Constrained variational quantum eigensolver: Quantum computer

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Quantum Electronic Structure

Constrained variational quantum eigensolver: Quantum computer search engine in the Fock space Ilya G. Ryabinkin, Scott N. Genin, and Artur F. Izmaylov J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.8b00943 • Publication Date (Web): 04 Dec 2018 Downloaded from http://pubs.acs.org on December 5, 2018

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Journal of Chemical Theory and Computation

Constrained variational quantum eigensolver: Quantum computer search engine in the Fock space Ilya G. Ryabinkin,†,‡ Scott N. Genin,¶ and Artur F. Izmaylov∗,†,‡ †Department of Physical and Environmental Sciences, University of Toronto Scarborough, Toronto, Ontario, M1C 1A4, Canada ‡Chemical Physics Theory Group, Department of Chemistry, University of Toronto, Toronto, Ontario M5S 3H6, Canada ¶OTI Lumionics Inc., 100 College St. #351, Toronto, Ontario M5G 1L5, Canada E-mail: [email protected]

December 3, 2018 Abstract Variational quantum eigensolver (VQE) is an efficient computational method promising chemical accuracy in electronic structure calculations on a universal-gate quantum computer. However, such a simple task as computing the electronic energy of a hydrogen molecular cation, H2+ , is not possible for a general VQE protocol because the calculation will invariably collapse to a lower energy of the corresponding neutral form, H2 . The origin of the problem is that VQE effectively performs an unconstrained energy optimization in the Fock space of the original electronic problem. We show how this can be avoided by introducing necessary constraints directing VQE toward the electronic state of interest. The proposed constrained VQE can find an electronic

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ACS Paragon Plus Environment

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state with a certain number of electrons, spin, or any other property. Moreover, the new algorithm naturally removes unphysical kinks in potential energy surfaces, which frequently appeared in the regular VQE and required significant additional quantum resources for their removal. We demonstrate performance of the constrained VQE by simulating potential energy surfaces of various states of H2 and H2 O on Rigetti Computing Inc’s 19Q-Acorn quantum processor.

1

Introduction

Quantum chemistry seeks the exact solution of the electronic Schrödinger equation, 1

ˆ e |Ψi (R)i = Ei (R) |Ψi (R)i , H

(1)

ˆ e is the electronic Hamiltonian of a molecule with a fixed nuclear configuration R, where H Ei (R) are its eigenvalues, also known as potential energy surfaces (PESs), and |Ψi (R)i are the corresponding electronic wavefunctions. Even though this is only the electronic part of the total molecular quantum problem, it determines systems’ properties crucial for designing new materials citeMartin:2004, Prasad:2013 and pharmaceutical compounds. 2 The main computational difficulty of tackling this problem is the exponential growth of complexity with the number of interacting particles (electrons). This exponential scaling makes it infeasible to obtain high accuracy for large systems (e.g. materials and proteins) on a classical computer. Various approximations compromising the accuracy become necessary. 1,3 There is a hope to overcome the exponential scaling by using a universal quantum computer. 4,5 The earliest proposal was the quantum phase estimation algorithm, 6–8 which was quite successful in terms of accuracy but placed strong requirements on quantum hardware to maintain coherence for a long time. As an alternative with reduced coherency requirements, the variational quantum eigensolver (VQE) has been suggested. 9–11 However, a unitary coupled cluster (UCC) ansatz with a fixed excitation rank used in VQE is not rigorously equivalent to the exact solution of

