Quantum equation of motion for computing molecular excitation energies on a noisy quantum processor

The computation of molecular excitation energies is essential for predicting photo-induced reactions of chemical and technological interest. While the classical computing resources needed for this task scale poorly, quantum algorithms emerge as promising alternatives. In particular, the extension of the variational quantum eigensolver algorithm to the computation of the excitation energies is an attractive choice. However, there is currently a lack of such algorithms for correlated molecular systems that is amenable to near-term, noisy hardware. Here, we introduce an efficient excited states quantum algorithm, that employs a quantum version of the well-established classical equation of motion approach, which allows the calculation of the excitation energies of a given system using an approximated description of its ground state wave function. We numerically test the algorithm for several small molecules and experimentally demonstrate the robustness of the algorithm by computing the excitation energies of a LiH molecule at varying noise levels.

Quantum computing is emerging as a new paradigm to solve a wide class of problems that show unfavorable scaling on conventional classical high performance computers [1,2].In particular, solving quantum chemistry and quantum physics problems using classical algorithms is hampered by the exponential growth of the resources (classical processors and memory) as a function of the number of degrees of freedom, N , (e.g., number of electrons or molecular basis functions) encoded in the system Hamiltonian.
The resources needed to compute the solution of the Schrödinger equation for molecular and solid state systems have a factorial scaling in the full Configuartion Interaction (full CI) representation of the ground state wave function [3] and O(N 10 ) for the Coupled Cluster (CC) expansion when truncated at the fourth order in the electronic excitation operator, named CCSDTQ (S stands for single, D for double, T for triple and Q for quadruple excitations).At present, the CCSD(T) expansion (that includes an approximated treatment of the triple excitations [4,5]) with a scaling O(N 7 ) is often considered to be the "gold standard" for quantum chemistry calculations.Energies computed at the CCSD(T) level of theory have an error that lies within the so-called chemical accuracy (errors less than 1 -2 kcal/mol = 0.043 -0.087 eV) for many systems (i.e., when no strong static correlation or multi-reference character of the ground state is present [6,7]).
Recently, the Variational Quantum Eigensolver (VQE) algorithm [8][9][10][11] was proposed for the efficient approximation of the electronic structure in near-term quantum computers.This algorithm is based on a parametrization of trial ground state wave functions.The parameters are encoded in single-qubit and two-qubit gate angles and are optimized self-consistently, using a classical processor, until the minimum ground state energy is reached.The energy corresponding to a given set of parameters is obtained by computing the expectation value of the system Hamiltonian and becomes therefore a function of the gate variables.The VQE has already been successfully applied to the simulation of the ground state properties of simple molecular systems on quantum hardware [12][13][14], and extended to more complex molecules in quantum simulators [15,16].
The calculation of molecular excited state properties constitutes an additional challenge for both classical and quantum electronic structure algorithms.In fact, in addition to calculating a well-converged ground state wave function, one needs to devise schemes for the evaluation of the higher energy states, which -in generalare not accessible through the optimization of a trial state.In classical computing, excited states are typically computed through perturbation theory (linear response, LR) or variational approaches (e.g., the equation of motion, EOM) starting from a ground state wave function optimized at a given level of theory (e.g., CC, multi-configurartional self-consistent field, configuration interaction, etc. [1]).In particular, CC theory was also extended to the calculation of excited state wave functions and energies using, for example, LR [17], EOM [18], state-universal multi-reference [19] and valence-universal multi-reference approaches [20].Alternatively, density functional theory (DFT) in its time-dependent formulation (namely, time-dependent density functional theory, TDDFT) or Green's function based techniques (like GW and the Bethe-Salpeter equation, BSE) can be used to evaluate excited state properties in the LR regime.
Independently of the approach used to compute the ground state (wave function, density, or Green's function based), LR theory makes it possible to identify excited states as the poles of the density-density response function (or the two-body Green's function in BSE), and requires therefore the evaluation of LR states, which are beyond the possibilities of currently available quantum algorithms for electronic structure calculations.In contrast, the EOM is based on a variational approach applied to the ground state wave function.Hence, it becomes the method of choice for the implementation of excited state properties within the VQE approach [21].
Other VQE based quantum algorithms for computing electronic transition energies were recently proposed.A useful overview can be found in [22].Among these algorithms the Quantum Subspace Expansion (QSE) [23] was used to compute the excited states of the H 2 molecule using two qubits [24].We show here that, compared to the QSE, the EOM strategy allows for a more robust computation of the excitation energies of the system.Furthermore the EOM method does not neglect contributions from de-excitations, which are important for correlated systems (see the Supplementary Materials).

I. THEORY
The EOM approach, first derived by Rowe [25], was extensively reviewed [26,27] and implemented in a series of electronic structure packages like Gaussian [28].Within this approach, excited states |n are generated by applying an excitation operator of the general form Ô † n = |n 0| to the ground state |0 of the system, where |n is the shorthand notation for the n-th excited state of the electronic structure Hamiltonian.Similarly, a deexcitation operator can be written as Ôn = |0 n|.
Taking the commutator of the Hamiltonian and the excitation operator and introducing the double commutator notation, [ Â, B, Ĉ] = 1  2 {[[ Â, B], Ĉ]+[ Â, [ B, Ĉ]]} [29], we can derive an expression for the excitation energies, The EOM approach aims at finding approximate solutions to Eq. ( 1) by expressing Ô † n as a linear combination of basis excitation operators with variable expansion coefficients.The excitation energies are then obtained through the minimization of Eq. (1) in the coefficient space.The simplest basis is composed of the Fermionic orbital creation and annihilation operators â † and â, where â † m âi represents the excitation of a single electron from an occupied orbital i to a virtual orbital m, and â † m â † n âi âj the double excitation of a pair of electrons from the occupied orbitals i, j to the virtual orbitals m, n.Calling α the degree of excitation, we can express Ô † n as where µ α is a collective index for all one-electron orbitals involved in the excitation.In this work, we will restrict our excitation operator basis to single (α = 1) and double (α = 2) excitations such that Ê(1) n âi âj , ( Ê(1) µ1 ) † = â † i âm and ( Ê(2) µ2 ) † = â † i â † j âm ân .By inserting the expansion of Eq. (2) into Eq.( 1) we obtain a parametric equation for the excitation energies.Applying the variational principle δ(E 0n ) = 0 in the parameter space spanned by the coefficients X (α) µα and Y (α) µα we obtain the following secular equation where Note that the rank of all these matrices equals the number of possible single and double excitations included in the active space that defines the operators in Eq. ( 2).Due to the limited size of this extended eigenvalue problem, the eigenvalues of Eq. ( 3) can be evaluated classically.In the quantum adapted version of EOM, named qEOM, the Jordan-Wigner transformation is used to map the commutators [( Ê(α) µα ) † , Ĥ, Ê(β) ν β ] and [( Ê(α) µα ) † , ( Ê(β) ν β ) † ], which are originally expressed in terms of the Fermionic creation and annihilation operators, into the qubits space.They are then evaluated using the ground state wave function obtained from, e.g., a VQE calculation, to compute the matrix elements of M, Q, V and W. From these measurements the secular equation (Eq.( 3)) is constructed.Its 2n eigenvalues are then classically solved to obtain the first n excitation (and corresponding de-excitation) energies.A graphical representation of the qEOM algorithm is given in Fig. 1a.

