Quantum Simulation of Geometry (and Arithmetics)

Transcription

1 Quantum Simulation of Geometry (and Arithmetics) José Ignacio Latorre Isfahan, September 204

2 Outline Motivation Quantum Simulation of Background Geometry Quantum Simulation of an Extra Dimension Quantum Simulation of Topology Primes

3 Quantum Computation What? Why? When? Where? Who?

4 Topological order NV Centers Artificial systems Optomechanics Sensors Control of positions Quantum Engineering Qubits superconductores Tensor Networks Ion traps Quantum Simulation Computation Control of time Cold Gases Precision Teleportation Non-locality Randomness Bell Quantum Information Philosophy Quantum Mechanics Elementary Particles, Nuclear Physics, Atomic&Molecular Physics, Condensed Matter, Quantum Field Theory, Astrophysics, Quantum Optics, Solid State, Description of Nature Fist Quantum Steps of Human Kind WHAT?

5 Validation of theory Exact Calculations Integrable models (from Hamiltonian), CFT (from symmetry), ADS/CFT (from conjecture) Approximate methods Perturbation theory, toy models, non-perturbative techniques,... Numerics Monte Carlo, Tensor Networks (MPS, PEPS, MERA,...) other powerful instruments?

6 Analogue Simulation

7 Theory of interest System under control H ({α }) H ' ({λ }) controlled parameters Analogies have to be analyzed very critically

8 Why? Quantum Simulation is an intelligent window to QC Quantum Computer Quantum Simulator Classical Computer General purpose quantum computation Shor s factorization algorithm Oracle problems, NAND trees, Few problems, few algorithms! Efficient analysis of specific quantum problems Explore new Physics Experimentally achievable Tensor Networks strategies: PEPS, MERA Monte Carlo Not sufficient

9 Quantum Simulation Why What? Q Simulation of models beyond classical simulation Q Simulation of criticality, frustration, topological order,... Q Simulation of non-abelian gauge theories Q Simulation of unphysical models, Klein paradox, Zitterbewegung,... Q Simulation of gravity, geometry, topology Ion traps Cold gases Molecules, solids, graphene, Where?

10 Quantum Simulation of Gravitational Backgrounds

11 Dirac equation on a square lattice 0 2 i(γ 0 + γ + γ 2) ψ=0 Can we simulate the Dirac equation on optical lattices? Can we simulate curved spaces? O. Boada, A. Celi, M. Lewenstein, JIL

12 Dirac Hamiltonian in 2+ dimensions i t ψ=h ψ= i γ 0 ( γ + γ 2 2) ψ i γ 0 γ = i γ2 =σ x i γ 0 γ 2 =i γ =σ y i γ γ 2=i γ0 =i σ z H ψ= ( σ x x +σ y y ) ψ=0 Η= dx dy ψ H ψ Discretized Dirac Hamiltonian Η= ( ψ ( σ ) ψ + ψ m +,n x m, n m,n + (σ y ) ψm,n )+h.c. m,n 2a SU(2) Fermi Hubbard model

13 ab Dμ= μ + wμ γ ab 2 Dirac equation in curved space-time μ μ a γ =e γ μ γ Dμ ψ=0 γ ab= [ γ a, γ b ] 2 If there exists a timelike Killing vector (time translation invariance in certain coordinates) there exists H conserved and well defined t gμ ν =0 Sufficient condition i ab t ab H = i γ t γ i + γ w i γ ab + γ w t γ ab 4 4 ( a i Η= g dx dy ψ γ 0 γ H ψ t )

14 Rindler space-time ds = (C x ) dt +dx + dy 0 e = C x dt e =dx 2 2 e =dy Steady Rindler observer is an accelerated Minkowski observer acceleration Cx Unruh effect temperature Rindler is the near horizon limit of Schwarzschild black hole Cx

15 For any metric of the form 2 ds = e Φ( x, y) 2 2 dt +dx + dy 2 The lattice version turns out to be Η= m,n J mn ( ψm+,n σ x ψm, n +ψm, n+ σ y ψm,n ) +h.c. 2a J mn =e Φ(am, an) geometry = energy cost for jumping to a nearest neighbor Site dependent couplings!

