Advanced Subjects SIMULATING CHEMISTRY AND PHYSICS …Importance of StaAc/Strong CorrelaAon in...
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Advanced Subjects
SIMULATING CHEMISTRY AND PHYSICS WITH QUANTUM COMPUTERS Francesco Evangelista | Emory University
Winter School on Quantum Compu3ng at Emory (WiSQCE) Emory University, January 10, 2020
Photo by Clyde He on Unsplash
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The Richness of Chemistry. Two Approaches
We want to predict and control properties, and design molecular structures with specific functionality
Modeling/Machine Learning
Fit energy to experimental and theoretical data
First principles methods
He nucleus (charge +2)
e–
e–
Coulomb law
Solve the molecular Schrödinger equation
~100 different elements and nearly unlimited ways to build molecules!
Fit
Validate
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Electronic Structure Scales ExponenAally With System Size
Molecule Electrons (N)/Orbitals(K) # of determinants
10/10 3 × 104
14/14 6 × 106
18/18 1 × 109
22/22 3 × 1011
26/26 5 × 1013Storing these many coefficients requires 360 TB!
# of determinants = fsymm ( KNα) ( K
Nβ)
Number of electron arrangements (determinants) that arise from the π/π* orbitals of polyacenes
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The Breakdown of Single-Reference Methods With the Onset of Degeneracy
Hartree-Fock theory (mean field)
|ΨHF⟩ = |Φ0⟩ = ∏i
a+i | − ⟩
N2 dissociation curve
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The Breakdown of Single-Reference Methods With the Onset of Degeneracy
Many-body methods (perturbative and non-perturbative) • Second-order PT (MP2) • Coupled cluster singles and doubles (CCSD) • CCSD + three-body terms [CCSD(T)]
|ΨCC⟩ = exp( T) |Φ0⟩
Hartree-Fock theory (mean field)
|ΨHF⟩ = |Φ0⟩ = ∏i
a+i | − ⟩
N2 dissociation curve
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What About DFT?
N2 dissociation curve
Many-body methods • Second-order PT (MP2) • Coupled cluster with single and double excitations (CCSD) • CCSD + three-body terms [CCSD(T)]
|ΨCC⟩ = exp( T) |Φ0⟩
Hartree-Fock theory (mean field)
|ΨHF⟩ = |Φ0⟩ = ∏i
a+i | − ⟩
Density functional theory
E = E[ρ] ρ = ∫ dx2⋯dxN |Φ0 |2
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Weak vs. Strong Electron CorrelaAon
ΔU
Energy gapElectron repulsion
(potential)
Weak or dynamical correlaAon
Strong or staAc correlaAonU/Δ
S
T
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Smaller S/T gap
Ground state changes nature
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µ
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S
T
−
In this regime electronic structure is simple. Very accurate approximate methods exists (DFT, coupled cluster theory)
In this regime electronic structure is difficult. The electrons are in a quantum entangled state.
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Importance of StaAc/Strong CorrelaAon in Quantum Chemistry
Global potenAal energy surfaces Conical intersections, multi-electron excited states, nonadiabatic dynamics
Small energy gaps
Electronic structure of transiAon metals, lanthanides, and acAnides Catalysis, red-ox reactions, molecular magnetism
Degeneracy and large e−-e− repulsion in d/f orbitals
Interface of chemistry and solid-state physics Polymers, spin-frustrated systems, lattice models
S
C6H13
SS
S
C6H13S
SN
N
O
O
N
N
O
OC6H13
C6H13
C6H13
C6H13
S
C6H13
SS
S
C6H13S
SN
N
O
ON
N
O
O
Degeneracy, dimensional-dependent screening,
geometric effects
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How Do We Simulate Chemistry and Physics on a Quantum Computer?
Schrödinger’s Equation
HΨ = EΨ
Ψ(r1, r2, …, rN)
HHamiltonian operator
Wave function
Hb
Map to problem to
qubits
Ψb(q1, q2, …, qK)K ≠ N
HbΨb = EΨb
Solve SE on quantum computer
⟨O⟩ = ⟨Ψb | Ob |Ψb⟩Compute observables (e.g. dipole, spin-spin correlaAon funcAon)
Research quesAons: 1. What is the most efficient
qubit representation? 2. What are the best quantum
algorithms for solving the SE?