2


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the electronic structure problem 12,13 but rather gives numerical results of chemical accuracy and is exponentially hard for a classical computer. Recent experimental work by Kandala et al. 14 demonstrated successful quantum simulations by means of the tailored VQE ansatz for PESs of few selected small molecules, H2 , LiH, and BeH2 . Despite the impressive results, there were still visible imperfections, “kinks”, in the simulated PESs whose origins were not clear. The authors attributed them to the limited accuracy of simulations and claimed that they could be removed by increasing resource requirements. Yet, it is still desirable to disentangle the difficulties related to experimental realization from a possible incompleteness of the employed formalism. One of the main goals of quantum chemistry is to produce smooth PESs that can be used further in modeling chemical dynamics. Therefore, having kinks is a significant drawback that cannot be left unresolved. Another problem that has not received enough attention is how to obtain information about electronic states with different numbers of electrons (e.g. cations and anions) or different spin (e.g. singlets, triplets, etc.). The key to understanding both problems, eliminating PES kinks and obtaining PESs for different charge and spin electronic states, is the first step in the formulation of the electronic structure problem for quantum computing. To encode the electronic Hamiltonian using qubits one needs to start not with the Hilbert space formulation (1) but rather with the so-called second-quantized formulation of the Hamiltonian, which operates in the Fock space. The Fock space for a particular molecule combines the Hilbert spaces of all molecular forms with all possible number of electrons. This leads to an interesting problem, namely: for molecule A there is only one Hamiltonian, whose eigenvalues are electronic energies of A, A± , A2± , etc. Since the electronic energy of a cation is always higher than that for a neutral—otherwise a molecule would be autoionizing—it becomes an excited state in the full spectrum. Any variational method aimed at unconditional minimization of energy will converge to the state of a neutral A, leaving A+ inaccessible. Even worse, for any molecule with a positive electron affinity the lowest-energy solution is

3



an anion rather than the neutral form. A similar situation is with the total electron spin, when the spin multiplicity of the lowest energy state could be different from the one that is of interest. The problem of finding excited states within VQE has already received some attention. Later on, several approaches, including both constraints and explicit parametrization of the electronic excitation manifolds, were considered in Refs. 15 and 16. However, these works did not make sharp distinction between excited states that have the same exact fermionic symmetries, such as the number of electrons and spin, and those that do not. This distinction is quite important though. The physical electronic Hamiltonian does not couple states with different spins and the number of electrons; thus any approximate method that introduces such a coupling must be remedied. We will show that in its current form, VQE leads to kinks in PESs due to variational instabilities that cause switches between electronic states of different symmetries within the Fock space. These issues also lead to the inability of current simulation protocols to compute PESs of molecular cations. A particularly simple example of this problem is H2+ , which is exactly solvable problem that dates back to the early days of quantum mechanics. 17,18 We resolve all these issues by introducing the constrained modification of the VQE, which is indispensable for quantum chemistry applications. The idea of symmetry constraints and constraint optimization was first discussed in Ref. 19 in the context of preserving correct symmetries after the Trotterization in the UCC method.

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2

Results

2.1

Operators in the Fock and qubit spaces

Formulation of the electronic structure problem for a quantum computer starts from the ˆ e in the second-quantized form electronic Hamiltonian H

ê = H

Nb X

N

hij a ˆ†i a ˆj

ij

b 1X + gijkl a ˆ†i a ˆ†k a ˆl a ˆj , 2 ijkl

(2)

where a ˆ†i and a î are fermionic creation and annihilation operators corresponding to a oneelectron state φi within an Nb one-electron basis set. 1 The coefficients, hij and gijkl , are oneand two-electron integrals, respectively. Nb determines a computational cost of solving the electronic Schrödinger equation because computational expenses grow exponentially with Nb if no approximations are made. A Hamiltonian of a free molecule in the absence of an external electromagnetic field forms ˆ , the z-projection of the a set of commuting operators with the electron number operator, N total molecular spin, Sˆz , and the square of the total spin, Sˆ2 . (The last two should not be confused with corresponding qubit operators.) All these operators have the second-quantized forms that can be found in Ref. 1. Quantum computers employ two-level systems (“qubits”) as the computational basis. Qubits can be thought of as spin-1/2 particles, although real quantum computers may not use genuine spins. 9,14 The fermionic Hamiltonian (2) is translated from fermionic to qubit representation by one of the fermion-to-qubit mappings, the Jordan–Wigner (JW) or more recent and resource-efficient the Bravyi–Kitaev (BK) transformation. 20,21 After the JW or ˆ e assumes BK transformations all operators become operators in qubit space, for instance, H the form ˆ = H

X

CI TÎ ,

(3)

I

where CI coefficients are functions of one- and two-electron integrals, and TÎ are products of 5



