Variational quantum eigensolver techniques for simulating carbon monoxide oxidation
Batched ADAPTVQE
The ADAPTVQE is a greedy iterative approach for building an efficient compact ansatz for approximating the molecular wavefunction^{27}. In the original implementation of ADAPTVQE^{27}, a single operator with the largest gradient is added at each iteration. Such a construction of the ansatz is based on the gradient criterion, which approximates the idea of recovering most correlation energy. Indeed, the numerical investigation of ADAPTVQE and randomly constructed ansatzes demonstrates significantly worse convergence of the latter with respect to parameter number in the quantum circuit. However, computations of the derivatives for the whole operator pool in order to add a single operator produce a notable overhead in the number of measurements. Due to the rather slow convergence of ADAPTVQE for strongly correlated systems and the increasing size of the operator pool, the application of ADAPTVQE for practicallyrelevant molecules is hindered.
In order to reduce the number of gradient computations, we introduce batched ADAPTVQE that adds multiple operators with the largest gradients simultaneously (see Fig. 1). This approach allows reducing the number of gradient computations while building a compact ansatz.
We apply batched ADAPTVQE with both fermionic and qubit pools. The quantum circuit in fermionic ADAPTVQE algorithm is obtained from the predefined operator pool, which consists of excitation operators. We note that here we define fermionic operators as in conventional UCCSD instead of originally used spinadapted operators (see Methods, Section titled Fermionic Pool). As the pool size scales as \({{{{{{{\mathcal{O}}}}}}}}({N}^{2}{n}^{2})\), fermionic ADAPTVQE introduces a polynomial overhead in the number of measurements compared to VQE with fixed circuits. Thus, adding multiple gradients to the pool has a practical advantage in both numerical simulations and experiments on a quantum computer.
In addition to fermionic ADAPTVQE, we also investigate qubit ADAPTVQE performance. Tang et al.^{28} proposed the qubit ADAPTVQE approach, which builds a qubit pool from individual Pauli strings. Unlike qubit coupled cluster approach^{35,36}, these Pauli strings are obtained by splitting each fermionic operator into subterms with all ZPaulis being removed. This method of qubit pool construction leads to an increase in the pool size compared to the fermionic one. However, it was demonstrated that such qubit pool contains redundant operators and can be reduced to a pool of linear size. In the present work, we study qubit pools of both polynomial and linear sizes. Our numerical investigation demonstrates that even though reduction of the qubit pool size aims to reduce the measurement overhead, in practice polynomial pools require fewer derivative evaluations due to the slow convergence of ADAPTVQE with a linear pool. Our numerical results illustrates that the use of the polynomial qubit pool with batched ADAPTVQE is a more efficient strategy compared to implementations of qubit ADAPTVQE with a linear pool.
Liu et al.^{30} added a fixed number of operators to improve fermionic ADAPTVQE slow convergence caused by inaccurate identification of operators due to Valdemoro’s reconstruction of 3RDM. The size of the batch in their implementation should be carefully tuned as inappropriate size leads to significant increases in the circuit depth. In our approach, we improve the original ADAPTVQE approach by adding batches of operators of the varying sizes at each algorithm step. Note that we add the operators to the ansatz following the order of computed gradients. The adaptive size of the batch allows adding more operators at the beginning of the ansatz growing procedure when the energy difference is significant, and on the contrary, reduce the batch size when the ansatz is closer to the desired state (see Supplementary Information, Supplementary Fig. 1). The numerical results demonstrate that such an implementation makes it possible to build an ansatz close to the original one while reducing the computational cost.
At each ADAPTVQE iteration, we pick all the gradients that differ from the largest by a ratio less than r, where r is incorporated as a hyperparameter (see Supplementary Information, Supplementary Fig. 2). We set r = 2 for all the considered molecules and our numerical results suggest that batched ADAPTVQE produces circuits close to the ones obtained with original ADAPTVQE. From the practical point of view, it is important that the hyperparameter value can be the same for different molecules.
Since batched ADAPTVQE adds multiple operators at each step, it requires sizably fewer iterations to build an ansatz, which considerably reduces the cost of computing gradients with respect to operators in the pool. This is also important for classical simulations since computations for CO_{2} molecule (19 qubits) take a significant amount of time. We note that in the case of experiments on existing quantum devices, the restriction of the operator batch size at each iteration can be useful due to the controllability of the quantum circuit size. The performance of ADAPTVQE under noisy conditions requires a separate study.
