Limitations of Hartree-Fock with quantum resources
HHF-QC
Limitations of Hartree-Fock with quantum resources
Sahil Gulania a) and James Daniel Whitfield b) Department of Chemistry, University of Southern California, Los Angeles, CA, 90089 Department of Physics and Astronomy, Dartmouth College, Hanover, NH 03755 (Dated: 21 July 2020)
The Hartree-Fock problem provides the conceptual and mathematical underpinning of a large portion of quantumchemistry. As efforts in quantum technology aim to enhance computational chemistry algorithms, the fundamentalHartree-Fock problem is a natural target. While quantum computers and quantum simulation offer many prospectsfor the future of modern chemistry, the Hartree-Fock problem is not a likely candidate. We highlight this fact froma number of perspectives including computational complexity, practical examples, and the full characterization of theenergy landscapes for simple systems.The present study aims to highlight the difficulty of con-ducting optimization problems in the context of electronicstructure, with and without quantum resources. Specifi-cally, we will focus on Hartree-Fock, an optimization prob-lem using the mean-field approximation. The Hartree-Fockproblem provides the mathematical setting for molecularorbitals widely used in chemistry and beyond. The ubiqui-tous self-consistent field (SCF) methodology used to solveHartree-Fock is also applied to most implementations of den-sity functional theory based on Kohn-Sham theory . Whilethe solution to an instance of the Hartree-Fock problem is of-ten insufficient for many applications, it often serves as thereference state for post-Hartree-Fock methods. Widely usedpost-Hartree-Fock methods include coupled cluster ansatz ,Møller–Plesset perturbation theory , equation of mo-tion , multi-reference configuration interaction andmany more. Here, we forgo improving the Hartree-Fockansatz and instead ask how difficult it is to find the trueHartree-Fock global minimum and its importance.Many instances of the Hartree-Fock problem can be solvedquickly using heuristic approaches. In practice, conventionalalgorithms for Hartree-Fock scales cubically with the numberof basis functions. However, this cost only reflects the cor-rect scaling if the number of iterations is bounded by a con-stant or if local minimums are acceptable in place of a globalsolution. Linear scaling methods avoid the diagonaliza-tion of hessian entirely and rely on localized properties of thesystem. This assumption may not be true generally for ev-ery case. Iterative procedures, regardless of cost per step, areprone to convergence issues. This article highlights the abovementioned properties in the context of standard numerical andhybrid-quantum approaches to the Hartree-Fock problem.Typical approaches to improve SCF convergence usesdirect inversion of the iterative subspace (DIIS), levelshifting , quadratically convergent Newton-Raphson tech-niques , or varying fractional occupation numbers , amongmany other approaches. The success of any above mentionedmethods varies at each instance and depends on the initial pa-rameters chosen, e.g. the size of the iterative subspace, andoften work well in combination leading to attempts to build a) Electronic mail: [email protected] b) Electronic mail: james.d.whitfi[email protected] black-box SCF procedures .Unfortunately, these methods cannot work in all cases with-out violating fundamental assumptions in the theory of com-putation. In previous work, arbitrary spin-glasses have beenmapped to instances of the Hartree-Fock problem . In ad-dition to other works , this shows that the Hartree-Fockproblem is difficult in the worst case setting. The complex-ity class of non-deterministic polynomial time (NP) problemsare the set of problems that can be solved efficiently witha hint . It is possible that every problem that can be solvedefficiently with a hint, can also be efficiently solved withoutthe hint. However, from all signs of practical experience sug-gest that problems in the NP class can do not admit efficientblack-box solutions. Thus, due to the NP-completeness of theHartree-Fock problem, it is unlikely that any classical algo-rithm can solve all instances in a time proportional a polyno-mial of the input size. Note that