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several spin operators, (I) (I) TÎ = · · · ω1 ω0 ,

(4) (I)

which we call “Pauli words” for the sake of brevity. Each of ωi

denotes one of the Pauli

matrices, xi , yi , or zi operating on the ith fictitious spin-1/2 particle that represents the ith ˆ is an Nb -qubit operator that has a 2Nb × 2Nb matrix representations. Importantly, qubit. H not only the matrix dimension but also the whole spectrum of the qubit Hamiltonian (3) is identical to its fermionic counterpart (2). Thus, finding the eigenstates of the qubit Hamiltonian (3) is equivalent to the solving the quantum chemistry problem. As an illustration, we consider the H2 molecule in the minimal STO-3G basis (Nb = 4). The fermionic Hamiltonian of this system describes 2Nb = 16 electronic states. Presence of states of different spin and number of electrons does not pose a difficulty in ordinary quantum chemistry on a classical computer because the electronic Hamiltonian is projected onto a Hilbert subspace corresponding to the electronic state of interest. This projection is done via ˆ i, Sz = hSˆz i use of Slater determinants. They implicitly fix the number of particles N = hN and even hSˆ2 i = S(S + 1) if appropriate combinations (configurations) of Slater determinants are chosen. Fig. 1 presents lowest singlet and triplet electronic PESs of the H2 molecule and the ground state PES of its cation obtained by the full configuration interaction approach. Using the BK transformation the electronic Hamiltonian in the STO-3G basis is mapped to the same number (Nb = 4) of qubits. The resulting Hamiltonian1 has 15 Pauli words multiplied by a coefficient inferred from one- and two-electron integrals (hij and gijkl ) at a 1

All Hamiltonians are generated using the OpenFermion software. 22

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−0.2

−0.4 +

H2 E, hartree

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−0.6

−0.8

H2, triplet

−1

−1.2

H2, singlet

0.5

1

1.5

2

2.5

R(H−H), Å Figure 1: Two lowest PESs of the H2 molecule and the ground state PES of the H+ 2 cation obtained using the full configuration interaction method in the STO-3G basis.

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given interatomic distance R. For example, at R = 0.75˚ A we have2 ˆ BK (R) H

R=0.75˚ A

= −0.109731

+ 0.169885 z0

+ 0.168212 z1

+ 0.169885 z1 z0

+ 0.0454429 x2 z1 x0

− 0.218863 z2

+ 0.0454429 y2 z1 y0

+ 0.120051 z2 z0

+ 0.165494 z2 z1 z0

+ 0.173954 z3 z1

+ 0.0454429 z3 x2 z1 x0

+ 0.120051 z3 z2 z0

+ 0.0454429 z3 y2 z1 y0

− 0.218863 z3 z2 z1

+ 0.165494 z3 z2 z1 z0 .

(5)

Diagonalization of this Hamiltonian in the 2Nb = 16-dimensional qubit space provides the same eigenvalues, but the information about the number of electrons N , or hSˆ2 i for corresponding eigenstates is now hidden. Let us consider the two lowest exact PESs of the H2 molecule that were calculated by ˆ BK (R) for different R (Fig. 2) We track the full diagonalization of the qubit Hamiltonian H physical nature of the solutions using properties corresponding to commuting observables: ˆ and Sˆ2 first, we constructed the BK-transformed operators N

ˆBK = 2 − (z0 + z1 z0 + z2 + z3 z2 z1 )/2, N

(6)

2 SˆBK = (6 − 3z1 + x2 x0 − x2 z1 x0 + y2 y0

+z2 z0 − z2 z1 z0 − 3z3 z1 − y2 z1 y0 +z3 x2 x0 − z3 x2 z1 x0 + z3 y2 y0 −z3 y2 z1 y0 + z3 z2 z0 − z3 z2 z1 z0 )/8

(7)

and then evaluated the mean values of these operators on the calculated exact states. First 2

The nuclear-nuclear repulsion energy 1/R = 0.705333 a.u. is included in this Hamiltonian.

