Quantum Simulation & Quantum Primes

Transcription

1 Quantum Simulation & Quantum Primes José Ignacio Latorre HRI, Allahabad, December 03

2 Outline Quantum Simulations Quantum Counting of Prime Numbers

3 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?

4 Analogue Simulation

5 H H ' (λ i ) controlled parameters Analogies have to be analyzed very critically

6 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: PEPS, MERA Monte Carlo Not sufficient

7 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?

8 Quantum Simulation of gravitational backgrounds Carriers in graphene are described by the Dirac equation Graphene acts a quantum simulator for Dirac fermions (γ 0 0+ γ + γ )ψ=0 Can we simulate the Dirac equation on optical lattices? Can we simulate curved spaces? O. Boada, A. Celi, M. Lewenstein, JIL (0)

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

10 Dirac equation in curved space-time μ γ Dμ ψ=0 ab Dμ= μ + wμ γ ab μ μ a γ =e γ γ ab= [ γ a, γ b ] 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 t H ψ )

11 Rindler space-time ds = (C x ) dt +dx + dy 0 e = C x dt e =dx 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

12 For any metric of the form ds = e Φ( x, y) dt +dx + dy 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. a J mn =e Φ(am, an) geometry = energy cost for jumping to a nearest neighbor Site dependent couplings!

13 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. a Experimental options superlattice techniques laser waist

14 Quantum Simulation of Extra Dimensions dimensions = connectivity D dimensions can be simulated in D- dimensions by tuning appropriately the nearest neighbor couplings O. Boada, A. Celi, Lewenstein, JIL (0)

15 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 (σ ) (σ +) (σ ) a r )+ h.c.

16 Bivolum State dependent lattice / On site dressed lattice Two 3D sub-lattices are connected via Raman transitions e J bilayer = Ω d x w ( x ) w( x r) g Exponential decay of Wannier functions suppresses undesired transitions Boada, Celi, Lewenstein, JIL PRL (0)

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18 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 interpolates between dimensions

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

20 -D optical lattice with species can be turned into a circle 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 Ladders + Species = Moëbius band

21 Torus vs Klein bottle: Hubbard model

22 Quantum Counting of Prime Numbers G. Sierra, JIL (Quant Comp. Comm. 03)

23 What will be a Quantum Computer used for? Build a Quantum Computer Break Classical Cryptography Use Quantum Cryptography Idle Quantum Computer? Could we use a Quantum Computer for pure Maths?

24 The Prime State P(n) = p n p< Primes π( ) n π (n ) is the prime counting function

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

26 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 x Li (x)= π (04 )= π (0 ) = ln(0 ) 4 dt x x log t ln x ln x Platt (0) Li (04 ) π (04 )=.7 09

27 If the Riemann Conjecture is correct, fluctuations are bounded ζ (s)= n ns If ζ ( s)=0 with 0 Real (s) then Real (s )= 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

28 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?

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

30 Using Euler totient function Twin primes mod 4 Chebyshev bias (i ) S (ρ ) log = n π() ( ) σ = π(n) () x n Δ ( ) σ() = z π(n ) Δ ( x )=π4,3 ( x) π4, (x ) Sub-series of primes, twin primes, etc. are amenable to measurements

31 Entanglement entropy of the Prime state ρn There is entanglement in the Primes Volume law scaling S.8858 n+const A. Monras, G. Sierra, JIL

32 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

33 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 π (n ) Prob( P (n) )=Prob(ancilla=0)= n n ln Efficient construction!

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

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

36 Construction of a Quantum Primality oracle An efficient Quantum Oracle can be constructed following classical primality tests Miller-Rabin primality test x x =s d Find Choose witness If a d mod x a 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

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

38 x3> x> x> x0=> Modular Exponenciation qubits Test s= d>= x3xx> Test s= d>= x3x> Test s=3 d>= x3> Test carriers Tests are conditioned to the actual value of x

39 x3> x> x> x0> Is x<? Acts only if x> > Modular Exponenciation qubits Test carrier Tests a=

40 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

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

42 π x Li ( x ) πqm (x ) < c ln x Error of counting is smaller than the bound for the fluctuations if Riemann conjecture is correct! π x < x ln x c ln x Best classical algorithm by Lagarias-Miller-Odlyzko (987) implemented by Platt (0) T x S x 4 A Quantum Computer could calculate the size of fluctuations more efficiently than a classical computer

43 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)

44 Conclusion I) Quantum Simulation of basic concepts 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

45 Other results Tensor Networks based on W-simplices span the ground state manifold frustrated systems Area law coefficient is sensitive to frustration S =α L+ ct Absolute Maximal Multipartite entangled states relates to Reed-Solomon codes, quantum sharing, multipartite teleportation,...