Each excitation operator Ê(α)
µα corresponds to a fixed number of Pauli-tensor-strings (PS), independent of the system size.In the worst case it is 4 for the single excitation (α = 1) and 16 for the double excitation (α = 2) operators.Hence, the number of measurements scales with the number of PS in the Hamiltonian: O(N 4 ).Moreover, the size of the M, V, Q and W matrices is O(O 2 V 2 ) where O and V are the number of occupied and virtual orbitals in the chosen active space, respectively.The overall scaling is O(O 2 V 2 N 4 ).To reduce the number of measurements we propose to group the PS such as in [12].
< l a t e x i t s h a 1 _ b a s e 6 4 = " x M m i Q 8 K O 1 w y p H W u N k 6 e A J I / i O 0 I = " > A A A D h H i c t V J b a 9 R A F J 5 N t F 1 X q 1 v 7 6 M v g 4 g X R J b E t 9 q V S F M G X Q h f c i 2 y W M J k 9 2 R 0 6 m Y S Z k 7 J L y B / x Z / n m v 3 H 2 A r V p o H 3 x w A w f 3 3 f u n C i T w q D n / W k 4 7 o O H O 7 v N R 6 3 H T / a e P m v v P x + Y N N c c + j y V q R 5 F z I A U C v o o U M I o 0 8 C S S M I w u v y 6 0 o d X o I 1 I 1 Q 9 c Z j B J 2 E y J W H C G l g r 3 G 7 + C C G Z C F V n C U I t F 2 a L W A o Q F F u s / i o v z s q S v K 1 y v L I O A 1 j j 3 3 t V 4 n 1 u y F Y C a X p e 5 u + y o L E N V X + T n S q o k P K X f w s J T 9 8 k 8 q G l x e D 3 Q h 4 q y m a j C D v 7 / S G G 7 4 3 W 9 t d H b w N + C D t n a R d j + H U x T n i e g k E t m z N j 3 M p w U T K P g E m y D u Y G M 8 U s 2 g 7 G F i i V g J s X 6 i E r 6 y j J T G q f a P o V 0 z f 4 b U b D E m G U S W U / b 3 9 x U t R V Z p 4 1 z j E 8 m h V B Z j q D 4 p l C c S 4 o p X V 0 k n Q o N H O X S A s a 1 s L 1 S P m e a c b R 3 2 7 J L 8 K s j 3 w a D j 1 3 f 6 / q 9 o 8 7 Z l + 0 6 m u Q F e U n e E p 9 8 I m f k O 7 k g f c K d h v P G 8 R z f 3 X H f u 4 f u 8 c b V a W x j D s g N c z / / B e t I I a g = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " x M m i Q 8 K O 1 w y p H W u N k 6 e A J I / i O 0 I = " > A A A D h H i c t V J b a 9 R A F J 5 N t F 1 X q 1 v 7 6 M v g 4 g X R J b E t 9 q V S F M G X Q h f c i 2 y W M J k 9 2 R 0 6 m Y S Z k 7 J L y B / x Z / n m v 3 H 2 A r V p o H 3 x w A w f 3 3 f u n C i T w q D n / W k 4 7 o O H O 7 v N R 6 3 H T / a e P m v v P x + Y N N c c + j y V q R 5 F z I A U C v o o U M I o 0 8 C S S M I w u v y 6 0 o d X o I 1 I 1 Q 9 c Z j B J 2 E y J W H C G l g r 3 G 7 + C C G Z C F V n C U I t F 2 a L W A o Q F F u s / i o v z s q S v K 1 y v L I O A 1 j j 3 3 t V 4 n 1 u y F Y C a X p e 5 u + y o L E N V X + T n S q o k P K X f w s J T 9 8 k 8 q G l x e D 3 Q h 4 q y m a j C D v 7 / S G G 7 4 3 W 9 t d H b w N + C D t n a R d j + H U x T n i e g k E t m z N j 3 M p w U T K P g E m y D u Y G M 8 U s 2 g 7 G F i i V g J s X 6 i E r 6 y j J T G q f a P o V 0 z f 4 b U b D E m G U S W U / b 3 9 x U t R V Z p 4 1 z j E 8 m h V B Z j q D 4 p l C c S 4 o p X V 0 k n Q o N H O X S A s a 1 s L 1 S P m e a c b R 3 2 7 J L 8 K s j 3 w a D j 1 3 f 6 / q 9 o 8 7 Z l + 0 6 m u Q F e U n e E p 9 8 I m f k O 7 k g f c K d h v P G 8 R z f 3 X H f u 4 f u 8 c b V a W x j D s g N c z / / B e t I I a g = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " x M m i Q 8 K O 1 w y p H W u N k 6 e A J I / i O 0 I = " > A A A D h H i c t V J b a 9 R A F J 5 N t F 1 X q 1 v 7 6 M v g 4 g X R J b E t 9 q V S F M G X Q h f c i 2 y W M J k 9 2 R 0 6 m Y S Z k 7 J L y B / x Z / n m v 3 H 2 A r V p o H 3 x w A w f 3 3 f u n C i T w q D n / W k 4 7 o O H O 7 v N R 6 3 H T / a e P m v v P x + Y N N c c + j y V q R 5 F z I A U C v o o U M I o 0 8 C S S M I w u v y 6 0 o d X o I 1 I 1 Q 9 c Z j B J 2 E y J W H C G l g r 3 G 7 + C C G Z C F V n C U I t F 2 a L W A o Q F F u s / i o v z s q S v K 1 y v L I O A 1 j j 3 3 t V 4 n 1 u y F Y C a X p e 5 u + y o L E N V X + T n S q o k P K X f w s J T 9 8 k 8 q G l x e D 3 Q h 4 q y m a j C D v 7 / S G G 7 4 3 W 9 t d H b w N + C D t n a R d j + H U x T n i e g k E t m z N j 3 M p w U T K P g E m y D u Y G M 8 U s 2 g 7 G F i i V g J s X 6 i E r 6 y j J T G q f a P o V 0 z f 4 b U b D E m G U S W U / b 3 9 x U t R V Z p 4 1 z j E 8 m h V B Z j q D 4 p l C c S 4 o p X V 0 k n Q o N H O X S A s a 1 s L 1 S P m e a c b R 3 2 7 J L 8 K s j 3 w a D j 1 3 f 6 / q 9 o 8 7 Z l + 0 6 m u Q F e U n e E p 9 8 I m f k O 7 k g f c K d h v P G 8 R z f 3 X H f u 4 f u 8 c b V a W x j D s g N c z / / B e t I I a g = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " x M m i Q 8 K O 1 w y p H W u N k 6 e A J I / i O 0 I = " > A A A D h H i c t V J b a 9 R A F J 5 N t F 1 X q 1 v 7 6 M v g 4 g X R J b E t 9 q V S F M G X Q h f c i 2 y W M J k 9 2 R 0 6 m Y S Z k 7 J L y B / x Z / n m v 3 H 2 A r V p o H 3 x w A w f 3 3 f u n C i T w q D n / W k 4 7 o O H O 7 v N R 6 3 H T / a e P m v v P x + Y N N c c + j y V q R 5 F z I A U C v o o U M I o 0 8 C S S M I w u v y 6 0 o d X o I 1 I 1 Q 9 c Z j B J 2 E y J W H C G l g r 3 G 7 FIG. 1: a. Graphical representation of the qEOM algorithm.b.Lower panels: Dissociation profile of the H 2 , the LiH and the H 2 O molecules.The gray lines represent the exact eigenenergies of the Hamiltonian obtained from its diagonalizaton.The coloured crosses depict the ground state (red), the first (green), the second (blue) and the third (purple) excited states obtained with the qEOM.Upper panels: Corresponding energy errors along the dissociation profile.The gray shaded area corresponds to the energy range within chemical accuracy.

II. SIMULATIONS OF THE QEOM ALGORITHM
To validate the performance and the accuracy of the qEOM approach, statevector-type simulations (where the exact unitary matrix representation of the circuit is applied on the state vector, no sampling or hardware noise is included) are performed.The algorithm is tested on three molecules, namely H 2 , LiH and H 2 O.We prepare the Hamiltonians by computing the one-and two-electron integrals using a Hartree-Fock/STO-3G calculation performed on Gaussian09 [28].To improve the efficiency of the ground state calculation, the resulting Hamiltonian is mapped into the particle-hole framework [30].In the case of LiH and H 2 O effective core potentials (ECPs) [31] are used to replace 1s core electrons.In this way, we are able to reduce the number of qubits to 4, 10 and 12 for H 2 , LiH and H 2 O, respectively.
In Fig. 1b we show the dissociation profile of H 2 , LiH and H 2 O obtained with qEOM (coloured crosses) and from the exact diagonalization of the Hamiltonian (gray lines).In the former case, the ground state is obtained with VQE using the UCCSD Ansatz (full active space), the Hartree-Fock initial state and the L-BFGS-B optimizer.The excited states shown in Fig. 1 are the 3 lowest lying excited states found by solving the qEOM pseudoeigenvalue equation.In this work, we choose to use only particle and spin conserving excitation operators Ê(α) µα .For this reason, the qEOM excited states displayed in Fig. 1 are not strictly the 3 lowest lying ones but rather the 3 lowest lying within this specific particle-and spinnumber subspace.By selecting a specific subset of the excitation operators it is therefore possible to address specific sectors of interest of the entire Hilbert space.
In the case of H 2 and LiH the accuracy of the ground state is excellent over the entire dissociation curve (see upper panels in Fig. 1), allowing us to compute accurate excited state energies within chemical accuracy (with errors < 0.015 Hartree, shaded area) for all geometries.For H 2 O, the VQE results are less accurate, leading to excited states slightly above chemical accuracy.
Assessing the error propagation from the ground state wave function calculation to the excited states energies is not straightforward and we propose a more detailed discussion in the Supplementary Materials.We found the error in the excitation energies to grow more slowly than the error in the ground state energy when adding an increasing error to the ground state parameters.The accuracy of the excitation energy is altered because of the propagation of errors through Eq. ( 3) due to the nature of the M, V, Q and W matrices.We expect the sampling noise to have a stronger effect on the results in strong correlation regimes.Moreover, the propagation of errors from the ground state to the excitation energies < l a t e x i t s h a 1 _ b a s e 6 4 = " 8 v e s t H t q 4 B W / M s S h g                was demonstrated to be weaker with qEOM than QSE.However, when excited state energies become almost degenerate, i.e., when one of the solution of the EOM generalized eigenvalue problem, Eq. ( 3), is close to zero, the conditioning of the corresponding matrices deteriorates, affecting the quality of the resulting excitation energies.
In preparation of computing molecular excited states on a quantum hardware, we implement the qEOM algorithm in the Qiskit software library [32].This enables us to perform realistic noisy simulations and model the performance of the algorithm on the IBM Q Poughkeepsie device.Using Qiskit, the molecular orbitals are computed with Hartree-Fock/STO-3G on the PySCF classical code [33].In this case the core orbitals are simply frozen (note that the absolute energies are shifted in comparison to using ECPs).The circuit depth is reduced as explained in the next section.We observe that around the optimal ground state parameter value the error in the excitation energies is about one order of magnitude smaller than in the ground state energy (see Supplementary Materials).In the next section the relative robustness of the experimentally obtained excitation energies measured at varying noise levels is studied.

