Chemical group theory for quantum simulation

Transcription

1 Title 1/19 Chemical group theory for quantum simulation James Daniel Whitfield U. Ghent September 28, 2015

2 Title 2/19 1. Computational chemistry 2. Symmetry considerations 3. Industrial grade quantum computing

3 Computational chemistry 2/19 Outline 1. Computational chemistry 2. Symmetry considerations 3. Industrial grade quantum computing

4 omputational chemistry 3/19 Electronic structure problem Molecular instance (Input to algorithm) Model Chemistry (algorithm specification) Approximation limit Reality Descriptive resolution DFT/B3LYP 6-31G** Sophistication Approximation Goal: Solve the Schrödinger equation: Quickly Accurately t Ψ t = iĥ Ψ t

5 omputational chemistry 4/19 Computational chemistry H problem instances Radical stabilization H energy C H + R H H H C H H H H + Cl H C C + R H H H H H + Cl barrier heights H H C C H H H H C C HH C C H H H H proton affinity H+Cl HCl reaction energies Cl +O 3 ClO+O 2 ClO+O 3 Cl + 2O 2

6 Computational chemistry 5/19 Quantum computer simulation algorithms Basic structure 1. Pick encoding 2. Initialize quantum state 3. Propagate to extract information Ideas from the literature First quantization (binary encoding) algorithms Second quantization (unary encoding) algorithms Black-box (sparse matrix encoding) algorithms Variational state parameterizations

7 Symmetry considerations 5/19 Outline 1. Computational chemistry 2. Symmetry considerations 3. Industrial grade quantum computing

8 Symmetry considerations 6/19 Symmetries Set of A such that [H, A] = 0... Γ 1 Γ 2 Forms the symmetry group of H Simplifies numerical simulations Γ 3 Lets one focus on one block Example: Number operator, M a i a i

9 ymmetry considerations 7/19 Antisymmetry Fermionic wave functions Ψ(x 1, x 2,..., x i,..., x j,..., x N ) = Ψ(x 1, x 2,..., x j,..., x i,..., x N ) Schmidt Decomposition Ψ = F i (r 1, r 2,...r N )Θ i (s 1,..., s N ) Energies Ψ H Ψ = ij F i H F j Θ i Θ j

10 Symmetry considerations 8/19 Permutation group and identical particle Hamiltonians Hamiltonian is invariant under any permutation of coordinates H = i T (x i ) + i V (x i ) + i>j W (x i, x j ) πh = i T (x π(i) ) + i V (x π(i) ) + i>j W (x π(i), x π(j) ) = i T (x i ) + i V (x i ) + i>j W (x i, x j ) π is from S N, permutations of N objects

11 ymmetry considerations 9/19 Permutation group and identical particle Hamiltonians [λ] = [2 2 1] = Symmetric group maximally commutes with unitary group U N elec is in unitary group Sectors with fixed S z and S 2 labelled by Young frame Both groups can be used to label eigenstates with same energy

12 Symmetry considerations 10/19 First quantization and Young s projections First quantization Ψ = r 1 s 1 r 2 s 2... r N s N Ψ [λ] = r 1 r 2... r Nelecs Each electron has its own grid Permuting electrons by permuting grids Young s tableaux label spin-free wave functions Abrams-Lloyd algorithm Drop spin degrees of freedom, lose N qubits

13 ymmetry considerations 11/19 Second quantization and Wigner projections 0 IRs F 1 G group algebra F G IRs Γ, ij Ψ D(g) Ψ Γ Second quantization Bacon, Chuang, and Harrow 2004 Qubit represent symmetry-adapted orbital Better initial states but no qubit savings Projection into a irreducible representation (IR) Deterministic construction with spin raising/lowering operators work by K. Sugisaki et al.

14 Symmetry considerations 12/19 Second quantization in a spatial basis Excited state In continuum limit, [W, V ] = 0 Enforcing commutation relation implies W ijkl = W (i, j)δ jk δ il V (i) and W (i, j) implemented based on occupancies m s = +1 m I = +1 m I = 0 Usual circuit templates for kinetic energy m 2γB I = -1 Cost(T + V + Ground W ) state =O(Morbs 2 ) + O(N elec) + O(Nelec 2 ) D m s = -1

15 Symmetry considerations 13/19 Blackbox simulation and the unitary group approach Oracular simulation based on determinant wave functions [Babbush+ 15, Toloui & Love 13] Configuration state function e.g. eigenfunctions of S 2 and S z Matrix elements from spin-free Hamiltonian H = hrs E rs hpqrs (E ps E qr δ qs E pr ) E mn = α,β σ a mσa nσ Simple rules for generating sparse elements of generators E i,i+1, Others via [E rs, E st ] = E rt

