Theory of analytical energy derivatives for the varia-tional quantum eigensolverKosuke Mitarai1,2, Yuya O. Nakagawa2, and Wataru Mizukami3

1Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama, Toyonaka, Osaka 560-8531, Japan.2QunaSys Inc., High-tech Hongo Building 1F, 5-25-18 Hongo, Bunkyo, Tokyo 113-0033, Japan3Interdisciplinary Graduate School of Engineering Science, Kyushu University, 6-1 Kasuga-Koen, Kasuga, Fukuoka,816-6580, Japan.June 10, 2019

The variational quantum eigensolver (VQE) and its variants, which is a methodfor finding eigenstates and eigenenergies of a given Hamiltonian, are appealingapplications of near-term quantum computers. Although the eigenenergies arecertainly important quantities which determine properties of a given system, theirderivatives with respect to parameters of the system, such as positions of nuclei ifwe target a quantum chemistry problem, are also crucial to analyze the system.Here, we describe methods to evaluate analytical derivatives of the eigenenergy ofa given Hamiltonian, including the excited state energy as well as the ground stateenergy, with respect to the system parameters in the framework of the VQE. Wegive explicit, low-depth quantum circuits which can measure essential quantities toevaluate energy derivatives, incorporating with proof-of-principle numerical simu-lations. This work extends the theory of the variational quantum eigensolver, byenabling it to measure more physical properties of a quantum system than beforeand to explore chemical reactions.

1 IntroductionThe variational quantum eigensolver (VQE) has attracted much attention as a potential ap-plication of near-term quantum computers [1]. The VQE is an iterative algorithm to constructa quantum circuit that outputs eigenstates and eigenenergy of a Hamiltonian which describesthe system under consideration. Originally, the method was devised for finding the groundstate of a system [1]. It has subsequently been extended for excited states by several propos-als [2–5]. From the generated eigenstates, one can measure its associated physical quantities,such as the particle densities and transition amplitudes between the different eigenstates.

Though eigenenergy and associated particle density are certainly important quantities,the wavefunction of a quantum system has valuable information besides those. Quantumchemistry calculations, which would be one of the most promising applications of the VQE,often utilize such information. Among such, we focus on energy derivatives in this work. Many

Kosuke Mitarai: u801032f@ecs.osaka-u.ac.jp

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time-independent physical/chemical properties can be defined as derivatives of the energy [6–9]. For example, the first derivatives of the energy with respect to nuclear coordinates give usthe forces acting on atoms, which can be utilized for the task of locating energy extrema onthe potential energy surface (i.e., geometry optimization) [10]. The second-order derivativesgive the force-constant matrix that not only helps to locate and verify transition states butalso allows us to compute vibrational frequencies and partition functions within the harmonicapproximation [10, 11]. The energy derivatives with respect to external fields have to dowith various spectroscopy: Intensities of infrared and Raman spectroscopy are proportionalto the cross derivatives with respect to vibrational normal modes and external electric fields[11]; NMR chemical shifts can be obtained using the cross derivatives with respect to nuclearspin and magnetic fields [12, 13]. Computing such derivatives is a core part of simulationsor analysis of molecular spectra. If quantum chemical methods were unable to extract thoseproperties, presumably they would not be widely used as it is today.

A simple way to compute energy derivatives is to use the finite difference method andcalculate them numerically. This approach, however, suffers from high computational costsas well as numerical errors and instabilities [7, 9]. Say, the number of energy points neededto evaluate the forces increases linearly to the number of atoms. This high computationalcost makes the numerical approach impractical in many cases. Moreover, with near-termquantum devices, where noise is inevitable, the numerical difference approach would give uspoor results. The other way – analytical approach – is therefore vital. The theory and programcodes of analytical energy derivatives indeed support the high practicality (and popularity)of today’s molecular electronic structure theory [9]. Methods to calculate the derivatives ofexcited state energy on classical computers has also been widely developed [14–34], but stillsuffers from its high computational cost and relatively low accuracy. The task of computingsuch derivatives on quantum computers has been addressed in the traditional methods whichutilize the quantum phase estimation [35], but not for the VQE.

In this work, we derive the analytical formulae and explicit quantum circuits to addressthe task of measuring the energy derivatives. More specifically, we describe the methodsto obtain the derivatives of the energy with respect to the system parameters up to thethird order, from which one can extract the physical properties. We also present a methodto extract the derivatives of excited state energy based on the technique presented in Refs.[3, 4]. Proof-of-principle numerical simulations are also performed to verify the correctnessof the derived equation and circuits. The presented methods extend the applicability of theVQE by enabling it to evaluate more physical properties than before.

2 Variational quantum eigensolverHere, we briefly review the algorithm of the VQE. In the VQE, we construct a parameterizedquantum circuit U(θ) and the corresponding ansatz state |ψ(θ)〉 = U(θ) |0〉⊗n, where |0〉⊗n isan initialized n-qubit state and θ = (θ1, . . . , θNθ) ∈ RNθ is a vector of parameters implementedon the circuit with Nθ being the number of them. A set of parameters θ are variationallyoptimized so that the expectation value E(θ) = 〈ψ(θ)|H |ψ(θ)〉 of a given Hamiltonian H isminimized. At the optimal point, one naturally expects

∂E(θ)∂θa

= 0, (1)

2

for all a. We denote such an optimal point by θ∗. Let

|∂aψ(θ)〉 = ∂

∂θa|ψ(θ)〉 . (2)

The condition of Eq. (1) reduces to

Re 〈ψ(θ∗)|H |∂aψ(θ∗)〉 = 0. (3)

Higher derivatives of the wave function will be denoted by

|∂a∂b · · · ∂cψ(θ)〉 = ∂

∂θa

∂

∂θb· · · ∂

∂θc|ψ(θ)〉 , (4)

to shorten the notation.As stated in the introduction, many physical properties of a quantum system are calculated

from the derivative of the energy with respect to the system parameter in a given Hamiltonian.The parameter can be, for example, the coordinates of atoms or the electric field applied tothe system. We denote such parameters by an Nx-dimensional vector x ∈ RNx . In this case,both of the Hamiltonian H and the optimal parameter θ∗ of the wave function at the specificvalue of x is also a function of x, which will be denoted by H(x) and θ∗(x), respectively.Corresponding to the change of this problem setting, we redefine the energy as,

E(θ, x) = 〈ψ(θ)|H(x)|ψ(θ)〉 . (5)

Let the optimal ground state energy be E∗(x), that is, E∗(x) = E(θ∗(x), x). In the

following sections, we show the analytical forms of the derivatives such as ∂E∗

∂xiand ∂2E∗

∂xi∂xj,

which are the essential quantities for extracting physical properties of the target system.