Complete pools
The initial formulation of qubit ADAPTVQE considers pools of Pauli strings originated from UCCSD excitations. As such strings are obtained by cutting each fermionic excitation and removing Z Pauli operators, the pool grows polynomially with the system size as \({{{{{{{\mathcal{O}}}}}}}}({n}^{4})\). Theoretically, this produces an \({{{{{{{\mathcal{O}}}}}}}}({n}^{8})\) overhead in the number of measurements in the straightforward implementation^{29,30}. That is why the idea of restricting the pool size seems to be attractive. We start our discussion with the papers on qubit ADAPTVQE^{28,29} that propose the concept of minimal complete pools (MCPs) and prove their existence.
MCPs include a set of operators, individual Pauli strings, that can transform a reference state to any real state in the Nqubit Hilbert space. Tang et al.^{28} obtained a 3qubit pool analytically and proved by induction that minimal complete pools exist for any N and have a size of 2N − 2. The authors derived a set of generators forming a theoretically complete pool, which performs successfully on random Hamiltonians. However, the derived pool is not appropriate for ADAPTVQE molecular simulations as the Pauli strings used as generators do not account for specific symmetries presented in the molecular Hamiltonian.
Further study of MCPs by Shkolnikov et al.^{29} focused mainly on MCPs for molecular simulations. According to their results, the size of a minimal complete pool for molecular Hamiltonians is even less than for random Hamiltonians as it should preserve the molecular symmetries. The authors of Ref. ^{29} deeply analyzed these symmetries from the molecular group point of view and demonstrated how they force additional restrictions on the structure of Pauli string generators which form a complete pool. The following criteria have been formulated for minimal complete pools in molecular case^{29}:

the number of electrons with a given spin changes by a multiple of 2;

each operator in the pool must conserve spatial parity;

the pool must contain enough «starters” for ADAPTVQE to start;

the pool generates the biggest subgroup and subalgebra of those generated by a general nonsymmetrypreserving MCP, that contain Pauli strings obeying conditions 1–2.
The most important challenge in practice is validating of the fourth criterion: the complexity of building Lie algebra scales exponentially with the number of qubits. This criterion can be replaced by checking the product group size and the inseparability condition: MCPs generators in a complete pool cannot be split into two mutually commuting sets. Additionally, a difficult part is the pool construction since in the original work authors performed manual analysis of molecular symmetries.
The practical implementation of qubit ADAPTVQE with MCPs requires simple and automated ways of building the complete pools. Here we demonstrate how to apply the proposed pool completeness criteria to the MCPs built in the reduced qubit space, i.e., with qubit tapering procedure (see Methods, Section titled Qubit pool completeness). Thus, we not only build a pool of linear size for the molecule, but also remove qubits from the simulation, which significantly reduces the required computational resources. Additionally, we propose a greedy algorithm for building complete molecular pools from a predefined large pool. In this work, we use qubit pools originated from tapered fermionic UCCSD pools.
The proofs for the complete pool construction combined with qubit tapering are presented in the Methods section. Here, we only highlight the main steps of building a complete pool. We propose to select Pauli strings from the set of tapered UCCSD operators instead of manually constructing them. First, the Pauli strings from UCCSD satisfy the complete pool conditions for the considered molecules (see Methods, Section titled Qubit pool completeness). Second, most of Pauli strings from UCCSD correspond to double excitations that are starters for ADAPTVQE, which allows satisfying criterion 3 without additional work. We propose the following scheme to build a complete pool in a reduced qubit space (a detailed scheme is illusrated in Fig. 2).

Taper off qubits based on Z_{2} symmetries for both molecular hamiltonian and UCCSD pool. This can be done using a builtin procedure in Qiskit or other frameworks for quantum computations. A more detailed explanation of the procedure can be found in Methods.

Cut the tapered UCCSD operators into individual Pauli strings. We additionally remove all ZPaulis from the obtained strings. Although it is not a necessary step, it makes circuits significantly shallower according to our numerical investigation. At this step, we also check the completeness of the pool, which is inexpensive from the computational point of view (see Gauss method in Methods).

Select 2(N − N_{s}) strings from the pool constructed in the previous step to form a complete pool. Here N_{s} is a number of tapered qubits. We propose a greedy approach to select the strings that form a complete pool.
