I. Kassal, J. D. Whitfield, A. Perdomo-Ortiz, M-H. Yung, A ... · Scott Aaronson, Quantum Computing since Democritus, CUP, 2013. 1950sLate Turingzoic. 1971sThe Cook-Levin Asteroid

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I. Kassal, J. D. Whitfield, A. Perdomo-Ortiz, M-H. Yung, A. Aspuru-Guzik,

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

Carlos L. Benavides-Riveros

Martes CuanticoUniversidad de Zaragoza,

19th November 2013

Carlos L. Benavides-Riveros Quantum Computers and Chemistry

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Simulating chemistry using quantum computers

:-(


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:-(Carlos L. Benavides-Riveros Quantum Computers and Chemistry

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Why quantum computers?


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Some of the Aaronson complexity historical epochs

Scott Aaronson, Quantum Computing since Democritus, CUP, 2013.

1950s Late Turingzoic.1971s The Cook-Levin Asteroid. (NP-complete problems)

Early 1970s The Karpian Explosion.1978 Early Cryptozoic.1980s Randomaceous Era.1994 Invasion of the Quantodactyls.


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1950s Late Turingzoic.

1971s The Cook-Levin Asteroid. (NP-complete problems)



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Early 1970s The Karpian Explosion.

1978 Early Cryptozoic.1980s Randomaceous Era.1994 Invasion of the Quantodactyls.


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Early 1970s The Karpian Explosion.1978 Early Cryptozoic.

1980s Randomaceous Era.1994 Invasion of the Quantodactyls.


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Early 1970s The Karpian Explosion.1978 Early Cryptozoic.1980s Randomaceous Era.

1994 Invasion of the Quantodactyls.


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Quantum chemical complexity

The “simple” system ∧3Hn of three electrons and an-dimensional one-particle Hilbert space...

≈ n3/24


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The “simple” system ∧3Hn of three electrons and an-dimensional one-particle Hilbert space... ≈ n3/24


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The configuration interaction (CI) wave function

|ψCI〉 =(1+

∑r,µcrµa†r aµ +

∑r <sµ<ν

crsµνa†r a†s aµaν + · · ·

)|ψ0〉

is exact in the full CI limit, but lacks size-extensivity with any

truncation of the configuration space.

“... traditional wave function methods, which providedthe required chemical accuracy, are generally limitedto molecules with a small number of chemically activeelectrons, N ≤ O(10)“.

W. Kohn, Nobel lecture, 1998.


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Quantum quantum-chemical complexity

Any implementation of a quantum-simulation algorithmrequires a mapping from the system wave function to the stateof qubits.


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Second quantization

In general the non-relativistic QM of an electronic system isdriven by the Hamiltonian

H = T +Vext +Vee =N∑i=1

−12∆~qi +

N∑i=1

V (~qi) +N∑i <j

1|~qi − ~qj |

.

Pure states γN := |ψ〉〈ψ| have skewsymmetric ψ(x1, . . . ,xN ),with xi = (~qi ,ςi), spatial and spin variables.

The general second-quantized chemical Hamiltonian hasO(n4) terms, where n is the dimension of the one-particleHilbert space. The Hamiltonian is:

H =n∑pq

hpqa†paq +

n∑pqrs

hpqrsa†pa†qar as where {a†p, aq} = δ

pq


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Simulation of time evolution

Any Hamiltonian

H =m∑i=1

Hi

can be simulated efficiently by a universal quantum computer.The key idea is based on the Trotter splitting of allnon-commuting operators,

e−iHt = limn→∞

(e−iH1t/n · · ·e−iHmt/n

)n.

The idea is to use the same formula for

e−ihpq a†p aqδt and e−ihpqrs a

†p a†q ar asδt .


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The Jordan-Wigner transformation

Expressing the Hamiltonian in second-quantized form allows astraightforward mapping of the state space to qubits. Thelogical states of each qubits are identified with the fermionicoccupancy of a single electron spin-orbital:

|0〉 = occupied

|1〉 = unoccupied.

The Jordan-Wigner transformation of the fermionic operatorsto spin variables is:

aj → 1⊗ · · · ⊗ 1⊗ σ+ ⊗ σ z ⊗ · · · ⊗ σ z

a†j → 1⊗ · · · ⊗ 1︸︷︷︸(j−1)times

⊗ σ− ⊗ σ z ⊗ · · · ⊗ σ z︸︷︷︸(N−j) times

,

where σ+ := 12 (σ

x + iσ y) = |0〉〈1| and σ− := 12 (σ

x − iσ y) = |1〉〈0|.Carlos L. Benavides-Riveros Quantum Computers and Chemistry

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The Jordan-Wigner transformation

Theσ+−

operators achieve the mapping of (un-)occupied states to thecomputational basis

|0〉 |1〉

while the other terms serve to maintain the requiredanti-symmetry of the wave function in the qubit representation.