3 Analytical expression of derivativesThe derivation of the formulae presented in this section is in the Appendix for completeness,or you can also refer to Ref. [36]. In the following, we use the following notation,

∂

∂θa

∂E(θ∗(x), x)∂xj

:= ∂

∂θa

∂E(θ, x)∂xj

∣∣∣∣∣θ=θ∗(x),x=x

. (6)

and likewise for terms similar to this.

3

3.1 Derivatives of ground state energyThe analytical expressions for the derivatives of ground state energy are the following,

∂E∗(x)∂xi

= 〈ψ (θ∗(x)) |∂H(x)∂xi

|ψ (θ∗(x))〉 , (7)

∂

∂xi

∂E∗(x)∂xj

=∑a

∂θ∗a(x)∂xi

∂

∂θa

∂E(θ∗(x), x)∂xj

+ 〈ψ(θ∗(x))| ∂∂xi

∂H(x)∂xj

|ψ(θ∗(x))〉 , (8)

∂

∂xi

∂

∂xj

∂E∗(x)∂xk

=∑a,b,c

∂

∂θa

∂

∂θb

∂E(θ∗(x), x)∂θc

∂θ∗a(x)∂xi

∂θ∗b (x)∂xj

∂θ∗c (x)∂xk

+ 〈ψ(θ∗(x))| ∂∂xi

∂

∂xj

∂H(x)∂xk

|ψ(θ∗(x))〉

+∑a,b

[∂θ∗a(x)∂xi

∂θ∗b (x)∂xj

∂

∂θb

∂

∂θa

∂E(θ∗(x), x)∂xk

+ ∂θ∗a(x)∂xk

∂θ∗b (x)∂xi

∂

∂θb

∂

∂θa

∂E(θ∗(x), x)∂xj

+∂θ∗a(x)∂xj

∂θ∗b (x)∂xk

∂

∂θb

∂

∂θa

∂E(θ∗(x), x)∂xi

]

+∑a

[∂θ∗a(x)∂xi

∂

∂θa

∂

∂xj

∂E(θ∗(x), x)∂xk

+ ∂θ∗a(x)∂xk

∂

∂θa

∂

∂xi

∂E(θ∗(x), x)∂xj

+∂θ∗a(x)∂xj

∂

∂θa

∂

∂xk

∂E(θ∗(x), x)∂xi

], (9)

where we assumed ∂E(θ∗(x),x)∂θ = 0. Note that, in general, the formulae for the d-th derivative

of E∗(x) contains θ-detrivatives up to the d-th in the form of ∂q

∂θq∂d−qE∂xd−q for q = 1, . . . , d. Also,

Wigner’s (2n+ 1)-rule [36] ensures that x-derivatives of the optimal parameter, θ∗(x), up tothe n-th are sufficient for calculating (2n + 1)-th derivative of E∗(x). In other words, ford-th derivative of E∗(x) we only need bd/2c-th derivative of θ∗(x), where byc is the floor

function denoting the greatest integer less than or equal to y. The term ∂θ∗a(x)∂xi

in the aboveequation can be calculated by solving the response equation, which we write down in the nextsubsection.

3.2 Derivatives of optimal parametersThe first and second derivatives of the optimal parameter, θ∗(x), can be obtained from thefollowing response equation.∑

b

∂

∂θa

∂E(θ∗(x), x)∂θb

∂θ∗b (x)∂xi

= − ∂

∂θa

∂E(θ∗(x), x)∂xi

, (10)

∑b

∂

∂θa

∂E(θ∗(x), x)∂θb

∂

∂xi

∂θ∗b (x)∂xj

= −γ(ij)a (θ∗(x), x), (11)

4

where,

γ(ij)c =

∑a,b

∂

∂θc

∂

∂θa

∂E

∂θb

∂θ∗a∂xi

∂θ∗b∂xj

+ 2∑a

∂

∂θc

∂

∂θa

∂E

∂xj

∂θ∗a∂xi

+ ∂

∂θc

∂

∂xi

∂E

∂xj. (12)

By Wigner’s (2n + 1)-rule [36], the above equations are enough to obtain derivatives of theenergy up to the 5th-order.

4 Measurement and calculation of derivatives of ground stateIn this section, we describe the methodology for calculating the derivatives of the groundstate energy, whose analytical expressions are shown in the previous section, when given anoptimal circuit parameter θ∗(x) at some x that gives the local minimum of E(θ, x).

4.1 Notations and assumptionsIn the VQE, we target a Hamiltonian which acts on an n-qubit system and is decomposedinto a sum of Pauli strings, P = I,X, Y, Z⊗n, as follows,

H(x) =∑P∈P

hP (x)P, (13)

where hP (x) ∈ R. hP (x) is assumed to be non-zero only for O(poly(n)) terms. For a quantumchemistry problem, the original Hamiltonian has the following form,

H(x) =∑i,j

hij(x)c†icj +∑i,j,k,l

hijkl(x)c†ic†jckcl, (14)

where c†i and ci are the fermion creation and annihilation operators acting on the i-th orbital,respectively. Eq. (14) is converted to the form of Eq. (13) by, for example, Jordan-Wigneror Braviy-Kitaev transformation [37, 38]. Note that because we always work in the secondquantization formalism, the effect of the change of the molecular orbital corresponding tothe change of molecular geometry is totally absorbed in the coefficients h(x). Therefore, thechange of the molecular orbital does not explicitly appear in the following discussion.

To calculate the energy derivatives of such Hamiltonian, first of all, we assume that thederivatives of Hamiltonian, ∂H∂xi ,

∂∂xi

∂H∂xj

, and so on, can be calculated by the classical computer,

e.g., the conventional libraries of quantum chemistry calculation. In other words, we are ableto calculate the quantities such as ∂hP (x)

∂xiand ∂

∂xi

∂hP (x)∂xj

. For quantum chemisry problems

given in terms of Hartree-Fock orbitals, these calculations correspond to solving the coupledperturbed Hartree-Fock equation [30, 36].

