Limitations of Quantum Hardware for Molecular Energy Estimation Using VQE

Supporting Information: Limitations of Quantum Hardware for Molecular Energy Estimation Using VQE Abel Carreras,∗,†Rom´an Or´us,†,‡,¶and David Casanova‡,¶ †Multiverse Computing, Donostia, 20014, Euskadi, Spain ‡Donostia International Physics Center (DIPC), 20018 Donostia, Euskadi, Spain ¶IKERBASQUE, Basque Foundation for Science, 48009 Bilbao, Euskadi, Spain E-mail: abelca[email protected] Contents S1 Effective Hamiltonian S3 S2 Compression of electronic Hamiltonians S4 S3 Fermion vs. Qubit operator pools S7 S4 Classical optimizer S9 S4.1 Stretched H4model................................ S11 S5 Circuit optimization S15 S6 Job Size Limitations in IBM Quantum Computers S15 S7 Quantum hardware experiments S17 S1 S8 Instruction execution times S18 S2 S1 Effective Hamiltonian To construct the effective Hamiltonian, we apply the frozen core approximation. In this approach, the molecular spin-orbitals are divided into three sets: –Core spin-orbitals: low-energy spin-orbitals that are treated as an effective potential in the Hamiltonian; –Active spin-orbitals: which are treated explicitly in the Hamiltonian; –Truncated spin-orbitals: corresponding to high-energy spin-orbitals that are not included in the Hamiltonian. In this work we used a spin-orbital basis, where spin-up (α) and spin-down (β) components are interleaved. In this basis the effective Hamiltonian is defined as ˆ Heff = act X pq ˜ hpqˆa† pˆaq+1 2 act X pqrs gpqrsˆa† pˆa† qˆarˆas+ˆ Veff,(S.1) where ˜ hpq are the effective one-electron integrals, defined as ˜ hpq =hpq + core X i (giqpi −giipq),(S.2) with hpq denoting the one-electron integrals from the original Hamiltonian; and the effective potential term ˆ Veff, accounting for the core spin-orbitals, is given by ˆ Veff = core X i hii +1 2 core X ij (gjiij −gjjii).(S.3) Note that indices i, j run over the core spin-orbitals, while p, q, r, s run over the active spin-orbitals. S3 S2 Compression of electronic Hamiltonians In this section, we analyze the impact of Hamiltonian compression on the ground state energy. Additionally, we examine how the size of the Hamiltonian evolves as the number of active space orbitals increases, considering different compression thresholds. For this study, we construct the Hamiltonian for increasing active space sizes, ranging from 2 to 12 molecular orbitals, and apply different compression thresholds (1e-2, 1e-3, and 1e-4). The results are calculated using canonical orbitals (Figures S3 and S3), and using natural orbitals S1 and S2. Focusing in natural orbitals, figure S1 (left) shows the number of Hamiltonian terms as a function of the active space size. As expected, the number of terms increases with the active space size. For small active spaces, there is almost no difference between the three compression thresholds. However, beyond 8 orbitals, a significant difference emerges between 1e-2 and the other two (1e-3 and 1e-4). In this regime, the number of terms is notably reduced for 1e-2, while for 1e-3 and 1e-4, the difference remains marginal. Figure S1 (right) illustrates the impact of compression from an energy perspective by plotting the ground state (GS) energy as a function of the number of Hamiltonian terms. The results indicate that a compressed Hamiltonian provides a better GS energy-to-terms ratio compared to the uncompressed Hamiltonian. This trend holds for all compression thresholds except 1e-2, where, around 1000 terms, the ratio deteriorates, leading to a GS energy higher than that of the uncompressed Hamiltonian for the same number of terms. In Figure S2, we analyze the evolution of the GS energy error (defined as the difference between the GS energy of the compressed and uncompressed Hamiltonians) with respect to both the active space size (left) and the number of Hamiltonian terms (right). Even for a compression threshold of 1e-2, the GS energy error remains within chemical accuracy up to an active space of 6 orbitals. For 1e-3 compression, the GS energy remains below chemical accuracy up to 12 molecular orbitals. Notably, the difference between 1e-3 and 1e-4 is minimal. S4 Examining the energy error as a function of the number of Hamiltonian terms, we observe that a 1e-3 compression is highly efficient up to 6 active orbitals, yielding a Hamiltonian with negligible loss relative to the uncompressed one. Beyond this point, the energy error increases exponentially with both the number of active orbitals and the number of Hamiltonian terms. These results suggest that as the active space size increases—and thus the accuracy of the GS energy improves—small terms become more relevant, limiting the extent to which compression can be applied. However, for active spaces of up to 6 orbitals, a compression threshold of 1e-2 appears sufficient to maintain the GS energy error below chemical accuracy. Figure S1: Analysis of the impact of Hamiltonian compression on the number of Hamiltonian terms and the ground state energy. On the left, we show the number of Hamiltonian terms as a function of the active space size for different Hamiltonian with three compression thresholds (1e-2, 1e-3, and 1e-4), compared to the uncompressed Hamiltonian. On the right, we plot the ground state (GS) energy as a function of the number of Hamiltonian terms. The GS energies have been computed using the FCI method for three different compression thresholds (1e-3, 1e-4, and 1e-5). The Hamiltonian is expressed in the basis of natural Orbitals. S5 Figure S2: Analysis of the energy error with respect to different Hamiltonian compression thresholds (1e-2, 1e-3, and 1e-4). On the left, we show the ground state energy error as a function of the active space size. On the right, we present the GS energy error as a function of the number of Hamiltonian terms. The ground state energy energies have been computed using the FCI method. The Hamiltonian is expressed in the basis of natural Orbitals. Examining the results obtained using canonical orbitals (Figures S3 and S4), we observe that the use of canonical orbitals is detrimental compared to the use of natural orbitals. The energy-to-number-of-terms ratio presented in Figure S3 (right) is significantly worse than that observed with natural orbitals, where compression provides little to no benefit at any level. Furthermore, we observe that the GS energy error with respect to the uncompressed Hamiltonian (Figure S4) is substantially higher when using canonical orbitals, frequently exceeding chemical accuracy. This suggests that the use of natural orbitals leads to Hamiltonians where the most important interactions are concentrated in a smaller number of terms, making the compression procedure more effective. However, it is important to note that as higher precision is required, the contribution of small Hamiltonian terms becomes more significant, and their impact on the ground state energy may no longer be negligible. S6 Figure S3: Analysis of the impact of Hamiltonian compression on the number of Hamiltonian terms and the ground state energy. On the left, we show the number of Hamiltonian terms as a function of the active space size for different Hamiltonian with three compression thresholds (1e-2, 1e-3, and 1e-4), compared to the uncompressed Hamiltonian. On the right, we plot the ground state (GS) energy as a function of the number of Hamiltonian terms. The GS energies have been computed using the FCI method for three different compression thresholds (1e-3, 1e-4, and 1e-5). The Hamiltonian is expressed in the basis of canonical orbitals. Figure S4: Analysis of the energy error with respect to different Hamiltonian compression thresholds (1e-2, 1e-3, and 1e-4). On the left, we show the ground state energy error as a function of the active space size. On the right, we present the GS energy error as a function of the number of Hamiltonian terms. The ground state energy energies have been computed using FCI method with a bond dimension of 200. The Hamiltonian is expressed in the basis of canonical Orbitals. S3 Fermion vs. Qubit operator pools The quantum circuits generated by fermionic ADAPT-VQE are constructed by mapping fermionic operators to qubit operators and implemented on quantum hardware using a firstorder Trotter approximation. We note that this Trotterization can potentially break some S7 symmetries. Figure S5: Ground state energy (in Hartrees) as a function of the number of iterations for the simulation of the benzene molecule with fermion (green) and qubit (blue) ADAPT-VQE algorithm. Figure S6: Comparison of the computational cost of VQE algorithms using fermion vs qubit operators for benzene molecule. On the left there is a break down of the total (accumulated) number of CNOTs used for all circuits in the full simulation. We separate this number in energy evaluations and gradient evaluations. On the right there is an analysis of the circuit depths of the circuits used to evaulate the energy and the gradients. We show the maxium circuit depth and the average circuit depth. S8 Figure S7: Comparison of the circuit depth (of the circuit used to evaluate the energy) vs energy obtained from an ADAPT-VQE simulation (exact). On the left fermion-ADAPTVQE, on the right qubit-ADAPT-VQE. S4 Classical optimizer Figure S8: Energy surface generated by the scan over the coefficients of a two operator qubit-ADAPT-VQE ansatz of H4molecule. Left figure shows the scan along coefficients 1 and 2 and right figure is the scan along coefficients 2 and 3 S9 both the number of two-qubit gates and the number of Pauli observables measured. The data shown in the plot below reflects the limitations observed at the time of writing this manuscript. A job was considered not allowed if the server returned an error at submission time, indicating that the circuit exceeded current resource or policy constraints. Figure S17: Maximum job size accepted by various IBM quantum computers, measured as the product of the number of two-qubit gates and the number of measured Pauli operators in the observable. IBM Brussels was tested 3 times S16 S7 Quantum hardware experiments Figure S18: Energy evaluations (in Ha) with Qubit-ADAPT ans¨atze with 0-7 qubit operators measured in the IBM Torino computer using 9000 shots per sample. The coefficients of the ansatz are obtained from an exact calculation. Horizontal lines indicate the exact HF (green) and FCI with 4 active orbitals and 4 active electrons (gray). The energy difference between HF and FCI (correlation energy) is 53.9 mHa. Error bars corresponds to 2σ. S17 S8 Instruction execution times Instructions times consider in our Qubit-ADAPT simulations with Qiskit Aer simulator are shown in Table S3, corresponding to: •Parameterized single-qubit rotation gates –U1: phase shift gate. –U2: single-qubit rotation with two parameters. –U3: general single-qubit unitary rotation with three parameters. •CX: CNOT gate. •Reset: reset operation initializing a qubit back to the |0⟩state. •Measure: the measurement operation collapses a qubit’s state into either |0⟩or |1⟩and records the result. Table S3: Instruction execution times (in ns) used in the noise model. instruction time U1 0 U2 50 U3 100 CX 300 Reset 1000 Measure 1000 S18