The Pauli string selection from the UCCSD pool (step 3) can be implemented in various ways. As mentioned above, to create a complete pool, we need to check the size of the product group and pool inseparability. The detailed instructions on checking the above conditions are given in the Methods. We use a greedy algorithm to construct a complete pool by iteratively adding strings to the pool one by one. We follow several rules to construct the pool: (i) to satisfy the group size criterion, we choose only the strings that increase the group size (i.e., linearly independent with already added strings); (ii) to meet the inseparability criterion, we select a string that does not commute with the largest subset of added strings; (iii) to make the operators more diverse, we try to add the Pauli strings, which act by X/Y on different sets of qubits. Although the other ways of pool construction are possible, the numerical results confirm that the performance of the obtained greedy MCP pools is at least on par with the ones tested in the original work^{29} for H_{4} and LiH molecule. The advantage of the proposed algorithm for obtaining MCP lies in the combination of qubit reduction and the simplicity of the pool construction: by using operators from tapered fermionic excitations, we account for all the necessary symmetries, while the greedy algorithm builds a pool of the required rank.
We perform simulations for the H_{4} molecule in the minimal basis set (STO3G) with the MCP pool given in ref. ^{29} (see MCP11 in Fig. 3) on 8 qubits for the bond length R being equal to 1.0 and 2.0 Å. We apply the tapering procedure to the given pool MCP11: we rotate the molecular hamiltonian and the MCP generators with a unitary transformation, such that the rotated generators act only by X or I on a subset of qubits and eliminate these qubits. The obtained pool contains Pauli strings acting on 5 qubits («MCP11; tapered”). Figure 3 shows that the tapered pool converges to the ground state with high precision (results for R = 1 Å are provided in Supplementary Information, Supplementary Fig. 3). The tapering procedure allows us to reduce the circuit depth significantly. We construct the «greedy MCP” pool consisting of 10 operators using the proposed computational scheme shown in Fig. 2. The greedy MCP pool demonstrates similar convergence with respect to the number of parameters compared to the tapered MCP11, but reduces the required circuit depth due to the absence of Z Paulis.
For the LiH molecule, after freezing the inner orbital, there are 2 electrons, which leads to only 10 excitations in VQEUCCSD after symmetry reduction (see Methods, Section titled Computational Details). The small size of the tapered fermionic pool leads to a special case, where the Pauli strings from UCCSD pool without Z Paulis do not form a complete pool. The rank of the pool is 10, while the required rank for a 6 qubit molecule should be 12. To overcome this difficulty, we first constructed the pool of 10 operators from the tapered UCCSD pool with removed Z paulis and then added two more operators with Zpaulis. The energy convergence with respect to the number of parameters and circuit depth for R = 3 Å is given in Fig. 4. As in the H_{4} case, tapering allows us to reduce the required circuit depth. The constructed pool «Greedy12” demonstrates the excellent convergence compared to tapered MCP14 and reduces the circuit depth.
The results for H_{4} and LiH are obtained at bond stretching when the correlation energy increases (the convergence of different pools for LiH molecule near the equilibrium is provided in Supplementary Information, Supplementary Fig. 4). Based on the provided numerical results, we expect that the proposed approach for pool construction is efficient for small molecules. Further, we use the pools constructed with the greedy algorithm to simulate larger molecules and compare the results to fermionic ADAPTVQE approach and polynomial qubit pools.
Simulation of molecules
Here we present the simulation results for a set of larger molecules. We perform simulations on the Qiskit statevector simulator^{37} using various methods, including our improved version of the ADAPTVQE approach – batched ADAPTVQE. We start with benchmarking the methods on H_{2}O at various bond lengths. Further, we present the simulation results for molecules involved in COoxidation: O_{2}, CO, and CO_{2}.
All the computations are performed in the minimal basis set using frozen core approximation and qubit tapering to reduce the required resources (see Methods, Section titled Computational Details). We compare numerical results obtained with trotterized (or disentangled) VQEUCCSD with SD ordering^{20}, ADAPTVQE, and their various implementations (fermionic/qubit and original/batched).
To compare the computational cost, we calculate the total number of 1parameter derivative computations N_{d1} during the simulation: we consider both the ansatz growing phase in ADAPTVQE case (i.e., derivatives with respect to each operator from the pool) and the VQE optimization procedure.