because it in the NP com-plexity class, the Hartree-Fock problem can be solved effi-ciently in polynomial time with a sufficiently strong hint e.g.as gleamed from experience or luck. It has been known sincethe late 1990’s that quantum computers can promise no morethan a quadratic speed up over their classical counterparts onsuch NP-complete problem .It has long been known that any classical algorithm can besimulated using quantum hardware. With the the heavy re-liance of the variational quantum eigensolvers on classical op-timization routines, it is marginal how the quantum approachdiffers from the conventional approaches. In the recent VQEstudy the optimization strategy used was an augmented Hes-sian approach rather than the typical DIIS . However, bothof these optimization strategies can be employed by conven-tional computers to optimize the orbitals.In this brief communication, we consider how the use ofquantum hardware enhances the ability of chemists to solveinstances of the Hartree-Fock problem. Quantum hardwarehas many prospects for applications to physical and chemicalsimulations , however the Hartree-Fock problem is not likelyto admit drastic advances using quantum computers.The present article is inspired by a recent study ofthe Google group and collaborators using a variationalquantum-classical hybrid formulation of the Hartree-Fockproblem. There the authors point out the primary purpose ofconsidering the Hartree-Fock problem was to benchmark theirquantum device. Here, we highlight obstructions to its use asa general purpose replacement for standard SCF solvers. We a r X i v : . [ phy s i c s . c o m p - ph ] J u l F-QC 2do so by first introducing the Hartree-Fock problem and char-acterizations of the solution landscape for an instance of theHartree-Fock problem. We then give examples that are (1)simple, (2) small, and (3) well-motivated instances of Hartree-Fock that present convergence problems for black-box ap-proaches.
I. HARTREE-FOCK THEORY
The Hartree-Fock ansatz is important for conventionalquantum chemistry and has been thoroughly developed overthe past 100 years . The Hartree-Fock problem can be statedsuccinctly as: minimize the electronic energy in the space ofsingle Fock states. E HF = min Ψ ∈ F (cid:104) Ψ | (cid:32) ∑ i j h i j a † i a j + ∑ i jkl h i jkl a † i a † j a k a l (cid:33) | Ψ (cid:105) (1)The space F is the set of all rank 1 (versus rank (cid:0) MN (cid:1) in thegeneral case) N -electron Fock states. Here and throughout, N is the number of electrons and M is the number of spatialbasis functions. Each single Fock state is of the form: Ψ = b †1 b †2 · · · b † N | Ω (cid:105) where we have [ b i , b † j ] + = b i b † j + b † j b i = δ i j and [ b j , b k ] + = { a j : [ a i , a † j ] + = (cid:82) dx φ i ( x ) φ ∗ j ( x ) = S i j , [ a i , a j ] + = } , then b † i = ∑ j W ji a † j . (2)Here we require that WW † = only when S = . For ap-plications to molecular physics, single-electron spin-orbitals φ i ( x ) corresponding to the fermionic operators a † j and a j areof Gaussian form e.g. STO-3G . A. Rotation of charge density matrix
In practice, most algorithms utilize the SCF method to solveeq. (1) using a effective potential term that takes into accountthe averaged two-body interaction (mean-field). In this ap-proach, the N -body problem is reduced to a non-linear singleparticle problem. At each iteration of the simplest implemen-tation, the Fock matrix, F , is formed as a function of a previ-ous bond density matrix ( D prev ) D prev = [ (cid:104) a † j ↑ a k ↑ + a † j ↓ a k ↓ (cid:105) ψ ] Mjk = C prev η C † prev (3)with η as the orbital occupancies written as a diagonal ma-trix. The new transformation matrix, C , is determined usingthe gradient of eq. (1) with respect to bond density matrix D .The new coefficient matrix is used to form a new bond den-sity matrix. At each iteration, the (real-valued) bond density matrix satisfies the following three properties: D = D T (4)Tr ( DS ) = N / D = DSD (6)We convert to an orthogonal basis using e.g. canonical orthog-onalization X canonical = U † S √ s with S = U S sU † S . converts fromthe non-orthogonal basis to an orthogonal one. e.g. canon-cially with X canonical = √ S , P = X † DX .The three properties of the bond-density matrix below im-ply, in an orthogonal basis, that P is a rank N / N /