8


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−0.2 H+2

E0 E1 EQMF

−0.4

E, hartree

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−0.6

−0.8

−1

−1.2

0.5

1

1.5

2

2.5

R(H−H), Å ˆ BK (R) (solid lines), and the PES Figure 2: Two lowest eigenstates of the Hamiltonian H corresponding to the minimum of the qubit mean field (QMF) functional, Eq. (12) (dashed line). Points with error bars correspond to QMF calculations performed on Rigetti’s 19QAcorn quantum computer. Error bars show the standard deviation over measured values.

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of all, for R ≤ 0.7˚ A the first excited state corresponds to the state of H2+ , N = 1, while for larger R it is a triplet state of the neutral molecule (N = 2, hSˆ2 i = 2). Thus, in the STO-3G basis set the triplet state changes its position in the spectrum as R varies. The cationic ground state is among the excited states and intersects with the triplet H2 state forming a kink due to the energy ordering of the electronic states. Therefore, one of the reasons for kinks in quantum calculations can be intersections of states that originally belonged to different Hilbert spaces of the fermionic problem and brought within the same qubit space by using the Fock-space second-quantized Hamiltonian (2).

2.2

Variational quantum eigensolver (VQE)

VQE relies on parametrization of a wave function that can be implemented as a quantum circuit. Originally, 9 it was proposed to use the UCC ansatz,

|Ψi = exp(Tˆ − Tˆ† ) |Φref i

(8)

where Tˆ is a sum of fermionic single-, double-, etc. excitation operators, and |Φref i is a suitably chosen reference function, usually the qubit image of the Hartree–Fock ground state. Contrary to the traditional coupled-cluster form, UCC leads to non-terminating Baker– Campbell–Hausdorff series 23 even at the fixed excitation rank, and hence, is exponentially hard for a classical computer, but only polynomially complex (via Trotterization 24 ) for a quantum computer. However, a large prefactor of the UCC form prompted Kandala et al. 14 to propose a more general, “hardware-efficient” alternative, which we consider below with some further simplifications. A trial wavefunction for an Nb -qubit problem is

|Ψ(Ω, τ )i =UENT (τ )UMF (Ω) |00 . . . 0i ,

(9)

where |00 . . . 0i is an initialized Nb -qubit state, UMF (Ω) is a product of rotations of individual 10


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q qubits, UMF (θ) = exp(−iθ1q zq ) exp(−iθ2q xq ) exp(−iθ3q zq ) where {θjq }3j=1 are three Euler angles

of the q th qubit. Ω is a subset of Euler angles for all qubits that transforms product states into each other up to a global phase. Below we show that Ω conveniently parametrizes the product of the so-called spin-coherent (Bloch) states. UENT (τ ) is a product of operators that entangle two or more qubits. Individual factors in UENT (τ ) are the exponent of Pauli words multiplied by the corresponding amplitude, exp(−iτI TÎ ); they are responsible for post meanfield treatment of electron-electron correlation effects. Kandala et al. 14 used only two-qubit Pauli words and fixed amplitudes to protect entanglement from noise and decoherence, and relied on single-qubit rotations interleaved with entanglers to provide variational parameters associated with multi-qubit transformations, but in general, amplitudes τ can be varied directly. Once the trial form (9) for a fixed set of parameters is implemented as a quantum circuit and executed, the expectation value of the Hamiltonian (3) is calculated from measurements of individual Pauli words, TÎ , as ˆ E(Ω, τ ) = hΨ(Ω, τ )|H|Ψ(Ω, τ )i X = CI hΨ(Ω, τ )|TÎ |Ψ(Ω, τ )i .

(10)

I

Finally, the function E(Ω, τ ) is minimized on a classical computer

E = min E(Ω, τ ), Ω,τ

(11)

giving the estimate of the lowest-energy eigenvalue of the qubit Hamiltonian (3). Such a two-step procedure allows for a feedback loop where the classical optimizer directs the quantum computer to encode trial wavefunctions that are closer in variational sense to the ground-state wavefunction of a system with the fermionic Hamiltonian (2). For a single-qubit rotations in UMF (Ω) of Eq. (9), only two out of the three Euler angles change the total energy, and one angle defines a global phase change, which does not affect 11