III. HARDWARE CALCULATION OF THE EXCITATION ENERGIES OF LIH
The UCCSD circuit for the optimization of the LiH ground state using a STO-3G basis set comprises over 12000 CNOT gates and 92 variational parameters.Given the limitations of state-of-the-art quantum hardware, we reduce the active space from 10 to 4 orbitals.The reduction of the active space in the quantum computing frame-work has already been discussed in literature [30,34].The orbitals composing the active space in LiH are selected according to their contribution to the CI expansion (see Supplementary Materials).The resulting active space is shown in Fig. 2 along with the layout of the 20 qubit superconducting processor IBM Q Poughkeepsie used for the experiment.In the 'conventional' quantum UCCSD, double excitations are encoded using 8 entangling blocks with different fixed pre/post rotations [30,35].Here, we replace the 8 blocks by a single one with variable pre/post rotations (in U ) where the angles are optimized in simulation to best approximate the exact UCCSD results (see Supplementary Materials).Due to this pre-processing procedure we are able to reduce the circuit, for LiH, to 6 CNOTs, 8 fixed single qubit rotations and a single variable qubit rotation R z (θ) as shown in Fig. 2.This modified variational UCC circuit used in the reduced active space can recover at least 56% of the correlation energy (and up to 87%, see the details in the Supplementary Materials) over the entire dissociation range considered, which corresponds to an energy error ≤ 7 mHa.Discrepancies are expected to be larger for long internuclear distances (> 2.5 Å) where strongly correlated effects become more important and a larger active space is therefore required.
Our experiments were performed on four qubits of the aforementioned IBM Q Poughkeepsie device (see Supplementary Materials for details).For the estimation of ground state and excitation energies, we employ the zero-noise extrapolation scheme proposed in [36,37] and recently implemented in [13].In this scheme, repeated measurements of expectation values at different noise levels are used to obtain a zero-noise estimate.The advantages of this are two-fold -while enabling error mitigated estimates to ground and excitation energies, it also enables us to test the robustness of the algorithm at varying noise levels.The noise levels are controllably varied by stretching the dynamics of the state preparation (see [13] and Supplementary Materials for details).
At each internuclear distance, the circuit parameter θ is swept, and energies are measured at three different stretch factors (1.0, 1.25 and 1.5), and a mitigated sweep is obtain using a linear extrapolation of these measurements.To reduce the sensitivity to any fluctuations, the optimal θ for the ground-state is obtained by fitting a quadratic curve to the mitigated sweep, as shown in Fig. 3a at the equilibrium bond length.The error mitigation protocol is then extended to each matrix element of the qEOM and the resulting "mitigated" secular equation is then solved classically, leading to the excitation energies E 0n with n = 1, . . ., 24.The absolute errors with respect to the statevector-type simulations are displayed on Fig. 3b.Error mitigation enables a gain in precision of approximately one order of magnitude in both the ground state and excitation energies.More importantly, we experimentally observe that the unmitigated results are more accurate for the excited states than the ground state (about 1e-2 Ha for the lowest energy states against an error superior to ∼1e-1 Ha for the ground state), from the runs at different stretch/noise amplification factors.Finally, we test our algorithm for varying internuclear distance, discussed in Fig. 3c (and Supplementary Materials).

IV. CONCLUSIONS
In this work, we introduced a quantum algorithm for the calculation of electronic excited state energies based on the equation of motion classical approach (EOM).The method, named qEOM, inherits the properties of the variational approaches used for ground state calculations.We tested the qEOM algorithm on three small molecules: H 2 , LiH and H 2 O, demonstrating that simulations can produce excitation energies within chemical accuracy (errors ≤ 1.5 mH).We studied the performance of our algorithm and showed that it is particularly well suited for calculations on state-of-the-art quantum device, manifesting robustness against hardware noise.While the ground state preparation remains the critical step in the process, we showed that our algorithm leads to more accurate results than the Quantum Subspace Expansion approach [23] especially when highly correlated ground state wave functions are involved.Finally, we adapted an error mitigation scheme to the qEOM approach and were able to compute the excitation energies of LiH on the IBM Q Poughkeepsie device.The stability of the qEOM algorithm, demonstrated in this work, opens up new avenues in the use of quantum computers for studying photochemical processes.
Supplementary Materials for: Quantum equation of motion for computing molecular excitation energies on a noisy quantum processor

I. ERROR PROPAGATION
Assessing the error propagation from the ground state wave function calculation to the excited state energies is not straightforward.To shed light on this issue, we compute the ground and excited states of a H 2 molecule.The Hamiltonian is prepared by computing the one-and two-electron integrals using a Hartree-Fock/STO-3G calculation performed on Gaussian09 [28].We compute the correlation energy by running a VQE with the UCCSD ansatz in a statevector-type manner (the exact unitary matrix of the circuit is applied on the quantum state vector).For this system, the VQE comprises three variational parameters.The excited state energies, E i , are computed by finding the excitation energies (energy gaps) with qEOM and add them to the ground state energy, E 0 .We add an 'ad-hoc' error, , to all parameters defining the ground state wave function and re-compute the ground state as well as the excited state energies as a function of .The absolute value of the corresponding ground and excited state energy variations Here, E 0 refers to the ground state energy and E i to the energy of state i.For all excited states calculations, Fig. S1b-d also report the ratio between the errors of the excited state i and the one obtained for the ground state as a function of .In all cases, the |∆E i ( )| (with i ∈ 0, 1, 2, 3) grow monotonically with the increase of .Moreover, the error slightly increases with the stretching of the bond.Interestingly, we also observe that the larger the value of is, the closer the errors in ground and excited states get (circles).This implies that the qEOM energy gaps are less sensitive to an error in the ground state than the ground state energy itself which, in this case, becomes the main source of error.As shown in Fig. S1b it was not possible to obtain energies for the first excited state for = 10 −1 and bond lengths larger than 1.4 Å.At these (a) < l a t e x i t s h a 1 _ b a s e 6 4 = " q Z / h y U B 1 8 < l a t e x i t s h a 1 _ b a s e 6 4 = " 3 x c j H N w u o L B r x a o i P s y q R h g j G / 4 = " G Y U S S F / J K 3 q x n 6 9 3 6 s D 5 n r S t W P n N E 5 m B 9 / Q I d 4 p k E < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " Y 0 l 8 V r m U 7 r J G Y U S S F / J K 3 q x n 6 9 3 6 s D 5 n r S t W P n N E 5 m B 9 / Q I d 4 p k E < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " Y 0 l 8 V r m U 7 r J G Y U S S F / J K 3 q x n 6 9 3 6 s D 5 n r S t W P n N E 5 m B 9 / Q I d 4 p k E < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " Y 0 l 8 V r m U 7 r J The legend shows the value of the added error.Squares: absolute difference between the energies computed with and without adding an error to the parameters.Circles: ratio between the absolute errors made on the excited state i and on the ground state.
< l a t e x i t s h a 1 _ b a s e 6 4 = " I d 5 L J X A a e c d 7 9 e r i C m B w k M 9    bond lengths the ground and the first excited state energies become almost degenerate.Within these conditions, the system to solve becomes ill-conditioned, and numerical instabilities appear.The same argument can also explain the slight deterioration of the first excited state energies for internuclear bond distances larger than 1.75 Å at all levels of noise.In the following we study the robustness of qEOM to statistical errors.In the second line of Fig. S2, we report the error of the norm of the matrices M, V and Q instead.Note that for H 2 , W is a null matrix.All absolute errors reported in Fig. S2 have been averaged over the aforementioned 100 realizations of the experiment.In general, we observe an increase of the absolute energy error with the statistical noise, i.e., with the decrease of NS.As mentioned above, each element of the M, V and Q matrices corresponds to a weighted sum of PS.The energy presents a large variance when the coefficients associated to the PS are large.In the case of the Q matrix, the PS coefficients are relatively small (< 10) across the whole range of internuclear distances.The matrix elements Q µαν β are therefore, weakly affected by the sampling error.The effects on the matrix V are more interesting.By definition, the coefficients weighing the PS of the V matrix elements do not depend on the bond length (they do not depend on the Hamiltonian).The error in the V matrix elements through the dissociation curve shows that the required number of shots for sampling the distribution increases with the bond length.With internuclear distance, the correlation effects increase and the wave function distribution broadens, requiring more shots to be accurately described.This is translated directly to the accuracy of the V matrix elements.Finally, the M matrix elements are weighted by large coefficients (∼ 50) at short bond lengths but they decrease as the internuclear distance increases.In this case the increase of the PS measurement errors and the decrease of the variance at large bond distances cancel out leading to a roughly constant accuracy for the matrix elements M µαν β over the entire dissociation range.We observe that the error in the excited state energies follow the error in the matrix norms (the V matrix, in this case, which is mostly affected).We also expect qEOM to be less accurate in strong correlation regimes (e.g. at large bond lengths), when the correct description of the ground state becomes difficult.In preparation of computing molecular excited states on a quantum hardware, we implement the qEOM algorithm in the Qiskit software library [32].This enables us to perform realistic noisy simulations and model the performance of the algorithm on the IBM Q Poughkeepsie device.Using Qiskit, the molecular orbitals of a H 2 and LiH molecules are computed with Hartree-Fock/STO-3G and the PySCF classical code [33].The core orbitals are frozen.The circuit depth is reduced as explained in the next section leading to circuits parametrized with a single angle θ for both molecules.The simulations are done using the qasm simulator with the Poughkeepsie noise model and 100K shots.We switch this parameter within the range [−π; π] and compute the ground and excited states of H 2 and LiH at equilibrium distance (0.75 Å and 1.6 Å respectively).The effects on the ground state energy as well as the excitation energies (energy gaps directly computed with qEOM) are shown in Fig. S3.Around the best θ value, the noise has the effect to shift up the computed ground state energy.On the other hand the qEOM is robust to the noise, leading to an error in the excitation energies (energy gaps) of about one order of magnitude smaller than for the ground state (see plots c and f of Fig. S3).The theta scans (plots a and d of Fig. S3) also help us determine a suitable region to inspect experimentally.In this region the energy change with θ should be higher than the fluctuations coming from the hardware noise but close enough to the bottom of the well such that the data can be fitted to an harmonic curve.Applying a fit to determine the bottom of the curve has the purpose of obtaining a value which is not biased by the fluctuations coming from the hardware (that are in the range of the energy change when approaching the optimal θ value).This region was selected to be θ ∈ [−1; 1].possible.We assess the performance of qEOM and QSE in computing the excitation energies (energy gaps which are in general of interest rather than absolute energies).These calculations are for the H 2 molecule (STO-3G basis set, 4 qubits).We consider the following ground state: where |Ψ 0 is the exact ground state obtained by diagonalization of the Hamiltonian.U is a random unitary matrix characterized by a density α of off-diagonal elements.The results are averaged over 10'000 trials.The left-hand side of Fig. S4 shows the ground state energy.For α = 0, the energy is exact (since U(0) is the identity matrix).For the qEOM the ground state energy corresponds to the expectation value of the Hamiltonian: Ψ| H |Ψ .In the case of QSE, the ground state energy is obtained after solving the pseudo-eigenvalue problem (see the right-hand side of Fig. S4).While for this system the QSE leads to more accurate absolute energies, the performance in computing the excitation energies is shown to be superior with qEOM than QSE when errors are made in the ground state preparation.I: The H 2 Hamiltonian at equilibrium bond length (0.75 Å).Listed are all the Pauli operators with the corresponding coefficients, not taking into account for the energy shifts due to the frozen core orbitals and the Coulomb repulsion between nuclei.Each column corresponds to a different tensor product basis set.X, Y , Z, I here stand for the Pauli matrices σ x , σ y , σ z and the identity operator on a single qubit subspace, respectively.The LiH Hamiltonian at equilibrium bond length (1.6 Å) after reduction of the active space.Listed are all the Pauli operators with the corresponding coefficients, not taking into account for the energy shifts due to the frozen core orbitals and the Coulomb repulsion between nuclei.Each column corresponds to a different tensor product basis set.X, Y , Z, I here stand for the Pauli matrices σ x , σ y , σ z and the identity operator on a single qubit subspace, respectively.The UCCSD circuit of LiH requires over 12000 CNOT gates and the optimization of 92 parameters.The circuit depth can, however, be reduced by following two steps.Firstly, the UCC ansatz is restricted to one excitation i.e., one variational parameter.This excitation is selected by computing the MP2 coefficients of the CI expansion,

