Simulating Chemistry using Quantum Computers, Annu. Rev. Phys. Chem. 62, 185 (2011).

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1 I. Kassal, J. D. Whitfield, A. Perdomo-Ortiz, M-H. Yung, A. Aspuru-Guzik, Simulating Chemistry using Quantum Computers, Annu. Rev. Phys. Chem. 6, 185 (011). Martes Cuántico Universidad de Zaragoza, 19th November 013

2 Simulating chemistry using quantum computers

3 Simulating chemistry using quantum computers :-(

4 Simulating chemistry using quantum computers

5 Why quantum computers?

6 Some of the Aaronson complexity historical epochs Scott Aaronson, Quantum Computing since Democritus, CUP, 013.

7 Some of the Aaronson complexity historical epochs Scott Aaronson, Quantum Computing since Democritus, CUP, s Late Turingzoic.

8 Some of the Aaronson complexity historical epochs Scott Aaronson, Quantum Computing since Democritus, CUP, s Late Turingzoic. 1971s The Cook-Levin Asteroid. (NP-complete problems)

9 Some of the Aaronson complexity historical epochs Scott Aaronson, Quantum Computing since Democritus, CUP, s Late Turingzoic. 1971s The Cook-Levin Asteroid. (NP-complete problems) Early 1970s The Karpian Explosion.

10 Some of the Aaronson complexity historical epochs Scott Aaronson, Quantum Computing since Democritus, CUP, s Late Turingzoic. 1971s The Cook-Levin Asteroid. (NP-complete problems) Early 1970s The Karpian Explosion Early Cryptozoic.

11 Some of the Aaronson complexity historical epochs Scott Aaronson, Quantum Computing since Democritus, CUP, s Late Turingzoic. 1971s The Cook-Levin Asteroid. (NP-complete problems) Early 1970s The Karpian Explosion Early Cryptozoic. 1980s Randomaceous Era.

Some of the Aaronson complexity historical epochs Scott Aaronson, Quantum Computing since Democritus, CUP, 013. 1950s Late Turingzoic.

12 Some of the Aaronson complexity historical epochs Scott Aaronson, Quantum Computing since Democritus, CUP, s Late Turingzoic. 1971s The Cook-Levin Asteroid. (NP-complete problems) Early 1970s The Karpian Explosion Early Cryptozoic. 1980s Randomaceous Era Invasion of the Quantodactyls.

13 Quantum chemical complexity The simple system 3 H n of three electrons and a n-dimensional one-particle Hilbert space...

14 Quantum chemical complexity The simple system 3 H n of three electrons and a n-dimensional one-particle Hilbert space... n 3 / 4

15 Quantum chemical complexity The configuration interaction (CI) wave function ( ψ CI = 1 + r,µ cµa r r a µ + r <s µ<ν ) cµνa rs r a s a µ a ν + ψ 0 is exact in the full CI limit, but lacks size-extensivity with any truncation of the configuration space.... traditional wave function methods, which provided the required chemical accuracy, are generally limited to molecules with a small number of chemically active electrons, N O(10). W. Kohn, Nobel lecture, 1998.

16 Quantum quantum-chemical complexity Any implementation of a quantum-simulation algorithm requires a mapping from the system wave function to the state of qubits.

17 Second quantization Paper In general the non-relativistic QM of an electronic system is driven by the Hamiltonian N H = T + V ext + V ee = 1 N N 1 q i + V ( q i ) + q i q j. i=1 Pure states γ N := ψ ψ have skewsymmetric ψ(x 1,...,x N ), with x i = ( q i,ς i ), spatial and spin variables. The general second-quantized chemical Hamiltonian has O(n 4 ) terms, where n is the dimension of the one-particle Hilbert space. The Hamiltonian is: i=1 i <j H = n n h pq â pâq + h pqrs â pâ qârâ s where {â p,â q } = δq p pq pqrs

18 Simulation of time evolution Any Hamiltonian H = can be simulated efficiently by a universal quantum computer. The key idea is based on the Trotter splitting of all non-commuting operators, ( n. e iht = lim e ih 1t/n e mt/n) ih n m i=1 H i The idea is to use the same formula for e ih pqâ pâqδt and e ih pqrsâ pâ qârâ s δt.