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the energy. A convenient basis for the Hilbert space of individual qubits representing these relations is a basis of spin coherent states, 25–28 {|Ωq i}, q = 1, 2, . . . , Nb where Ω = (φ, θ) encodes a position of a qubit orientation on the Bloch sphere (see Methods for more formal definitions). The direct product of individual qubit coherent states forms the Nb -qubit mean-field solution |Ωi = UMF (Ω) |00 . . . 0}i. The optimal values of Ω can be obtained using | {z Nb

the variational principle, 28 if E0 is the ground state energy for the Hamiltonian in Eq. (3) ˆ ˆ then E0 ≤ hΩ|H|Ωi = EQMF (Ω). Therefore, hΩ|H|Ωi defines the qubit mean-field energy functional, EQMF (Ω) =

X

(I)

(I)

CI FI (n1 , . . . nNb ),

(12)

I (I) (I) where each FI is obtained from TÎ in Eq. (3) by substitution ωi → ni and converting (I)

operator into products of functions. ni

(I)

is a shorthand notation for the ωi

component of

the unit vector on a Bloch sphere: n = (cos φ sin θ, sin φ sin θ, cos θ). The ground-state mean-field solution for H2 , EQMF (Fig. 2), which behaves like the restricted Hartree–Fock (RHF) curve for small R, has a second type of kink near R ≈ 1.6 ˚ A. This kink is due to switching of a mean-field minimum from a singlet (S = 0) to a triplet (S = 1) solution. To expose the second type of kink in quantum computing for PESs in their most vivid form we avoided using entanglers. Generally, entanglers are supposed to bring the mean-field solution closer to the exact one. The later is smooth in this case (Fig. 2), and therefore, if the entanglers fully accomplished the task there would be no kink. However, in practical calculations there is no general prescription how to choose the entangler that rigorously guarantees convergence to the exact answer. Thus, any approximate entangler can make kinks originating in mean-filed solutions less pronounced but still existent.

12


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2.3

Constrained Variational Quantum Eigensolver

To modify the variational procedure to include information about N and hSˆ2 i of the target state we use the constrained optimization. A constrained minimization is readily applicable to VQE by adding a penalty to the energy functional, namely

ˆ E(Ω, τ , µ) = hΨ(Ω, τ )|H|Ψ(Ω, τ )i i2 X h ˆ i |Ψ(Ω, τ )i − Oi , + µi hΨ(Ω, τ )|O

(13)

i

ˆ i are constraining operators (e.g. N ˆ , Sˆ2 , etc.), Oi are their desired mean values, and where O µi are large fixed numbers. 29 Comparing operators in Eqs. (5) and (6) one can see that they ˆ i |Ψ(Ω, τ )i can be formed share the same Pauli words, which means that the value hΨ(Ω, τ )|O reusing values of hΨ(Ω, τ )|TÎ |Ψ(Ω, τ )i at zero additional cost. To obtain lowest energy mean-field solutions of neutral H2 with well-defined electron spin we minimize the following functional h

2 ES (Ω, µ) = EQMF (Ω) + µ hΩ|SˆBK |Ωi − S(S + 1)

i2

,

(14)

where S is either 0 (singlet) or 1 (triplet). The PESs of such constrained minimizations do not exhibit any kinks and retain their target spin values at all R (Fig. 3). The constrained mean-field singlet PES coincides with the RHF curve and demonstrates the same asymptotic behavior by going to the incorrect dissociation limit that is exactly in between purely covalent H· + H· and ionic H+ + H− solutions. To correct for this behavior requires an addition of an entangler. On the other hand, the triplet mean-field counterpart reproduces the exact triplet PES because there is no electron-electron correlation for the triplet state in this minimal basis setup.

13



0 H2, singlet H2, triplet H+2 EQMF, S2=0

−0.2 −0.4 E, hartree

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−0.6 −0.8 −1 −1.2

0.5

1

1.5

2

2.5

R(H−H), Å Figure 3: Three constrained mean-field PESs for lowest singlet (S = 0) and triplet (S = 2) electronic states of H2 and the ground electronic state of H+ 2 (N = 1). The exact PESs for all but the singlet H2 state coincide with the constraint mean-field solutions. Points with error bars correspond to constrained QMF calculations performed on Rigetti’s 19Q-Acorn quantum computer. Error bars show the standard deviation over measured values.