III. MOLECULAR HAMILTONIANS AND CIRCUITS
where, h ijkl are the two-electron integrals and e i is the energy of orbital i.We select the excitation with the largest MP2 coefficient.By including only the relevant orbitals, we also reduce the 10-qubit LiH active space to a 4-qubit space.Indeed, we consider the active space to be represented by a 4-qubit quantum register while the inert space is mapped to a 6-qubit register, the active and inert registers are uncorrelated and therefore, where, Â and Î are operators acting on the active and inert space respectively.In the inert register, since the qubits are all in the |0 state, Z = I = 1 and X = Y = 0.It follows that only the active register has to be modeled in quantum hardware.Let us illustrate the previous paragraph with a short example.In the 10-qubit LiH, the four first qubits represent the active space while the six last ones are inert.Let's consider IZIZIIXIII to be one of the PS composing the Hamiltonian.In this example, IZIZ is measured on the active register while IIXIII is evaluated on the inert register.Since one of the qubits of the inert register is measured in the X basis, IIXIII inert = 0 and therefore, IZIZIIXIII full = 0.This PS can be set to 0. On the other hand, if we consider IZIZIIZIII, IIZIII inert = 1 and the value of the PS can be set to that measured on the active register: IZIZIIZIII full = IZIZ active .Thus, by reducing the active space we can measure the 10-qubit LiH Hamiltonian on a 4-qubit register.
The second step to reduce the circuit depth consists of modifying the UCC circuit.In the regular UCC, the circuit (derived by applying the Jordan-Wigner transformation on the classical UCC Ansatz) representing a single excitation is made up of four entangling blocks, with different pre/post rotations.Those four blocks are parametrized with the same angle θ.For the double excitations, a similar construction is used and extended to eight entangling blocks.All pre/post rotations are fixed and different for each entangling block.We propose to replace the four and eight entangling blocks by a single block in which the pre/post rotations angles {Φ} are optimized on a small system (e.g., H 2 is used to optimize the pre/post rotation angles of the double excitation blocks), see Fig. S5.For the 4-qubit LiH circuit, this allows us to further reduce the circuit from 48 to 6 CNOTs without loss of accuracy.This circuit can recover from 57% of the correlation energy at 1.0 Å up to 80% at 2.0 Å (the amount of correlation energy captured by this circuit along the dissociation curve, is given in Table III).We assume this approximation of the ground state to be good enough to compute accurate excitation energies.Regardless of the method chosen to approximate the ground state, we want to compute the excited states of the LiH  molecule in the STO-3G (10-qubit) basis.The ground state wave function is evolved in the reduced active space, and the previously described measurement method is applied to compute the qEOM operators and reconstruct the full pseudo-eigenvalue problem.The expected accuracy, given the reduced circuit, is depicted in Fig. S6 and lies between 1 and 10 mHa.

IV. EXPERIMENTAL DETAILS
The experiments presented in this work used 4 superconducting qubits (Q0, Q1 ,Q2 and Q5).See Figure 2 in the main text for the connectivity of the 20 qubit processor IBM Q Poughkeepsie.The qubit frequencies are in the range 4.8-5 GHz, with relaxation and coherence times of T 1 and T 2,echo ∼ 40 − 110µs.The single and two-qubit gates are implemented by all-microwave drives.Every trial circuit is composed of 6 CNOT gates, implemented using cross-resonance pulses and single qubit gates.The shortest single qubit gates used for the experiments are of duration 103 ns, and the shortest gate times for CNOT 21 , CNOT 10 and CNOT 05 are 1278 ns, 1210 ns and 1448 ns respectively.To improve the quality of the computation, we use the error mitigation scheme previously implemented in [13].Here, expectation values of interest are re-measured under amplified noise strengths in order to then extrapolate to the zero-noise limit.Under the assumption of time invariant noise, this noise amplification is achieved by stretching in time the single and two qubit gates that constitute the quantum circuit of interest.In this work, we employ stretch factors of c = 1, 1.25, 1.5 and use linear extrapolation for obtaining zero-noise estimates.An important consideration for noise amplification by stretching the gates is the introduction of undesired coherent errors, that could result in unphysical extrapolations.In this context, we employ a four-pulse echo sequence for the construction of the ZX 90 gate that serves as the primitive for realizing a CNOT.Similar sequences have in the past been employed to mitigate the effect of spectator interactions in Parity measurements for quantum error correction [41].At each stretch factor, the gates are characterized by randomized benchmarking and the obtained fiedlities are reported in Table 1.The average readout assignment errors for the four qubits were r ∼ 0.05.As discussed in [12,13] all measured expectation values were corrected for assignment infidelity using a readout calibration of all the basis states.The energy of the gaps grows from left to right.The gray shaded area corresponds to the energy range within chemical accuracy.The error bars are computed using 50 numerical experiments obtained by bootstrapping of the experimental data points, and depict the range between the 1st and 3rd quantile.For all the bond lengths, the excitation energies are seen to be more robust than the ground state energies, at various noise levels (i.e.stretch factors).
r 5 1 g / 6 q T C h U n C I r P F / U T S T G i 0 z h o T 2 j g K M e W M K 6 F v Z X y I d O M o w 2 t Y E P w F l 9 e J o 3 z s u e W v d u L U u U 6 i y N P j s k J O S M e u S Q V U i U 1 U i e c P J J n 8 k r e n C f n x X l 3 P u a t O S e b O S R / 4 H z + A H N J l u A = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " d w R 5 6 Z 9 Y l y o 8 0 c t s k o y I 7 v 9 I 9 L 4 r 5 1 g / 6 q T C h U n C I r P F / U T S T G i 0 z h o T 2 j g K M e W M K 6 F v Z X y I d O M o w 2 t Y E P w F l 9 e J o 3 z s u e W v d u L U u U 6 i y N P j s k J O S M e u S Q V U i U 1 U i e c P J J n 8 k r e n C f n x X l 3 P u a t O S e b O S R / 4 H z + A H N J l u A = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " d w R 5 6 Z 9 Y l y o 8 0 c t s k o y I 7 v 9 I 9 L 4 r 5 1 g / 6 q T C h U n C I r P F / U T S T G i 0 z h o T 2 j g K M e W M K 6 F v Z X y I d O M o w 2 t Y E P w F l 9 e J o 3 z s u e W v d u L U u U 6 i y N P j s k J O S M e u S Q V U i U 1 U i e c P J J n 8 k r e n C f n x X l 3 P u a t O S e b O S R / 4 H z + A H N J l u A = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " d w R 5 6 Z 9 Y l y o 8 0 c t s k o y I 7 v 9 I 9 L 4 r 5 1 g / 6 q T C h U n C I r P F / U T S T G i 0 z h o T 2 j g K M e W M K 6 F v Z X y I d O M o w 2 t Y E P w F l 9 e J o 3 z s u e W v d u L U u U 6 i y N P j s k J O S M e u S Q V U i U 1 U i e c P J J n 8 k r e n C f n x X l 3 P u a t O S e b O S R / 4 H z + A H N J l u A = < / l a t e x i t > ✓ 0 < l a t e x i t s h a 1 _ b a s e 6 4 = " H z I Y P j v j / j Q d s Q i B F x 6 m B O X J E E A = " > A A A B 7 3 i c b V D L S g N B E O y N r x h f U Y 9 e B o P g K e y K o M e g F 4 8 R z A O S J c x O e p M h s w 9 n e o U Q 8 h N e P C j i 1 d / x 5 t 8 4 S f a

