ϵ_{6} and ϵ_{8} are not only more accurate than the UCCSD but also have significantly fewer parameters. However, in the case of H_{6}, the QEBADAPTVQE on average requires more parameters than the UCCSD. The reason for this is that H_{6} is more strongly correlated than LiH and BeH_{2}, so even an optimally constructed ansatz would require more variational parameters than the UCCSD, to accurately approximate the ground state of H_{6}.
An interesting observation is the abrupt changes in the number of variational parameters used by the QEBADAPTVQE for H_{6} at bond distances of around 1, 2, and 2.75 Å. The reason for these changes are molecular structure transformations, where different eigenstates of H become lowest in energy (energylevel crossings).
Energy convergence
In this section, we compare the QEBADAPTVQE against the fermionicADAPTVQE and the qubitADAPTVQE using energyconvergence plots (see Fig. 6). To ensure a fair comparison we choose the following settings for the three protocols: We perform the QEBADAPTVQE for n = 1, using an ansatzelement pool of all unique single and doublequbitexcitation evolutions. The fermionicADAPTVQE is performed as in ref. ^{36}, using an ansatzelement pool of all unique single and double spincomplement fermionicexcitation evolutions. For the qubitADAPTVQE, we use an ansatz element of all evolutions of XYPauli strings of length 2 and 4 that have an odd number of Ys. This pool consists of O(N^{4}) Pauli string evolutions that can be combined to obtain all qubitexcitation evolutions in the ansatz element of the QEBADAPTVQE (see the section “Ansatz elements”). Because of this, the comparison between the QEBADAPTVQE and qubitADAPTVQE, in terms of ansatzcircuit efficiency, can be considered fair. We note that the authors of ref. ^{37} proved that the qubitADAPTVQE actually can construct an ansatz that exactly recovers the FCI wavefunction, using a reduced ansatzelement pool of only 2N_{MO} − 2 Pauli string evolutions. This reduced pool can decrease the number of quantum computer measurements required to evaluate the energy gradients at each iteration (see step 2 of the QEBADAPTVQE) from \(O({N}_{MO}^{8})\) to \(O({N}_{MO}^{5})\). However, the reduced ansatzelement pool will also result in a slower and less circuitefficient ansatz construction, so using this reduced pool in the comparison with the QEBADAPTVQE would not be fair.
We compare the three protocols in terms of three cost metrics, required to construct an ansatz to achieve a specific accuracy: (1) the number of iterations; (2) the number of variational parameters; and (3) the number of CNOTs. The number of iterations and the number of variational parameters (the number of iterations is the same as the number of variational parameters for the fermionicADAPTVQE and the qubitADAPTVQE, but not for the QEBADAPTVQE) determine the total number of quantum computer measurements (see Supplementary Note 4). The CNOT count of the ansatz circuit is approximately proportional to its depth. Hence, the CNOT count can be used as a measure of the run time of the quantum subroutine of the VQE, which also reflects the error accumulated by the quantum hardware. Due to the limited coherence times of NISQ computers, the CNOT count is considered as a primary cost metric.
Figure 6 shows energyconvergence plots, obtained with the three ADAPTVQE protocols, for LiH, H_{6}, and BeH_{2} at bond distances of r_{Li−H} = 1.546 Å, r_{H−H} = 1.5 Å, and r_{Be−H} = 1.316 Å, respectively. All energyconvergence plots are terminated at ϵ = 10^{−12} Hartree.
In Fig. 6a–c, we notice that the QEBADAPTVQE and the fermionicADAPTVQE perform similarly in terms of the number of iterations. This implies that the QEBADAPTVQE and the fermionicADAPTVQE use approximately the same number of the qubit and fermionicexcitation evolutions, respectively, when constructing their respective ansätze. This result is expected because the two types of excitation evolutions perform similarly in constructing electronic wavefunction ansätze. Since qubitexcitation evolutions are implemented by simpler circuits than fermionicexcitation evolutions, the QEBADAPTVQE systematically outperforms the fermionicADAPTVQE in terms of CNOT count in Fig. 6g–i.