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Mapping a General Quantum Problem to Qubits
H = −12 ∑
i
∇2i + ∑
i
V(ri) + ∑i<j
g(ri, rj)
First-quantization formulation of quantum many-body problems
r1
r2
E.g. map position of a particle on a grid specifying the coordinate with qubits
r1 → (q1q2q3) = {0,1,2,3,4,5,6,7}r2 → (q4q5q6) = {0,1,2,3,4,5,6,7}
Not very practical. Have to enforce correct symmetry
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Second QuanAzaAon: CreaAon and AnnihilaAon Operators
No particles |Ψ⟩ = |0⟩
One particle |Ψ⟩ = a† |0⟩ = |1⟩a† =CreaAon operator
Destroy one particle a =AnnihilaAon
operator
|Ψ⟩ = a |1⟩ = |0⟩
Quantum state
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Physics in Second QuanAzaAon
Consider quantum particles on a 1D lattice
No particles0 1 32
2 particles on sites 0 and 2
Quantum state
|Ψ⟩ = |0000⟩
|Ψ⟩ = a†2 a†
0 |0000⟩ = |1010⟩
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Physics in Second QuanAzaAon
Hopping probabilities (kinetic + potential energy)
0 1 32Quantum state
|Ψ⟩ = |0100⟩
a†2 a1
0 1 32
|Ψ⟩ = |0010⟩
Hop from 1 to 2 with probability proportional toh21 a†2 a1 h21
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Physics in Second QuanAzaAon
Interactions (e.g. Coulomb repulsion)
Interaction between particles on sites 1 and 2 scatters them to 0 and 3
0 1 32Quantum state
|Ψ⟩ = |0110⟩
a†3 a†
0 a2 a1
0 1 32
|Ψ⟩ = |1001⟩
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Bosons/Fermions
a2 a1 = + a1 a2
0
1
2
3
Bosons
Ψ(r1, r2, …) = + Ψ(r2, r1, …)
⋮
One site can have any integer number of particles
0
1
Fermions
Ψ(r1, r2, …) = − Ψ(r2, r1, …)
a2 a1 = − a1 a2
At most one electron per site
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What’s the Point of Second QuanAzaAon?
0 1 32
H = ∑ij
hij a†i aj
1-body
+ ∑ijkl
hijkl a†i a†
j al ak
2-body
+ …
General framework to represent many-body problems
Introduce a finite computational basis (e.g. expansion in Fourier series, lattice sites, Gaussians, …)
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Fermionic Mapping to Qubits (Jordan–Wigner)
Create one particle |Ψ⟩ = a† |0⟩ = |1⟩a† =CreaAon operator
Destroy one particle a =AnnihilaAon
operator
|Ψ⟩ = a |1⟩ = |0⟩
a† |0⟩ = |1⟩a |1⟩ = |0⟩
Looks like the X Pauli (NOT) gate
X |0⟩ = |1⟩X |1⟩ = |0⟩
a = ( a†)†
a†p →
12
(Xp + iYp) ⊗ Zp−1 ⊗ …Z0
ap →12
(Xp − iYp) ⊗ Zp−1 ⊗ …Z0
Jordan–Wigner mapping
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Example0 1 32
a†2 a1
0 1 32
a†2 a1 =
X2 + iY2
2X1 − iY1
2
(X2 + iY2)/2(X1 − iY1)/2
Not unitary!
exp[iα( a†2 a1 + a†
1 a2)]
Quantities like these appear in unitary operators, e.g.
