||Quantum Information Processing: ImplementationsStudent Presentation: Quantum chemistry with SC qubits

Physical Review X 6, 031007 (2016)P. J. J. O’Malley, R. Babbush, I. D. Kivlichan, J. Romero, J. R. McClean, R. Barends, J. Kelly, P. Roushan,A. Tranter, N. Ding, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, A. G. Fowler, E. Jeffrey,E. Lucero, A. Megrant, J. Y. Mutus, M. Neeley, C. Neill, C. Quintana, D. Sank, A. Vainsencher, J. Wenner,T. C. White, P. V. Coveney, P. J. Love, H. Neven, A. Aspuru-Guzik, and J. M. Martinis

09/Apr/2018 1

Scalable Quantum Simulation of Molecular Energies

Christopher Santek & Florian Koch

||Quantum Information Processing: ImplementationsStudent Presentation: Quantum chemistry with SC qubits 09/Apr/2018Christopher Santek & Florian Koch 2

Process followed in the experiment

𝐻𝐻 = −�𝑖𝑖

∇𝑅𝑅𝑖𝑖2

2𝑀𝑀𝑖𝑖−�

𝑖𝑖

∇𝑟𝑟𝑖𝑖2

2−�

𝑖𝑖,𝑗𝑗

𝑍𝑍𝑖𝑖𝑅𝑅𝑖𝑖 − 𝑟𝑟𝑖𝑖

+ �𝑖𝑖,𝑗𝑗>𝑖𝑖

𝑍𝑍𝑖𝑖𝑍𝑍𝑗𝑗𝑅𝑅𝑖𝑖 − 𝑅𝑅𝑗𝑗

+ �𝑖𝑖,𝑗𝑗>𝑖𝑖

1𝑟𝑟𝑖𝑖 − 𝑟𝑟𝑗𝑗

𝐻𝐻𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 = 𝐻𝐻𝑛𝑛𝑛𝑛𝑛𝑛𝑡𝑡𝑛𝑛𝑖𝑖 ⊗ 𝐻𝐻𝑛𝑛𝑡𝑡𝑛𝑛𝑛𝑛𝑡𝑡𝑟𝑟𝑡𝑡𝑛𝑛

𝐻𝐻 is in units of hartree ℏ2

𝑚𝑚𝑒𝑒𝑛𝑛2𝑡𝑡02

𝐻𝐻 = �𝑝𝑝𝑝𝑝

ℎ𝑝𝑝𝑝𝑝𝑎𝑎𝑝𝑝†𝑎𝑎𝑝𝑝 +

12�𝑝𝑝𝑝𝑝𝑟𝑟𝑝𝑝

ℎ𝑝𝑝𝑝𝑝𝑟𝑟𝑝𝑝𝑎𝑎𝑝𝑝†𝑎𝑎𝑝𝑝

†𝑎𝑎𝑟𝑟𝑎𝑎𝑝𝑝

||Quantum Information Processing: ImplementationsStudent Presentation: Quantum chemistry with SC qubits

𝐻𝐻 = 𝑔𝑔0𝟙𝟙 + 𝑔𝑔1𝑍𝑍0 + 𝑔𝑔2𝑍𝑍1 + 𝑔𝑔3𝑍𝑍0𝑍𝑍1 + 𝑔𝑔4𝑌𝑌0𝑌𝑌1 + 𝑔𝑔5𝑋𝑋0𝑋𝑋1

{𝑋𝑋𝑖𝑖 ,𝑌𝑌𝑖𝑖 ,𝑍𝑍𝑖𝑖}are Pauli matrices

{𝑔𝑔𝛾𝛾}are real numbers

Replaces Jordan-Wigner𝒪𝒪 𝑛𝑛 vs.𝒪𝒪 log𝑛𝑛

09/Apr/2018Christopher Santek & Florian Koch 3

Bravyi-Kitaev transformation

||Quantum Information Processing: ImplementationsStudent Presentation: Quantum chemistry with SC qubits 09/Apr/2018Christopher Santek & Florian Koch 4

Variational Quantum Eigensolver

||Quantum Information Processing: ImplementationsStudent Presentation: Quantum chemistry with SC qubits 09/Apr/2018Christopher Santek & Florian Koch 5

Unitary coupled cluster ansatz

||Quantum Information Processing: ImplementationsStudent Presentation: Quantum chemistry with SC qubits 09/Apr/2018Christopher Santek & Florian Koch 6

Variational Quantum Eigensolver

||Quantum Information Processing: ImplementationsStudent Presentation: Quantum chemistry with SC qubits 09/Apr/2018Christopher Santek & Florian Koch 7

Algorithm

||Quantum Information Processing: ImplementationsStudent Presentation: Quantum chemistry with SC qubits 09/Apr/2018Christopher Santek & Florian Koch 8

Algorithm

||Quantum Information Processing: ImplementationsStudent Presentation: Quantum chemistry with SC qubits 09/Apr/2018Christopher Santek & Florian Koch 9

Algorithm

||Quantum Information Processing: ImplementationsStudent Presentation: Quantum chemistry with SC qubits 09/Apr/2018Christopher Santek & Florian Koch 10

