advanced subjects simulating chemistry and physics …importance of staac/strong correlaaon in...

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

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

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

The Breakdown of Single-Reference Methods With the Onset of Degeneracy

Hartree-Fock theory (mean field)

|ΨHF⟩ = |Φ0⟩ = ∏i

a+i | − ⟩

N2 dissociation curve

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⟩


|ΨHF⟩ = |Φ0⟩ = ∏i

a+i | − ⟩


What About DFT?


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⟩


|ΨHF⟩ = |Φ0⟩ = ∏i

a+i | − ⟩

Density functional theory

E = E[ρ] ρ = ∫ dx2⋯dxN |Φ0 |2

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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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.

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

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?

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

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

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⟩


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


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⟩

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

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, …)

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

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

Using Quantum Computers To Represent Molecular Wave FuncAons

|Ψ⟩

|0⟩|0⟩

Orbitals

|00⟩

H1

2( |00⟩ + |10⟩)

+ +

1

2( |01⟩ + |10⟩)

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


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⟩

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

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

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

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)

Examples of Exact and Not Exact FactorizaAons of UCC

1234

1234

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⟩)

− +

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)

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)

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)

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)

The Effect of Troaer ApproximaAon and Comparison With Other Methods

Trotter approximation to exp(-iHt) → ADAPT-VQEClassical adaptive algorithm

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)

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

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

advanced subjects simulating chemistry and physics …importance of staac/strong correlaaon in...

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