W HY Q UANTUM C HEMISTRY ? Understanding Control T HE E LECTRONIC S - PowerPoint PPT Presentation

Q UANTUM C OMPUTATION FOR C HEMISTRY AND M ATERIALS Jarrod R. McClean Alvarez Fellow - Computational Research Division Lawrence Berkeley National Laboratory

W HY Q UANTUM C HEMISTRY ? Understanding Control

T HE E LECTRONIC S TRUCTURE P ROBLEM “The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble.” -Paul Dirac

G RAND SOLUTIONS FROM A GRAND DEVICE Fertilizer N 2 + 3 H 2 → 2 NH 3 Humans: Haber Process Nature: Nitrogenase “FeMoco” 25 ° C & 1 atm 400 ° C & 200 atm 1-2% of ALL energy on earth, used on Haber process Beyond all current classical methods Both electronic structure and substrate attachment almost totally unknown Classically – No clear path to accurate solution Quantum Mechanically – 150-200 logical qubits for solution

A SIDE : P ROBABILITY D ISTRIBUTIONS P 1 (Store i ) P 2 (Store j ) P 12 (Store i , Store j ) = P 1 (Store i ) P 2 (Store j ) O ( N P ) O ( PN ) Technical caveat: our “probability distributions’’ may be complex valued

T HE EXPONENTIAL PROBLEM M M M 2 M = 100 D = 100 80 = 10 160 D = M N N = 80 Electrons: ✓ M ◆ ✓ M One ¡mole ¡ Particles ¡in ¡universe ◆ D = 10 23 10 80 N α N β

LCAO AND M OLECULAR O RBITALS

S IMPLE BUT NOT GOOD ENOUGH

C LASSICAL P RE -C ALCULATIONS Second-Quantized Electronic Hamiltonian a † a † a † H e ( R ) = h pq ( R )ˆ p ˆ a q + h pqrs ( R )ˆ p ˆ q ˆ a r ˆ a s Atom Centered Basis Hartree-Fock (Mean-Field) Molecular Orbitals H e

B EYOND THE MEAN F IELD Virtual Occupied | Ψ i = c 0 + c 1 + c 2 + c 3 c i 1 i 2 ...i N | i 1 i 2 ...i N i X | Ψ i = i 1 i 2 ...i N

B ETWEEN M EAN -F IELD AND E XACT DFT (Density Function Theory): Errors in transition states, Charge transfer excitations, anions, Bond breaking MP2 – Second order perturbation theory, Good for hydrogen bonding, failing for Weakly bond systems and bond breaking QMC – Quantum Monte Carlo, Stochastic, accuracy depends on trial function CCSD(T) (Coupled Cluster single doubles excitations with perturbative triples) – “Gold Standard” for weakly bound systems, fails for multiple bond breaking Exact (Full Configuration Interaction) M. Head-Gordon, M. Artacho, Physics Today 4 (2008)

Q UANTUM S IMULATION – T HE Q UANTUM A DVANTAGE Quantum Simulation Abstraction Quantum Computation Factoring Products of Two Large Primes • Linear Partial Differential Equations • Solution of Linear Equations • Prep | ψ � {| Ψ i � , E i } Evolution Measurement

Q UANTUM H ARDWARE 4p 4.5 mm 4.88 µm

Q UANTUM C OMPUTING A BSTRACTION ✓ 0 ✓ 1 ◆ ◆ 1 | 0 i = X = NOT = σ x = 1 0 0 ✓ 0 ◆ X | 0 i = | 1 i | 1 i = 1 X | 1 i = | 0 i

C HALLENGES IN Q UANTUM S IMULATION Prep | ψ � {| Ψ i � , E i } Evolution Measurement Information Extraction Number of Qubits +New input/output spec Coherence Time & Fidelity +Scalable manufacture +Full readout loses advantage +Robust control & stable qubits +(N-1) qubit problem +Algorithm timescale problem Co-Design Better Algorithms Better Hardware Previous: Coherence time flexible – VQE Future: Improved coherence time flexibility, novel • property extraction, and demonstration – QSE Qubit number flexible algorithms and • larger demonstrations

A New Co-design Perspective Currently: Given a task, design quantum circuit (or computer) to perform it. 42 Problem: General or optimal solution can require millions of gates. Alternative: Given a task and the current architecture, find the best solution possible. 42.02 Peruzzo † , McClean† , Shadbolt, Yung, Zhou, Love, Aspuru-Guzik, O’Brien. Nature Communications , 5 (4213):1– 7, 2014. † Equal Contribution by authors

E ASY T ASK F OR A Q UANTUM C OMPUTER h σ z i i h σ z n i 1 σ z 2 .... σ z • Efficient to perform on any prepared quantum state • In general, it may be very hard to calculate this expectation value for a classical representation, containing an exponential number of configurations c i 1 i 2 ...i N | i 1 i 2 ...i N i X | Ψ i = i 1 i 2 ...i N

Back to Basics h Ψ | H | Ψ i Variational Formulation: Minimize Decompose as: By Linearity: Easy for a Quantum Computer: Easy for a Classical Computer: Suggests Hybrid Scheme: • Parameterize Quantum State with Classical Experimental Parameters • Compute Averages using Quantum Computer • Update State Using Classical Minimization Algorithm (e.g. Nelder-Mead)

Computational Algorithm Algorithm 2 Algorithm 1 QPU CPU classical feedback decision quantum state preparation quantum module 1 classical adder quantum module 2 quantum module 3 quantum module n Adjust the parameters for the next input state

E SSENTIALS OF A QUANTUM A DVANTAGE All Possible Quantum States “Classically Easy” “Easy” Quantum States Quantum States

