FORMULATING LATTICE FIELD THEORY FOR A QUANTUM COMPUTER* *The accidental discovery of a Quantum Algorithm for Lattice Gauge theories: Circa 1998: Rich Brower — Boston University EDS Blois June 28, 2019 ICISE Quy Nhon, Vietnam SOME REFERENCES ON EUCLIDEAN QUANTUM LINKS (AKA D-THEORY) • D. Horn, Finite Matrix Model with Continuous Local Gauge Inv. Phys. Lett. B100 (1981) • P. Orland, D. Rohrlich, Lattice Gauge Magnets: Local Isospin From Spin Nucl. Phys. B338 (1990) 647 • S. Chandrasekharan, U-J Wiese Quantum links models: A discrete approach to gauge theories Nucl. Phys. B492 (1997) • R. C. Brower, S. Chandrasekharan, U-J Wise , QCD as quantum link model, Phys. Rev D 60 (1999). • R. C. Brower, The QCD Abacus: APCTP-ICPT Conference, Seoul, Korea, May (1997) • R. C. Brower, S. Chandrasekharan, U-J Wiese, D-theory: Field quantization … discrete variable Nucl. Phys. B (2004) • David Berenstein. R. C. Brower Real Time Gauge Theory: Programming on U(1) theory on Quantum Computer. (To be pub 2019) From Bits to Qubits ? QCD Fermionic Qubit ? * See THE QCD ABACUS: A New Formulation for Lattice Gauge Theories R. C. Brower https://arxiv.org/abs/hep-lat/9711027 Lecture at ”APCTP-ICTP Joint International Conference ’97 on Recent Developments in Non-perturbative Method” May, 1997, Seoul, Korea. MIT Preprint CTP 2693. OUTLINE 1. The quantum link (or QCD abacus) Hamiltonian was introduced as a classical algorithm representing both gauge and matter fields by single bit fermion operators in an extra dimension. 2. This formalism is recast for quantum computing, as a Hamiltonian in Minkowski space for real time Qubit simulations. 3. The advantages of pseudo-fermions to implement the Jordan Wigner transformation and the Trotter expansion in local gauge invariant local kernels is discussed. 4. For U(1) compact QED the kernels on a triangular lattice are defined and a Qubit circuit implementation given to test on existing hardware. WILSON’S LATTICE QCD* 6 † g2 Tr[U⇤ + U ] Zwilson = dU e ⇤ ⇤ ZHaar P ij U ij(x, x + µ)=eiagAµ (x) i, j =1, 2, 3 SU(3) Gauge Transport on each link. Exact per site gauge invariance U⇤µ⌫ (x)=[U(x, x + µ)U(x + µ, x + µ + ⌫)][U(x, x + ⌫)U(x + ⌫,x+ ⌫ + µ)]† 1+a2iF (a4/2)F 2 + ' µ⌫ − µ⌫ ··· 45 Years Ago * K.G. Wilson, Phys. Rev. D 10 (1974), 2445. THE CLASSICAL COMPETITION Ken Wilson Lattice 1989 Capri “One lesson is that lattice gauge theory could also require a 10^8 increase in computer power AND specular algorithmic advance before useful interaction with experiment” • ab initio Chemistry • ab initio QCD 1. 1930+50 = 1980 vs 1. 1980 + 50 = 2030?* 2. 0.1 flops ➔ 10 Mflops 2. 