H_{2}O molecule
For H_{2}O, we simulate symmetrical stretching of two OH bonds with a fixed angle of 104.51^{∘} (see Fig. 5). The UCCSD pool after symmetry reduction consists of 30 excitations acting on 8 qubits (see Methods, Table 1). Close to the equilibrium geometry (bond length R = 1.0 Å) and at R = 1.5 Å, fermionic VQEUCCSD provides accurate results for the ground state energy (see fUCCSD in Fig. 5b, c).
With the bond stretching and increase in the correlation energy at R = 2.0 Å, VQEUCCSD is not sufficient to reproduce the energy with the required accuracy. Fermionic ADAPTVQE improves energy compared to VQEUCCSD. At R = 1.0 Å and 1.5 Å fermionic ADAPTVQE reaches the chemical accuracy threshold with 17 and 18 parameters, respectively, and increases N_{d1} more than in 4 times (Fig. 5a). As the HO distance increases, the number of required parameters grows to 29 parameters at R = 2.0 Å, which almost equals the size of the UCCSD pool. The total number of computed derivatives is higher in this case by order of magnitude.
Fermionic batched ADAPTVQE reaches the threshold with a slight overhead in the number of parameters at these bond lengths — 22, 24, and 32 — which leads to an increase in the circuit depth by less than 20% compared to the original ADAPTVQE (Fig. 5a). At the same time, fermionic batched ADAPTVQE requires at least 3 times fewer derivative computations (N_{d1}, derivatives with respect to a single parameter) in total for each bond length. It is worth noting that at R = 2.0 Å batched ADAPTVQE initially produces energies noticeably worse than the original implementation. Still, over a single iteration it improves the energy by 20 mHartree and at 22 parameters catches up with the original ADAPTVQE values and further provides similar energies.
The observed results demonstrate that close to the VQEUCCSD result, batched fermionic ADAPTVQE is close to VQEUCCSD in terms of computed derivatives N_{d1}, while the original ADAPTVQE approache increases N_{d1} significantly. Even though the convergence slows down near the threshold in the presence of strong correlation, batched fermionic ADAPTVQE allows reducing the number of computed derivatives significantly.
To implement qubit ADAPTVQE, we start by constructing a minimal complete pool with the proposed greedy approach. For H_{2}O we obtain the pool of 16 operators (Greedy16), which includes no more than a single Pauli string from each initial fermionic excitation after removing Z Paulis. At R = 1.0 Å, Greedy16 demonstrates significantly slower convergence compared to fermionic ADAPTVQE and reaches the energy threshold with 118 parameters. Batched ADAPTVQE, in this case, reaches the threshold with 115 parameters and requires 1.8 times fewer derivatives. For R = 1.5 Å the situation changes: original ADAPTVQE requires 108 parameters to reach the threshold, while batched implementation produces a large overhead and converges at 123 parameters. However, from the Fig. 5b, one can see that batched ADAPTVQE still reduces the number of derivative computations by a half. Moreover, batched ADAPTVQE does not lose in terms of the circuit depth here due to the usage of lightweight Pauli strings with no Z chains as well as Qiskit’s heavy transpilation of circuits. Finally, at R = 2.0 Å batched ADAPTVQE performs worse than the original one and converges only with 101 parameters instead of 74 for original ADAPTVQE and computing about 10% more derivatives.
One can see that in comparison to fermionic ADAPTVQE, qubit ADAPTVQE with greedy MCP produces a significant overhead in the number of derivative computations required to obtain accurate results. To verify that MCP in practice reduces the measurement overhead in qubit ADAPTVQE, we compare the Greedy16 performance with a qubit pool of size 30. This pool is obtained by the same greedy algorithm, but we pick 30 operators — at least a single operator from each UCCSD excitation after removing Z paulis — instead of 16. We refer to such pools as «Greedy sdN”, where «sd” refers to picking operators from each UCCSD excitation. Thus, such pool is at least of the size of the UCCSD pool and satisfies the completeness criteria.