Notice that under this assumption, we only need to consider how to evaluate quantitieswhich involve the differentiation with respect to θ such as ∂

∂θa∂∂θb

∂E∂xi

, because the expectation

values such as ∂∂xi

∂∂xj

∂E∂xk

= 〈ψ(θ)| ∂∂xi

∂∂xj

∂H(x)∂xk

|ψ(θ)〉 at θ = θ∗(x), can be evaluated with

the exactly same procedure as the usual VQE, i.e., we can measure the expectation valueof each Pauli term which appears in ∂

∂xi∂∂xj

∂H(x)∂xk

. After the measurement of all quantities

which appear in Eqs. (7), (8), and (9), one can compute the energy derivative using a classicalcomputer by summing up the terms.

5

The parameterized quantum state, |ψ(θ)〉, is constructed by applying a parameterizedunitary matrix, that is, a parameterized quantum circuit, U(θ) to an initialized state; |ψ(θ)〉 =U(θ) |0〉⊗n. Following [39], we assume U(θ) to be a product of unitary matrices each with oneparameter,

U(θ) = UNθ(θNθ) · · ·U2(θ2)U1(θ1). (15)We define each unitary Ua(θa) to be generated by a generator Ga as Ua = eiθaGa , which canbe decomposed into a sum of Pauli strings;

Ga =∑µ

ga,µPa,µ, (16)

where ga,µ ∈ R and Pa,µ ∈ P. We often use this form of parametrized quantum circuits, forexample see Ref. [40–43].

4.2 Overview of the algorithmThe presented algorithm in this work for evaluating the d-th derivative is the following:

1. Perform the VQE and obtain the optimal parameter θ∗(x).

2. Compute derivatives of the Hamiltonian, ∂qH(x)∂xq , for q = 1, . . . , d on a classical computer.

3. Evaluate x-derivatives of E, ∂qE∂xq , for q = 1, . . . , d at θ = θ∗(x) by the method described

above (Sec 4.1).

4. Evaluate terms involving differentiations with respect to θ, ∂q

∂θq∂d−qE∂xd−q and ∂qE

∂θq , for q =1, . . . , d at θ = θ∗(x) on a quantum device by the method described in the followingsubsections (Secs. 4.3-4.5).

5. Solve the response equations to obtain ∂qθ∗

∂xq for q = 1, . . . , bd/2c. (For d ≤ 3, solvingEq.(10) suffices.)

6. Substitute all terms into Eqs. (7), (8), and (9) to obtain the energy derivatives.

Let us recall NH and Nθ be the number of terms in the target Hamiltonian and thenumber of the VQE parameters. The cost to evaluate d-th derivatives by the above algorithmis roughly O(NHN

dθ ) when we ignore the cost of the task performed on a classical computer.

This is because the most time-consuming part of the algorithm is Step 3 and 4 on a quantumcomputer, which take O(NHN

dθ ) as shall be clear in Secs. 4.3-4.5 and 5.

4.3 Measurement of ∂∂θa

∂E∂θb

The key quantities for evaluating ∂∂xi

∂E∗

∂xjand higher order derivatives are ∂

∂θa∂∂θb· · · ∂E∂θc and

∂∂θa

∂∂θb· · · ∂

∂θc∂E∂xi

. Note that the first order derivative, ∂E∂θa

can be evaluated by the method

presented in Ref. [44]. First, we show how to measure ∂∂θa

∂E∂θb

. A detailed expression of ∂∂θa

∂E∂θb

is,

∂

∂θa

∂E(θ∗(x), x)∂θb

= 2Re [〈∂a∂bψ(θ∗(x))|H(x)|ψ(θ∗(x))〉+ 〈∂aψ(θ∗(x))|H(x)|∂bψ(θ∗(x))〉] .

(17)

6

|∂aψ(θ)〉 can be expressed as,

|∂aψ(θ)〉 = i∑µ

ga,µUN (θN ) · · ·Pa,µUa(θa) · · ·U2(θ2)U1(θ1) |0〉⊗n , (18)

and |∂a∂bψ(θ)〉 is,

|∂a∂bψ(θ)〉 = −∑µ,ν

ga,µgb,νUN (θN ) · · ·Pa,µUa(θa) · · ·Pb,νUb(θb) · · ·U2(θ2)U1(θ1) |0〉⊗n . (19)

For convinience, we define,

|φ(a,µ),(b,ν),···(c,ρ)(θ)〉 := UN (θN ) · · · (iPa,µ)Ua(θa) · · · (iPb,ν)Ub(θb) · · · (iPc,ρ)Uc(θc) · · ·U1(θ1) |0〉⊗n .(20)

Then,

∂

∂θa

∂E(θ∗(x), x)∂θb

= 2∑µ,ν

∑Q∈P

hQ(x)ga,µgb,νRe[〈φ(a,µ),(b,ν)(θ∗(x))|Q |ψ(θ∗(x))〉+ 〈φ(a,µ)(θ∗(x))|Q|φ(b,ν)(θ∗(x))〉

].

(21)

In Fig. 1 (a), we show quantum circuits for evaluation of each of the terms in the aboveequation, which is a variant of the circuit presented in Ref. [39]. From the outputs of thecircuits in Fig. 1 (a), each term of Eq. (21) can be calculated by,

Re[〈φ(a,µ),(b,ν)(θ∗(x))|Q|ψ(θ∗(x))〉] = 〈ZancQ〉0,(a,µ),(b,ν)

Re[〈φ(a,µ)(θ∗(x))|Q|φ(b,ν)(θ∗(x))〉] = 〈ZancQ〉1,(a,µ),(b,ν) (22)

The strategy proposed in Ref. [45] gives low-depth versions of the circuits to measure thesame quantities. We show the low-depth quantum circuit in Fig. 1 (b). From the measurementof 〈Q〉(a,µ,±),(b,ν,±) with the circuit in Fig. 1 (b), we can evaluate each term of Eq. (21) by thefollowing formula,

2Re[〈φ(a,µ),(b,ν)(θ∗(x))|Q |ψ(θ∗(x))〉+ 〈φ(a,µ)(θ∗(x))|Q|φ(b,ν)(θ∗(x))〉

]= 〈Q〉(a,µ,+),(b,ν,+) + 〈Q〉(a,µ,−),(b,ν,−) − 〈Q〉(a,µ,−),(b,ν,+) − 〈Q〉(a,µ,+),(b,ν,−). (23)

Notice that the low-depth version doubles the number of the circuit runs, and therefore,if the device can execute the circuit in Fig. 1 (a) while maintaining sufficient overall fidelity,it is advantageous to utilize Fig. 1 (a) and Eq. (22). Otherwise the low-depth version shouldbe used to obtain meaningful result.