10 Mflops ➔ 1000 Tflops 3. Gaussian Basis functions 3. Clever Collective Variable? Lattice QCD has arrived 10 years early ! TODAY Oak Ridge National Laboratory's 200 petaflop supercomputer “Lattice Gauge Theory Machine” 200,000,000,000,000,000 Floats/sec 9,216 IBM POWER9 CPUs and 27648 NVIDIA GPUs Each GPU has 5120 Cores and total of 580,608,000,000,000 transistors BUT WILSON’S LGT CAN NOT DO IT ALL: (SOME OUT STANDING PROBLEMS!) • Problem 1: LGT on curved Riemann Manifolds*: e.g. CFT on boundary of Anti-de-Sitter space. Quantum Finite Elements should solve this? • Problem 2: Chiral Gauge Theories#: Many be topological methods in extra dimension? • Problem 3 : Minkowski Space is exponential hard @: e.g.real time jets, scattering, sign problem of baryon chemical pot at finite T etc. *https://arxiv.org/abs/1610.08587 Dirac Fermions on Simplicial Manifold https://arxiv.org/abs/1803.08512 Phi 4th on Riemannian Manifold # https://arxiv.org/pdf/1809.11171 A non-Perturbative Definition of the Standard Model ? @ This workshop? Richard Feynman On quantum physics and computer simulation “There is plenty of room to make [computers] smaller. nothing that I can see in the physical laws . says the computer elements cannot be made enormously smaller than they are now. In fact, there may be certain advantages. —1959” 60 years ago! “trying to find a computer simulation of physics seems to me to be an excellent program to follow out. the real use of it would be with quantum mechanics. Nature isn’t classical . and if you want to make a simulation of Nature, you’d better make it quantum mechanical, and by golly it’s a wonderful problem, because it doesn’t look so easy. —1981” TODAY Superconducting QIS plattform Intel 49 qubit device Rigetti 19 qubit device IBM 50 qubit device Cloud quantum computing Google 72 qubit device Resources: 100,000,000,000,000 VND Nikolai Lauk, Aspen 2019 Universality == Many equivalent LFT • Different space-time + field discretizations define exactly the same continuum quantum field theory • eg. Lattice different lattice give identical c = 1/2 CFT — square, triangle or spherical lattice! • Fields: Continuum phi 4th field and single bit Ising fields are equivalent s 1 φx R 2 ± () 2 • Bosonic Sine Gordon Theory = Fermionic Thirring Theory • N = 4 SUSY is dual AdS Gravity WARM UP: O(3) HEISENBERG SPIN MODEL Classical O(3) MODEL Quantum antiferromagnet O(3) MODEL β 1 ~ ~ Z = dS exp[ S~ S~ ] g2 x,µ x x+µ g2 x · y Tr[e− · ] Z x,y ! hXi P Warm up with Fermionic Quantum Operator ij ij S~ a†(x)σ a (x) or Sˆ (x)=a†(x)a (x) x ! i j i j a (x),a†(x) = δ δ , a†(x)a†(x) =0 , a (x),a (x) =0 { i j } ij xy { i j } { i j } Z =Trexp( βHˆ ). Hˆ = Tr[Sˆ Sˆ ] ,Tr[Sˆ ]=0 − x y x x,y hXi Global Rotation: J~ = Tr[~ Sˆ ]= [J,~ Hˆ ]=0 B.B. Beard and U-J Wiese x x ) Local Fermion No: Fˆ = Tr[Sˆ ]= [Fˆ , Hˆ ]=0 https://arxiv.org/abs/cond-mat/9602164 Px x ) x KOGUT SUSKIND HAMILTONIAN 2 g 2 2 1 H = (Tr[E (x, µ)] + Tr[E (x, µ)]) Tr[U + U † ] 2 L R − 2g2 ⇤ ⇤ x,x+µ h X i X⇤ d Example: U(1) Abelian P/Q symplectic E = i ,U= exp[i✓] operators in Q-basis are d✓ U(N) generalization of Gauge Algebra is ij ij ↵ ↵ β ↵γ ↵ E λ↵ E = [E ,E ]=2if E ⌘ ) [E ,U]= E U,[E ,U]= UE L − L R − R ER = U †ELU,[U, U †]=0 UU† =1 D-THEORY: FERMIONS FOR GAUGE, SCALAR AND DIRAC LATTICE FIELDS! d-dimensional ordinary lattice field theory Z =Trexp( βHˆ ). − (d+1)-dimensional D-theory Original Motivation: MAKE Easy Bosonic into Hard Fermionic Theories may have smart cluster (aka worm) algorithms? ⇠ =1/am exp[cβ/g2] ⇠ Asymptotic Freedom of 2d spin and 4d gauge implies exponential dimensional reduction as β !1 see Square-Lattice Heisenberg Antiferromagnet at Very Large Correlation Lengths B. B. Beard, R. J. Birgeneau, M. Greven, and U.-J. Wiese Phys. Rev. Lett. 80, 1742 (1998) OPERATOR QUANTUM GAUGE LINK On each link (x,x + µ)introduce2Nccomplex fermion a,ai† right(+) moving and bj,bj† left(-) moving fluxon a LINK: i bj† U (x, x + µ) Uˆ = a (x) b†(x + µ) ij ! ij i j x x + µ a ,a† = δ b ,b† = δ { i j} ij { i j} ij Local Gaue Operators ⌦ (x)=a†(x)a (x)+ ij i j ··· aia† aib† For SU(3) QCD have SU(6) per Link Lie Algebra j j "biaj† bibj† # Hilbert Space is a large qubit array of color vectors in 4d space-time QC MINKOWSKI DYAMICS • Change 4-th axis to real Time for Hamiltonian. • Let short 5-th axis be field space (like “Domain Wall Fermion flavors) x +ˆn β LINK ˆ Uij [Eˆ , Uˆ]= Uˆ ± Eˆij(x, x +ˆn) ± Time nˆ =1, 2, 3 x 1 2 2 2 ˆ ˆ ˆ † ˆ ˆ H0 = Tr[U⇤ + U ]+g Tr[E+(x, y)+E (x, y] + quarks −4g2 ⇤ − x,y X⇤ hXi FERMIONIC QUBIT GATES Each Fermion On Each qubit a†(↵ 1 + β 0 )=β 1 a(↵ 1 + β 0 )=↵ 0 | i | i | i | i | i | i 01 2 1 11 a† + a = X = (a + a†) 1= H = ) 10 − ) p2 11 − 0 i 10 (a† + a)/i = Y = − (2a†a 1) = Z = ) i 0 − ) 0 1 − A A | i | i (1 a†a)+a†a(b + b†)= ContrNOT B A B − ) | i | ⇥ i (1 a†ab†b)+a†ab†b(c† + c)= To↵oli − ) Classical Reversible Computing Conserving Energy R. Landauer IBM J journal of Research and Development, vol. 5, pp. 183-191, 1961 WHAT ABOUT ANTI-SYMMETRIC FERMIONIC FOCK SPACE? Step #1: PARA-STATISTICS: D-Theory only require anti-commutator within each a’s and b’s set 0 [ai†aj,ap† bq]=[ai†aj,ap† ] bq + ap† [ai†aj,bq] = [E ,U]= E U same for a —> b ) L − L Step #2; Jordan-Wigner: Apply to Locally to each set of 3 a's and b's. + z + z z + a† = σ ,a† = σ σ ,a† = σ σ σ same for a —> b 1 1 2 − 1 2 3 1 2 3 PARA-STATISTICS & JORDAN-WIGNER TO THE RESCUE STARTING TO TEST REAL TIME QUBIT ALGORITHM FOR U(1) QUANTUM LINK GAUGE THEORY Joint Work with D. Berenstein (UCSB), Cameron Valier (BU) & H. Kawai (BU) 2 e z z 2 + + Hˆ = (σ + σ ) + ↵ [σ σ− + σ− σ ] 2 s s+1 s ⌦ s+1 s ⌦ s+1 links,sX links,sX 1 + + + [σ σ σ + σ− σ− σ−] − 2e2 s ⌦ s ⌦ s s ⌦ s ⌦ s ,s X4 Few very simple kernels in Trotter factorization into Gauge invariant Unitary operators with very few Qubit width Choose 2 + 1 on U(1) Hamiltonian on a Triangular spacial lattice To evaluate lattice: alternate between two coloring of triangles.