14


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Similarly, the constrained minimization of the functional h i2 ˆBK |Ωi − 1 EN (Ω, µ) = EQMF (Ω) + µ hΩ|N

(15)

that imposes the N = 1 constraint has been employed to extract the lowest PES of H2+ (Fig. 3). The resulting curve is smooth and coincides with the exact H+ 2 PES due to the absence of electron-electron correlation. The constrained methodology can be easily extended to larger systems where constraints become especially useful due to increasing density of electronic states. As an example, we consider the ground singlet state of the water molecule in the STO-3G basis. The water qubit Hamiltonian contains 193 Pauli words. It simulates 6 electrons (the remaining 4 electrons are frozen) on 4 orbitals (3 occupied and 1 virtual). Figure 4 shows its PES obtained by changing a single OH-bond length from the symmetric equilibrium configuration with R(OH) = 0.9605˚ A and ∠HOH = 104.95◦ . Here, we used both spin and number of electrons constraints to obtain the mean-field solution. The spin constraint is invaluable for avoiding convergence to the closely spaced triplet solution (Fig. 4). Another advantage of using the constraints in quantum computing is their capacity to reduce the noise in the measurement process. 30 Generally, there are two sources of uncertainty when the Hamiltonian components are measured on the quantum computer: 1) the wavefunction encoded in qubits is not an eigenfunction of a particular Pauli word (TI ) and hence there is an intrinsic quantum uncertainty for the TI measurement, 2) uncontrolled interactions of qubits with their environment that introduce noise unrelated to the ideal, quantum uncertainty. A typical measurement on a quantum computer produces an eigenstate of a measured Pauli word, TI . We found that re-weighting results of the measurement based on overlaps of a collapsed wavefunction with eigenfunctions of the property operators (e.g. ˆ and Sˆ2 ) with target eigenvalues (fixed charge and spin) strongly reduces the noise coming N from the second source, and does not alter the statistics originating from the truly quantum

15



−70 −71 −72 E, hartree

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−73 −74 −75 −76

0.5

1

1.5 R(OH−H), Å

2

2.5

Figure 4: Restricted Hartree–Fock PESs for the lowest singlet (blue line) and triplet (red line) electronic state of H2 O obtained on a classical computer. Points with error bars correspond to the constrained QMF calculations for the lowest singlet state performed on the Rigetti quantum computer. Error bars show the standard deviation over measured values.

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uncertainty. Our constraint based post-processing scheme is detailed in the Methods section.

2.4

Discussion

We have proposed a simple constrained VQE approach that is indispensable if one seeks a solution of a quantum chemistry problem for an electronic state with a well-defined electronic spin, charge, or any other property of interest. The corresponding procedure requires minimal modification of the current VQE protocol and incurs virtually no additional quantum costs. Using the constrained VQE not only allows one to target specific states but also removes kinks in PESs arising due to numerical instabilities associated with the root switching. In the current study only the electron number and the total spin operators have been used for imposing constraints. The z-projection of the total electron spin (Sˆz ) has not been constrained, although it is another symmetry that one can use. We found that restriction of Sz was not giving any improvements in the considered cases. Moreover, the operators of conserved quantities can be used to post-process the results of measurements done on the quantum computer to reduce the noise due to qubit interactions with the environment. This post-processing does not affect the statistics of the true quantum distribution arising from the ideal projective measurement.

3 3.1

Methods Spin coherent states

A spin coherent state, also known as a “Bloch state”, for a single particle with spin j (j ≥ 0 is integer or half-integer) is defined by the action of an appropriately scaled exponent of the lowering operator sˆ− on the normalized eigenfunction of sˆz operator, sˆz |jmi = m |jmi, with

17



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maximal projection m = j: 28 θ θ iφ exp tan e sˆ− |jji |Ωi = cos 2 2 1/2 j X 2j = m+j m=−j θ i(j−m)φ θ j−m j+m sin e |jmi , × cos 2 2 2j

(16)

where the |jmi states are normalized as |jmi =

2j m+j

1/2

[(j − m)!]−1 sj−m |jji . −

(17)

{|Ωi} constitute an overcomplete non-orthogonal set of states on a unit Bloch sphere parametrized by spherical angles Ω = (φ, θ), 0 ≤ φ < 2π, 0 ≤ θ ≤ π.