S TATE A NSATZ Quantum Hardware Ansatz : “Any Quantum Device with knobs” Use the complexity of your device to your advantage = | Ψ ( { θ i } ) � Coherence time requirements are set by the device, not algorithm | Ψ i = e T − T † | Φ 0 i Unitary Coupled Cluster Ansatz (A-)diabatic State Preparation A (0) = 0 H ( s ) = [1 − A ( s )] H i + A ( s ) H p A (1) = 1

V ARIATIONAL E RROR S UPPRESSION McClean, J.R., Romero, J., Babbush, R, Aspuru-Guzik, A. “The theory of variational hybrid quantum-classical algorithms” ArXiv e-prints (2015) arXiv: 1509.04279 [quant-ph]

"K ILLER A PP ”: Q UANTUM C HEMISTRY Current experimental literature state of the art: Quantum Phase Variational Quantum Estimation Eigensolver H 2 NMR (Jiangfeng Du et al. 2010) Photonic chips (B. P. Lanyon et al. 2010) Superconducting qubits Superconducting qubits (P. J. J. O’Malley, Babbush, (P. J. J. O’Malley, Babbush, McClean et McClean et al. 2015) al. 2015) HeH + NV centers Photonic chips (Ya Wang et al. 2015) (Peruzzo, McClean et al. 2014) Trapped ions (Yangchao Shen et al. 2015) Theoretical and Algorithmic (2016): [1] McClean et al., N. J. Phys 18 023023 (2016) [2] Sawaya and McClean et al, JCTC - in press (2016) [3] McClean, Schwartz, Carter, de Jong ArXiv:1603.05681 [quant-ph] (2016) [4] Reiher et al. ArXiv:1605.03590 [quant-ph] (2016) [5] Babbush et al. N. J. Phys. 18 (3), 033032 (2016)

S CALABLE S IMULATION OF M OLECULAR E NERGIES IN S UPERCONDUCTING Q UBITS P.J.J. O’Malley, R. Babbush,…, J.R. McClean et al. “Scalable Simulation of Molecular Energies” ArXiv e-prints (2015) arXiv: 1512.06860 [quant-ph]

V ARIATIONAL E RROR S UPPRESSION 0 . 2 Exact Energy VQE Experiment 0 . 0 Total Energy (Hartree) PEA Experiment − 0 . 2 − 0 . 4 − 0 . 6 − 0 . 8 − 1 . 0 − 1 . 2 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0 Bond Length (Angstrom)

V ARIATIONAL E RROR S UPPRESSION 0 . 2 Exact Energy VQE Experiment 0 . 0 Total Energy (Hartree) PEA Experiment − 0 . 2 0 . 12 − 0 . 4 Error at Experimental Angle − 0 . 6 Error at Theoretical Angle 0 . 10 − 0 . 8 Local Error (Hartree) − 1 . 0 − 1 . 2 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0 0 . 08 Bond Length (Angstrom) 0 . 06 0 . 04 0 . 02 equilibrium 0 . 00 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0 Bond Length (Angstrom)

Q UANTUM S UBSPACE E XPANSION (QSE) Expand to Linear Response (LR) Subspace Quantum State on Quantum Device Extra Quantum Measurements − 0 . 80 Exact − 0 . 85 − 0 . 90 E (E h ) − 0 . 95 − 1 . 00 − 1 . 05 − 1 . 10 − 1 . 15 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0 R ( ˚ A ) Classical Generalized Eigenvalue Problem HC = SCE Excited State Energy and Properties Hybrid Quantum-Classical Hierarchy for Mitigation of Decoherence and Determination of Excited States McClean, J.R. , Schwartz, M.E, Carter, J., de Jong, W.A. ArXiv:1603.05681 [quant-ph] (2016)

E XPANSION M ITIGATES N OISE Exact − 0 . 6 AP AP LR − 0 . 7 AP LR ( S 2 = 0) − 0 . 8 E (E h ) − 0 . 9 − 1 . 0 Subspace expansion restores symmetry − 1 . 1 OC = SC Λ 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0 R ( ˚ A ) H ij ˜ (Hamiltonian projected into symmetry subspace) kl

E XCITED S TATES AND E RROR S UPPRESSION HC = SCE 1 . 0 − 0 . 6 Exact Exact ( N e = 2) AP LR AP LR 0 . 5 − 0 . 7 AP LR ( S 2 = 0) − 0 . 8 E (E h ) E (E h ) 0 . 0 − 0 . 9 − 0 . 5 − 1 . 0 − 1 . 0 − 1 . 1 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0 R ( ˚ A ) R ( ˚ A ) Experimental demonstration in progress!

E XPANSION FORMS EXACT HIERARCHY | Ψ i LR k = 1 QR k = 2 ... Exact k = N e

W HY N OW ? *http://web.physics.ucsb.edu/~martinisgroup/

G RAND SOLUTIONS FROM A GRAND DEVICE Fertilizer N 2 + 3 H 2 → 2 NH 3 Humans: Haber Process Nature: Nitrogenase “FeMoco” 25 ° C & 1 atm 400 ° C & 200 atm 1-2% of ALL energy on earth, used on Haber process Beyond all current classical methods Both electronic structure and substrate attachment almost totally unknown Classically – No clear path to accurate solution Quantum Mechanically – 150-200 logical qubits for solution

S UMMARY 150-200 Logical Qubits | Ψ i k = 1 k = 2 k = N e

Acknowledgements LBNL: Wibe A. De Jong Jonathan Carter Harvard University: Alán Aspuru-Guzik Google Quantum AI Labs Ryan Babbush Peter O’Malley John Martinis UC Berkeley: Irfan Siddiqi Mollie Schwartz