7

:1 ,

0 ⊗0 anc anc , , , , : +1 , : +1

:1 ,±0 ⊗ : +1 ,± : +1 , ,± , , ,±

(a)

(b)

Figure 1: Quantum circuit to evaluate ∂∂θa

∂∂θb

E(θ∗(x), x). Ua:b = Ua · · ·Ub+1Ub. (a) Ancilla-basedapproach. This circuit is a variant of the one presented in Ref. [39]. In the figure, Zanc is the Pauli Zoperator acting only on the ancilla qubit, and s ∈ 0, 1. See Eq. (22) for detailed procedure. (b) Low-depthversion of (a), derived with the strategy presented in Ref. [45]. In the figure, R±

a,µ = exp(±iπPa,µ/4). SeeEq. (23) for detailed procedure.

4.4 Evaluation of ∂∂θa

∂∂θb

∂E∂θc

Here, we show how to evaluate ∂∂θa

∂∂θb

∂E∂θc

. It should be clear how one can extend the method

to the order higher than the third. ∂∂θa

∂∂θb

∂E∂θc

can be written down as,

∂

∂θa

∂

∂θb

∂E(θ∗(x), x)∂θc

= 2∑µ,ν,ρ

∑Q∈P

hQ(x)ga,µgb,νgc,ρRe[〈φ(a,µ),(b,ν),(c,ρ)(θ∗(x))|Q |ψ(θ∗(x))〉

+ 〈φ(a,µ),(b,ν)(θ∗(x))|Q|φ(c,ρ)(θ∗(x))〉+ 〈φ(a,µ),(c,ρ)(θ∗(x))|Q|φ(b,ν)(θ∗(x))〉

+ 〈φ(b,ν),(c,ρ)(θ∗(x))|Q|φ(a,µ)(θ∗(x))〉]. (24)

Each term can be evaluated with the circuit in Fig. 2 (a):

Re[〈φ(a,µ),(b,ν),(c,ρ)(θ∗(x))|Q |ψ(θ∗(x))〉

]= 〈ZancQ〉(a,µ,0),(b,ν,0),(c,ρ,0),

Re[〈φ(a,µ),(b,ν)(θ∗(x))|Q |φ(c,ρ)(θ∗(x))〉

]= 〈ZancQ〉(a,µ,0),(b,ν,0),(c,ρ,1),

Re[〈φ(a,µ),(c,ρ)(θ∗(x))|Q |φ(b,ν)(θ∗(x))〉

]= 〈ZancQ〉(a,µ,0),(b,ν,1),(c,ρ,0),

Re[〈φ(b,ν),(c,ρ)(θ∗(x))|Q |φ(a,µ)(θ∗(x))〉

]= 〈ZancQ〉(a,µ,1),(b,ν,0),(c,ρ,0). (25)

The circuits can also be reduced to the low-depth version using the same strategy [45]. Thelow-depth circuit for ∂

∂θa∂∂θb

∂E∂θc

is shown in Fig. 2 (b). The circuit can evaluate each term ofthe summation by the following formula.

− 2Re[〈φ(a,µ),(b,ν),(c,ρ)(θ∗(x))|Q|ψ(θ∗(x))〉+ 〈φ(a,µ),(b,ν)(θ∗(x))|Q|φ(c,ρ)(θ∗(x))〉

+ 〈φ(a,µ),(c,ρ)(θ∗(x))|Q|φ(b,ν)(θ∗(x))〉+ 〈φ(b,ν),(c,ρ)(θ∗(x))|Q|φ(a,µ)(θ∗(x))〉]

= 〈Q〉(a,µ,+),(b,ν,+),(c,ρ,+) − 〈Q〉(a,µ,−),(b,ν,−),(c,ρ,−)

+ 〈Q〉(a,µ,−),(b,ν,−),(c,ρ,+) + 〈Q〉(a,µ,−),(b,ν,+),(c,ρ,−) + 〈Q〉(a,µ,+),(b,ν,−),(c,ρ,−)

− 〈Q〉(a,µ,−),(b,ν,+),(c,ρ,+) − 〈Q〉(a,µ,+),(b,ν,−),(c,ρ,+) − 〈Q〉(a,µ,+),(b,ν,+),(c,ρ,−) (26)

8

:1 ,

0 ⊗0 anc anc , , , , , , ,, : +1 , : +1

:1 ,±0 ⊗ : +1 ,± : +1 , ,± , , ,± , ,,±

(a)

(b)

: +1,

,± : +1Figure 2: Quantum circuit to evaluate ∂

∂θa

∂∂θb

∂∂θc

E(θ∗(x), x). For the definition of the notations, refer toFig. 1. (a) Ancilla-based approach. See Eq. (25) for detailed procedure. (b) Low-depth version of (a). SeeEq. (26) for detailed procedure.

Notice the pattern in the signs associating 〈Q〉(a,µ,±),(b,ν,±),(c,ρ,±), that is, the signs are de-termined in the parity of ± appearing in the subscript. Higher order derivatives of E withrespect to θ can be evaluated by following the strategy described in this and the previoussection.

Similar to the previous case, the low-depth version doubles the number of the circuit runs,and therefore, the same argument about the tradeoff between the noise and the number ofthe circuit runs also applies in this case.

4.5 Evaluation of ∂∂θa

∂∂θb· · · ∂

∂θc∂E∂xi

∂∂θa

∂∂θb· · · ∂

∂θc∂E∂xi

can be measured with the same protocol as described in the previous sections;

one can substitute hQ with∂hQ∂xi

. With this substitution, Eqs. (21) and (24) give us the

analytical formula for ∂∂θa

∂∂θb· · · ∂

∂θc∂E∂xi

, where each term in the summation can be evaluated

with the same procedure. Also, ∂∂θa

∂∂xi

∂E∂xk

, which appears in Eq. (9), can be measured usingthe same strategy.

