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Quantum Orbital-Optimized Unitary Coupled Cluster Methods in the Strongly Correlated Regime: Can Quantum Algorithms Outperform Their Classical Equivalents? Igor O

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Quantum Orbital-Optimized Unitary Coupled Cluster Methods in the Strongly Correlated Regime: Can Quantum Algorithms Outperform Their Classical Equivalents? Igor O Quantum Orbital-Optimized Unitary Coupled Cluster Methods in the Strongly Correlated Regime: Can Quantum Algorithms Outperform their Classical Equivalents? Igor O. Sokolov,1, a) Panagiotis Kl. Barkoutsos,1 Pauline J. Ollitrault,1, 2 Donny Greenberg,3 Julia Rice,4 Marco Pistoia,3, 5 and Ivano Tavernelli1, b) 1)IBM Research GmbH, Zurich Research Laboratory, S¨aumerstrasse 4, 8803 R¨uschlikon, Switzerland 2)ETH Z¨urich,Laboratory of Physical Chemistry, Vladimir-Prelog-Weg 2, 8093 Z¨urich, Switzerland 3)IBM Thomas J. Watson Research Center, Yorktown Heights, New York 10598, USA 4)IBM Almaden Research Center, San Jose, California 95120, USA 5)JPMorgan Chase & Co, New York, New York 10172, USA (Dated: 7 October 2020) The Coupled Cluster (CC) method is used to compute the electronic correlation energy in atoms and molecules and often leads to highly accurate results. However, due to its single-reference nature, standard CC in its projected form fails to describe quantum states characterized by strong electronic correlations and multi- reference projective methods become necessary. On the other hand, quantum algorithms for the solution of many-electron problems have also emerged recently. The quantum UCC with singles and doubles (q-UCCSD) is a popular wavefunction Ansatz for the Variational Quantum Eigensolver (VQE) algorithm. The variational nature of this approach can lead to significant advantages compared to its classical equivalent in the projected form, in particular for the description of strong electronic correlation. However, due to the large number of gate operations required in q-UCCSD, approximations need to be introduced in order to make this approach implementable in a state-of-the-art quantum computer. In this work, we evaluate several variants of the standard q-UCCSD Ansatz in which only a subset of excitations is included. In particular, we investigate the singlet and pair q-UCCD approaches combined with orbital optimization. We show that these approaches can capture the dissociation/distortion profiles of challenging systems such as H4,H2O and N2 molecules, as well as the one-dimensional periodic Fermi-Hubbard chain. These results promote the future use of q-UCC methods for the solution of challenging electronic structure problems in quantum chemistry. Keywords: Unitary Coupled Cluster, Orbital Optimization, VQE, Strong Correlation, Quantum Computing I. INTRODUCTION method allows to efficiently compute ( (N 4)) an ap- proximation of the ground state which doesO not include 8 One of the main goals of computational quantum any electronic correlation effects . The correction to the chemistry is the design of efficient methods for the calcu- energy can then be computed with the so-called post- lation of the ground and excited state properties of molec- HF methods. Among the most popular ones, we find Møller-Plesset (MP) perturbation theory9{12, Configura- ular systems. When combined with experimental results, 13 14{16 first-principle (or ab-initio) calculations enable the inves- tion Interaction (CI) and Coupled Cluster (CC) . tigation of chemical and industrial processes (e.g. catal- In particular, in CC the exponential Ansatz allows the ysis, electrochemistry, polymerization, and photochem- systematic introduction of higher order configurations istry to mention only a few) as well as the discovery of and makes the approach size extensive. The CC method 1{6 including single, double and an approximate treatment arXiv:1911.10864v2 [quant-ph] 5 Oct 2020 new materials and catalysts . of the triple excitations, named CCSD(T)17, scales as The ground state energy of a molecular or solid state 7 16 system can be obtained from the solution of the corre- (N ) and is often regarded as the gold standard for quantumO chemistry calculations. Usually the projective sponding Schr¨odingerequation. However, the solution 14 space (i.e., the Hilbert space) grows exponentially with CC equations are iteratively solved using, for instance, the quasi-Newton and direct inversion iterative sub-space the system size, N, (e.g. the number of electrons and 7 basis functions), making the exact solution of this prob- methods . The single-reference formulation has the dis- lem intractable for systems with more than a few atoms7. advantage to be non-variational and has been shown to Relying on a mean field approach, the Hartree-Fock (HF) fail in the limit of strongly correlated regimes of diatomic molecules such as N2 for which static correlations (with their multi-reference character) are known to play a de- cisive role18{21. While a variational version of CC was proposed18, the high computational