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 x 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
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) a (x) or Sˆ (x)=a†(x)a (x) x ! i j i j a (x),a†(x) = , a†(x)a†(x) =0 , a (x),a (x) =0 { i j } ij xy { i j } { i j }
Z =Trexp( Hˆ ). 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 ↵ ↵ ↵ ↵ E ↵ 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( Hˆ ).
(d+1)-dimensional D-theory Original Motivation: MAKE
Easy Bosonic into Hard Fermionic Theories may have smart cluster (aka worm) algorithms? ⇠ =1/am exp[c /g2] ⇠
Asymptotic Freedom of 2d spin and 4d gauge implies exponential dimensional reduction as !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† = b ,b† = { 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
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 + 0 )= 1 a(↵ 1 + 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† = ,a† = ,a† = 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 = ( + ) + ↵ [ + ] 2 s s+1 s ⌦ s+1 s ⌦ s+1 links,sX links,sX 1 + + + [ + ] 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.
+ + 3 2 + h.c
+ 1
Total: about 15-20 consecutive gate operations (coherence time) per qubit per Trotter step
Estimate of current machines: 3 Trotter steps on Lattice Extra dimension builds local field rep. from XYZ ferromagnetic chain
Using + + 1 x x y y + = ( + ) ⌦ ⌦ 2 ⌦ ⌦ ( z 1+1 z)2 = 2(1 + z z) ⌦ ⌦ ⌦ The each links between two triangle are coupled by 2 Qubit ferromagnetic interaction operator to align them.
x x y y z z Halign ↵align ( + ) align
XY couoling of gauge fixed extra + dimension squares! E^2 coupling term s Plaquette
Still gauge invariant if broken vertically: few operations per qubit. (Not all to all) Parameter fitting Two Triangle couple Hamiltonian needing 6 Qubits & eigenvalues of 64x64 matrices
10
5
-1.0 -0.8 -0.6 -0.4 -0.2 Just XXZ piece: need to avoid level crossings, -5 close to XXX is better
8
6 Together with plaquette operator: 4 Gap persists: suggests simulation will 2 not be too polluted by UV
0.2 0.4 0.6 0.8 1.0
-2
-4 21 a. General k-link Wilson loop: In general any k-link loop Hermitian operator when expo- nentiated results in a simple single qubit unitary rotation between fully occupied (right) or fully empty (left) flux states.
+ + + i↵[ + ] U = e 1 2 ··· k 1 2 ··· k (A11)
This is a rotation by ↵ in a+ two dimensional+ + subspace in the 00 0 - 11 1 .
X X • X X • • • R • ✓ • The first step is to turn the state to a ++ state, while keeping the +++ Depending on details| i of architecture: |it can i take anywhere between | i invariant. This digital flipping5 computing is done with cycles CNOT and and20 (depth). NOT gates (the NOT gate is iden- tified withFor the experts:X Pauli May matrix). be done That effi wayciently we with have ancillas concentrated if Toffoli the gates problem available. of classifying the two states of interest to the last qubit. Now, we only do rotations in the plane of interest if the first two qubits are set to +. Finally, we undo the bit flipping operators to return it to the correct form in a computational basis. Its is clear from the diagram that this can be done for an arbitrary plaquette and the number of bit flip operations scales like the number of quibts in the plaquette. The only di↵erence is the number of control qubits that are required before we perform the wanted rotation. An algorithm to perform the C2(U)canbe found in [? ], figure 4.8, pp 182, and an ancilla assisted computation that performs Cn(U) in figure 4.10, pp 184. Both the number of ancillas add the number of operations scales like the number of qubits in the initial state that we want to operate on.