2. We use D to denote an arbitrary bond density ma-trix and P as a bond density matrix in an orthogonal basis.Next, we consider transformation between bond densitymatrices. A convenient parameterization of the set of bonddensity matrices that avoids redundancy is P ( A ) = e − A block P e A block (7)with A block = P A ( − P ) + ( − P ) AP for arbitrary skew-symmetric A = − A T .Once an instance of a Hartree-Fock problem has been spec-ified, the objective for the optimization problem is given bythe energy matrix functional E [ P ] = [ hP ] + Tr [ G P ] . (8)Here the mean-field, G , is a function of the bond density ma-trix: G = G [ P ] µν = M ∑ κ , λ h A κµνλ P κλ (9)with the antisymmetrized two-electron integral over spatialdegrees of freedom defined by h A µκλν = h µκλν − h µκνλ . Notethat expression eq. (8) follows directly from evaluating theenergy’s expectation value in eq. (1) using e.g. Slater-Condonrules . The matrix derivative of eq. (8) gives the Fock opera-tor F i j = ( h i j + G i j ) . (10)By rotating the charge density matrix with all possible rota-tions, we are able to do brute force exploration of whole spacefor some small examples. Below we expand about P = P core where the N lowest eigenmodes of H core = h i j a † i a j are occu-pied. Before turning to examples, we will introduce the quan-tum ansatz for the Hartree-Fock method. II. QUANTUM CIRCUIT ANSATZ FOR HARTREE-FOCK
The quantum circuit for creating the Hartree-Fock ansatzstate was applied in the context of VQE . The descrip-tion of the ansatz circuit will be aided through the use of QRdecomposition .F-QC 3 = ( θ ) − sin ( θ )
00 sin ( θ ) cos ( θ )
00 0 0 1 = Measurement θ θ θ θ | i| i| i FIG. 1: Quantum circuit for the two fermion and threemolecular orbital (for example H + in STO-3G basis).The Givens rotations provide a useful canonical characteri-zation of an arbitrary orthogonal matrix, W . The QR decom-position of a real n × n orthogonal matrix W can be done using T = n ( n − ) / W = G G G ... G T D (11)When W has determinant of one, D is just the identity matrix.Each Givens rotation, G i , is of the form G i = g ( a , b , θ ) with g kk = k is either a or b when instead g kk = cos ( θ ) . Alloff-diagonal elements are zero except g ab = − g ba = − sin ( θ ) .Applications of the Givens decomposition to fermionic or-bital rotations has been worked out elsewhere resulting ina quantum circuit that is able to prepare arbitrary Slater deter-minants following the parameters of the QR decomposition.By ordering the QR decomposition appropriately, a fermionicswap network can be used to rotate each pair of orbitals us-ing the appropriate Givens rotation parameters. This resultsin an efficient state preparation circuit of the form depicted infig. 1. The full compilation down to gates including hardwareoptimization is given elsewhere .Our characterization of the fermionic space in eq. (7) givesus a set of parameters, Θ that also characterizes the mixingbetween pairs of orbitals. The resulting orthogonal transfor-mation W ( Θ ) is then given to the QR decomposition and for-warded to the quantum circuit construction. III. CALCULATIONS AND RESULTS
All calculations of the molecular system are done in theSTO-3G basis . Energies are reported in Hartrees, angles ofrotation in radians, and bond lengths in Angstroms.The Hartree-Fock energy surfaces (HES) were computedusing PySCF . In this paper we only consider RestrictedHartree-Fock (RHF) solutions where the alpha and beta spa-tial orbitals are restricted to be identical. The quantum op-timization routines were that of OpenFermion-Cirq and weonly modify the initial state routines and the input moleculardata . The data that support the findings of this study areavailable from the authors upon reasonable request. A. Landscape analysis
We consider H , H + as minimal basis model systems whoseHartree-Fock instances we can completely characterize. We begin with the H example.When considering H in the minimal basis with there isonly a single orbital mixing parameter. In fig. 2, we haveplotted the 1D HES surface as a function of bond length forH . The number of minimums in HES ( θ ) changes with bondlength. Before a bond length of approximately 1 .