cost associated to a) Electronic mail: [email protected] the numerical optimization of the CC parameters has b)Electronic mail: [email protected] 2 limited its applications. tization is given by Interestingly, the unitary variant of CC (UCC)22{25 ^ X ^ X y H = r h s a^r,σa^s,τ (1) can naturally be mapped to a quantum circuit for the rs h j j i στ preparation of corresponding wavefunction in a digi- 1 X X y y 26{33 + rs g^ tu a^ a^ a^u,ν a^t,µ + ENN ; tal quantum computer . Moreover, variational ap- 2 h j j i r,σ s,τ proaches and in particular the Variational Quantum rstu στνµ 26 Eigensolver (VQE) algorithm in combination with ^ truncated wavefunction expansions (with polynomial where the one-electron integrals, r h s , and the two- electron integrals, rs g^ tu , are givenh j j ini Appendix A; number of parameters) appear, at present, the most y h j j i promising way of solving chemistry problems on near- a^r,σ (^ar,σ) represent the fermionic creation (annihilation) term quantum hardware34{38. Hence, in this work we operators for electrons in HF spin-orbitals, φr,σ(~r) with study the performance of the variational implementation spatial component (molecular orbitals, MOs) φr(~r) and 26 spin σ ; . ENN describes the nuclear repulsion of UCC as a quantum algorithm (q-UCC) in comput- 2 f" #g ing the ground state of molecules and lattice models for energy. Here and for the remainder of this work we use which the classical CC formulation is known to break. indices r; s; t; u to label general MOs; i; j; k; l for occupied For the practical implementation of the q-UCC (i.e. re- MOs; m; n; p; q for virtual ones; σ; τ; µ, ν for spin compo- duction of the circuit depth) we also study two variations nents of spin-orbitals. The same spatial orbitals are used of the Ansatz, namely pair CC doubles (pCCD)19,20,39,40 for both spin-up and spin-down spin-orbitals. and singlet CC doubles (CCD0)19,21. These alternatives were previously developed to address the breakdown of B. Hubbard Hamiltonian the standard CC theory, in particular when strong cor- relation effects become important. Classically, the CCD and pCCD Ans¨atzewere shown to provide more accurate The repulsive N-site Fermi Hubbard Hamiltonian is results when combined with orbital optimization (OO) defined as 41{44 within the Lagrangian CCD-Λ formulation . Here N;f";#g we investigate the implementation of this class of trun- ^ X y y H = t (^ar+1,σa^r,σ +a ^r,σa^r+1,σ) (2) cated CC expansions into the corresponding quantum al- − r,σ gorithms. In particular, the OO procedure is embedded N as an extension of the VQE algorithm, which we name X + U n^r;"n^r;#; ooVQE. Its performance in terms of accuracy and effi- r ciency is studied in combination with the series of q-UCC Ans¨atzeintroduced above. The paper is organized as fol- where t is the energy associated with electron hopping, U lows. In SectionII, we define the molecular and Hubbard is the on-site electronic repulsion andn ^r,σ is the number y Hamiltonians and recall the theory of the classical CC operatorsa ^r,σa^r,σ. The index r labels the lattice sites and UCC methods. We also discuss the q-UCC and the each of which is divided into two sub-sites for spin up ( ) " quantum equivalents of pCCD and CCD0, i.e. q-pUCCD and down ( ). This implies that the representation of # and q-UCCD0, as new parametrized wavefunctions. Sec- this Hamiltonian on a lattice requires at least 2N qubits. tion III describes the implementation of these methods within the framework of the VQE and the ooVQE algo- rithms. In SectionIV, we apply q-UCCSD, q-pUCCD C. Classical Coupled Cluster and Unitary Coupled Cluster and q-UCCD0 (with and without the use of OO) to the Ans¨atze H4,N2,H2O molecules and the one-dimensional Hubbard chain with 6 sites (2 spins), in which strong correlation A non-linear parametrization of the system wavefunc- effects may induce the failure of the standard CC theory. tion is given by the CC Ansatz Conclusions are presented in SectionV. T^(θ~) Ψ(θ~) = e Φ0 ; (3) j i j i where Φ0 is the Hartree-Fock state, θ~ is the CC am- j i plitudes vector and T^(θ~) is the full excitation operator, defined as n II. THEORY X T^(θ~) = T^k(θ~); (4) k=1 A. Molecular Hamiltonian th with T^k(θ~) being the excitation operator of k order. The calculation of the CC amplitudes θ~ is commonly per- Within the Born-Oppenheimer approximation, the formed self-consistently, solving the projective CC equa- non-relativistic molecular Hamiltonian in second quan- tions7 but may lead to non-variational energies. 3 The UCC Ansatz defined as energy within chemical accuracy (i.e., with an error less −3 than 1 kcal/mol or 1:6 10 Hartree) for the H2 molecule. T^(θ~)−T^y(θ~) · Ψ(θ~) = e Φ0 : (5) For the case of q-UCCSD in small systems, there is evi- j i j i dence29 that the error can be `absorbed' and distributed is of particular importance for quantum computing since over the entire set of q-UCCSD parameters during the ^ ~ ^y ~ the eT (θ)−T (θ) operator is unitary and therefore can be VQE optimization.
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