19 The Jordan-Wigner transformation Expressing the Hamiltonian in second-quantized form allows a straightforward mapping of the state space to qubits. The logical states of each qubits are identified with the fermionic occupancy of a single electron spin-orbital: 0 = occupied 1 = unoccupied. The Jordan-Wigner transformation of the fermionic operators to spin variables is: â j 1 1 ˆσ + ˆσ z ˆσ z â j 1 } {{ } 1 ˆσ ˆσ z ˆσ z, } {{ } (j 1) times (N j)times where ˆσ + := 1 ( ˆσ x + i ˆσ y ) = 0 1 and ˆσ := 1 ( ˆσ x i ˆσ y ) = 1 0.

20 The Jordan-Wigner transformation The ˆσ + operators achieve the mapping of (un-)occupied states to the computational basis 0 1 while the other terms serve to maintain the required anti-symmetry of the wave function in the qubit representation.

21 Exponentiation of the Hamiltonian The goal is to implement efficiently with a reasonable number of logic gates. The Hamiltonian H can be used to generated a unitary operator Û, with E mapped to the phase of its eigenvalue e πiφ, in the following way: Û Ψ = e ihτ Ψ = e πiφ Ψ with E = πφ τ. Then, the goal is to use a modified phase-estimation algorithm. A. Aspuru-Guzik, et alt, Science, 309, 1704, 005.

22 Phase-estimation algorithm Suppose a unitary operator Û has an eigenvector u with eigenvalue e πiφ, where the value φ is unknown. The goal of the phase-estimation algorithm is to estimate this value.

23 Phase-estimation algorithm PEA procedure 0 u 1 t 1 t t=0 1 t t 1 t=0 j Uj u = 1 t t 1 t=0 j eπijφ u φ u φ initial state j u create superposition apply inverse Fourier transform. measure first register.

24 A quantum simulator for H The two 1s -type orbitals are combined to form the bonding g and antibonding u molecular orbitals

25 A quantum simulator for H The six two-electron configurations are: Φ 1 = 1 ( g g g g ), Φ = 1 ( g u u g ), Φ 3 = 1 ( g u u g ), Φ 4 = 1 ( g u u g ), Φ 5 = 1 ( g u u g ), Φ 6 = 1 ( u u u u ).

26 A quantum simulator for H The six two-electron configurations are: Φ 1 = 1 ( g g g g ), Φ = 1 ( g u u g ), Φ 3 = 1 ( g u u g ), Φ 4 = 1 ( g u u g ), Φ 5 = 1 ( g u u g ), Φ 6 = 1 ( u u u u ). In this basis the Hamiltonian is block-diagonal: Φ 1 H Φ Φ 1 H Φ 6 0 Φ H Φ Φ H = 3 H Φ 3 Φ 3 H Φ Φ 4 H Φ 3 Φ 4 H Φ Φ 5 H Φ 5 0 Φ 6 H Φ Φ 6 H Φ 6

27 A quantum simulator for H The six two-electron configurations are: Φ 1 = 1 ( g g g g ), Φ = 1 ( g u u g ), Φ 3 = 1 ( g u u g ), Φ 4 = 1 ( g u u g ), Φ 5 = 1 ( g u u g ), Φ 6 = 1 ( u u u u ). In this basis the Hamiltonian is block-diagonal: Φ 1 H Φ Φ 1 H Φ 6 0 Φ H Φ Φ H = 3 H Φ 3 Φ 3 H Φ Φ 4 H Φ 3 Φ 4 H Φ Φ 5 H Φ 5 0 Φ 6 H Φ Φ 6 H Φ 6 The states Φ 1, Φ 6 and 1 ( Φ 3 Φ 4 ) are eigenvalues of Ŝ and Ŝ z with (j,m) = (0,0).

28 A quantum simulator for H The Hartree-Fock calculations were carried out on a classical computer using the STO-3G basis. The software used was the PyQuante quantum chemistry package version 1.6. The experimental setup is:

29 A quantum simulator for H The low eigenvalue of H (1,6)

30 Summary and beyond Paper

31 D-wave Paper

Accelerating QMC on quantum computers. Matthias Troyer

Accelerating QMC on quantum computers Matthias Troyer International Journal of Theoretical Physics, VoL 21, Nos. 6/7, 1982 Simulating Physics with Computers Richard P. Feynman Department of Physics, California