16 ymmetry considerations 14/19 Symmetry considerations Summary First quantization Second quantization Oracular methods arxiv: arxiv: Unpublished arxiv: In progress Outlook Molecular problems have symmetries Applications to solid-state simulations Constant factor improvements needed for working algorithms

17 Industrial grade quantum computing 14/19 Outline 1. Computational chemistry 2. Symmetry considerations 3. Industrial grade quantum computing

18 QC companies Industrial grade quantum computing 15/19

19 ndustrial grade quantum computing 16/19 Future of quantum computing in chemistry Validation of a test set in quantum hardware Establishes approach agnostic metric Realistic comparisons to existing techniques Performance tracking

20 Future of quantum computing Industrial grade quantum computing 17/19

21 Industrial grade quantum computing 17/19 Future of quantum computing Can t beat em... Quantum co-processor for short simulations Constructing classical-quantum interfaces Embeddings of quantum simulation into existing algorithms Quantum region in molecular dynamics A high-accuracy region in DFT embeddings

22 ndustrial grade quantum computing 18/19 Simulation cost tradeoffs Renewed push for quantum algorithms Slow algorithm, Low accuracy (Pointless) Slow algorithm, High accuracy Fast algorithm, Low accuracy Fast algorithm, High accuracy (Idealized) Faster quantum technologies Refocus from scaling size to scaling speed Sufficient large that simulation is somewhat difficult classically

23 losing 19/19 Acknowledgements Thanks to... Alán Aspuru-Guzik (PhD supervisor) Frank Verstraete (Current supervisor) Collaborators and colleagues Organizers for the invitation Funding Postdoctoral Vienna Center for Quantum Science and Technology Postdoctoral Fellowship Ford Foundation Fellowship PhD studies Harvard Graduate Prize Fellowship Army Research Office

24 1/21 Dynamic approaches to measurement Phase estimation J(ω) = C eiωt ψ(t) ψ(0) dt = F[ ψ e iht ψ ](ω) Shor 94, Kitaev 95, Cleve et al. 97

25 2/21 Dynamic approaches to measurement Heller. Accounts of Chemical Research (1981) Absorption spectroscopy ɛ(ω) = Cω ei(ω+e 0)t ψ(t) ψ(0) dt = F[ ψ e iht ψ ](ω)

26 3/21 Implementing Wigner Projectors 0 IRs F 1 G group algebra F G IRs Γ, ij Ψ D(g) Ψ Γ Bacon, Chuang, and Harrow 2004 A, 11 Ψ Γ, ij Pij Γ Ψ ij Γ Measurement projects into irreducible subspace

27 4/21 Experimental quantum computation for chemistry Humble beginnings H 2 Lanyon [Optics] Du [NMR] HeH + Puzzuro [Optics] Wang [NV] Isomerization reaction Lu [NMR] co-authored with JDW

28 5/21 Spin-adapted states What F i are compatible with prescribed spin? Involves symmetric group, Young frames, and corresponding projecto [λ] = E [λ] i E [λ] i = = NP ( E [λ] i ) 2 Ψ = F [λ] i Θ [λc ] i i λ and λ c must be correlated such that Ψ is antisymmetric After [λ c ] = is determined by S 2 and S z, the spin-adapted state is specified by F [λ] i = E [λ] i F

29 6/21 Unitary group and change of one-particle basis The permutation and unitary groups (with elements U U... U) with U an M M matrix where UU = 1 The irreducible representation of U(M) along with the irreducible representation labels for S N uniquely label all states with a fixed energy

30 7/21 Young Tableaux and the symmetric group Partitions N objects into groups Labels the irreducible representations of symmetric group Symbolizes cycle structure of permutations

31 8/21 Correspondence to elements of symmetric group Corresponding permutations: (123)(45), (124)(35), (134)(25), (125)(34), (135)(24) Cycle notation: (ijk) = i goes to j, j goes to k and k goes to i (1324) abcd = dcab

32 9/21 Standard Tableaux Standard Tableaux increasing along row increasing down column Standard tableaux are 1-1 with basis functions of irrep.