5 t b x e 3 S
z u 7 e / k H 5 8 K h p k k w L b I h E J b o d c I N K x t g g S Q r b q U Y e B Q p b w e h 2 5 r e e U B u Z x A 8 0 T t G P + C C W o R S c r N T u 0 h C J 9 9 x e u e J W 3 T n Y K v F y U o E c 9 V 7 5 q 9 t P R B Z h T E J x Y z q e m 5 I / 4 Z q k U D g t d T O D K R c j P s C O p T G P 0 P i T + b 1 T d m a V P g s T b S s m N l d / T 0 x 4 Z M w 4 C m x n x G l o l r 2 Z + J / X y S i 8 9 i c y T j P C W C w W h Z l i l L D Z 8 6 w v N Q p S Y 0 u 4 0 N L e y s S Q a y 7 I R l S y I X j L L 6 + S 5 k X V c 6 v e / W W l d p P H U Y Q T O I V z 8 O A K a n A H d W i A A A X P 8 A p v z q P z 4 r w 7 H 4 v W g p P P H M M f O J 8 / y + O P y w = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " H z I 7 H s 0 Z I o S w z N L M F H M 3 o r I G C t M j M 2 o Y k P w l l 9 e J e 2 L h u c 2 v P v L W v O m i K M M J 3 A K d f D g C p p w B y 3 w g Y C A Z 3 i F N 0 c 5 L 8 6 7 8 7 F o L T n F z D H 8 g f P 5 A 3 q K j 4 c = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 8 v e s t H t q 4 B W / M s S h g I b K i c a 3 X b 0 7 H s 0 Z I o S w z N L M F H M 3 o r I G C t M j M 2 o Y k P w l l 9 e J e 2 L h u c 2 v P v L W v O m i K M M J 3 A K d f D g C p p w B y 3 w g Y C A Z 3 i F N 0 c 5 L 8 6 7 8 7 F o L T n F z D H 8 g f P 5 A 3 q K j 4 c = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 8 v e s t H t q 4 B W / M s S h g I b K i c a 3 X b 0 7 H s 0 Z I o S w z N L M F H M 3 o r I G C t M j M 2 o Y k P w l l 9 e J e 2 L h u c 2 v P v L W v O m i K M M J 3 A K d f D g C p p w B y 3 w g Y C A Z 3 i F N 0 c 5 L 8 6 7 8 7 F o L T n F z D H 8 g f P 5 A 3 q K j 4 c = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 8 v e s t H t q 4 B W / M s S h g I b K i c a 3 X b 0 7 H s 0 Z I o S w z N L M F H M 3 o r I G C t M j M 2 o Y k P w l l 9 e J e 2 L h u c 2 v P v L W v O m i K M M J 3 A K d f D g C p p w B y 3 w g Y C A Z 3 i F N 0 c 5 L 8 6 7 8 7 F o L T n F z D H 8 g f P 5 A 3 q K j 4 c = < / l a t e x i t > U ( 2) < l a t e x i t s h a 1 _ b a s e 6 4 = " K 4 S A J j Y n 7 + Q X q C 2 p z T d F f / B B u 7 o = " > A A A B 8 H i c b V B N T w I x E J 3 F L 8 Q v 1 K O X R m K C F 7 J L S P R I 9 O I R E x c w s C H d 0 o W G t r t p u y Z k w 6 / w 4 k F j v P p z v P l v L L A H B V 8 y y c t 7 M 5 m Z F y a c a e O 6 3 0 5 h Y 3 N r e 6 e 4 W 9 r b P z g C H w g I e I Z X e H O U 8 + K 8 O x / L 1 o K T z 5 z C H z i f P 3 w P j 4 g = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " K 4 S A J j Y n 7 + Q X q C 2 p z T d F f / B B u 7 o = " > A A A B 8 H i c b V B N T w I x E J 3 F L 8 Q v 1 K O X R m K C F 7 J L S P R I 9 O I R E x c w s C H d 0 o W G t r t p u y Z k w 6 / w 4 k F j v P p z v P l v L L A H B V 8 y y c t 7 M 5 m Z F y a c a e O 6 3 0 5 h Y 3 N r e 6 e 4 W 9 r b P z g C H w g I e I Z X e H O U 8 + K 8 O x / L 1 o K T z 5 z C H z i f P 3 w P j 4 g = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " K 4 S A J j Y n 7 + Q X q C 2 p z T d F f / B B u 7 o = " > A A A B 8 H i c b V B N T w I x E J 3 F L 8 Q v 1 K O X R m K C F 7 J L S P R I 9 O I R E x c w s C H d 0 o W G t r t p u y Z k w 6 / w 4 k F j v P p z v P l v L L A H B V 8 y y c t 7 M 5 m Z F y a c a e O 6 3 0 5 h Y 3 N r e 6 e 4 W 9 r b P z g C H w g I e I Z X e H O U 8 + K 8 O x / L 1 o K T z 5 z C H z i f P 3 w P j 4 g = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " K 4 S A J j Y n 7 + Q X q C 2 p z T d F f / B B u 7 o = " > A A A B 8 H i c b V B N T w I x E J 3 F L 8 Q v 1 K O X R m K C F 7 J L S P R I 9 O I R E x c w s C H d 0 o W G t r t p u y Z k w 6 / w 4 k F j v P p z v P l v L L A H B V 8 y y c t 7 M 5 m Z F y a c a e O 6 3 0 5 h Y 3 N r e 6 e 4 W 9 r b P z g C H w g I e I Z X e H O U 8 + K 8 O x / L 1 o K T z 5 z C H z i f P 3 w P j 4 g = < / l a t e x i t > U ( 3) < l a t e x i t s h a 1 _ b a s e 6 4 = " + 8 X s d 8 C N s 3 L S 0 B 8 U y v 8 O Y o 5 8 V 5 d z 4 W r Q U n n z m G P 3 A + f w B 9 l I + J < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " + 8 X s d 8 C N s 3 L S 0 B 8 U y v 8 O Y o 5 8 V 5 d z 4 W r Q U n n z m G P 3 A + f w B 9 l I + J < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " + 8 X s d 8 C N s 3 L S 0 B 8 U y v 8 O Y o 5 8 V 5 d z 4 W r Q U n n z m G P 3 A + f w B 9 l I + J < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " + 8 X s d 8 C N s 3 L S 0 B 8 U y v 8 O Y o 5 8 V 5 d z 4 W r Q U n n z m G P 3 A + f w B 9 l I + J < / l a t e x i t > U † ( 3) < l a t e x i t s h a 1 _ b a s e 6 4 = " C 2 c 7 G 4 i 7 f k X z L z L f s p C J N z J f W + g = " > A A A B / H i c b V B N S 8 N A E N 3 4 W e t X t E c v i 0 W o l 5 K o o M e i F 4 8 V T F t o Y t h s N u n S z S b s b o Q Q 6 l / x 4 k E R r / 4 Q b / 4 b t 2 0 O 2 v p g 4 P H e D D P z g o h H S C C s d F 5 1 H Y K 9 + P I y 6 Z 2 1 b a t t 3 1 0 0 O 9 d V H D V w B I 5 B C 9 j g E n T A L e g C B 2 B Q g G f w C t 6 M J + P F e D c + 5 q 0 r R j X T A H 9 g f P 4 A 2 L W U O g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " C 2 c 7 G 4 i 7 f k X z L z L f s p C J N z J f W + g = " > A A A B / H i c b V B N S 8 N A E N 3 4 W e t X t E c v i 0 W o l 5 K o o M e i F 4 8 V T F t o Y t h s N u n S z S b s b o Q Q 6 l / x 4 k E R r / 4 Q b / 4 b t 2 0 O 2 v p g 4 P H e D D P z g o h H S C C s d F 5 1 H Y K 9 + P I y 6 Z 2 1 b a t t 3 1 0 0 O 9 d V H D V w B I 5 B C 9 j g E n T A L e g C B 2 B Q g G f w C t 6 M J + P F e D c + 5 q 0 r R j X T A H 9 g f P 4 A 2 L W U O g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " C 2 c 7 G 4 i 7 f k X z L z L f s p C J N z J f W + g = " > A A A B / H i c b V B N S 8 N A E N 3 4 W e t X t E c v i 0 W o l 5 K o o M e i F 4 8 V T F t o Y t h s N u n S z S b s b o Q Q 6 l / x 4 k E R r / 4 Q b / 4 b t 2 0 O 2 v p g 4 P H e D D P z g o h H S C C s d F 5 1 H Y K 9 + P I y 6 Z 2 1 b a t t 3 1 0 0 O 9 d V H D V w B I 5 B C 9 j g E n T A L e g C B 2 B Q g G f w C t 6 M J + P F e D c + 5 q 0 r R j X T A H 9 g f P 4 A 2 L W U O g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " C 2 c 7 G 4 i 7 f k X z L z L f s p C J N z J f W + g = " > A A A B / H i c b V B N S 8 N A E N 3 4 W e t X t E c v i 0 W o l 5 K o o M e i F 4 8 V T F t o Y t h s N u n S z S b s b o Q Q 6 l / x 4 k E R r / 4 Q b / 4 b t 2 0 O 2 v p g 4 P H e D D P z g o FIG. 2: a. Active space and corresponding molecular orbitals of LiH. b.IBM Q Poughkeepsie device layout.The active orbitals are mapped onto the coloured qubits.c.UCC-inspired circuit of 4 qubits.The sets Φ i of angles are: Φ 0 = {π/2, 0.0, 0.930}, Φ 1 = {−π/2, π, −1.207}, Φ 2 = {−π/2, −π, 1.310}, Φ 3 = {−π/2, 0.0, 1.877}.