5 Computational costHere, we give a comparison between the presented algorithm and the numerical differentiationto take d-th energy derivatives. We specifically discuss the cost for each step in the overallalgorithm presented in 4.2 for quantum chemistry problem, where a problem Hamiltonian andits derivatives always has O(n4) terms as expressed by Eq. (14), as it is the main target of theproposed algorithm. This analysis should be easy to be extended to more general cases. Wesafely assume that classical computational cost involved in algorithms is much smaller thanthe quantum ones and thus can be ignored, since usual classical computation is much fasterthan sampling from the NISQ devices.

First, we ignore the cost of performing the VQE (Step 1), since both of the numericaldifferentiation and the presented algorithm need this step. Step 2, 5, and 6 is merely classical,and therefore, we also ignore the cost for this part. Step 3 of the presented algorithm requiresus to run a quantum computer O(n4/ε2) times to estimate each term up to an additionalerror of ε. For Step 4, it takes O(n4∑d−1

k=0(Nd−kθ )/ε2) = O(n4Nd

θ /ε2) runs. Note that even

for the calculation of the terms involving x- and θ-differentiation a the same time, such as∂∂θa

∂E∂xi

, we only need to perform measurement of the θ-derivatives of each term, such as

9

∂∂θa〈ψ(θ)| c†ic

†jckcl |ψ(θ)〉, in the Hamiltonian on the quantum device and then combine them

by∑ijkl

∂hijkl(x)∂x

∂∂θa〈ψ(θ)| c†ic

†jckcl |ψ(θ)〉 where x-differentiated h(x) is computed classically.

Therefore, Nx does not appear in the number of NISQ runs. Overall, the cost for the quantumpart of the computation scales as O(n4Nd

θ /ε2).

Let us compare this cost against the numerical differentiation. More specifically, we con-sider the case of evaluating the second derivative by the formula,

∂2E∗(x)∂x2

i

≈ E∗(x+ h) + E∗(x− h)− 2E∗(x)h2 , (27)

where h > 0. Let E∗(x) be the estimate of E∗(x) obtained by measuring the state |ψ(θ∗(x))〉.If |E∗(x)− E∗(x)| ≤ εE , the precision of Eq. (27) is [46],∣∣∣∣∣∂2E∗(x)

∂x2i

− E∗(x+ h) + E∗(x− h)− 2E∗(x)h2

∣∣∣∣∣ ≤ O(h2) +O(εE/h2). (28)

For classical computation where the source of εE is mainly the round-off error, the secondterm of the right-hand side is usually negligibly small for a decent h. On the other hand, forthe VQE, this term is the leading factor for the precision, since E∗(x) has to be calculatedby sampling. To achieve |E∗(x) − E∗(x)| ≤ εE with high probability, we need to run thequantum computer for O(1/ε2E) times. Therefore, if we want to achieve the precision of ε for∂2E∗(x)∂x2i

, we need O(1/(ε2h4)) samples from |ψ(θ∗(x))〉, and overall cost is O(n4N2x/(ε2h4)).

This scaling with respect to the error and the dimension of x is clearly worse than that of thepresented method, whose scaling is O(1/ε2).

6 Derivatives of excited state energyThe generation of excited states can be a powerful application of the VQE, because theclassical computation, despite the recent significant improvement in the theory and the com-putational power, still suffers in the calculation of them [47]. Among the several proposals[2–5] to generate excited states with the VQE, we adopt the one proposed in Refs. [3, 4] tocompute the derivatives of the excited state energy.

The algorithm [3, 4] works as follows. First, we find the ground state of the given Hamil-tonian H0(x), which we denote by |ψ(0)(θ(0)(x))〉, where θ(0)(x) is the optimal parameter forthe ground state (θ∗(x) in the previous sections). Then, we iteratively define a HamiltonianHr(x) for r = 1, 2, · · · as,

Hr(x) := H0(x) +r−1∑s=0

βs |ψ(s)(θ(s)(x))〉 〈ψ(s)(θ(s)(x))| , (29)

where |ψ(r)(θ(r)(x))〉 is the ground state ofHr(x). If βs is sufficiently large, each Hr(x) has r-thexcited state of the original Hamiltonian, H0(x), as its ground state. Therefore, by finding theground state of each Hr(x), one can generate the series of excited states of H0(x). We assume|ψ(r)(θ)〉 = U (r)(θ) |0〉, where U (r)(θ) has the same structure as U(θ) in previous sectionsIn this algorithm, it is also assumed that we have a device which can measure the overlapbetween |ψ(r)(θ)〉 and |ψ(s)(ϕ)〉, that is, we assume we can measure | 〈ψ(r)(θ)|ψ(s)(ϕ)〉 |2. Let

10

the expectation value of Hr(x) with respect to the state |ψ(r)(θ)〉 be Er(θ, x); Er(θ, x) =〈ψ(r)(θ)|Hr(x)|ψ(r)(θ)〉. We define the optimal energy by E∗r (x) = Er(θ(r)(x), x).

The task is to compute the derivatives such as ∂2E∗r

∂xi∂xj. Since E∗r is the ground state energy

for Hr, the formulae in the previous sections, Eqs. (7), (8), and (9) can be adapted for thistask. The only difference from that of the actual ground state is that the derivative of theHamiltonian, such as ∂2Hr

∂xi∂xj, cannot be computed classically. For example, the expression of

the first derivative of the Hamiltonian is,

∂Hr

∂xi(x) = ∂H0

∂xi(x) +

r−1∑s=0

βs

(∂ |ψ(s)(θ(s)(x))〉

∂xi〈ψ(s)(θ(s)(x))|+ h.c.

)

= ∂H0∂xi

(x) +r−1∑s=0

∑a

βs∂θ

(s)a

∂xi(x)

(|∂aψ(s)(θ(s)(x))〉 〈ψ(s)(θ(s)(x))|+ h.c.