Our results demonstrate that such pool converges faster than MCP with respect to the number of parameters, total number of derivatives, and circuit depth. For R = 1.0 Å qubit ADAPTVQE with Greedy sd30 pool converges at a significantly smaller number of parameters (27 for both original and batched) compared to R = 2.0 Å (76 and 67 for original and batched, respectively), which matches the expectations as the correlation energy is small near the equilibrium. These results indicate that the completeness criterion is insufficient to construct an efficient pool for qubit ADAPTVQE computations. Batched ADAPTVQE with Greedy sd pool performs similar to the original ADAPTVQE in terms of the parameter number and circuit depth (see Fig. 5), while reducing the total number of derivatives N_{d}1 significantly: from 2.2 fewer derivative evaluations at R = 1.5 up to 5 times at R = 2.0 Å. One can see that by using batched qubit ADAPTVQE with Greedy sd pool we can reduce N_{d1} by 1–2 orders of magnitude compared to the original qubit ADAPTVQE with greedy MCP.
Considering our results, enlarging the qubit pool in the combination with batched ADAPTVQE procedure appears to be an efficient implementation of qubit ADAPTVQE. This observation contradicts the idea that the construction of a minimal linear pool reduces the measurement overhead. Numerically, it appears that limitation of the pool size leads to a significant increase in the number of iterations in the ansatz construction procedure.
O_{2} molecule
For the molecules involved in CO oxidation, we perform simulations at equilibrium geometries, precomputed classically with CCSD/STO3G. The ground state of the molecular oxygen is triplet, with two electrons remaining unpaired on the π orbital. This is not a typical case for neutral molecules and thus a good testing case for ADAPTVQE on openshell systems. For O_{2}, we use the unrestricted HartreeFock (UHF) state as a reference state in ADAPTVQE and VQEUCCSD computations.
With the use of the qubit tapering procedure, the Hamiltonian can be reduced from 16 to 11 qubits, while the operator pool size becomes relatively small for O_{2} molecule and consists of 24 excitations. VQEUCCSD converges to the energy slightly above the chemical accuracy threshold (around 2 mHartree). Fermionic ADAPTVQE (original and batched) converges to the fermionic VQEUCCSD energy at 24 parameters, meaning no significant benefit in the parameter number observed in this case (Fig. 6a). Fermionic ADAPTVQE reaches the chemical accuracy threshold with 27 parameters for both original and batched implementations. Batched ADAPTVQE converges very similar to the original implementation producing similar quantum circuits and reducing the number of total derivative evaluations by 1.5 times at 27 parameters.
Greedy MCP for O_{2} requires 22 operators (Greedy22) and reaches the chemical accuracy threshold with 65 and 66 parameters for original and batched ADAPTVQE, respectively, which reduces the circuit depth from about 2200 in fermionic ADAPTVQE to 300. Batched ADAPTVQE requires 2.7 times less derivative evaluations than the original one. Due to the small size of the tapered UCCSD pool we cannot pick a single operator from each excitation (after removing Z Paulis) and form a complete pool from such operators. For this reason, we add two extra operators to ensure the pool completeness. The obtained pool contains 26 operators (Greedy sd26), which is very close to the MCP size. Qubit ADAPTVQE converges slowly for both Greedy22 and Greedy sd26. Nevertheless, adding a few operators to the pool can improve the convergence in terms of both the circuit depth and the total number of derivatives. Batched qubit ADAPTVQE with Greedy sd26 pool performs better than other qubit ADAPTVQE implementations in this case. Comparing these two extreme cases – batched ADAPTVQE in combination with Greedy sd26 pool and original ADAPTVQE with MCP pool – we can reduce the number of total derivative evaluations by 5 times. It confirms our idea that decreasing the pool size is not sufficient to reduce the number of required measurements. On the contrary, O_{2} results follow the pattern observed for H_{2}O for a larger operator pool: the convergence of qubit ADAPTVQE improves while the computational cost reduces.
CO molecule
Carbon monoxide is a more complicated case compared to oxygen due to the larger number of electronic excitations. In computational chemistry, carbon monoxide is a known example when density functional theory (DFT) fails, which is appeared in the incorrect prediction of the CO adsorption on the transition metal surface^{38,39}. The observed inconsistencies between classical simulations and experimental results make quantum simulations of reactions involving carbon monoxide an attractive problem with the proper development of quantum technologies. At this stage, we are only able to simulate an isolated CO molecule. However, even in the minimal basis set, it represents an interesting test case for quantum methods.