FIG. 3 :
FIG. 3: a. Ground state correlation energy versus the variational parameter θ at equilibrium bond length (1.6 Å).The energies computed with statevector-type simulations are shown together with the experimental results obtain for each of the three stretch factors and after mitigation.The yellow dotted curve displays the fit to the mitigated values.b.Error in the n energy gaps at equilibrium bond length (1.6 Å).The error is computed with respect to the results obtained by statevector-type simulation of the reduced circuit.The results are shown for each of the three stretch factors and after mitigation.The energy of the gaps grows from left to right.c.Lower panel: Dissociation profile of the five lowest-lying electronic transition energies of the LiH molecule.The markers show the mitigated experimental results.The dashed lines are the qEOM results from statevector-type simulations (using the reduced 4-qubit active space).Inset: Dissociation profile of the 6 lowest-lying electronic states of the LiH molecule.The qEOM transition energies are added to the ground state energy (red).Upper pannel: Corresponding energy errors along the dissociation profile.The gray shaded area corresponds to the energy range within chemical accuracy.The errorbars are computed using 50 numerical experiments obtained by bootstrapping of the experimental data points, and depict the range between the 1st and 3rd quantile.
FIG. S1:Error in a ground state, b first, c second and d third excited state energies when adding an error to the VQE optimized ground state parameters.The legend shows the value of the added error.Squares: absolute difference between the energies computed with and without adding an error to the parameters.Circles: ratio between the absolute errors made on the excited state i and on the ground state.
s u r P S 3 U j H h l g v 0 h J e 2 e I 6 e T M 0 l W q W w L i 4 / d q K w E 3 0 9 b B 1 / W R D a J u / I A f l A I n J E j s k Z u S B d I s h P 8 p t c k 5 v g V / A v u A 3 u H l o 3 g s X M P l m q 4 P 4 / O K y x 7 g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " I d 5 L J X A a e c d 7 9 e r i C m B w k M 9 1 Z P 4 j h I O d i y s c w 8 N J w D S 4 u 5 9 + t 6 H v v D O k o s / 4 Z p H P 3 8 U T J t X M z n f p O z X H i V r P a f C o b F D j 6 F J f S 5 A W C E Q + L R o W i m N G a H R 1 K C w L V z A s u r P S 3 U j H h l g v 0 h J e 2 e I 6 e T M 0 l W q W w L i 4 / d q K w E 3 0 9 b B 1 / W R D a J u / I A f l A I n J E j s k Z u S B d I s h P 8 p t c k 5 v g V / A v u A 3 u H l o 3 g s X M P l m q 4 P 4 / O K y x 7 g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " I d 5 L J X A a e c d 7 9 e r i C m B w k M 9 1 Z P 4 = " > A A A C Q X i c b Z D P a h s x E M a 1 a d o k 7 p 8 4 z b E X E V P o w Z j d E k g v h d A Q y C W Q Q h 0 b r O 2 i l c e 2 s K R d p N l Q s + w T 9 T H 6 B D 0 2 O f f Q W 8 g 1 l 2o d H 2 I 7 A 4 K P 7 5 t h R r 8 0 V 9 J h G P 4 J N p 5 t P n + x t b 3 T e P n q 9 Z v d 5 t 7 b S 5 c V V k B X Z C q z / Z Q 7 U N J A F y U q 6 O c W u E 4 V 9 N L p S Z 3 3 r s A 6 m Z l v O M s h 1 n x s 5 E g K j t 5 K m q f n S c l 0 w U x R 0 c + U p Z a X Y U U H p 3 O 3 + l 6 y I R + P w V Z t h v A D y 7 O q T e v M t 8 c J y z W b A v q B p N k K O + G 8 6 L q I F q J F F n W RN P + y Y S Y K D Q a F 4 s 4 N o j D H u O Q W p V B Q N V j h I O d i y s c w 8 N J w D S 4 u 5 9 + t 6 H v v D O k o s / 4 Z p H P 3 8 U T J t X M z n f p O z X H i V r P a f C o b F D j 6 F J f S 5 A W C E Q + L R o W i m N G a H R 1 K C w L V z A s u r P S 3 U j H h l g v 0 h Je 2 e I 6 e T M 0 l W q W w L i 4 / d q K w E 3 0 9 b B 1 / W R D a J u / I A f l A I n J E j s k Z u S B d I s h P 8 p t c k 5 v g V / A v u A 3 u H l o 3 g s X M P l m q 4 P 4 / O K y x 7 g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " I d 5 L J X A a e c d 7 9 e r i C m B w k M 9 1 Z P 4= " > A A A C Q X i c b Z D P a h s x E M a 1 a d o k 7 p 8 4 z b E X E V P o w Z j d E k g v h d A Q y C W Q Q h 0 b r O 2 i l c e 2 s K R d p N l Q s + w T 9 T H 6 B D 0 2 O f f Q W 8 g 1 l 2 o d H 2 I 7 A 4 K P 7 5 t h R r 8 0 V 9 J h G P 4 J N p 5 t P n + x t b 3 T e P n q 9 Z v d 5 t 7 b S 5 c V V k B X Z C q z / Z Q 7 U N J A F y U q 6 O c W u E 4 V 9 N L p S Z 3 3 r s A 6 m Z l v O M s h 1 n x s 5 E g K j t 5 K m q f n S c l 0 w U x R 0 c + U p Z a X Y U U H p 3 O 3 + l 6 y I R + P w V Z t h v A D y 7 O q T e v M t 8 c J y z W b A v q B p N k K O + G 8 6 L q I F q J F F n W R N P + y Y S Y K D Q a F 4 s 4 N o j D H u O Q W p V B Q N V j h I O d i y s c w 8 N J w D S 4 u 5 9 + t 6 H v v D O k o s / 4 Z p H P 3 8 U T J t X M z n f p O z X H i V r P a f C o b F D j 6 F J f S 5 A W C E Q + L R o W i m N G a H R 1 K C w L V z As u r P S 3 U j H h l g v 0 h J e 2 e I 6 e T M 0 l W q W w L i 4 / d q K w E 3 0 9 b B 1 / W R D a J u / I A f l A I n J E j s k Z u S B d I s h P 8 p t c k 5 v g V / A v u A 3 u H l o 3 g s X M P l m q 4 P 4 / O K y x 7 g = = < / l a t e x i t >V µ⌫ = h0| [E † µ , E ⌫ ] ± |0i < l a t e x i t s h a 1 _ b a s e 6 4 = " v E 8 I f g P V R 2 c 4 F W Y v q g b w R B B J R P M = " > A A A C O H i c b Z D N S g M x F I U z / t b / q k s 3 w S K 4 k D I j g i 4 U i i K 4 r G B b o T M O m f S 2 D U 0 y Q 5 I R y j A v 4 2 P 4 B G 5 1 5 8 6 N i F u f w E z t Q l s v B A 7 n n s u 9 + a K E M 2 1 c 9 9 W Z m Z 2 b X 1 g s L S 2 v r K 6 t b 5 Q 3 t 5 o 6 T h W F B o 1 5 r G 4 j o o E z C Q 3 D D I f b R A E R E Y d W N L g o + q 1 7 U J r F 8 s Y M E w g E 6 U n W Z Z Q Y a 4 X l 0 2 a Y + S L 1 Z Z r j M + x H i m R u j t u X I z e / y / w O 6 f V A 5 Q e 4 s G w q C P 1 E + A M w N h e W K 2 7 V H R W e F t 5 Y V N C 4 6 m H 5 3 e / E N B U g D e V E 6 7 b n J i b I i D K M c s i X / V R D Q u i A 9 K B t p S Q C d J C N f p n j P e t 0 c D d W 9 k m D R + 7 v i Y w I r Y c i s k l B T F 9 P 9 g r z v 1 4 7 N d 2 T I G M y S Q 1 I + r O o m 3 J s Y l w g w x 2 m g B o + t I J Q x e y t m P a J I t R Y s H + 2 W H y W T M H F m 6 Q w L Z q H V c + t e t d H l d r 5 m F A J 7 a B d t I 8 8 d I x q 6 A r V U Q N R 9 I C e 0 D N 6 c R 6 d N + f D + f y J z j j j m W 3 0 p 5 y v b 3 S P r h A = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " v E 8 I f g P V R 2 c 4 F W Y v q g b w R B B J R P M = " > A A A C O H i c b Z D N S g M x F I U z / t b / q k s 3 w S K 4 k D I j g i 4 U i i K 4 r G B b o T M O m f S 2 D U 0 y Q 5 I R y j A v 4 2 P 4 B G 5 1 5 8 6 N i F u f w E z t Q l s v B A 7 n n s u 9 + a K E M 2 1 c 9 9 W Z m Z 2 b X 1 g s L S 2 v r K 6 t b 5 Q 3 t 5 o 6 T h W F B o 1 5 r G 4 j o o E z C Q 3 D D I f b R A E R E Y d W N L g o + q 1 7 U J r F 8 s Y M E w g E 6 U n W Z Z Q Y a 4 X l 0 2 a Y + S L 1 Z Z r j M + x H i m R u j t u X I z e / y / w O 6 f V A 5 Q e 4 s G w q C P 1 E + A M w N h e W K 2 7 V H R W e F t 5 Y V N C 4 6 m H 5 3 e / E N B U g D e V E 6 7 b n J i b I i D K M c s i X / V R D Q u i A 9 K B t p S Q C d J C N f pn j P e t 0 c D d W 9 k m D R + 7 v i Y w I r Y c i s k l B T F 9 P 9 g r z v 1 4 7 N d 2 T I G M y S Q 1 I + r O o m 3 J s Y l w g w x 2 m g B o + t I J Q x e y t m P a J I t R Y s H + 2 W H y W T M H F m 6 Q w L Z q H V c + t e t d H l d r 5 m F A J 7 a B d t I 8 8 d I x q 6 A r V U Q N R 9 I C e 0 D N 6 c R 6 d N + f D + f y J z j j j m W 3 0 p 5 y v b 3 S P r h A = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " v E 8 I f g PV R 2 c 4 F W Y v q g b w R B B J R P M = " > A A A C O H i c b Z D N S g M x F I U z / t b / q k s 3 w S K 4 k D I j g i 4 U i i K 4 r G B b o T M O m f S 2 D U 0 y Q 5 I R y j A v 4 2 P 4 B G 5 1 5 8 6 N i F u f w E z t Q l s v B A 7 n n s u 9 + a K E M 2 1 c 9 9 W Z m Z 2 b X 1 g s L S 2 v r K 6 t b 5 Q 3 t 5 o 6 T h W F B o 1 5 r G 4 j o o E z C Q 3 D D I f b R A E R E Y d W N L g o + q 1 7 U J r F 8 s Y M E w g E 6 U n W Z Z Q Y a 4 X l 0 2 a Y + S L 1 Z Z r j M + x H i m R u j t u X I z e / y / w O 6 f V A 5 Q e 4 s G w q C P 1 E + A M w N h e W K 2 7 V H R W e F t 5 Y V N C 4 6 m H 5 3 e / E N B U g D e V E 6 7 b n J i b I i D K M c s i X / V R D Q u i A 9 K B t p S Q C J C N f pn j P e t 0 c D d W 9 k m D R + 7 v i Y w I r Y c i s k l B T F 9 P 9 g r z v 1 4 7 N d 2 T I G M y S Q 1 I + r O o m 3 J s Y l w g w x 2 m g B o + t I J Q x e y t m P a J I t R Y s H + 2 W H y W T M H F m 6 Q w L Z q H V c + t e t d H l d r 5 m F A J 7 a B d t I 8 8 d x q 6 A r V U Q N R I C e 0 D N 6 c R 6 d N + f D + f y J z j j j m W 3 0 p 5 y v b 3 S P r h A = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " v E 8 I f g P V R 2 c 4 F W Y v q g b w R B B J R P M = " > A A A C O H c b Z D N S g M x F U z / t b / q k s 3 w S K 4 k D I j g i 4 U i i K 4 r G B b o T M O m f S 2 D U 0 y Q 5 I R y j A v 4 2 P 4 B G 5 1 5 8 6 N i F u f w E z t Q l s v B A 7 n n s u 9 + a E M 2 1 c 9 9 W Z m 2 b X 1 g s L S 2 v r K 6 t b 5 Q 3 t 5 o 6 T h W F B o 1 5 r G 4 j o o E z C Q 3 D D I f b R A E R E Y d W N L g o + q 1 7 U J r F 8 s Y M E w g E 6 U n W Z