x y x y Alternatively as above we may expand ± = i ,intothe , basis as a sum of ⌥ k 1 y 2 commuting terms. This follows from Hemiticity which implies an even number of factors and as a consequence that any two terms have an even number of missmatched x- y x y pairs. To prove this consider two string A vs B. Suppose that A has Nx/Ny factors of / TROTTER VS EXACT 2 COUPLED TRIANGLES
n Real (0) U(t) (0) vs Real (0) [Uxy(t/n)UE(t/n)U (t/n)] (0) h | | i h | 4 | i
2) Questions? BACK UP SLIDES Quantum Link form: THE QCD ABACUS
ˆ ˆ ˆ ˆ ˆ ˆ ˆ H = Tr[Ux,µUx+ˆµ,⌫ Ux†+ˆ⌫,µUx,† ⌫ ]+ [detUx,µ +detUx,µ† ] x,µ=⌫ x,µ X6 X circa 2400 b.c Abacus
circa 20xx a.d. UNIVERSALITY: Tools & Tricks
1. Lattice: Discretization: (Graphs & Simplicial Structure)
• Subgroups of Space-Time Isometries
2. “Field Truncation” (Group Algebra & IR expansion)
• Discrete Manifolds for Global/Gauge/Spinors
3. “Dualities” (Hodge star & Dim Reduction)
•Gauge/Spin & Ads/CFT & Gauge/Gravity etc EQUILATERAL TRIANGULATION
Triangle case p = 3 p =3 Preserves Discrete Subgroup of Isometries q
1 1 2 + > 1/2 de Sitter S vertex q =3, 4, 5 p q
1 1 2 + =1/2 flat T vertex q =6 p q
1 1 2 + < 1/2 Hyperbolic AdS vertex q =7, 8, 9, p q ··· The CIO’s Guide to Quantum Computing https://www.gartner.com/smarterwithgartner/the-cios-guide-to-quantum-computing/ COMPANY/ COMPUTER TYPE NUMBER OF PREFERRED UNIVERSITY QUBITS METRIC
Google Super- 72 Quantum supremacy conducting Super- IBM 20 Quantum volume conducting
Super- Rigetti Computing 16 Quantum advantage conducting
University of Trapped ions 5 Benchmark Maryland comparison
Mircosoft Topological Qubits ? ? From Bits to Qubits ?
QCD Abacus* Fermionic Qubit Algorithm ?
?
* 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. DISCRETE ISOMETRIES & THE TRIANGLE GROUP
> ⇡ Postive curvature ⇡ ⇡ ⇡ + + = ⇡ Zero curvature p q r 8 < < ⇡ Negative Curvature :
(2, 3, 6) (2, 3, 7) (2, 3, 5) Subgroup of Modular 120 element Triangle Lattice 2 2 Icosahedral in O(3) on Euclidean R Group on H https://en.wikipedia.org/wiki/(2,3,7)_triangle_group Hyperbolic (e.g. Poincare Disk) and Global AdS q = 7
Triangle Group Tessellation: Preserve Finite subgroup of the Modular Group 1/2 + 1/3 + 1/q < 1 az + b z ad bc =1 ! cz + d a, b, c, d Z mod q 2
Are these Tessellation “Tensor Networks” ? YES: See Daniel Harlow’s Slide from Wednesday
Can we do QC lattice Field Theories in AdS? Classical YES /QC Maybe
Triangle Group Tiling (r, p,q) - (2,3,q) • Reflection: a, b, c Equilateral Case • Rotations: A = bc, B = ca, C= AB =ba • Two generators
a b
c q way vertex A
r = 2
(p, q, r)= a, b, c a2 = b2 = c2 = Ap = Bq = Cr { | } 3 Spheres and 4D Radial Simplicial Lattices 3 3 S = R S ) ⇥
Fast Code Domains of Aristotle’ s 2% Error! Regular 3D Grids on Refinement (2⇡ 5ArcCos[1/3])/(2⇡)=0.0204336
The full symmetry group of the 600-cell is the Weyl group of H4. This is a group of order 14400. It consists of 7200 rotations and 7200 rotation-reflections. The rotations form an invariant subgroup of the full symmetry group.