VQEUCCSD simulations of CO molecule with STO3G basis set have been reported previously^{15}. This simulation uses 20 qubits since the symmetry reduction and the orbital freezing are not applied. The reported VQEUCCSD energy (−111.363 Hartree) matches the classical CCSD result (−111.362 Hartree), which is about 10 mHartree higher than the exact one. This is a sizable deviation compared to small molecules, where VQEUCCSD at equilibrium geometry accurately recovers correlation energy. Our simulations of CO require only 12 qubits, which greatly reduces the computational cost. VQEUCCSD with 12 qubits for CO molecule fits previously reported results.
ADAPTVQE convergence to the ground energy is presented in Fig. 6b. Fermionic ADAPTVQE reaches VQEUCCSD energy with 72 and 75 parameters (original and batched implementations), which is significantly less than the UCCSD pool size (85 operators). This allows reducing the circuit depth by about 30% compared to VQEUCCSD. Fermionic batched ADAPTVQE requires about 9 times fewer derivatives to reach VQEUCCSD energy compared to the original one. When reaching the chemical accuracy threshold, batched ADAPTVQE performs better as well: it converges with 128 parameters instead of 134 and with about 7 times fewer derivative computations.
Thus, fermionic batched ADAPTVQE rapidly converges to VQEUCCSD energy and has a reasonable overhead in the number of derivative evaluations (about 3500 N_{d1} for batched fermionic ADAPTVQE compared to 1275 N_{d1} for fermionic VQEUCCSD), while original ADAPTVQE requires about 30,100. The observed results match the ones obtained for H_{2}O: batched implementation of fermionic ADAPTVQE can achieve VQEUCCSD results more efficiently than VQEUCCSD while significantly reducing the computational cost compared to original ADAPTVQE. However, beyond this point the further energy improvement becomes slow from the optimization point of view, with the number of computed derivatives increasing to around 75,000 for batched ADAPTVQE and 555,000 for original ADAPTVQE. The original fermionic ADAPTVQE produces a huge overhead in the number of derivative evaluations.
We perform qubit ADAPTVQE computations for Greedy and Greedy sd pools for the CO molecule. Greedy MCP contains 24 operators (Greedy24) for 12qubit CO molecule, while the Greedy sd pool consists of 85 Pauli strings (Greedy sd85). For Greedy24 we perform original ADAPTVQE computations up to 140 parameters to see if batched ADAPTVQE matches its convergence curve.
With batched qubit ADAPTVQE, we do not achieve chemical accuracy due to the high computational cost, but we achieve the level of the fermionic VQEUCCSD energy. Both pools allow reducing the circuit depth significantly compared to UCCSD. Qubit ADAPTVQE with Greedy sd85 pool achieves VQEUCCSD result with 164 and 156 parameters with original and batched implementations, respectively. Batched ADAPTVQE reduces the number of computed derivatives by a factor of 5. Qubit ADAPTVQE with Greedy sd pool turns out to be the most efficient in terms of circuit depth. However, the price we pay for the circuit reduction is the increase in the number of derivative evaluations by two orders of magnitude compared to VQEUCCSD.
At the same time, batched ADAPTVQE with Greedy24 requires 413 parameters to reach VQEUCCSD energy. The number of derivative evaluations is about 23 times higher for the MCP pool compared to Greedy sd85 when computed with batched ADAPTVQE.
The observed results demonstrate a vastly worse convergence of greedy pool than greedy sd, which again confirms that in practice, the linear pool does not reduce the measurement overhead, especially with the increase of the molecule complexity and size.
CO_{2} molecule
According to precise classical ab initio calculations, an accurate description of reaction energies involving carbon dioxide requires incorporating triple and even higher order of excitations in coupled cluster ansatz^{40}. Such a slow convergence arises from the two degenerated πmolecular orbitals.
The full simulation of carbon dioxide in STO3G basis set requires 30 qubits, which can be further reduced to 19 by freezing core orbitals and tapering 5 qubits considering the symmetry point group. To the best of our knowledge, it is the first such sizeable numerical computation of CO_{2} ground state energy using quantum computing devcies.
Due to the complex correlation effects, CCSD significantly deviates from the exact ground state energy at the equilibrium geometry. Energy recovered by fermionic VQEUCCSD for carbon dioxide is close to the classical CCSD and is about 25 mHartree higher than the exact energy.