). (30)

In the analytical expression for ∂E∗r

∂xi(Eq. (7)), ∂Hr

∂xi(x) appears as the expectation value with

respect to |ψ(r)(θ(r)(x))〉; 〈ψ(r)(θ(r)(x))|∂Hr∂xi(x)|ψ(r)(θ(r)(x))〉. As for ∂2E∗

r∂xi∂xj

(Eq. (8)), it ap-

pears as Re[〈ψ(r)(θ(r)(x))|∂Hr∂xi

(x)|∂aψ(r)(θ(r)(x))〉]. The quantities that cannot be computed

classically in Eqs. (7) to (9) are the terms which involves the inner product between the states,such as Re[〈ψ(r)(θ(r)(x))|∂aψ(s)(θ(s)(x))〉 〈ψ(s)(θ(s)(x))|ψ(r)(θ(r)(x))〉]. However, notice that ifthe condition 〈ψ(r)(θ(r)(x))|ψ(s)(θ(s)(x))〉 = 0 holds for all x, which we naturally expect atthe optimal parameter, we obtain by differentiating with respect to x the both hand side ofthe equation,

∑a

∂θ(s)a

∂xi(x)Re[〈ψ(r)(θ(r)(x))|∂aψ(s)(θ(s)(x))〉 〈ψ(s)(θ(s)(x))|ψ(r)(θ(r)(x))〉] = 0. (31)

We can therefore ignore the inner-product terms for the evaluation of derivatives of excitedstate energy and utilize the same procedure as the ground state energy in this case. Likewise,the inner-product terms that appear in the higher-order derivatives can also be ignored whenthe orthogonality condition 〈ψ(r)(θ(r)(x))|ψ(s)(θ(s)(x))〉 = 0 is satisfied. The so-called sub-space search VQE method [2], which guarantees the orthogonality condition of the resultantstates, can be advantageous to obtain such condition.

7 Numerical simulationWe provide proof-of-principle numerical simulations using the electronic Hamiltonian of thehydrogen molecule. The Hamiltonians are calculated with the open source library PySCF [48]and OpenFermion [49]. The simulation of quantum circuits are performed with Qulacs [50].

7.1 Approximation of the potential energy surfaceFirst, we perform a simulation of the VQE and the methods described in Sec. 4 with theHamiltonian of a hydrogen molecule, calculated with the STO-3G basis set, at the bondingdistance r = 0.735 A. We use the ansatz circuit shown in Fig. 3, which is a variant of so-calledhardware efficient circuit [41]. The result of the simulation is shown as Fig. 4. The ansatz

11

0

000

Figure 3: Ansatz used in simulations. Rx and Ry is single-qubit x and y rotation gate, respectively. Theparameters θ are implemented as rotation angles of Rx and Ry.

0.4 0.6 0.8 1.0 1.2H-H distance (Å)

1.1

1.0

0.9

Ener

gy (h

artre

e)

Full-CIHarmonic approx.3rd-order approx.Energy minimum

Figure 4: The harmonic and third-order approximation of the energy curve of the hydrogen molecule at thebonding distance, determined by the simulation of the VQE and the proposed method.

used in this simulation could achieve the exact ground state, which is called full configurationinteraction (Full-CI) state in the context of chemistry, and therefore, we could draw theharmonic and the third-order approximation of the Full-CI energy as shown in Fig. 4. Theharmonic approximation can be used to calculate the vibrational spectra of a molecule, andthe third order approximation can be utilized for more accurate description of the vibration.

7.2 Continuous determinination of the optimal parameterThe response equation, Eq. (10), can be used to determine the optimal paramter θ∗(x) fromthe one at the slightly different system parameter, θ∗(x+ δx), that is, to the first order,

θ∗(x+ δx) ≈ θ∗(x) + ∂θ∗

∂x(x)δx, (32)

holds up to the additive error of O(δx2). One can iteratively use the equation above, whichresembles the Euler method, to obtain the optimal parameter θEuler(x) ≈ θ∗(x) from someθ∗(x0) in a range of x around x0 where the error term is sufficiently small.

We demonstrate this parameter update in Fig. 5, by plotting the energy expectation valueof |ψ(θEuler(r))〉 where θEuler(r) is determined with the iterative use of Eq. (32), starting fromthe optimal parameter at r = 1.5 A with the step size δr = 0.02 A. Eq. (32) resembles the

12

0.5 1.0 1.5 2.0H-H distance (Å)

1.1

1.0

0.9

Ener

gy (h

artre

e)

E( Euler(r), r)Full-CIOptimized point

Figure 5: The evolution of the energy expectation value when the circuit parameters are evolved accordingto Eq. (32), starting from the optimized parameter at interatomic distance of 1.5 A.

Euler method, therefore we refer to the update of θ according to Eq. (32) as so in Fig. 5. Itcan be seen that in the range where r is sufficiently close to the optimized point, r = 1.5 A, theparameter determined from Eq. (32) is almost optimal, but as r goes far from the optimizedpoint, they deviate fast from the optima. This is because, as the error of the each updateaccumulates, the state becomes a non-variational state, that is, ∂E

∂θ 6= 0, which results in the

breakdown of the response equation, Eq. (10), where we assumed ∂E∂θ = 0. This method

should be useful for drawing potential energy surfaces using the VQE, because it can reducethe time for the optimization of the circuit parameter by predicting the optimal parameterfrom the one determined with the slightly different system. We belive it can also be combinedwith the parameter interpolation approach proposed in Ref. [51].

8 ConclusionWe have described a methodology for computing the derivatives of the ground state andthe excited state energy. We have shown the straight-forwardly constructed quantum circuitto measure the quantities necessary for the calculation of energy derivatives, and also thelow-depth version of the circuit. The low-depth protocol for the measurement makes themsuitable for the use in NISQ devices. We also performed numerical simulations of the proposedmethod, which validates the correctness. This work enables to extract the physical propertiesof the quantum system under investigation, thus widening the application range of the NISQdevices.

Note added - During the final revision of this article, we have become aware of recentarticle [52] which also describes a methodology for evaluating energy derivatives. Their workis based on perturbative sum-over-state approach and differs from our methods which providesa way to evaluate analytic derivatives.

13

AcknowledgmentsKM thanks for METI and IPA for its support through MITOU Target program. KM issupported by JSPS KAKENHI No. 19J10978. WM thanks for KAKENHI No. 18K14181.This work is also supported by MEXT Q-LEAP.