While the QEBADAPTVQE and the fermionicADAPTVQE require similar numbers of iterations (Fig. 6a–c), the QEBADAPTVQE requires up to twice as many variational parameters (Fig. 6d–f). This difference is due to the fact that the QEBADAPTVQE assigns one parameter to each qubitexcitation evolutions in its ansatz, whereas the fermionicADAPTVQE assigns one parameter to a pair of spincomplement fermionicexcitation evolutions.
Figure 6a–d shows that the QEBADAPTVQE converges faster, requiring systematically fewer iterations and variational parameters than the qubitADAPTVQE. As suggested in the section “Ansatz elements”, this result is due to the fact that single and doublequbitexcitation evolutions correspond to combinations of 2 and 8 Pauli string exponentials.
In terms of CNOT count (Fig. 6g–i), the qubitADAPTVQE is more efficient than the QEBADAPTVQE at low accuracies. However, for higher accuracies, and correspondingly larger ansätze, the QEBVQEADAPT starts to systematically outperform the qubitADAPTVQE in terms of CNOT efficiency. This result can be attributed to the fact that qubit evolutions allow for the local circuit optimizations introduced in ref. ^{47}, whereas Pauli string evolutions, albeit more variationally flexible, do not allow for any local circuit optimizations.
As a side point, it is interesting to note that when the fermionicADAPTVQE is performed with a pool of independent single and doublefermionic evolutions (Figs. 3 and 4) it is able to converge, albeit more slowly, to higher final accuracies than when it is performed with a pool of spincomplement pairs of single and doublefermionic evolutions (Fig. 6). This is owing to the fact that the pool of independent fermionicexcitation is more variationally flexible.
Discussion
In this work, we investigated the use of qubit excitations to construct electronic VQE ansätze. We demonstrated numerically that in general an ansatz of qubitexcitation evolutions can approximate a molecular electronic wavefunction almost as accurately as an ansatz of fermionicexcitation evolutions. However, fermionicexcitationbased ansätze were found to be a slightly more accurate per number of excitation evolutions when approximating strongly correlated states. These results suggest that, on their own, the Pauliz strings, which measure the parity of the state and account for the anticommutation of the fermionicexcitation operators, play little role in the variational flexibility of an electronic wavefunction ansatz. These results agree with previous findings in refs. ^{37,45}. Another advantage of fermionicexcitation evolutions is that they can form spincomplement pairs of fermionicexcitation evolutions. Such spincomplement pairs can then be used to enforce parity conservation and thus reduce the number of variational parameters of an ansatz by up to a factor of 2. Nonetheless, fermionicexcitation evolutions are implemented by circuits whose size, in terms of CNOT count, scales linearly (logarithmically) in the Jordan–Wigner (Bravyi–Kitaev) encoding with the system size, as opposed to qubitexcitation evolutions, which enjoy the quantumcomputational benefit of being implemented by fixedsize circuits. Therefore, for NISQ devices, where the number of CNOTs is a primary cost factor, qubitexcitation evolutions are more suitable for constructing electronic ansätze.
Motivated by the accuracy and circuit efficiency of qubitexcitationbased ansätze, we introduce the qubitexcitationbased adaptive variational quantum eigensolver (QEBADAPTVQE). The QEBADAPTVQE simulates molecular electronic ground states with a problemtailored ansatz, grown iteratively by appending single and doublequbitexcitation evolutions. We benchmarked the performance of the QEBADAPTVQE with classical numerical simulations for LiH, H_{6}, and BeH_{2}. In particular, we compared the QEBADAPTVQE to the original fermionicADAPTVQE, and its more slowly converging, but a more circuitefficient cousin, the qubitADAPTVQE. Compared to the fermionicADAPTVQE, the QEBADAPTVQE requires up to twice as many variational parameters. However, the QEBADAPTVQE requires asymptotically fewer CNOTs, owing to its use of qubitexcitation evolutions.