exp(−itH)
For example, the time evolution operator
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Using Quantum Computers To Represent Molecular Wave FuncAons
|Ψ⟩
|0⟩|0⟩
Orbitals
|00⟩
H1
2( |00⟩ + |10⟩)
+ +
1
2( |01⟩ + |10⟩)
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The State-of-the-Art in Quantum Algorithms for Molecular Problems
Quantum Phase EsAmaAon (QPE)Perform unitary evolution and measure the spectrum
t = 2ke−iHtk
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The State-of-the-Art in Quantum Algorithms for Molecular Problems
The VariaAonal Quantum Eigensolver (VQE)1
E = minθ
⟨Ψ(θ) | H |Ψ(θ)⟩
1. Peruzzo, et al. Nat Comms. (2014) 2. Wecker, Hastings, Troyer, Phys. Rev. A (2015) 3. McClean, Romero, Babbush, Aspuru-Guzik, New J. Phys. (2016)
4. Barkoutsos, et al. Phys. Rev. A (2018) 5. Ryabinkin, Yen, Genin, Izmaylov, J. Chem. Theory Comput. (2018) 6. Grimsley, Economou, Barnes, Mayhall, Nat Comms. (2019)
Implementations of VQE have focused on unitary CC, generalized UCC, and variations on UCC1–3
Several recent developments extended VQE to more general wave function forms ๏ Qubit coupled cluster method4 ๏ q-UCC5 ๏ ADAPT-VQE algorithm6
|Ψ(θ)⟩ → |ΨUCCSD⟩ = e T− T† |Φ⟩
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The State-of-the-Art in Quantum Algorithms for Molecular Problems
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Slater determinant
Basics of UCC
|ΨUCC⟩ = U |Φ⟩ = e σ |Φ⟩
τμ = aab⋯ij⋯ = a†
a a†b⋯ aj ai
ExcitaAon/subsAtuAon operator
σ = ∑μ
tμ( τμ − τ†μ) = ∑
μ
tμ κμ
The unitary coupled cluster ansatz Electron configurations are generated by operators that excite electrons out of a reference configuration
Occupied (holes, i, j,…)
Virtual (parAcles, a,b,…)
aabij aij
ab
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Unitary Coupled Cluster on a Quantum Computer
1. Peruzzo, et al. Nat Comms. (2014)
In current literature, the Trotterized form is viewed as an approximation to UCC. Using a finite value of n introduces a Trotterization error.
The UCC ansatz cannot be directly implemented on a quantum computer (operators don’t commute)
Lie–Troaer–Suzuki formula
e A+B = limm→∞ (e
Ame
Bm)
m
e σ |Φ⟩ = e ∑μ tμ κμ |Φ⟩
Unitary coupled cluster
e σ |Φ⟩ ≈ (∏μ
etμm κμ)m |Φ⟩
Trotterized unitary coupled cluster1
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What We Do NOT Know About UCC?
๏ Is full UCC exact? Limited numerical evidence suggests yes, but no studies have probed the strong correlation regime extensively. A formal proof is lacking.
๏ What errors are introduced when UCC is approximated via “Trotterization"?
๏ Are UCC ansätze used in quantum computing (generalized products) exact? Are they the most efficient?
Collaborators Garnet Chan Gustavo Scuseria
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An Exact Disentangled Form of UCC
We can write a product form of the UCC ansatz that uses NFCI − 1 parameters and is exact
μi → ( κμ1, κμ2
, …, κμNFCI−1)
All particle-hole operators are used only once
|Φ⟩ = (∏j
etμjκμj)† |Ψ⟩
Proof: Construct sequences of particle-hole operators that transform any state back to a single determinant
|Ψ⟩ =NFCI
∏j
etμjκμj |Φ⟩ = etμ1
κμ1 etμ2κμ2⋯ |Φ⟩
1) Evangelista, Chan, Scuseria, J. Chem. Phys. (2019)
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Examples of Exact and Not Exact FactorizaAons of UCC
1234
1234
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A Toy Problem: Two Electrons in Two Orbitals
|Ψ⟩ =1
2( |ai⟩ + | ia⟩) i
a+
Factorized Unitary CC
| ii⟩exp(− π
4 κaaii ) 1
2( | ii⟩ − |aa⟩)
exp( π2 κa
i ) 1
2( | ia⟩ + |ai⟩)
− +
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Exactness of an Infinite Factorized Form of UCCSD
|Ψ⟩ =∞
∏j
etμjκ(1,2)μj |Φ⟩
We can break down any quantum state into a infinite a sequence of one- and two-body particle-hole operators applied to a single Slater determinant
κ(1,2)μ = {
κai = aa
i − aia
κabij = aab
ij − aijab
One- and two-body antihermitian particle-hole operators