0 0.5 1 1.5 2 2.5 3 1 2 3 θ r HH -1-0.5 0 E ne r g y ( a . u . ) FIG. 2: H HES ( θ ; r HH ). Each fixed value of the nuclearseparation, r HH , generates a Hartree-Fock instancecharacterized by a single the orbital rotation parameter, θ .Notice the appearance of a second HES minimum at a higherenergy around r ≈ . + . Now, instead of a 1D HES, we now have twoparameters that mix the one occupied orbital with the two vir-tual orbitals. We plot the HES in fig. 3 for a linear configura-tion with hydrogen atoms separated by 2.5 Å. There are threeminimums for HES ( θ , θ ). In fig. 4 we give the HES of H + at 4.36 Å where there are several minimums with the sameglobally optimal value.FIG. 3: H + HES ( θ , θ ), showing three different minimums(global minimum - ◦ , second minimum - (cid:78) and thirdminimum - × ) at r HH = . P = P core .In both the case of H + and H , there is a single occupied(spatial) orbital occupied and m virtual orbitals. For m = + HES ( θ , θ ) surface for fixed bond length atr HH = .
36 Å. Here the surface is expanded about P = P core .this leads to an A block generator of the form A block = − θ − θ θ θ (12)The eigenvalues of this matrix are λ ± = { , ± i (cid:113) θ + θ } .Since the matrix exponential of A block merely exponentiatesthe eigenvalues, when λ ± = i π , the rotation acts trivially onthe density matrix. This underlies periodicity to the plots seenin 3 and 4.We can explain the periodicity in terms of this invari-ant by converting to polar coordinates where θ = R cos φ and θ = R sin φ . Now the nontrivial eigenvalue is λ ± = ± iR and we can express the periodicity of the plots asHES ( θ , θ ) =HES ( R , φ ) = HES ( R + n π , φ ) with n an integer.There is a nice generalization of this fact. For a single spa-tial orbital that is doubly occupied with two electrons and m virtual orbitals, the generalization of eq. (12) is A block = − θ − θ − θ . . . − θ m θ . . . θ . . . θ . . . θ m . . . (13)It is straightforward to calculate that the eigenvalues of thismatrix are zero except λ ± = ± i (cid:113) θ + θ + θ + ... + θ m .Following the same argument as in the m = ( Θ ) = HES ( R , Φ ) = HES ( R + n π , Φ ) (14)where R = (cid:113) θ + ... + θ m .Therefore, the range of the minimal required search spacefor each θ j is restricted to a hyper-sphere with radius π of di-mension m . But, the default search space was a hyper cubeof dimension m with side 2 π . Now, the ratio of minimal re-quired search space with default search goes to zero as m tendsto infinity. This is a well known consequence of the vanish-ing ratio of the volume of a hyper-sphere to the volume of thecorresponding hyper-cube . B. Convergence analysis
We used the quantum algorithm outlined in ref. for ob-taining RHF solutions for four examples. Depending on initialguess it may converge to local rather global solutions.The different initial guess were generated using the Givensrotations corresponding to different minimums in figs. 2 and 3,respectively. The results for H , converging to two differentminimums on quantum simulator is shown in fig. 5. Simi-larly results for H + , converging to two different minimums onquantum simulator is shown in fig. 6. In fig. 5, the values of -0.8-0.4 0 0 5 10 15 20 25 30 35 E ne r g y ( a . u . ) Iterationsglobal minimumlocal minimum λ = 1.0 λ = 1.2 FIG. 5: Convergence of the HF quantum optimization todifferent minimums for the Hartree-Fock instance of H atr HH = . λ , to the global ground state. The value used in theOpen-Fermion implementation was λ = . λ = λ = . λ , the convergence to the minimum is highly likelyso long as the system does not climb uphill in energy sincethe initial state has energy less than all minimums except theglobal minimum. -0.8-0.7-0.6-0.5-0.4 5 10 15 20 25 30 35 40 E ne r g y ( a . u . ) Iterationsglobal minimumlocal minimum λ = 0.3 λ = 0.6 λ = 0.7 λ = 1.0 FIG. 6: Convergence of the HF quantum optimization todifferent minimums for the Hartree-Fock instance of H + atr HH = . λ , to the global ground state.As final examples, we choose diatomic carbon and itscation. We also consider C and C + as instances that arecommonly known to confound solvers due to the appearanceof saddle points with in the optimization landscape. To illus-F-QC 5trate the complications of convergence we modified