16 Discretized Dirac equation in a Rindler space Η= m,n c m ( ψm +,n σ x ψm, n+ ψm,n + σ y ψm, n ) + h.c. 2a Experimental options superlattice techniques laser waist

17 Quantum Simulation of an extra dimension

18 Connectivity dimensions = connectivity D+ dimensions can be simulated in D dimensions by tuning appropriately the nearest neighbor couplings O. Boada, A. Celi, Lewenstein, JIL

19 Dimension D + H= J q j= a q + u a q +h.c. j q =( r, σ) D + Species D H= J r, σ j= (a r +u a r +a r (σ ) j jump (σ ) (σ +) (σ ) a r )+ h.c. change species

20 Bivolum State dependent lattice / On site dressed lattice Two 3D sub-lattices are connected via Raman transitions e J bilayer = Ω d 2x w ( x ) w( x r) 2 g Exponential decay of Wannier functions suppresses undesired transitions Boada, Celi, Lewenstein, JIL PRL

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22 Single particle observable: Kaluza-Klein modes Single particle correlators take contributions from jumps back and forth to other dimensions in the form of exponential (KK) massive corrections Many-body observables: Shift of phase transition point interpolates between dimensions

23 Quantum Simulation of topology Quantum Simulation of boundary conditions Non-local interactions can be artificially generated O. Boada, A. Celi, M. Lewenstein, J. Rodríguez-Laguna, JIL

24 Boundary conditions H = J i=,..., n σ i σ i + + J ' σ σ n x x x x Three ways of create the boundary term: Bend the physical system on itself Create a non-local interaction Add an extra dimension Adding extra dimensions (as a species) retains locality of interactions

25 local interaction local interaction nearest neighbor interaction species 2 species same site

26 -D optical lattice with 2 species can be turned into species on a circle cos(θ) change of species (ex: local action of Raman lasers at the boundaries) Ex: Frustration from boundary condition on a chain n n H= σ i σ n + cos(θ)σn σ + λ σ i x x x x z Entaglement entropy jumps

27 Ladders + Species = Moëbius band Or 4 species

28 Torus vs Klein bottle: Hubbard model On one atom?

29 Proposals for Quantum Simulation of basic Geometrical concepts Geometry Dimensionality Topology Site-dependent coupling Connectivity via change of species Site depending species coupling Quantum Simulation is the natural avenue for QM now

30 What will a full Quantum Computer be used for? Build a Quantum Computer Break Classical Cryptography Use Quantum Cryptography Idle Quantum Computer??

31 Quantum Counting for Arithmetics?

32 The Prime State P(n) = π (2n ) π( 2 ) n p< 2 Primes p n is the Prime Counting Function G. Sierra, JIL

33 Quantum Mechanics allows for the superposition of primes implemented as states of a computational basis P( n) = + π( 2 ) n p< 2 Primes p n + P(3) = ( ) 4 +

34 Gauss, Legendre Sieve of Eratosthenes x π (x ) ln x Prime Number Theorem π (x ) Li (x ) Gauss, Riemann Hadamard, de la Vallée Poussin Density of primes /log x 24 x Li (x)= 2 π (0 )= π (0 ) = ln (0 ) 24 dt x x log t ln x ln x Platt (202) Li (024 ) π (024 )=.7 09

35 If the Riemann Conjecture is correct, fluctuations are bounded ζ (s)= n ns If ζ ( s)=0 with 0 Real (s) then Real (s )= 2 Li ( x ) π( x ) < x ln x 8π The prime number function will oscillate around the Log Integral infinitely many times Littlewood, Skewes A first change of sign is expected for some x <e

36 Could the Prime state be constructed? Does it encode properties of prime numbers? What are its entanglement properties? Could it provide the means to explore Arithmetics?