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Exponentiation of the Hamiltonian

The goal is to implement efficiently with a reasonable numberof logic gates.

The HamiltonianH can be used to generated a unitaryoperator U, with E mapped to the phase of its eigenvalue e2πiφ,in the following way:

U|Ψ 〉 = eiHτ |Ψ 〉 = e2πiφ|Ψ 〉 with E =2πφτ.

Then, the goal is to use a modified phase-estimation algorithm.A. Aspuru-Guzik, et alt, Science, 309, 1704, 2005.


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Phase-estimation algorithm

Suppose a unitary operator U has an eigenvector |u〉 witheigenvalue e2πiφ, where the value φ is unknown. The goal ofthe phase-estimation algorithm is to estimate this value.


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Phase-estimation algorithm

PEA procedure

|0〉|u〉 initial state

→ 1√2t

∑2t−1t=0 |j〉|u〉 create superposition

→ 1√2t

∑2t−1t=0 |j〉U j |u〉

= 1√2t

∑2t−1t=0 |j〉e2πijφ|u〉

→ |φ〉|u〉 apply inverse Fourier transform.

→ φ measure first register.


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A quantum simulator for H2

The two |1s〉-type orbitals are combined to form the bonding |g〉and antibonding |u〉molecular orbitals


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The six two-electron configurations are:

|Φ1〉 = 1√2(|g↑g↓〉 − |g↓g↑〉), |Φ2〉 = 1√

2(|g↑u↑〉 − |u↑g↑〉),

|Φ3〉 = 1√2(|g↑u↓〉 − |u↓g↑〉), |Φ4〉 = 1√

2(|g↓u↑〉 − |u↑g↓〉),

|Φ5〉 = 1√2(|g↓u↓〉 − |u↓g↓〉), |Φ6〉 = 1√

2(|u↑u↓〉 − |u↓u↑〉).

In this basis the Hamiltonian is block-diagonal:

H =

〈Φ1|H|Φ1〉 0 0 0 0 〈Φ1|H|Φ6〉0 〈Φ2|H|Φ2〉 0 0 0 00 0 〈Φ3|H|Φ3〉〈Φ3|H|Φ4〉 0 00 0 〈Φ4|H|Φ3〉〈Φ4|H|Φ4〉 0 00 0 0 0 〈Φ5|H|Φ5〉 0

〈Φ6|H|Φ1〉 0 0 0 0 〈Φ6|H|Φ6〉

The states |Φ1〉, |Φ6〉 and 1√

2(|Φ3〉 − |Φ4〉) are eigenvalues of S2 and Sz with

(j,m) = (0,0).


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|Φ1〉 = 1√2(|g↑g↓〉 − |g↓g↑〉), |Φ2〉 = 1√

2(|g↑u↑〉 − |u↑g↑〉),

|Φ3〉 = 1√2(|g↑u↓〉 − |u↓g↑〉), |Φ4〉 = 1√

2(|g↓u↑〉 − |u↑g↓〉),

|Φ5〉 = 1√2(|g↓u↓〉 − |u↓g↓〉), |Φ6〉 = 1√

2(|u↑u↓〉 − |u↓u↑〉).


H =


〈Φ6|H|Φ1〉 0 0 0 0 〈Φ6|H|Φ6〉

The states |Φ1〉, |Φ6〉 and 1√2(|Φ3〉 − |Φ4〉) are eigenvalues of S2 and Sz with

(j,m) = (0,0).


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|Φ1〉 = 1√2(|g↑g↓〉 − |g↓g↑〉), |Φ2〉 = 1√

2(|g↑u↑〉 − |u↑g↑〉),

|Φ3〉 = 1√2(|g↑u↓〉 − |u↓g↑〉), |Φ4〉 = 1√

2(|g↓u↑〉 − |u↑g↓〉),

|Φ5〉 = 1√2(|g↓u↓〉 − |u↓g↓〉), |Φ6〉 = 1√

2(|u↑u↓〉 − |u↓u↑〉).


H =


〈Φ6|H|Φ1〉 0 0 0 0 〈Φ6|H|Φ6〉

The states |Φ1〉, |Φ6〉 and 1√

2(|Φ3〉 − |Φ4〉) are eigenvalues of S2 and Sz with

(j,m) = (0,0).


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The Hartree-Fock calculations were carried out on a classicalcomputer using the STO-3G basis. The software used was thePyQuante quantum chemistry package version 1.6.

The experimental setup is:


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The low eigenvalue ofH(1,6)


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Summary and beyond


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D-wave


Documents

I. Kassal, J. D. Whitfield, A. Perdomo-Ortiz, M-H. Yung, A ... · Scott Aaronson, Quantum Computing since Democritus, CUP, 2013. 1950sLate Turingzoic. 1971sThe Cook-Levin Asteroid