Exact representation of any state
This justifies recent work on VQE where UCC is replaced with general 1- and 2-body operators
1) Evangelista, Chan, Scuseria, J. Chem. Phys. (2019)
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A Third Way: Quantum Subspace DiagonalizaAon
Prepare a complicated nonorthogonal basis of many-body states
|ψα⟩ = Uα |Φ0⟩, α = 1,…, N
Sαβ = ⟨ψα |ψβ⟩
Compute overlap and Hamiltonian matrices
Hαβ = ⟨ψα | H |ψβ⟩
Hc = ScEObtain the energy via solution of a generalized eigenvalue problem
1) McClean, Kimchi-Schwartz, Carter, de Jong, Phys. Rev. A (2017) 2) Motta et al., (arXiv:1901.07653)
3) Parrish, McMahon, (arXiv:1909.08925) 4) Huggins, Lee, Baek, O'Gorman, Whaley (arXiv:1909.09114)
Several recent work have explored this strategy (1-4)
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Exploring Hilbert Space With Real-Time Dynamics
|Ψ(t)⟩ = e−iℏtH |Ψ(0)⟩
Propagation of a wave function in time is a natural process to simulate on a quantum computer
iℏddt
|Ψ(t)⟩ = H |Ψ(t)⟩
Animation: The Little Prince by Ann Segeda1) Stair, Huang, Evangelista (arXiv:1911.05163) 2) Parrish, McMahon, (arXiv:1909.08925)
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A Quantum Subspace DiagonalizaAon Algorithm Based on Real-Time EvoluAon
− 3.2
− 3.0
− 2.8
− 2.6
− 2.4
Ener
gy/E
h
0.6 1.0 1.4 1.8r / Å
10− 4
10− 3
10− 2
10− 1
Ener
gyEr
ror/
Eh
MRSQK(m=1)
MRSQK(m=2)
MRSQK(m=4)
MRSQK(m=8)
RHF
FCI
MRSQK(m=8)
MRSQK(4)
MRSQK(m=2)MRSQK(m=1)
MP2
RHF
MP2
CCSD
MRSQK(m=∞)
MRSQK(m=∞)
CCSD
Chemicalaccuracy
0
12
3
4
51011
7
69
8
C
|Ψ⟩ =d
∑I=1
s
∑n=0
c(n)I |ψI(tn)⟩
Build the wave function as a combination of time-evolved states
Dissociation curve of linear H6 (simulation of a quantum computer)
1) Stair, Huang, Evangelista (arXiv:1911.05163)
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The Effect of Troaer ApproximaAon and Comparison With Other Methods
Trotter approximation to exp(-iHt) → ADAPT-VQEClassical adaptive algorithm
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Enabling Large-Scale ApplicaAons of Quantum CompuAng to Chemistry
Deploy the exponential scaling part of the computation to a quantum computer!
Theory challenges: ๏ Efficient electronic structure quantum
algorithms ๏ Hybrid quantum-classical methods
that are quantum computer friendly
virtual
active
core
(1) Diagonalization of active
space Hamiltonian
(2) DSRG Hamiltonian
Downfolding
(3) Relaxation of active
space wave function
Effective
Hamiltonian
...QuantumComputer
Activeorbitals
1) Bauer, Wecker, Millis, Hastings, Troyer, Phys. Rev. X (2016) 2) Wecker, Hastings, Wiebe, Clark, Nayak, Troyer, Phys. Rev. A (2015) 3) Bauman, Bylaska, Krishnamoorthy, Low, Wiebe, Granade, Roetteler, Troyer, and Kowalski J. Chem. Phys. (2019)
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Conclusions
๏ Solving quantum many-body problems on QCs requires a mapping to qubits and quantum algorithms
๏ Fermions naturally map to qubits via the Jordan–Wigner representation
๏ State-of-the-art algorithms for solving eigenvalue problems: quantum phase estimation (QPE) and the quantum variational eigensolver (VQE)
๏ Our contribution: Better understanding of wave function guesses used in VQE methods and new algorithms for ground states based on real-time dynamics
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Acknowledgements
SofwareFunding
The group
Thank You for Your AaenAon!
QCQC Project Collaborators
McClean
Thrust IDownfolding
for Quantum Computers
Thrust IIQuantum-Classical
Embedding Theories
QCQC Team and research synergisms QCQC Research Thrusts
QCQC Collaborators
Babbush
Thrust IVBenchmarks and Software
for Quantum Computers
Thrust IIIQuantum Algorithms
for Quantum ChemistryEvangelista
(Lead)
Shiozaki Whitfield
Aspuru-Guzik Chan
ZgidScuseria Mario Moaa (IBM)
Interested in doing research at the intersecAon of chemistry and quantum compuAng? Contact me at
francesco.evangelista-at-emory.edu