37 Entanglement: single qubit reduced density matrices P(n) = π( 2 ) n i n,..., i,i 0=0, pi n...i i 0 i n,...,i,i 0 { p=i n 2 n +...+i 0= prime otherwise pi...i i = 0 n () ρab = n π(2 ) i n,...,i 2, i 0= 0, pi n 0,...,i 2, a,i 0 pi n,...,i 2, b,i 0 n () 00 ρ = π 4, (2 ) n π (2 ) n () ρ = + π 4,3 (2 ) n π(2 ) () ρ0 = n π() (2 ) 2 n π(2 ) Each element counts sub-series of primes and twin primes πa,b(x ) () π 2 ( x) counts the number of primes equal to a mod b counts twin primes equal to mod 4

38 Entanglement entropy of the Prime state ρn 2 There is entanglement in the Primes Volume law scaling S.8858 n+const

39 Scaling of entanglement entropy S n const Random states S.8858 n+const Prime state S n d d + const Area law in d-dimensions c S log n+ const 3 Critical scaling in d= at quantum phase transitions S log(ξ)=const Finitely correlated states away from criticality

40 Construction of the Prime state H H H H H H P( n) Primality 0 U primality x x 0 = P(n) 0 + c composite c n π (2 ) Prob ( P(n) )=Prob ( ancilla=0)= n n ln 2 2 Efficient construction!

41 Grover construction of the Prime state M ψ0 = x<2 n x = ( p primes p + c composites c ) n π (2 ) N # calls to Grover R( n) π 4 ψf = P( n) N π n ln 2 M 4 We need to construct an oracle!

42 Construction of a Quantum Primality oracle An efficient Quantum Oracle can be constructed following classical primality tests Miller-Rabin primality test s Find x x =2 d Choose witness If a d mod x a 2 d mod x r a x then x is composite 0 r s If any test fails, x may be prime or composite: liars Eliminate strong liars checking less than ln(x) witnesses

43 Utests Utests Test a Test a2 Test am carrier carrier m global Structure of the quantum primality oracle

44 Quantum Counting of Prime numbers quantum oracle + quantum Fourier transform = quantum counting algorithm Brassard, Hoyer, Tapp (998) Counts the number of solutions to the oracle

45 We want to count M solutions out of N possible states We know an estimate M π π M < M M + 2 c c Bounded error in quantum counting using c N calls Brassard, Hoyer, Tapp (998) Bounded error in the quantum counting of primes 2 2π x Li ( x ) πqm ( x ) < c ln 2 x

46 Li ( x ) πqm (x ) < 2 2π x c ln 2 x Error of counting is smaller than the bound for the fluctuations if Riemann conjecture is correct! Best classical algorithm by Lagarias-Miller-Odlyzko (987) implemented by Platt (202) T x 2 S x 4 A Quantum Computer could calculate the size of fluctuations more efficiently than a classical computer

47 Conclusion I) Quantum Simulation of Geometry and Topology Geometry Site-dependent coupling Dimensions Connectivity Boundary conditions Non-local coupling II) Quantum Simulation of Arithmetics Superposition of series of numbers using appropriate q-oracles Measurements of arithmetic functions Verification of Riemann Hypothesis

48 Construction of twin primes U +2 P(n) 0 = p primes p +2 0 U primality U +2 P(n) 0 = p, p+2 primes p p +2 primes p +2 n Prob ( twin primes )= π 2 (2 ) π(2 n) (n ln 2)2 Efficient construction!

49 x3> x2> x> x0=> Modular Exponenciation qubits Test s= d>= x3x2x> Test s=2 d>= x3x2> Test s=3 d>= x3> Test carriers Tests are conditioned to the actual value of x

50 x3> x2> x> x0> Is x<22? Acts only if x>22 > Modular Exponenciation qubits Test carrier Tests a=2

51 Beyond Prime numbers: the q-functor f : S X S = S = χ (x) x x X S S x x S S χ S (x)= { 0 x S x S Primes Average (Cramér) primes Dirichlet characters... Only needs the construction of a quantum oracle for χ S(x)