The simulations also showed that the qubitADAPTVQE is more CNOTefficient than the QEBADAPTVQE in achieving low accuracies that correspond to small ansatz circuits. However, for higher accuracies and correspondingly larger ansatz circuits, the QEBADAPTVQE systematically outperformed the qubitADAPTVQE in terms of CNOT efficiency. The primary reason for this is that qubit evolutions allow for local circuit optimizations, while the more rudimentary Pauli string evolutions, utilized by the qubitADAPTVQE, do not. In practice, we are often just interested in reaching chemical accuracy. Therefore, one might question what is the usefulness of constructing more CNOTefficient ansätze with the QEBADAPTVQE for accuracy higher than chemical accuracy. Although the numerical results presented here are not sufficient to draw a general conclusion, they indicate that the CNOT efficiency of the QEBADAPTVQE becomes more evident for larger ansatz circuits. Therefore, for larger molecules, the QEBADAPTVQE will likely be able to reach chemical accuracy using fewer CNOTs than the qubitADAPTVQE. Our simulation results also demonstrated that in terms of convergence speed, the QEBADAPTVQE requires fewer variational parameters, and correspondingly fewer ansatzconstructing iterations, than the qubitADAPTVQE.
These results imply that the QEBADAPTVQE is more circuitefficient and converges faster than the qubitADAPTVQE, which to our knowledge was the previously most circuitefficient, scalable VQE protocol for molecular modeling. We do remark though, that in our comparison of the QEBADAPTVQE and the qubitADAPTVQE, we ignored the fact that the latter protocol can use a reduced ansatz element of O(N_{MO}) Pauli string evolutions, as shown in ref. ^{37}. Using a reduced ansatzelement pool would decrease the number of required quantum computer measurements, but will also result in a slower and less efficient ansatz construction. Moreover, the complexity of a single iteration of both the QEBVQEADAPT and the qubitADAPTVQE, might actually be dominated by running the VQE (see Supplementary Note 4). Therefore, reducing the size of the ansatzelement pool might not affect the overall complexity of the protocol. We also note that, in theory, hardwareefficient ansätze and the ansätze of the IQCC protocol suggested in refs. ^{39,40} can be implemented by shallower circuits than the ansätze constructed by the QEBADAPTVQE. However, hardwareefficient ansätze and the IQCC are unlikely to be scalable for large systems: the optimization of hardwareefficient ansätze is likely to become intractable for large systems; and the IQCC requires evaluating a number of expectation values, exponential in the number of variational parameters.
As further work, three potential upgrades to the QEBVQEADAPT can be considered. First, the ansatzelement pool of the QEBVQEADAPT can be expanded to include nonsymmetrypreserving terms as suggested in ref. ^{64}. Potentially, this expanded pool could further improve the speed of convergence and boost the resilience to symmetrybreaking errors of the QEBVQEADAPT. Second, methods from ref. ^{41} can be used to “prune”, from the already constructed ansatz, qubitexcitation evolutions that have little contribution to the energy reduction. This could potentially optimize further the constructed ansatz. Third, the QEBVQEADAPT functionality can be expanded to enable estimations of energies of lowlying excited states. This will be the topic of another work (see ref. ^{65} for a preprint).
Data availability
Data generated during the study is available upon request from the authors (Email: yy387@cam.ac.uk or drma2@cam.ac.uk).
Code availability
The code used to perform the numerical simulations presented in this paper is publicly available at https://github.com/JordanovSJ.
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Acknowledgements
The authors wish to thank K. Naydenova and J. Drori for their useful discussions. Y.S.Y. acknowledges financial support from the EPSRC and Hitachi via CASE studentships RG97399. D.R.M.A.S. was supported by the EPSRC, Lars Hierta’s Memorial Foundation, and Girton College.
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Y.S.Y., D.A.S. and V.A. conceived the project. Y.S.Y. wrote the code, performed the calculations, and wrote the paper. D.A.S. and C.H.W.B. supervised the project.
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Yordanov, Y.S., Armaos, V., Barnes, C.H.W. et al. Qubitexcitationbased adaptive variational quantum eigensolver. Commun Phys 4, 228 (2021). https://doi.org/10.1038/s42005021007300
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DOI: https://doi.org/10.1038/s42005021007300
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