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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
Paper
Simulating chemistry using quantum computers
:-(
Carlos L. Benavides-Riveros Quantum Computers and Chemistry
Paper
Simulating chemistry using quantum computers
:-(Carlos L. Benavides-Riveros Quantum Computers and Chemistry
Paper
Simulating chemistry using quantum computers
Carlos L. Benavides-Riveros Quantum Computers and Chemistry
Paper
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.
Carlos L. Benavides-Riveros Quantum Computers and Chemistry
Paper
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.
Carlos L. Benavides-Riveros Quantum Computers and Chemistry
Paper
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.
Carlos L. Benavides-Riveros Quantum Computers and Chemistry
Paper
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.
Carlos L. Benavides-Riveros Quantum Computers and Chemistry
Paper
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.
Carlos L. Benavides-Riveros Quantum Computers and Chemistry
Paper
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.
Carlos L. Benavides-Riveros Quantum Computers and Chemistry
Paper
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.
Carlos L. Benavides-Riveros Quantum Computers and Chemistry
Paper
Quantum chemical complexity
The “simple” system ∧3Hn of three electrons and an-dimensional one-particle Hilbert space...
≈ n3/24
Carlos L. Benavides-Riveros Quantum Computers and Chemistry
Paper
Quantum chemical complexity
The “simple” system ∧3Hn of three electrons and an-dimensional one-particle Hilbert space... ≈ n3/24
Carlos L. Benavides-Riveros Quantum Computers and Chemistry
Paper
Quantum chemical complexity
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.
Carlos L. Benavides-Riveros Quantum Computers and Chemistry
Paper
Quantum quantum-chemical complexity
Any implementation of a quantum-simulation algorithmrequires a mapping from the system wave function to the stateof qubits.
Carlos L. Benavides-Riveros Quantum Computers and Chemistry
Paper
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
Carlos L. Benavides-Riveros Quantum Computers and Chemistry
Paper
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 .
Carlos L. Benavides-Riveros Quantum Computers and Chemistry
Paper
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
Paper
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.
Carlos L. Benavides-Riveros Quantum Computers and Chemistry
Paper
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.
Carlos L. Benavides-Riveros Quantum Computers and Chemistry
Paper
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.
Carlos L. Benavides-Riveros Quantum Computers and Chemistry
Paper
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.
Carlos L. Benavides-Riveros Quantum Computers and Chemistry
Paper
A quantum simulator for H2
The two |1s〉-type orbitals are combined to form the bonding |g〉and antibonding |u〉molecular orbitals
Carlos L. Benavides-Riveros Quantum Computers and Chemistry
Paper
A quantum simulator for H2
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).
Carlos L. Benavides-Riveros Quantum Computers and Chemistry
Paper
A quantum simulator for H2
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).
Carlos L. Benavides-Riveros Quantum Computers and Chemistry
Paper
A quantum simulator for H2
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).
Carlos L. Benavides-Riveros Quantum Computers and Chemistry
Paper
A quantum simulator for H2
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:
Carlos L. Benavides-Riveros Quantum Computers and Chemistry
Paper
A quantum simulator for H2
The low eigenvalue ofH(1,6)
Carlos L. Benavides-Riveros Quantum Computers and Chemistry