For fermionic ADAPTVQE, we perform 55 iterations to check if batched ADAPTVQE fits the original convergence curve (see Fig. 6c). Based on the previous results, we expect batched ADAPTVQE to perform similarly to original ADAPTVQE in circuit size. Fermionic batched ADAPTVQE requires only 4 iterations and 176 parameters to achieve fermionic VQEUCCSD energy. The number of parameters is thus significantly reduced compared to UCCSD, which has 204 excitations. The number of computed derivatives in batched fermionic ADAPTVQE procedure is almost the same as in VQEUCCSD (3900 and 4300 N_{d1}) in this case, due to the great parameter savings. As it is observed for other molecules, the convergence slows down significantly when we try to improve energy beyond VQEUCCSD and approach the exact result. We perform simulations up to doubling the parameter number compared to the UCCSD pool size for fermionic and qubit batched ADAPTVQE. For fermionic batched ADAPTVQE, we achieved energy about 2.5 mHartree higher than the exact result, which is very close to the chemical accuracy threshold.
Since greedy MCP is not efficient for H_{2}O and CO molecules, we run the simulation only for the Greedy sd pool due to the high computational cost of CO_{2} simulation. For original qubit ADAPTVQE, we perform around 175 iterations to reach the gradient norm less than 10^{−1} (see Methods, Section titled Convergence Criteria). The energy difference between batched and original qubit ADAPTVQE is less than 1 mHartree at a fixed number of parameters confirming batched ADAPTVQE’s decent performance. Batched qubit ADAPTVQE approaches VQEUCCSD with 390 parameters (see Fig. 6c) and the circuit depth of 1560 in contrast to 24,800 for the fermionic VQEUCCSD. However, such circuit reduction goes at the price of 40 times more derivative computations.
Electronic energy of reaction
Since relative energies but not their absolute values are crucial to determine the accuracy of ab initio methods, we can compare the electronic energy of COoxidation reaction computed by the studied methods:
$${{\mbox{CO}}}+\frac{1}{2}{{{\mbox{O}}}}_{2}\to {{{\mbox{CO}}}}_{2}.$$
(1)
Electronic energy of this reaction is given by the following formula:
$$\Delta {E}_{r}={E}_{{{{{{{{{\rm{CO}}}}}}}}}_{2}}{E}_{CO}\frac{1}{2}{E}_{{{{{{{{{\rm{O}}}}}}}}}_{2}}$$
(2)
Table 2 shows the electronic energy of reaction computed by different methods. Classical coupled cluster (CCSD) only slightly improves the uncorrelated Hartree Fock (HF) result. The inclusion of triplet excitations in coupled cluster is necessary for accurate estimation of the electronic reaction energy.
We compute the electronic energy of the reaction using only the batched implementation of ADAPTVQE because CO_{2} molecule is too computationally expensive for the original ADAPTVQE simulation. We compare energies obtained with different stopping criteria. By setting the maximum derivative value to 10^{−3} (or alternatively the gradient norm to 10^{−2}) we could obtain accurate results for batched fermionic ADAPTVQE. When stopping the simulation after reaching \(\max {g}_{i}\le 1{0}^{2}\), batched ADAPTVQE obtains energies close to VQEUCCSD, but with shallower circuits. Therefore, the provided results for fermionic batched ADAPTVQE can be regarded as a theoretical benchmark of the adaptive ansatz for reaching exact energy.
Qubit batched ADAPTVQE converges significantly slower, which leads to a poor description of the reaction’s electronic energy. It is worth noting that energy underestimation also relates to the fact that qubit ADAPTVQE quickly converges for O_{2} molecule compared to CO and CO_{2}. It should be noted that both the maximum derivative value and gradient change nonmonotonically, leading to a significant variance in the estimated reaction energy. As we do not achieve \(\max {g}_{i}\le 1{0}^{3}\) for CO_{2}, we estimate the «best” energy by taking the results for O_{2} and CO at the number of parameters two times larger than in the UCCSD pool. The best achieved energy for qubit ADAPTVQE only slightly improves HF result. Improving VQEprocedure is necessary for qubit ADAPTVQE to speed up the convergence and obtain an accurate result.
From the energies obtained with various stopping criteria, one can see sluggish convergence of qubit ADAPTVQE on molecules with complicated electronic structures. Even stricter convergence criteria are required to further energy improvement, which would lead to a significant increase in the computational cost. Our simulation could reach fermionic VQEUCCSD energy, which is already a promising result, as we tested it on complex and relatively large molecules.