3.2

Classical minimization

Constrained [Eqs. (14) and (15)] and unconstrained [Eq. (12)] minimizations on a classical computer were done using the sequential quadratic programming (SQP) algorithm as implemented by the fmincon routine of the matlab 31 software. All mean-field solutions were obtained by minimizing the corresponding energy functions with respect to all 4 × 2 = 8 Bloch angles.

3.3

Quantum computer simulation details

We performed the simulations on the Rigetti 19Q-Acorn quantum processor unit (QPU) 32 using pyQuil and Forest API. 33 Our wavefunction ansatz was obtained performing RZ and RX gate operations on individual qubits. This corresponds to mean-field rotations, i.e. without qubit entanglement. Qubits were selected based on ensuring the one gate fidelity of the qubits were greater than 0.99.

18


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After the BK transformation, terms of the resulting Hamiltonian that form a mutually commutative set of operators simultaneously diagonalizable by single qubit rotations are grouped together to perform a measurement on all of them at the same time. Since all the operators within a commutative set share eigenfunctions this procedure reduces the spread of measurement results due to general non-commutativity of the BK Hamiltonian and its terms. The expectation value of each commutative group was obtained by averaging over 1000 and 10000 measurements for H2 and H2 O, respectively. A post-processing procedure removing results with incorrect electron numbers and spin evaluated for each read was used for these measurements. It is shown below that this post-processing only removes results that appear due to experimental noise and does not alter quantum distributions of measurements. Upon assembling the expectation value of the total Hamiltonian from the expectation values of commutative groups, this procedure was repeated 20 (4) times for H2 (H2 O) to obtain representative statistics for the Hamiltonian expectation values. Averages and standard deviations calculated over these 20 (4) Hamiltonian expectation values are reported as the final averages and standard deviations obtained on the QPU. Since our time was limited on the Rigetti system, we were not able to perform a more in-depth sampling procedure. All four experiments were performed in 24 hours over the course of 5 sessions. The classical optimization step for the VQE, we implemented the Nelder-Mead (NM) algorithm. 34 Methods such as conjugate gradient descent were tried, but the NM algorithm demonstrated more robustness to the noise that was generated by errors.

3.4

Post-processing procedure

Description of the procedure. For the illustration purpose, let us assume that our ˆ = Aˆ + B, ˆ [A, ˆ B] ˆ 6= 0 (e.g. Aˆ and B ˆ are Hamiltonian has two non-commuting parts, H ˆ on a wavefunction |Ψi is two non-commuting Pauli words). Estimating the average of H ˆ ˆ ˆ done by adding the averages from non-commuting parts, hΨ|H|Ψi = hΨ|A|Ψi + hΨ|B|Ψi. The averages for both parts are computed by doing repetitive measurements, but these 19



Page 20 of 27

ˆ do not share measurements collapse |Ψi to different eigenfunctions because 1) Aˆ and B ˆ If we denote the eigenfunctions, and 2) |Ψi is not generally an eigenfunction of Aˆ or B. ˆ as |fn i(|gn i) and an (bn ), respectively, then the eigenfunctions and eigenvalues of Aˆ (B) Hamiltonian average is

ˆ hΨ|H|Ψi =

X

an | hΨ|fn i |2 + bn | hΨ|gn i |2 .

