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Quantum Chemistry Algorithms: Classical vs Quantum
Dr. David P. Tew
School of Chemistry, Bristol !
Apr. 2015
http://www.chm.bris.ac.uk/pt/tew/ [email protected]
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Quantum Chemistry on a Quantum Computer
Why? !1. Curiosity !“A Quantum machine may be more efficient at simulating a quantum system than a classical machine.” Feynman !!
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Quantum Chemistry on a Quantum Computer
Why? !1. Curiosity
2. Difficulty !“The fundamental laws necessary for the mathematical treatment of a large part of physics and the whole of chemistry are thus completely known, and the difficulty lies only in the fact that application of these laws leads to equations that are too complex to be solved.” ! Dirac
i~@ @t
= H
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Quantum Chemistry on a Quantum Computer
Why? !1. Curiosity
2. Difficulty
3. Importance !!
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Quantum Chemistry Programs on CPUs (80 and counting)
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The problem
!Schrödinger Equation !!!!!!!!!!Water: N = 3 n = 10 !Protein: N = 10000 n = 50000
H = �X
i
1
2mir2
i +X
i>j
ZiZj
rij
~ = 4⇡✏0 = me = e = 1
i@
@t= H
(r1, r2, . . . , rn,R1,R2, . . . ,RN , t)
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The problem
H = �X
i
1
2mir2
i +X
i>j
ZiZj
riji@
@t= H
(r1, r2, . . . , rn,R1,R2, . . . ,RN , t)
H =X
pq
hpqa†paq +
X
pqrs
gpqrsa†pa
†qaras
some steps
H| i = E| i
| i =X
P
CP |P i
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The problem
Step 1. Adiabatically separate electronic and nuclear motion !!
!Yields the time-independent Schrödinger Equation for the electrons !!!!!!!!
H = E
(r1, r2, . . . , rn)
(r,R, t) ! e(r;R) n(R, t)
H = �1
2
X
i
r2i �
X
I,i
ZI
riI+X
i>j
1
rij
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The problem
Step 2. Select a (finite) basis of 1-p functions (LCAO, PW)
!
!
• Mean field approximation (independent particle model)
!
!
!
Defines a set of one-particle states
and an n-particle Hilbert space
HF(r1, r2, . . . , rn) = AY
i
�i(ri)
F�p(r) = "p�p(r)
H =X
pq
hpqa†paq +
X
pqrs
gpqrsa†pa
†qaras
�µ(r) = x
iy
jz
ke
�↵r2 �p(r) =X
µ
cµp�µ(r)
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The problem
Step 3. Find the eigenstates
!
!
!
Dimension of n-p Hilbert space is combinatorial in the number of electrons (n) and available1-p states (m)
!
!
!
Water: m = 30 n = 10 : 1010 !Protein: m = 150000 n = 50000 : 1010000
H =X
pq
hpqa†paq +
X
pqrs
gpqrsa†pa
†qaras
✓mn
◆
H| i = E| i
| i =X
P
CP |P i
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Three layers of approximation
Simulation window
1-p Representation cut-off
n-p Representation cut-off
(r, t)
basis set incompleteness
H| i = E| i
| i =X
P
CP |P i
polynomial number of parameters
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The problem
Step 3. Find approximations to the eigenstates
!
!
!
!
!
!
!
• is acceptable
• Provided “Chemical accuracy’’
!
H =X
pq
hpqa†paq +
X
pqrs
gpqrsa†pa
†qarasH| i = E| i
E + ✏
✏/n <
Exact
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Classical Algorithm 1: Coupled Cluster
!
!
!
• Factorised many-body expansion
!
!
• Obtain energy and coefficients via projection (like PT)
H =X
pq
hpqa†paq +
X
pqrs
gpqrsa†pa
†qaras
T =X
ai
tai a†aai +
X
abij
tabij a†aa
†bajai| i = eT |0i
0 T1 T2
n2m4H
i
a
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Classical Algorithm 1: Coupled Cluster
For many cases, convergence with respect to truncation of many-body expansion is near exponential
0.01
0.1
1
10
100
1000
CCSD CCSDT CCSDTQ
chemical accuracy
Erro
r| i = eT1+T2+T3+...|0i
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Coupled Cluster State of the art
• For insulators, the interactions are short range: polynomial number of parameters and operations: O(n)
!
m > 8800
n > 900
N > 450
!
time 106 seconds
(2 weeks, 1 CPU)
!