Algorithm

||Quantum Information Processing: ImplementationsStudent Presentation: Quantum chemistry with SC qubits

→Unitary operator 𝑈𝑈 with eigenvalue 𝑒𝑒2𝜋𝜋𝑖𝑖𝜋𝜋 and eigenstate |𝑢𝑢⟩←Phase 𝜑𝜑 of eigenstate |𝑢𝑢⟩ Algorithm initial state 0 |𝑢𝑢⟩ create superposition 1

2𝑡𝑡∑𝑗𝑗=02𝑡𝑡−1 𝑗𝑗 |𝑢𝑢⟩

apply 𝑗𝑗-controlled unitary 𝑈𝑈 to superposition state

1

2𝑡𝑡∑𝑗𝑗=02𝑡𝑡−1 𝑗𝑗 𝑈𝑈𝑗𝑗|𝑢𝑢⟩ = 1

2𝑡𝑡∑𝑗𝑗=02𝑡𝑡−1 𝑒𝑒2𝜋𝜋𝑖𝑖𝑗𝑗𝜋𝜋𝑢𝑢 𝑗𝑗 |𝑢𝑢⟩

apply IQFT to first register to optain �𝜑𝜑𝑛𝑛 |𝑢𝑢⟩ measure first register

09/Apr/2018Christopher Santek & Florian Koch 11

Phase Estimation Algorithm

||Quantum Information Processing: ImplementationsStudent Presentation: Quantum chemistry with SC qubits 09/Apr/2018Christopher Santek & Florian Koch 12

Phase Estimation Algorithm

Fig. 5.2 of Quantum computation and quantum information / Michael A. Nielsen & Isaac L. Chuang. Reprinted. Cambridge: Cambridge University Press; 2010

+

+

+

+

||Quantum Information Processing: ImplementationsStudent Presentation: Quantum chemistry with SC qubits

𝑒𝑒−𝑖𝑖𝑖𝑖𝑡𝑡 = 𝑒𝑒−𝑖𝑖𝑡𝑡 ∑𝛾𝛾 𝑔𝑔𝛾𝛾𝑖𝑖𝛾𝛾 ≈ 𝑈𝑈𝑇𝑇𝑟𝑟𝑡𝑡𝑡𝑡 𝑡𝑡 ≡ �𝛾𝛾

𝑒𝑒−𝑖𝑖𝑔𝑔𝛾𝛾𝑖𝑖𝛾𝛾𝑡𝑡

𝜌𝜌

𝜌𝜌

𝑒𝑒−𝑖𝑖𝑖𝑖𝑡𝑡 𝜙𝜙 = �𝑛𝑛

𝑒𝑒−𝑖𝑖𝐸𝐸𝑛𝑛𝑡𝑡 𝑛𝑛 𝑛𝑛 𝜙𝜙 = �𝑛𝑛

𝑎𝑎𝑛𝑛𝑒𝑒−𝑖𝑖𝐸𝐸𝑛𝑛𝑡𝑡|𝑛𝑛⟩

09/Apr/2018Christopher Santek & Florian Koch 13

Trotterized time evolution

||Quantum Information Processing: ImplementationsStudent Presentation: Quantum chemistry with SC qubits 09/Apr/2018Christopher Santek & Florian Koch 14

Trotterized Phase Estimation

||Quantum Information Processing: ImplementationsStudent Presentation: Quantum chemistry with SC qubits

variational approach yields upper bound VQE is more accurate PEA has more random fluctuations

09/Apr/2018Christopher Santek & Florian Koch 15

Comparison PEA & VQE

||Quantum Information Processing: ImplementationsStudent Presentation: Quantum chemistry with SC qubits 09/Apr/2018Christopher Santek & Florian Koch 16

Conclusions

||Quantum Information Processing: ImplementationsStudent Presentation: Quantum chemistry with SC qubits 09/Apr/2018Christopher Santek & Florian Koch 17

Scalable Quantum Simulation of Molecular Energies

Physical Review X 6, 031007 (2016)P. J. J. O’Malley, R. Babbush, I. D. Kivlichan, J. Romero, J. R. McClean, R. Barends, J. Kelly, P. Roushan,A. Tranter, N. Ding, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, A. G. Fowler, E. Jeffrey,E. Lucero, A. Megrant, J. Y. Mutus, M. Neeley, C. Neill, C. Quintana, D. Sank, A. Vainsencher, J. Wenner,T. C. White, P. V. Coveney, P. J. Love, H. Neven, A. Aspuru-Guzik, and J. M. Martinis

Further sources:Nielsen, M. A., & Chuang, I. L. (2010). Quantum Computation and Quantum Information. Cambridge University Press. http://doi.org/10.1017/CBO9780511976667Tranter, A., Sofia, S., Seeley, J., Kaicher, M., McClean, J., Babbush, R., … Love, P. J. (2015). The Bravyi-Kitaev transformation: Properties and applications. International Journal of Quantum Chemistry, 115(19), 1431–1441. http://doi.org/10.1002/qua.24969Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J. M., & Gambetta, J. M. (2017). Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. Nature, 549(7671), 242–246. http://doi.org/10.1038/nature23879