Iterative qubit-excitation based variational quantum eigensolver

Molecular simulations with the variational quantum eigensolver (VQE) are a promising application for emerging noisy intermediate-scale quantum computers. Constructing accurate molecular ans¨atze that are easy to optimize and implemented by shallow quantum circuits is crucial for the successful 10 implementation of such simulations. Ans¨atze are, generally, constructed as series of fermionic- 11 excitation evolutions. Instead, we demonstrate the usefulness of constructing ans¨atze with “qubit- excitation evolutions”, which, contrary to fermionic excitation evolutions, obey “qubit commutation 13 relations”. We show that qubit excitation evolutions, despite the lack of some of the physical features 14 of fermionic excitation evolutions, accurately construct ans¨atze, while requiring asymptotically fewer 15 gates. Utilizing qubit excitation evolutions, we introduce the iterative qubit excitation based VQE 16 (IQEB-VQE) algorithm. The IQEB-VQE performs molecular simulations using a problem-tailored 17 ansatz, grown iteratively by appending evolutions of single and double qubit excitation operators. By 18 performing numerical simulations for small molecules, we benchmark the IQEB-VQE, and compare 19 it against other competitive VQE algorithms. In terms of circuit eﬃciency and time complexity, we 20 ﬁnd that the IQEB-VQE systematically outperforms the previously most circuit-eﬃcient, practically 21 scalable VQE algorithms.

INTRODUCTION 23 Quantum computers are anticipated to enable simulations of quantum systems more efficiently and accurately than 24 classical computers [1,2]. A promising algorithm to perform this task on emerging noisy intermediate-scale quantum 25 (NISQ) [3][4][5] computers is the variational quantum eigensolver (VQE) [6][7][8][9][10][11][12]. The VQE is a hybrid quantum-classical 26 algorithm that estimates the lowest eigenvalue of a Hamiltonian H by minimizing the energy expectation value E(θ θ θ) = As already mentioned N MO is the number of molecular spin-orbitals, a † i and a i are fermionic creation and annihilation 99 operators, corresponding to the i th molecular spin-orbital, and the factors h ij and h ijkl are one-and two-electron 100 integrals, written in a spin-orbital basis [15]. The Hamiltonian expression in equation (1) can be mapped to quantum 101 gate operators using an encoding method, e.g. the Jordan-Wigner (JW) [47] or the Bravyi-Kitaev [48] methods.
where 108 We refer to Q † i and Q i as qubit creation and annihilation operators, respectively. They act to change the occupancy 110 of spin-orbital i. The Pauli-z strings, in equations (3) and (4), compute the parity of the state and act as exchange 111 phase factors that account for the fermionic anticommutation of a † and a. Substituting equations (3) and (4) 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 114 real scalar coefficient. The number of terms in equation (7) scales as O(N 4 MO ).

121
Iterative VQE algorithms construct their ansätze on the go, taking into account details of the system of interest.

122
At the m th iteration in algorithms like our IQEB-VQE and the ADAPT-VQE, one or several unitary operators, r )}, which we refer to as ansatz elements, are appended to the left of the already existing ansatz, U (θ θ θ (m−1) ): The ansatz elements, U  This, iteratively greedy, strategy results in an ansatz that is tuned specifically to the system being simulated, and 128 can approximate the ground eigenstate of the system with considerably fewer variational parameters and a shallower 129 ansatz circuit, compared to general-purpose fixed ansätze, like the UCCSD.

130
In the ADAPT-VQE, the ansatz element pool P is the set of all spin-complement pairs of single and double fermionic 131 excitation evolutions. In the qubit-ADAPT-VQE, P is a set of parametrized exponentials of XY -Pauli strings of 132 lengths 2 or 4 that have an odd number of Y s. The growth strategy of the ADAPT-VQE and the qubit-ADAPT-VQE 133 is to add, at each iteration, the ansatz element with the largest energy gradient magnitude where |ψ (m−1) is the trial state at the (m − 1) th iteration. For detailed descriptions of the ADAPT-VQE and the 135 qubit-ADAPT-VQE, we refer the reader to the original Refs. [35] and [36], respectively. Single and double fermionic excitation operators, are defined, respectively, by the skew-Hermitian operators Single and double fermionic-excitation evolutions are thus given, respectively, by the unitaries Using equations (3) and (4), for i < j < k < l, A ik and A ijkl can be expressed in terms of quantum gate operators as As seen from equations (14) and (15), fermionic excitation evolutions act on a number of qubits that scales as O(N MO ).

143
Therefore, they are implemented by circuits whose size (in terms of number of CN OT s) also scales as O(N MO ). We 144 provided a CN OT -efficient method to construct circuits for fermionic excitations evolutions in Ref. [46]. The circuits 145 for a single and a double fermionic excitation evolutions have CN OT counts of 2(k − i) + 1 and 2(l + j − i − k) + 9, 146 respectively.

147
Qubit excitation operators are defined by the qubit annihilation and creation operators, Q i and Q † i [equations (5) 148 and (6)], which satisfy the qubit commutation relations operators are given, respectively, by the skew-Hermitian operators Thus, single and double qubit-excitation evolutions are given, respectively, by the unitary operators Using equations (5) and (6),Ã ik andÃ ijkl can be re-expressed in terms of quantum gate operators as As seen from equations (21) and (22) can be performed by the circuit in Fig. 2, which we introduced in Ref. [46], with a CN OT count of 13.
For larger systems, qubit excitation evolutions are increasingly more CN OT -efficient compared to fermionic exci-

170
Global circuit optimization is beyond the scope of this paper. classical computer [6].

179
Second, we define an ansatz element pool P(Ã, N MO ) of all unique single and double qubit excitation evolutions, Here, || · || denotes a set's cardinality.