Chemical accuracy for e.g. binding energies
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Coupled Cluster success and failure
Simple example of H2 with varying bond length
-1.5
-1
-0.5
0
0.5
0 1 2 3 4 5 6 7 8
Ener
gy (
E h)
Bond Length (a0)
(b)
|00i
|11i
|00i+ |11i
|1i|0i
Works well Fails
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Classical Algorithm 2: DMRG
Tensor train factorisation of the CI vector
!
!
!
!
!
!
!
State-of-the-art
!
m = 64 n = 30 : 1017
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Classical Algorithm 3: FCI-QMC
A stochastic realisation of the imaginary-time Schrödinger Equation in n-particle Hilbert-space
!
!
!
The CI coefficients are represented through a population of walkers in Hilbert space. After reaching steady state, energies and properties are extracted through time-averaging
!
!
!
State-of-the-art : > 1020
i@
@t= H @
@⌧= �H
�@CP
@⌧= (HPP � E)CP +
X
Q 6=P
HPQCQ
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Classical Algorithm 4: Density Functional Theory
There is an existence proof that there is a one-to-one mapping between the wave function and the electron density
!
!
for n-representable densities
!
Kohn-Sham: search over non-interacting mean-field states
!
!
(r1, r2, . . . , rn) $ ⇢(r)
min⇢
E[⇢]
AY
i
�i(ri) ! ⇢(r)
E[⇢] = Ts
+ V [⇢] + J [⇢] + Vxc
[⇢]
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Approximate Density Functionals in G09
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State-of-the-art for DFT
Accuracy - twice “Chemical Accuracy” if the molecule under investigation resembles those the functionals were parameterised to get right. Else …
(Important, but shrinking, class of problems for which DFT fails)
N = 16000 !m = 100000 !n = 50000 !!1000 time steps !Blue Gene/Q
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Summary
For a wide class of molecules, the electronic structure of the undistorted ground state is relatively easy. Weakly correlated.
DFT and CCSD(T) hit different sweet spots of accuracy vs cost
!
An important class of systems have difficult electronic structure, usually characterised by many degenerate or near degenerate states and a poor mean field solution. Strongly correlated.
!
We don’t know how to solve these problems efficiently and reliably.
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Quantum Chemistry on a Quantum Computer
Exploit the mapping of Fermionic creation and annihilation operators onto qubit operations
!
!
!
Unitary-type operations can be used to prepare a state, perform QFT, and evolve a state according to a Hamiltonian
!
!
Trotter expansion makes it possible to decompose general angle unitaries from into a sequence of local angle unitaries.
H =X
pq
hpqa†paq +
X
pqrs
gpqrsa†pa
†qaras
eiHt| i| i = U |0i
eiHt
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Quantum Algorithm 1: Phase Estimation
!
!
!
!
Requires that has a large overlap with the true eigenstate
For the easy cases, where CC works, this is probably possible
Open question: How to prepare good states for hard cases?
!
!
!
Prepare
| i = U |0iQFT
E
| i
Prepare QFT
E
Adiabatic map
|0i ei((1�t)F+tH)|0i
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Quantum Algorithm 2: Variational Approach
Decompose the Hamiltonian into a sum of unitary operations
H =X
pq
hpqa†paq +
X
pqrs
gpqrsa†pa
†qaras
=X
i
ciUi
Prepare
| i = U |0iMeasure
E
Refine U
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Quantum Algorithm 2: Variational Approach
!
!
!
!
!
!
We will only have access to a limited space of unitaries
!
Questions:
Is unitary truncated coupled cluster better than regular?
How easy or hard is the refinement of U?
Prepare
| i = U |0iMeasure
E
Refine U
U = eT�T †
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Quantum Algorithm 2: Variational Approach
Numerical experiments for a 1-d periodic Hubbard Hamiltonian
!
!
!
!
!
!
4 1-particle states for each spin
!
Half-filled case:
2 up spin particles, two down spin particles: 36 states
H = �tX
hi,ji�
a†i�aj� + UX
i
ni"ni#
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Quantum Algorithm 2: Variational Approach
Numerical experiments for a 1-d periodic Hubbard Hamiltonian
0 2 4 6 8 10
-4
-3
-2
-1
0
U/t
E FCI
CISD
UCC
| i = eT�T †|0i T =
X
ai
tai a†aai +
X
abij
tabij a†aa
†bajai
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Summary
Classical algorithms are efficient when the electronic structure is well approximated by one occupation number state
!
Classical algorithms struggle when many occupation number states are required for a qualitatively correct ground state. This is where quantum algorithms will probably have the biggest impact.
!
This situation occurs in e.g. superconducting materials and clusters of transition metal atoms in the body
!
Many important topics have not been mentioned:
Excited states for Fermionic systems
Bosonic Hamiltonians for QM of nuclei