the initialparameters following the method used in . Namely, we be-gin with the solution provided by the classical SCF solver,perturb from those solutions and observe if the quantum algo-rithm still converges to the correct minimum.This example allows us to highlight the importance of usingthe information of Hessian to avoid saddle points during SCFoptimization in either quantum or classical methods. In figs. 7and 8, convergence results are shown for C and C + , respec-tively. In each of the plots, we have plotted the performanceof the quantum circuit optimization routine and the classicaloptimization routines as implemented in PySCF. At an inter-nuclear separation of 1 . -74.34-74.31 2 4 6 8PySCF -74.4-74.3-74.2 0 40 80 E ne r g y ( a . u . ) Iterationslocal minimumsaddle point λ = 0.1 FIG. 7: SCF convergence for different optimizers for theminimal basis C at r CC = . P core . -72.8-72.4 5 10 E ne r g y ( a . u . ) Iterationslocal minimumsaddle point λ = 0.1 -72.95 2 4 6PySCF FIG. 8: Optimization to local minimums for C + on quantumsimulator, while avoiding convergence to saddle point atr CC = . P core .The use of the orbital Hessian helps solvers avoid saddle points where the gradient may vanish at a non-optimal values.The solver uses Hessian which provides a notion of the curva-ture of the landscape. This allows the solver to avoid saddlepoints. The augmented Hessian conjugate-gradient method was used for the quantum optimization of the Hartree-Fockcircuit. This allows the solver to avoid convergence to sad-dle points as illustrated in figs. 7 and 8. Many of the quantumchemistry do not check the the RHF solutions by default. Thismakes them prone to failure by convergence to a saddle pointrather than the RHF solution. IV. DISCUSSION
In this study, we showed that the convergence to a localminimum or global minimum is a function of initial guess.The proposed algorithm on quantum simulator carries this fea-ture from classical algorithm. There is no a priori guaranteethat it will find global solution.Since the Hartree-Fock problem is an optimization prob-lem, quantum computing via a modified Grover search can be used to find the global solution quadratically faster thanthe classical brute force approaches. However, in both con-ventional and quantum solvers, local searches are employedfor local search whereby no guarantees on finding the glob-ally optimal solutions are given.In analyzing the Hartree-Fock functional for H , H + , C ,and C + we note that the number of critical points in the so-lution space changes as the nuclear separation changes. Thisalso implies that there are additional difficulties in applyingHartree-Fock for nuclear dynamics or other non-equilibriumconfigurations. Most solvers for the Hartree-Fock problem donot explore the entire space and usually only choose a singlestarting point (rather than multiple starting points). While thisis not often an issue, it can cause serious pathology when usedin post-Hartree-Fock methods e.g. coupled cluster method.This will be true for both on conventional computers and inits unitary formulation for quantum computers. V. CONCLUSIONS
The recent application of quantum technology to theHartree-Fock problem may serve as a hardware benchmarkbut is unlikely to have a dramatic impact on the practical ap-proaches to this problem.While the application to Hartree-Fock is not likely tochange the workhorse routines used on conventional comput-ers, there are still interesting use cases for the results fromRef. . The projection methods and purity extrapolation pre-sented in will still be useful.Hartree-Fock is NP-hard and is not likely to admit morethan a quadratic speed up. When considering application ar-eas of quantum computers, it is far more likely to make majorbreakthroughs when considering time-dependent phenomena.For example, the quantum-classical hybrid algorithm for ob-taining the Kohn-Sham potential of time-dependent densityF-QC 6functional , also requires measuring the bond density ma-trix.This article highlights the wide body of knowledge on theSCF method, its difficulty, and shows the lack of verificationof the solution are material in both the quantum and conven-tional computing domains. VI. ACKNOWLEDGEMENTS
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