Z Q Y a 4 X l 0 2 a Y + S L 1 Z Z r j M + x H i m R u j t u X I z e / y / w O 6 f V A 5 Q e 4 s G w q C P 1 E + A M w N h e W K 2 7 V H R W e F t 5 Y V N C 4 6 m H 5 3 e / E N B U g D e V E 6 7 b n J i b I i D K M c s i X / V R D Q u i A 9 K B t p S Q C d J C N f p n j P e t 0 c D d W 9 k m D R + 7 v i Y w I r Y c i s k l B T F 9 P 9 g r z v 1 4 7 N d 2 T I G M y S Q 1 I + r O o m 3 J s Y l w g w x 2 m g B o + t I J Q x e y t m P a J I t R Y s H + 2 W H y W T M H F m 6 Q w L Z q H V c + t e t d H l d r 5 m F A J 7 a B d t I 8 8 d I x q 6 A r V U Q N R 9 I C e 0 D N 6 c R 6 d N + f D + f y J z j j j m W 3 0 p 5 y v b 3 S P r h A = < / l a t e x i t >Q µ⌫ = h0| [E † µ , H, E † ⌫ ] ± |0i < l a t e x i t s h a 1 _ b a s e 6 4 = " r U 2 f U G Q I H W 4 l 7 m + 9 h 2 f i J C E C P H A = " > A A A C T X i c b Z B B S x t B F M d n Y 1 u t 1 p r q s Z e h Q e i h h l 0 R 2 o s g S s G j A a N C Z l 1 m J y 9 x y M z s M v N W G p b 9 X H 4 M z x 4 8 V j + B N x F n Y w 6 J + m D g z / / / H u / N L 8 2 V d B i G N 0 F j 4 c P H T 4 t L n 5 d X v q x + X W t + W z 9 x W W E F d E W m M n u W c g d K G u i i R A V n u Q W u U w W n6 e i g z k 8 v w T q Z m W M c 5 x B r P j R y I A V H b y X N T i c p m S 6 Y K S q 6 S 7 d Y a n k Z V r T 3 d 2 J X 5 y X r 8 + E Q b P W L M o R / W B 5 6 V Y d m N o w T l m s 2 A v S z S b M V t s N J 0 b c i m o o W m d Z R 0 v z P + p k o N B g U i j v X i 8 I c 4 5 J b l E J B t c w K B z k X I z 6 E n p e G a 3 B x O f l 6 R T e 9 0 6 e D z P p n k E 7 c 2 Y m S a + f G O v W d m u O F e 5 3 V 5 n t Z r 8 D B n 7 i U J i 8 Q j H h Z N C g U x Y z W H G l f W h C o x l 5 w Y a W / l Y o L b r l A T 3 t u i 0 f q y d R c o t c U 3 o q T 7 X Y U t q P O T m t v f 0 p o i X w n P 8 h P E p H f Z I 8 c k i P S J Y J c k V t y R + 6 D 6 + A h e A y e X l o b w X R m g 8 x V Y / E Z R R C 1 1 A = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " r U 2 f U G Q I H W 4 l 7 m + 9 h 2 f i J C E C P H A = " > A A A C T X i c b Z B B S x t B F M d n Y 1 u t 1 p r q s Z e h Q e i h h l 0 R 2 o s g S s G j A a N C Z l 1 m J y 9 x y M z s M v N W G p b 9 X H 4 M z x 4 8 V j + B N x F n Y w 6 J + m D g z / / / H u / N L 8 2 V d B i G N 0 F j 4 c P H T 4 t L n 5 d X v q x + X W t + W z 9 x W W E F d E W m M n u W c g d K G u i i R A V n u Q W u U w W n 6 e i g z k 8 v w T q Z m W M c 5 x B r P j R y I A V H b y X N T i c p m S 6 Y K S q 6 S 7 d Y a n k Z V r T 3 d 2 J X 5 y X r 8 + E Q b P W L M o R / W B 5 6 V Y d m N o w T l m s 2 A v S z S b M V t s N J 0 b c i m o o W m d Z R 0 v z P + p k o N B g U i j v X i 8 I c 4 5 J b l E J B t c w K B z k X I z 6 E n p e G a 3 B x O f l 6 R T e 9 0 6 e D z P p n k E 7 c 2 Y m S a + f G O v W d m u O F e 5 3 V 5 n t Z r 8 D B n 7 i UJ i 8 Q j H h Z N C g U x Y z W H G l f W h C o x l 5 w Y a W / l Y o L b r l A T 3 t u i 0 f q y d R c o t c U 3 o q T 7 X Y U t q P O T m t v f 0 p o i X w n P 8 h P E p H f Z I 8 c k i P S J Y J c k V t y R + 6 D 6 + A h e A y e X l o b w X R m g 8 x V Y / E Z R R C 1 1 A = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " r U 2 f U G Q I H W 4 l 7 m + 9 h 2 f i J C E C P H A = " > A A A C T X i c b Z B B S x t B F M d n Y 1 u t 1 p r q s Z e h Q e i h h l 0 R 2 o s g S s G j A a N C Z l 1 m J y 9 x y M z s M v N W G p b 9 X H 4 M z x 4 8 V j + B N x F n Y w 6 J + m D g z / / / H u / N L 8 2 V d B i G N 0 F j 4 c P H T 4 t L n 5 d X v q x + X W t + W z 9 x W W E F d E W m M n u W c g d K G u i i R A V n u Q W u U w W n 6 e i g z k 8 v w T q Z m W M c 5 x B r P j R y I A V H b y X N T i c p m S 6 Y K S q 6 S 7 d Y a n k Z V r T 3 d 2 J X 5 y X r 8 + E Q b P W L M o R / W B 5 6 V Y d m N o w T l m s 2 A v S z S b M V t s N J 0 b c i m o o W m d Z R 0 v z P + p k o N B g U i j v X i8 I c 4 5 J b l E J B t c w K B z k X I z 6 E n p e G a 3 B x O f l 6 R T e 9 0 6 e D z P p n k E 7 c 2 Y m S a + f G O v W d m u O F e 5 3 V 5 n t Z r 8 D B n 7 i U J i 8 Q j H h Z N C g U x Y z W H G l f W h C o x l 5 w Y a W / l Y o L b r l A T 3 t u i 0 f q y d R c o t c U 3 o q T 7 X Y U t q P O T m t v f 0 p o i X w n P 8 h P E p H f Z I 8 c k i P S J Y J c k V t y R + 6 D 6 + A h e A y e X l o b w X R m g 8 x V Y / E Z R R C 1 1 A = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " d A 5 a g 2 p j I L i X y o I y d y 4 O S 9 C Y X X s = " > A A A B 7 H i c b Z D N S g M x F I X v 1 L 9 a q 9 a 1 m 2 A R X J U Z N 7 o U 3 L i s Y H + g H U o m c 6 c N z W S G 5 I 5 Q h r 6 A W 5 / A n f h G P o D v Y f q z s K 0 H A o d z E u 7 N F + V K W v L 9 b 6 + y t 3 9 w e F Q 9 r p 3 U a 6 d n 5 4 1 6 1 2 a F E d g R m c p M P + I W l d T Y I U k K + 7 l B n k Y K e 9 H 0 c d H 3 X t F Y m e k X m u U Y p n y s Z S I F J x e 1 R 4 2 m 3 / K X Y r s m W J s m r D V q / A z j T B Q p a h K K W z s I / J z C k h u S Q u G 8 N i w s 5 l x M + R g H z m q e o g 3 L 5 Z p z d u 2 S m C W Z c U c T W 6 Z / X 5 Q 8 t X a W R u 5 m y m l i t 7 t F + F 8 3 K C i 5 D 0 u p 8 4 J Q i 9 W g p F C M M r b 4 M 4 u l Q U F q 5 g w X R r p d m Z h w w w U 5 M h t T I s O n S H O H J d i G s G u 6 t 6 3 A b w X P P l T h E q 7 g B g K 4 g w d 4 g j Z 0 Q E A M b / D u l d 6 H 9 7 n C V / H W H C 9 g Q 9 7 X L 0 o o k n 0 = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " L f P a u I W L B f p 0 z l N k N 5 6 C q z 3 4 s L U = " > A A A C Q n i c b Z B B a x s x E I V n 0 z R N 3 D R 1 e + 1 F 1 B R 6 a M 1 u L + k l E C g F H 2 O o k 4 C 1 W b T y 2 B G W t I s 0 G 2 K W / V 3 9 G T 3 n k G P b X 9 B b K d E 6 P s R x H w g e 7 2 k Y 6 c t L r T z F 8 U 2 0 9 W T 7 6 c 6 z 3 b 3 O 8 / 0 X B y + 7 r / Z P f V E 5 i S N Z 6 M K d 5 8 K j V h Z H p E j j e e l Q m F z j W T 7 / 0 v Z n V + i 8 K u w 3 W p S Y G j G z a q q k o B B l 3 e E w q 7 m p u K 0 a d s Q + 8 t y J O m 7 Y + O s y b i 5 q P h G z G b r m A + O E 1 1 Q P g m t L + 7 B M M 1 4 a P k c K s 1 m 3 F / f j p d i m S V a m B y u d Z N 2 f f F L I y q A l q Y X 3 4 y Q u K a 2 F I y U 1 N h 1 e e S y F n I s Z j o O 1 w q B P 6 + X X G / Y u J B M 2 L V w 4 l t g y f T h R C + P 9 w u T h p h F 0 6 R 9 3 b f i / b l z R 9 H N a K 1 t W h F b e L 5 p W m l H B W o 5 s o h x K 0 o t g h H Q q v J X J S + G E p E B 7 b U t A G s i 0 X J L H F D b N 6 a d + E v e T Y Q y 7 8 A b e w n t I 4 B C O Y Q A n M A I J 3 + E W f s H v 6 E f 0 J / p 7 T 3 A r W q F 8 D W u K / t 0 B d I e 1 G w = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " L f P a u I W L B f p 0 z l N k N 5 6 C q z 3 4 s L U = " > A A A C Q n i c b Z B B a x s x E I V n 0 z R N 3 D R 1 e + 1 F 1 B R 6 a M 1 u L + k l E C g F H 2 O o k 4 C 1 W b T y 2 B G W t I s 0 G 2 K W / V 3 9 G T 3 n k G P b X 9 B b K d E 6 P s R x H w g e 7 2 k Y 6 c t L r T z F 8 U 2 0 9 W T 7 6 c 6 z 3 b 3 O 8 / 0 X B y + 7 r / Z P f V E 5 i S N Z 6 M K d 5 8 K j V h Z H p E j j e e l Q m F z j W T 7 / 0 v Z n V + i 8 K u w 3 W p S Y G j G z a q q k o B B l 3 e E w q 7 m p u K 0 a d s Q + 8 t y J O m 7 Y + O s y b i 5 q P h G z G b r m A + O E 1 1 Q P g m t L + 7 B M M 1 4 a P k c K s 1 m 3 F / f j p d i m S V a m B y u d Z N 2 f f F L I y q A l q Y X 3 4 y Q u K a 2 F I y U 1 N h 1 e e S y F n I s Z j o O 1 w q B P 6 + X X G / Y u J B M 2 L V w 4 l t g y f T h R C + P 9 w u T h p h F 0 6 R 9 3 b f i / b l z R 9 H N a K 1 t W h F b e L 5 p W m l H B W o 5 s o h x K 0 o t g h H Q q v J X J S + G E p E B 7 b U t A G s i 0 X J L H F D b N 6 a d + E v e T Y Q y 7 8 A b e w n t I 4 B C O Y Q A n M A I J 3 + E W f s H v 6 E f 0 J / p 7 T 3 A r W q F 8 D W u K / t 0 B d I e 1 G w = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 9 s u c O w 5 h o s S Y s 1 z I 2 d S c P R W 0 h s O k 5 r p i p m q o Z / o I U s t r 8 O G T r 8 u 7 e Z 7 z T K e 5 2 C b D 5 Q FIG.S2: Top: absolute error in the a first, b second and c third excited state energies due to shot noise.The error corresponds to the difference between energies computed with and without sampling.Bottom: absolute error in the norm of the d M, e V and f Q matrices due to finite number of shots.The error corresponds to the difference between the matrix norms computed with and without sampling.
FIG.S3: Left: Correlation energy of the ground state computed with (markers) and without (dashed lines) noise as a function of the variational parameter θ for a H 2 and d LiH at equilibrium distance.Middle: Corresponding excitation energies computed with qEOM with (markers) and without (dashed lines) noise.For H 2 b the three excitation energies are shown.For LiH e the excitation energies corresponding to the five lowest energy states (degeneracies are not taken into account) are displayed.Right: Absolute error in the noisy simulations with respect to the reference for the ground state and corresponding excitation energies with a zoom in the region θ ∈ [−1; 1] for c H 2 and f LiH.
FIG. S5: Reduction of the usual eight blocks of the double excitation part of the UCCSD circuit to a single block.

