A blessing and a curse of dimensionality: using quantum computers to simulate strongly correlated fermionic systems. Open Access
Stair, Nicholas (Summer 2021)
Abstract
In this dissertation, we develop and demonstrate need for novel quantum computational algorithms for simulating molecular systems with strong electronic correlations. First we explore the limitations of classical algorithms for strongly correlated electronic structure problems. To this end, we test the performance of selected configuration interaction (sCI), singular value decomposition full CI (SVD-FCI), and the density matrix renormalization group (DMRG) using a novel set of one, two, and three-dimensional hydrogen models. We find that although a significant reduction in pre-factor can be achieved, a reduction from exponential to polynomial scaling was not observed for any of the three methods in the general case, motivating the development of quantum algorithms for strong correlation. We then introduce a novel quantum Krylov (QK) algorithm, amenable to the simulation of molecular systems. The algorithm uses unitary Hamiltonian time evolution to build a basis of states (stored on a quantum computer) that are equivalent to a classical Krylov basis in a small time-step limit. We apply the QK algorithm, and a selected multi-reference variant (MRSQK) to H6, H8, and BeH2 and find that chemical accuracy can be achieved with a Krylov basis orders of magnitude smaller than that of the full CI (FCI) space. We then introduce another novel quantum simulation algorithm deemed the projective quantum eigensolver (PQE), which is amenable to noisy intermediate-scale quantum (NISQ) hardware. The approach seeks to solve the Schrödinger equation by zeroing its projections onto an orthogonal space of excited determinants. We compare PQE to the variational quantum eigensolver (VQE), both using disentangled unitary coupled cluster ansatz and find that the former is more computationally resource efficient when tested on H6 and BeH2. We also introduce a selected ansatz variant of PQE and show that it compares favorably to classical sCI and DMRG. Finally, we showcase our novel open-source quantum simulation package QFORTE which facilitated the work presented in this dissertation and will be used for future algorithmic developments and comparative studies.
Table of Contents
1 Introduction................................... 1
1.1 Introduction................................. 1 1.2 The electronic structure problem...................... 2 1.2.1 The antisymmetry of fermionic wave functions . . . . . . . . . . 3 1.2.2 Slater determinants as mean-field wave functions . . . . . . . . . 5 1.2.3 Second quantization......................... 7 1.3 Electron correlation............................. 9 1.3.1 Exact diagonalization and the curse of dimensionality . . . . . . . 10 1.3.2 Static and dynamical correlation.................. 12 1.3.3 Classical methods for strong correlation . . . . . . . . . . . . . . 14 1.4 Quantum computers: the blessing of dimensionality . . . . . . . . . . . . 18 1.4.1 Quantum bits............................ 19 1.4.2 Quantum circuits.......................... 20 1.4.3 Determination of expectation values................ 22 1.4.4 The fermionic encoding problem.................. 24 1.4.5 Operators in the qubit basis..................... 25 1.5 Quantum algorithms for electronic structure . . . . . . . . . . . . . . . . 261.5.1 Algorithms based on Hamiltonian dynamics . . . . . . . . . . . . 27
1.5.2 Quantum variational optimization algorithms . . . . . . . . . . . 33
1.6 Prospectus.................................. 37
2 Exploring Hilbert space on a budget...................... 55
2.1 Introduction................................. 55 2.2 Theory.................................... 60 2.2.1 Definition of the accuracy volume ................. 60 2.2.2 Overview of the computational methods . . . . . . . . . . . . . . 61 2.2.3 Metrics of strong electronic correlation. . . . . . . . . . . . . . . 66 2.3 Computational details............................ 70 2.4 Results.................................... 72 2.4.1 Ground and low-lying electronic states. . . . . . . . . . . . . . . 73 2.4.2 Spin correlation and frustration................... 76 2.4.3 Performance of sCI, SVD-FCI, and DMRG . . . . . . . . . . . . 78 2.4.4 Comparison with other electronic structure methods . . . . . . . . 82 2.5 Scaling of the accuracy volume and size consistency. . . . . . . . . . . . 84 2.6 Conclusions and future work........................ 873 A quantum Krylov algorithm for strongly correlated electrons . . . . . . . 104
3.1 Introduction................................. 104 3.2 Theory.................................... 108 3.2.1 Choice of the unitary operators................... 109 3.2.2 Efficient evaluation of off-diagonal matrix elements . . . . . . . . 111 3.2.3 Reference selection......................... 114 3.2.4 Analysis of computational cost................... 116 3.3 Computational details............................ 116 3.4 Numerical studies and discussion...................... 117 3.5 Conclusions................................. 1224 Simulating many-body systems with a projective quantum eigensolver . . 132
4.1 Introduction................................. 132 4.2 Theory.................................... 135 4.2.1 The projective quantum eigensolver approach . . . . . . . . . . . 135 4.2.2 Traditional and disentangled unitary coupled-cluster ansätze . . . 137 4.2.3 Numerical solution of the UCC-PQE amplitude equation . . . . . 140 4.2.4 Efficient measurement of the residual elements . . . . . . . . . . 141 4.2.5 Efficient operator selection..................... 142 4.2.6 Outline of the selected PQE algorithm . . . . . . . . . . . . . . . 145 4.3 Results and Discussion........................... 147 4.3.1 Comparison of PQE and VQE with a disentangled UCC ansatz . . 147 4.3.2 Effect of stochastic errors on the convergence of PQE and VQE. . 149 4.3.3 SelectedPQEbasedonafulldUCCoperatorpool. . . . . . . . . 151 4.4 Conclusions................................. 158 4.5 Appendix.................................. 160 4.5.1 Gradient of the PQE energy..................... 160 4.5.2 Derivation of the UCC-PQE update equations . . . . . . . . . . . 162 4.5.3 Additional numerical comparison of PQE and VQE . . . . . . . . 163 4.5.4 Formal comparison of PQE and VQE ............... 163 4.5.5 Reduced-cost estimation of the approximate residual in selected PQE................................. 1675 QForte...................................... 177
5.1 Introduction................................. 177 5.2 Key data structures............................. 180 5.2.1 The state-vector simulator...................... 181 5.2.2 The QubitOperator and SQOperator classes . . . . . . . . . . . . 184 5.2.3 The molecule class......................... 185 5.3 Overview of algorithms implemented ................... 186 5.3.1 dUCC variational quantum eigensolver. . . . . . . . . . . . . . . 186 5.3.2 dUCC projective quantum eigensolver . . . . . . . . . . . . . . . 188 5.3.3 Quantum imaginary time evolution................. 190 5.3.4 Quantum Krylov .......................... 192 5.4 Conclusion................................. 1936 Conclusions and future directions . . . . . . . . . . . . . . . . . . . . . . . 198
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