A Derivative of the optimal parameter

We derive the expression for the derivatives of the optimal parameter, such as ∂θ∗a(x)∂xi

, byTaylor expansion. Assume that at x = α, we know the optimal parameter θ∗(α) We performTaylor expansion of |ψ(θ∗(x))〉 and H(x) around α. For H(x), we obtain,

H(α+ x) = H(α) +∑i

∂H(α)∂xi

xi + 12∑i

∂

∂xi

∂H(α)∂xj

xixj + · · · . (33)

For |ψ(θ∗(x))〉, we can expand it as follows.

|ψ(θ∗(α+ x))〉

= |ψ

θ∗(α) +∑i

∂θ∗(α)∂xi

xi + 12∑i,j

∂

∂xi

∂θ∗(α)∂xj

xixj + · · ·

〉= |ψ (θ∗(α))〉

+∑a

|∂aψ (θ∗(α))〉

∑i

∂θ∗a(α)∂xi

xi + 12∑i,j

∂

∂xi

∂θ∗a(α)∂xj

xixj + · · ·

+ 1

2∑a,b

|∂a∂bψ (θ∗(α))〉

∑i

∂θ∗a(α)∂xi

xi + 12∑i,j

∂

∂xi

∂θ∗a(α)∂xj

xixj + · · ·

∑

i

∂θ∗b (α)∂xi

xi + 12∑i,j

∂

∂xi

∂θ∗b (α)∂xj

xixj + · · ·

+ · · · .

14

When grouped by the order of x,

|ψ(θ∗(α+ x))〉= |ψ (θ∗(α))〉

+∑a,i

∂θ∗a(α)∂xi

|∂aψ (θ∗(α))〉xi

+ 12∑i,j

∑a

∂

∂xi

∂θ∗a(α)∂xj

|∂aψ (θ∗(α))〉+∑a,b

∂θ∗a(α)∂xi

∂θ∗b (α)∂xj

|∂a∂bψ (θ∗(α))〉

xixj+ 1

6∑i,j,k

∑a

∂

∂xi

∂

∂xj

∂θ∗a(α)∂xk

|∂aψ (θ∗(α))〉+ 3∑a,b

∂

∂xi

∂θ∗a(α)∂xj

∂θ∗b (α)∂xj

|∂a∂bψ (θ∗(α))〉

+∑a,b,c

∂θ∗a(α)∂xj

∂θ∗b (α)∂xj

∂θ∗c (α)∂xk

|∂a∂b∂cψ (θ∗(α))〉

xixjxk+ · · · . (34)

We can derive a similar expression for |∂aψ(θ∗(α+ x))〉. Now, we use the condition of Eq. (1)to derive the expression of the derivatives of θ∗(α). Plugging Eqs. (33) and (34) into Eq. (1)and imposing the coefficient of each order in x to be zero, we get the analytical expressionfor the derivatives of θ∗(α). In the following two subsections, we derive the expression for thefirst and second derivatives of θ∗(α).

A.1 First derivativeFor the first order in x, we get

Re[∑a

(〈∂bψ (θ∗(α)) |H(α)|∂aψ (θ∗(α))〉+ 〈ψ (θ∗(α)) |H(α)|∂b∂aψ (θ∗(α))〉) ∂θ∗a(α)∂xi

+ 〈∂bψ (θ∗(α)) |∂H(α)∂xi

|ψ (θ∗(α))〉]

= 0. (35)

Note that the first term of Eq. (35) includes the Hessian of E(θ, x), that is,

∂

∂θb

∂E(θ∗(α), α)∂θa

= 2Re [(〈∂bψ (θ∗(α)) |H(α)|∂aψ (θ∗(α))〉+ 〈ψ (θ∗(α)) |H(α)|∂b∂aψ (θ∗(α))〉)] .

(36)Also, notice that,

∂

∂θb

∂E(θ, x)∂xi

= 2Re[〈∂bψ (θ) |∂H

∂xi(x)|ψ (θ)〉

]. (37)

We define ∇dθ∂E∂xi

likewise. Finally, we obtain,∑a

∂

∂θb

∂E(θ∗(α), α)∂θa

∂θ∗a(α)∂xi

= − ∂

∂θb

∂E(θ∗(x), x)∂xi

, (38)

which is Eq. (10) of the main text. Note that we expect the matrix ∂∂θb

∂E(θ∗(α),α)∂θa

to bepositive definite because θ∗(α) is a local minimum, and thus Eq. (38) is solvable to obtain∂θ∗a(α)∂xi

.

15

A.2 Second derivativeThe second derivative, ∂

∂xi∂θa∂xj

, is derived from the second order in x of Eq. (1). We have,

from Eq. (1), for all i, j and c,

Re

12∑a,b

∂θ∗a(α)∂xi

∂θ∗b (α)∂xj

(〈∂cψ (θ∗(α)) |H(α)|∂a∂bψ (θ∗(α))〉+ 〈∂a∂b∂cψ (θ∗(α)) |H(α)|ψ (θ∗(α))〉

+ 〈∂a∂cψ (θ∗(α)) |H(α)|∂bψ (θ∗(α))〉+ 〈∂b∂cψ (θ∗(α)) |H(α)|∂aψ (θ∗(α))〉)

+ 12∑a

∂

∂xi

∂θ∗a(α)∂xj

(〈∂cψ (θ∗(α)) |H(α)|∂aψ (θ∗(α))〉+ 〈∂a∂cψ (θ∗(α)) |H(α)|ψ (θ∗(α))〉)

+∑a

∂θ∗a(α)∂xj

(〈∂cψ (θ∗(α)) |∂H(α)

∂xj|∂aψ (θ∗(α))〉+ 〈∂a∂cψ (θ∗(α)) |∂H(α)

∂xj|ψ (θ∗(α))〉

)

+12 〈∂cψ (θ∗(α)) | ∂

∂xi

∂H(α)∂xj

|ψ (θ∗(α))〉]

= 0. (39)

∂∂θa

∂∂θb

∂E∂θc

, ∂∂θa

∂E∂θb

can be used to greatly simplify the above equation, which gives us,

14∑a,b

∂θ∗a(α)∂xi

∂θ∗b (α)∂xj

∂

∂θc

∂

∂θa

∂E(θ∗(α), α)∂θb

+ 14∑a

∂

∂xi

∂θ∗a(α)∂xj

∂

∂θc

∂E(θ∗(α), α)∂θa

+ 12∑a

∂θ∗a(α)∂xi

∂

∂θc

∂

∂θa

∂E(θ∗(α), α)∂xj

+ 12Re

[〈∂cψ (θ∗(α)) | ∂

∂xi

∂H(α)∂xj

|ψ (θ∗(α))〉]

= 0.