(18)

n

The probabilities | hΨ|fn i |2 and | hΨ|gn i |2 are not measured but instead emerge as a result ˆ The post-processing procedure of collecting results of individual measurements of Aˆ and B. simply removes the eigenvalues an and bn if corresponding eigenfunctions |fn i and |gn i, available as read-outs of QPU, are orthogonal to those for the number of electrons operator. This procedure can be extended to involving the spin operator but its eigenstates are more complicated and were not used in this work. In the case when eigenfunctions |fn i (or |gn i) have non-zero overlap with eigenfunctions of symmetry operators, the corresponding results are not altered. As shown below, the re-weighting does not change the correct statistics, it only removes contributions of wrong symmetry. Practically, the Rigetti 19Q-Acorn quantum processor provides access to read-outs using the read and measure routine. 35 This routine generates bit strings of 0 and 1’s encoding a wavefunction resulted from the original wave function collapse onto one of the eigenstates of the measured operator. Simultaneously, one can measure only Pauli words that commute qubit-wise, extension of this restrictive protocol has been recently suggested. 36 For the post-processing, we used a group of Pauli words that commute with the electron number operator because its eigenstates has either zero or unit overlap with the eigenstate of the correct number of electrons. Those measurements that correspond to the zero overlap are discarded.

Invariance of the quantum average to the post-processing procedure. The postprocessing procedure based on the electron number operator is equivalent to introducing 20


Page 21 of 27 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


a projector PˆN = |N i hN | to the eigen-subspace corresponding to the correct number of ˆ |N i = N |N i, electrons, N

ˆ PˆN |Ψi = hΨ|PˆN H

X

an | hΨ|PˆN |fn i |2

n

+bn | hΨ|PˆN |gn i |2 .

(19)

Indeed, eigenvalues associated with | hΨ|PˆN |fn i |2 and | hΨ|PˆN |gn i |2 will not contribute if corresponding eigenfunctions are orthogonal to the N -subspace: PˆN |fn i = 0 and PˆN |gn i = 0. On the other hand, assuming that |Ψi is within the N -subspace, hΨ| PˆN = hΨ|, it ˆ PˆN |Ψi = hΨ|H|Ψi, ˆ is straightforward to see that hΨ|PˆN H and hence, the Hamiltonian expectation values are not affected by the post-processing procedure.

Noise reduction. To understand how the post-processing reduces the noise related to uncontrolled qubit interactions with its environment, it is convenient to employ the density matrix formalism. Let us denote the ideal pure-state density as ρ0 = |Ψ0 i hΨ0 |, while the real P mixed-state density, appearing due to spurious interactions, is ρ = i=0 ωi |Ψi i hΨi |. Some components associated with the noise ({|Ψi i}i6=0 ) violate the correct value for the number operator, and therefore, PˆN |Ψi i = 0, we will refer to the associated part of the density as reducible, ρR , while the rest of ρ will be referred as irreducible, ρI = PˆN ρPˆN . Substituting ˆ ˆ the Hamiltonian average with ρ, E¯ = Tr[ρH]/Tr[ρ], by that with ρI , E¯I = Tr[ρI H]/Tr[ρ I] makes results more accurate because of removing the density part associated with the wrong electron numbers, ρR . As in the pure state consideration, the post-processing procedure is equivalent to considering the following average ˆ ˆ PˆN ] Tr[PˆN ρPˆN H] Tr[ρPˆN H = Tr[ρPˆN ] Tr[PˆN ρPˆN ] ˆ Tr[ρI H] = Tr[ρI ]

21


(20) (21)


and hence it removes the reducible part of the density in the Hamiltonian average and produces less noisy result. In Eq. (20) we used the projector property PˆN = PˆN2 and the invariance of trace with respect to cyclic permutations.

3.5

Acknowledgements

The authors thank P. Brumer, M. Mosca, and V. Gheorghiu for stimulating discussions. The authors are grateful to the support that Will Zeng, Nick Rubin, and Ryan Karle provided in terms of discussion of applying the theory onto their hardware and for access to the 19Q-Acorn Quantum computer. OTI Lumionics Inc. would like to acknowledge the support of the Creative Destruction Lab provided in facilitating the interbusiness collaborations between OTI Lumionics Inc. and Rigetti Computing Inc. A.F.I. acknowledges financial support from Natural Sciences and Engineering Research Council of Canada through the Engage grant.

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Page 270of 27

E, hartree

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17


H2, singlet H2, triplet H+2 EQMF, S2=0

−0.2 −0.4 −0.6 −0.8 −1 −1.2

0.5


1

1.5

R(H−H), Å

2

2.5

Constrained variational quantum eigensolver: Quantum computer

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