FIG. S6 :
FIG. S6: Lower panel: Dissociation profile the LiH molecule.The grey lines represent the exact eigenstates of the Hamiltonian obtained from its diagonalizaton.The coloured-dashed lines show the six lowest-lying electronic states obtained with VQE and qEOM in statevector simulation with the reduced circuit.Upper panel: Corresponding energy errors along the dissociation The gray shaded area corresponds to the energy range within chemical accuracy FIG.S7: Error in the experimentally obtained ground state energies (n = 0) and excitation energies up to n = 24, for several internuclear distances.The error is computed with respect to the results obtained by statevector-type simulation of the reduced circuit.The results are shown for each of the three stretch factors and after mitigation.The energy of the gaps grows from left to right.The gray shaded area corresponds to the energy range within chemical accuracy.The error bars are computed using 50 numerical experiments obtained by bootstrapping of the experimental data points, and depict the range between the 1st and 3rd quantile.For all the bond lengths, the excitation energies are seen to be more robust than the ground state energies, at various noise levels (i.e.stretch factors).
The ground state wave function of the H 2 molecule optimized with a statevector-type VQE (as explained in the previous paragraph) is used to compute the excited state energies with qEOM by introducing this time a statistical error: the expectation value of each matrix element, M µαν β , Q µαν β , V µαν β , and W µαν β , is obtained by projecting the state and averaging over a given number of shots, NS ∈ {8192, 4096, 2048, 1024}.For each different choice of NS we perform 100 computations of excited state energies and compute the error, |∆E|, with the corresponding values obtained without statistical noise.In Fig.S2(top), we plot the average of this error for each bond length for a the first, b the second, and c the third excited state.

TABLE III :
Percentage of correlation energy recovered by the reduced 4-qubit active space UCC-inspired circuit for LiH through the dissociation profile.The VQE results are obtained with statevector-type simulations, the reduced circuit and the COBYLA optimizer.The exact results are obtained by diagonalization of the full HF/STO-3G Hamiltonian.

TABLE IV :
Gate characterization Single and two qubit gate fidelities for the gates employed in this work, for the various stretch factors, estimated by randomized benchmarking.