33 10/21 Weyl tableaux versus Young tableaux Young [λ] labels irrep Frame has N boxes Fill with items from 1..N duplicates not allowed Weyl [λ] labels irrep Frame has N boxes Fill with items from 1..M duplicates allowed

34 11/21 Weyl tableaux W (K) p = (K, T p ) = i k m j l T p = K = (i, j, k, l, m) k i {1..M} k i k i+1 Number of Weyl tableaux: D(M, N, S) = 2S + 1 ( )( ) M + 1 M + 1 N M S + 1 N 2 S

35 12/21 Example: HeH + beyond minimal basis (N = 2, M = 3) Without symmetry: D = With symmetry: ( ) Mspin 2 = ( ) 6 = 15 determinants 2 d S=0 = D(M = 3, N = 2, S = 0)d = 6 d S=1 = D(M = 3, N = 2, S = 1)d = 3 D(M = 3, N = 2, S = 0) = 1 ( 4 )( 4 ) = 6 D(M = 3, N = 2, S = 1) = 3 ( 4 )( 4 ) = 3 d = ( 2 ) ( 2 2 ) = 2 1 = d = ( 2 ) ( 2 2 ) = 1 0 =

36 13/21 Example: LiH beyond minimal basis (N = 3, M = 4) Without symmetry: D = With symmetry: ( ) Mspin 2 = ( ) 8 = 56 determinants 3 d S=1/2 = D(M = 3, N = 2, S = 1/2) d = 3 d S=3/2 = D(M = 3, N = 2, S = 3/2) d = 6 D(M = 3, N = 2, S = 1/2) = 1 ( 4 )( 4 ) = 6 D(M = 3, N = 2, S = 3/2) = 3 ( 4 )( 4 ) = 3 d = ( 2 ) ( 2 2 ) = 2 1 = d = ( 2 ) ( 2 2 ) = 1 0 =

37 14/21 Subduction chain For Weyl tableaux we get a subduction chain by removing the orbital indicies one at a time e.g. for M = 5, K = (1, 1, 2, 3, 4), 1 2 and T = we get the chain: (1) Summarized using a Gel fand tableau.

38 15/21 Subduction chain Gel fand tableaux (2) Summarized using a Gel fand tableaux which describes the subduction to subgroups: λ 1 = 2 λ 2 = 2 λ 3 = 1 λ 4 = 0 λ 5 = 0 λ 1 = 2 λ 2 = 2 λ 3 = 1 λ 4 = 0 λ 1 = 2 λ 2 = 2 λ 3 = 0 λ 1 = 2 λ 2 = 1 λ 1 = 2

39 16/21 Abrams-Lloyd anti-symmetrization Abrams-Lloyd algorithm 1. Prepare N registers of log 2 N qubits in state B = 1 ( N ) i N! = 1 N! i=1 N N 1 b 1 =1 b 2 =1.. ( N 1 1 b N =1 i=1 i ) b 1 b 2...b N... ( ) ( 1 ) 2. Use reversible classical algorithm: (b 1 b 2...b N ) (b 1 b 2...b N ) 2.1 b 1 = b b i = the bth i number not in b 1 b 2...b i 1 3. Coherently sort (with phases) b in parallel with function register

40 17/21 Abrams-Lloyd example Reversible permutation generator algorithm 1. b 1 = b 1 2. b m= the b th m number not in b 1 b 2...b m 1 N = 3 b 1 b 2 b 3 b 1 b 2 b

41 18/21 Abrams-Lloyd anti-symmetrization φ 1 φ 2... φ N B = 1 N! 1 N! 1 N! N N 1 b 1 =1 b 2 =1.. 1 b N =1 π S N φ 1...φ N π(1...n) φ 1...φ N b 1 b 2...b N π S N sgn(π 1 ) π 1 (φ 1...φ N ) 1...N = A(φ 1...φ N ) 1...N

42 19/21 Abrams-Lloyd anti-symmetrization φ 1 φ 2... φ N B = 1 N! 1 N! 1 N! N N 1 b 1 =1 b 2 =1.. 1 b N =1 π S N φ 1...φ N π(1...n) φ 1...φ N b 1 b 2...b N π S N sgn(π 1 ) π 1 (φ 1...φ N ) 1...N = A(φ 1...φ N ) 1...N Note: requires a third register for failure cases

43 20/21 Symmetry projections E [3 D2] = A(14)A(25)S(123)S(45) φ i φ j φ k φ l φ m A(il) A(jm) S(ijk) A(lm) E [32] φ i φ j φ k φ l φ m φ n

44 21/21 Wigner projection operators Weighted sum of unitaries P Γ κλ = i c i U i gf k = j F jd jk (g) For G = S N, d [λ] = ( N ) ( N N ) 2 S N ; h = N! 2 S 1