(40)

This is equivalent to Eq. (11)

B Derivatives of the ground state energyB.1 First derivativeThe first derivative of the energy is calculated as,

∂E∗

∂xi(x) = ∂

∂xi〈ψ(θ∗(x))|H(x) |ψ(θ∗(x))〉 ,

= 2Re[〈ψ(θ∗(x))|H(x)∂ |ψ(θ∗(x))〉

∂xi

]+ 〈ψ(θ∗(x))| ∂H

∂xi(x) |ψ(θ∗(x))〉 ,

= 2∑a

∂θ∗a(x)∂xi

Re [〈ψ(θ∗(x))|H(x) |∂aψ(θ∗(x))〉] + 〈ψ(θ∗(x))| ∂H∂xi

(x) |ψ(θ∗(x))〉 ,

and first term of the above equation vanishes by Eq. (1) of the main text. Thus, we get,

∂E∗

∂xi(x) = 〈ψ (θ∗(x), x) |∂H

∂xi(x)|ψ (θ∗(x), x)〉 , (41)

which is Eq. (7) of the main text

16

B.2 Second derivativeHere, we derive the expression for the second derivative.

∂

∂xi

∂E∗

∂xj(x)

= ∂

∂xi〈ψ(θ∗(x))|∂H

∂xj(x)|ψ(θ∗(x))〉

= 2Re[〈ψ(θ∗(x))| ∂H

∂xj(x)∂ |ψ(θ∗(x))〉

∂xi

]+ 〈ψ(θ∗(x))| ∂

∂xi

∂H

∂xj(x) |ψ(θ∗(x))〉

= 2∑a

Re[〈ψ(θ∗(x))| ∂H

∂xj(x) |∂aψ(θ∗(x))〉

]∂θ∗a∂xi

(x) + 〈ψ(θ∗(x))| ∂∂xi

∂H

∂xj(x)|ψ(θ∗(x))〉

=∑a

∂θ∗a∂xi

(x) ∂

∂θa

∂E

∂xj(θ∗(x), x) + 〈ψ(θ∗(x))| ∂

∂xi

∂H

∂xj(x) |ψ(θ∗(x))〉 . (42)

This is Eq. (8) of the main text.

B.3 Third derivativeThird derivative can be calculated as follows.

∂

∂xi

∂

∂xj

∂E∗

∂xk(x)

= ∂

∂xi

(∑a

∂θ∗a∂xj

(x) ∂

∂θa

∂E

∂xk(θ∗(x), x) + 〈ψ(θ∗(x))| ∂

∂xj

∂H

∂xk(x) |ψ(θ∗(x))〉

)

=∑a

∂

∂xi

∂θ∗a∂xj

(x) ∂

∂θa

∂E

∂xk(θ∗(x), x) +

∑a

∂θ∗a∂xj

(x) ∂

∂xi

(∂

∂θa

∂E

∂xk(θ∗(x), x)

)

+ 2Re[∂ 〈ψ(θ∗(x))|

∂xi

∂

∂xj

∂H

∂xk(x) |ψ(θ∗(x))〉

]+ 〈ψ(θ∗(x))| ∂

∂xi

∂

∂xj

∂H

∂xk(x) |ψ(θ∗(x))〉

=∑a

∂

∂xi

∂θ∗a∂xj

(x) ∂

∂θa

∂E

∂xk(θ∗(x), x) +

∑a,b

∂θ∗a∂xj

(x)∂θ∗b

∂xi(x)

(∂

∂θb

∂

∂θa

∂E

∂xk(θ∗(x), x)

)

+ 2∑a

∂θ∗a∂xj

(x)Re[〈∂aψ(θ∗(x))| ∂

∂xi

∂H

∂xk|ψ(θ∗(x))〉

]

+ 2∑a

∂θ∗a∂xi

(x)Re[〈∂aψ(θ∗(x))| ∂

∂xj

∂H

∂xk(x) |ψ(θ∗(x))〉

]+ 〈ψ(θ∗(x))| ∂

∂xi

∂

∂xj

∂H

∂xk(x) |ψ(θ∗(x))〉 .

(43)

This expression (Eq. (43)) can be simplified so as to avoid the explicit calculation of ∂∂xi

∂θ∗a

∂xj.

By multiplying ∂θa∂xk

to Eq. (10) and ∂∂xi

∂θa∂xj

to Eq. (11), and combining them, we obtain,

∂

∂xi

∂θa∂xj

∂

∂θa

∂E

∂xk=

∑b,c

∂

∂θa

∂

∂θb

∂E

∂θc

∂θa∂xk

∂θb∂xi

∂θc∂xj

+ 2∑b

∂

∂θa

∂

∂θb

∂E

∂xj

∂θb∂xi

∂θa∂xk

+ 2Re[〈∂aψ|

∂

∂xi

∂H

∂xj|ψ〉]∂θa∂xk

. (44)

17

This yields,

∂

∂xi

∂

∂xj

∂E∗

∂xk(x)

=∑a,b,c

∂

∂θa

∂

∂θb

∂E

∂θc(θ∗(x), x)∂θa

∂xi(x) ∂θb

∂xj(x) ∂θc

∂xk(x) + 〈ψ(θ∗(x))| ∂

∂xi

∂

∂xj

∂H

∂xk(x) |ψ(θ∗(x))〉

+∑a,b

[∂θ∗a∂xi

(x)∂θ∗b

∂xj(x) ∂

∂θb

∂

∂θa

∂E

∂xk(θ∗(x), x) + ∂θ∗a

∂xk(x)∂θ

∗b

∂xi(x) ∂

∂θb

∂

∂θa

∂E

∂xj(θ∗(x), x)

+∂θ∗a∂xj

(x) ∂θ∗b

∂xk(x) ∂

∂θb

∂

∂θa

∂E

∂xi(θ∗(x), x)

]

+ 2∑a

[∂θ∗a∂xi

(x)Re[〈∂aψ(θ∗(x))| ∂

∂xj

∂H

∂xk(x) |ψ(θ∗(x))〉

]

+ ∂θ∗a∂xk

(x)Re[〈∂aψ(θ∗(x))| ∂

∂xi

∂H

∂xj(x) |ψ(θ∗(x))〉

]

+∂θ∗a∂xj

(x)Re[〈∂aψ(θ∗(x))| ∂

∂xk

∂H

∂xi(x) |ψ(θ∗(x))〉

]], (45)

which is Eq.(9) of the main text.

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