182
Third, we choose an initial reference state |ψ 0 . For faster convergence, |ψ 0 should have a significant overlap with 183 the unknown ground state, |E 0 . In the classical numerical simulations presented in this paper we use the conventional 184 choice of the Hartree-Fock state [50]. 185 Fourth, we initialize the iteration number to m = 1, and the ansatz to the identity U → U (0) = I. Then, we initiate 186 the IQEB-VQE iterative loop. We start by describing the six steps of the m th iteration of the IQEB-VQE. We then 187 comment on these steps. 188 1. Prepare state |ψ (m−1) = U (θ θ θ (m−1) )|ψ 0 , with θ θ θ (m−1) as determined in the previous iteration. 189 2. For each qubit excitation evolutionÃ p (θ p ) = e θpTp ∈ P(Ã, N MO ), calculate the energy gradient: 3. Identify the set of n qubit excitation evolutions,Ã (m) (n), with largest energy gradient magnitudes.  Else: 6. Enter the m + 1 iteration by returning to step 1 206 We now provide some more information and observations about the steps of our algorithm. The IQEB-VQE loop devices are available, step 3 can also be parallelized.

228
If it is known, a priori, that the ground state of the simulated system has spin zero, as the Hartree-Fock state does, we 229 assume that qubit-excitation evolutions come in spin-complement pairs. Hence, we append the spin-complementary of 230Ã (m) (θ (m) ),Ã (m) (θ (m) ) (step 5) to the ansatz. However, unlike the ADAPT-VQE, our algorithm assigns independent 9 variational parameters to the two spin-complement excitations. The reason for this is that qubit excitation evolutions minimization method [54] from Scipy [55]. We also supply to the BFGS an analytically calculated energy gradient 246 vector (see Sec. III of the supplementary information), for a faster optimization. We note that in the presence of 247 high noise levels, gradient-descend minimizers are likely to struggle to find the global energy minimum [56,57], while 248 direct search minimizers [58] are likely to perform better [59,60]. The energy convergences of the two algorithms for LiH and BeH 2 are given in Fig. 3. The two algorithms converge 259 remarkably similarly, with the one using fermionic excitation evolutions converging slightly faster for large ansatz 260 sizes. Figure 3 suggests the remarkable result, that the more complex and physically motivated fermionic excitation 261 evolutions have little or no advantage over the qubit excitation evolutions in approximating electronic wavefunctions.

262
This means that the Pauli-z strings in equations (14) and (15) Hartree. We also include dissociation curves obtained with three other standard methods: the full-configuration-269 interaction (FCI) method [61]; the Hartree-Fock (HF) method [50]; and the VQE using an untrotterized UCCSD  While the IQEB-VQE and the ADAPT-VQE require similar numbers of iterations (Fig. 5a,b), the IQEB-VQE 302 requires up to twice as many variational parameters (Fig. 5e,f). This difference is due to the fact that the IQEB-303 VQE assigns one parameter to each qubit excitation evolutions in its ansatz, whereas the ADAPT-VQE assigns one 304 parameter to a pair of spin-complement fermionic excitation evolutions. where the number of CNOTs is a primary cost factor, qubit excitations are more suitable for constructing VQE 326 electronic ansätze than fermionic excitations. 327 molecules. In particular, we compared the IQEB-VQE to other competitive iterative VQE algorithms, in particular the 332 original ADAPT-VQE, and its more slowly converging, but more circuit-efficient younger cousin, the qubit-ADAPT-333 VQE. Compared to the ADAPT-VQE, the IQEB-VQE requires approximately twice as many variational parameters, 334 but constructs systematically more CN OT -efficient ansätze, owing to the use of qubit excitation evolutions. We also 335 found that the IQEB-VQE outperforms even the qubit-ADAPT-VQE, in constructing more CN OT -efficient ansätze.

336
The primary reason for this is that qubit-excitation evolutions allow for efficient local circuit optimizations, whilst the 337 more rudimentary Pauli string exponentials, utilized by the qubit-ADAPT-VQE, do not. Furthermore, the IQEB-338 VQE converges faster, requiring systematically fewer variational parameters, and correspondingly fewer iterations, 339 than the qubit-ADAPT-VQE. These results imply that the IQEB-VQE is not only more circuit-efficient, but also 340 faster than the qubit-ADAPT-VQE, which to our knowledge was previously the most circuit-efficient, scalable VQE 341 algorithm for molecular simulations.

342
As further work, three potential upgrades to the IQEB-VQE can be considered. First, the ansatz element pool of 343 the IQEB-VQE can be expanded to include non-symmetry-preserving terms as suggested in Ref. [62]. Potentially, this 344 expanded pool could further improve the speed of convergence and boost the resilience to symmetry-breaking errors 345 of the IQEB-VQE. Second, we suggest the use of methods from Ref. [40] to "prune", from the already constructed 346 ansatz, ansatz elements that have little contribution to the energy reduction. This could potentially optimize further 347 the constructed ansatz. Third, we suggest the expansion of the IQEB-VQE functionality to enable estimations of 348 energies of low lying excited states.

349
Finally, we note that there might be hardware-efficient ansätze that require shallower circuits than the ansätze 350 constructed by the IQEB-VQE. However, as pointed out in the introduction of this paper, the classical optimization 351 of such not-physically-motivated ansätze, is unlikely to be tractable for large molecules. To the best of our knowledge, 352 the IQEB-VQE is the most circuit-efficient, scalable VQE algorithm for molecular simulations. We believe it 353 represents a significant step towards implementing molecular simulations on NISQ computers.