martes cuantíco Quantum Algorithms for Quantum Field Theories David Zueco ICMA-Unizar)

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martes cuantíco David Zueco (ARAID @ ICMA-Unizar)

es/~zueco/ Quantum Algorithms for Quantum Field

1 martes cuantíco David Zueco ICMA-Unizar) Quantum Algorithms for Quantum Field Theories Stephen P. Jordan, 1 * Keith S. M. Lee, 2 John Preskill 3 [Published in Science (June 2012)]

2 Disclaimer: I am not an author (neither a referee) of the paper I am not an expert. It was hard for me My presentation will be biased.

[Int J Theor Phys (1982)] Simulating Physics with Computers Richard P. Feynman Department of Physics, California Institute of Technology, Pasadena, California 91107 Received May 7, 1981 1.

I do not intend in any way to suggest what should be in this meeting as a keynote of the subjects or anything like that.

3 [Int J Theor Phys (1982)] Simulating Physics with Computers Richard P. Feynman Department of Physics, California Institute of Technology, Pasadena, California Received May 7, INTRODUCTION On the program it says this is a keynote speech--and I don't know what a keynote speech is. I do not intend in any way to suggest what should be in this meeting as a keynote of the subjects or anything like that. I have my own things to say and to talk about and there's no implication that anybody needs to talk about the same thing or anything like it. So what I Now, what kind of physics are we going to imitate? First, I am going to describe the possibility of simulating physics in the classical approximation, a thing which is usuauy described by local differential equations. But the physical world is quantum mechanical, and therefore the proper problem is the simulation of quantum physics--which is what I really want to talk about, but I'U come to that later. So what kind of simulation do I mean? There is, of course, a kind of approximate simulation in which you design numerical algorithms for differential equations, and then use the computer to compute these algorithms and get an approximate view of what ph2csics ought to do. That's an interesting subject, but is not what I want to talk about. I want to talk about the possibility that there is to be an exact simulation, that the computer will do exactly the same as nature. If this is to be proved and the type of computer is as I've already explained, then it's

4 analog vs digital (q)

Institute, Garching Michel Devoret, Yale University Nicolas Gisin, University of Geneva David Gross University of Santa Barbara Serge

5 a little of motivation... Invited speakers: Rainer Blatt, University of Innsbruck, Immanuel Bloch, Max Planck Institute, Garching Ignacio Cirac, Max Planck Institute, Garching Michel Devoret, Yale University Nicolas Gisin, University of Geneva David Gross University of Santa Barbara Serge Haroche, College de France Atac Imamoglu, ETH, Zurich Jeff Kimble, Caltech Michael Lukin, Harvard University David Mermin, Cornell University Bill Phillips, NIST, US John Preskill, Caltech Peter Zoller, University of Innsbruck Steven Chu, Stanford University

6 (SC) Transmission Lines lumped c [ Zaragoza] l = Continuum limit ( L= = ( ) ( line velocity ( + + ) Covariant form ) L= µ d ( =(, ) µ ) )

7 4 -model, say in 1+1 L = dx( µ ) 2 µ ± illustrated in Figure 2. V Figure 2. The classical e ective potential in 4 -theory illustrated for µ (red) and µ 2 0 < 0(blue)showingthetwo possible ground states for the latter case. [From Milsted arxiv (2013)]

8 lattice model H = X l H l H = a l 1 2 (x l) ( (x l) (x l+1 ) 2 + µ 0 (x l ) 2 + (x l ) 4

9 discretized in space (lattice) (x j ) a j + a j and discretized at each site max/ =2 n y n +2 n 1 y n y 0 ' n log 2 ( max / ) # qubits to encode the field

10 max (x) 2 (x) 2 E/a In the Sup Mat the authors estimate in # qubits

11 e iht? Efficient algorithm (digital): Running in polinommial time, i.e. polinommial number of gates E.g.: = 1-qubit 2-qubit = =, =,

12 Trotter decomposition (efficient): H = X l H l U( t) = Y l e ih l t iterative application of local gates In their case: H l = (x l ) ( (x l) (x l+1 ) 2 + µ 0 (x l ) 2 + (x l ) 4

13 wave packet creation (from free theory) a wp X a d '(x j )a x easy use trotter make it unitary! (with ancilla) H = a wp 1ih0 +h.c e ih wp /2 vaci 0i = ia wp vaci 1i

14 Preparation (from free theory) 0(= 0) f Adiabatic (Trotter) [Galindo / Pascual QM] Evolution within the interacting theory (Trotter) Measurement phase estimation (free theory)

15 [Nielsen & Chuang (2000)] +( )+( )+... = # gates classical algorithm

16 Concluding remarks. The most computationally costly part of the algorithm is either adiabatic state preparation or preparation of the free vacuum, depending on the asymptotic scaling considered and the number of spatial dimensions (33). Because a ~ e 1/2,preparationofthefree vacuum has complexity ~ V ~ a 2.376d ~ a good motivation for your next paper in analog q-sims ;)

crudely to be on the order of 1000 to 10,000, corresponding to e ranging from 10% to 1% (33).

17 g solely from violating operand quantum caling of their veals that the e as a 2 in D = Idominate.In hscaleasa 2.) tors and their scale of the the EFT, alare no longer the number of e scattering in =1,2spatial momentum of l 0 (the disand n out (the umber of out- vial simulation in D = 2 is estimated necessarily crudely to be on the order of 1000 to 10,000, corresponding to e ranging from 10% to 1% (33). We have shown that quantum computers can efficiently calculate scattering probabilities in f 4 theory to arbitrary precision at both weak and strong coupling. Known classical algorithms take exponential time to do this in the strong-coupling and high-precision regimes. In addition to establishing a new exponential quantum speedup, our results lead the way toward a quantum algorithm for simulating the Standard Model of particle physics, which has new features (not exhibited by f 4 theory) such as chiral fermions and gauge interactions. Such an algorithm would establish that, except for quantum-gravity effects, the standard quantum circuit model suffices to capture completely the computational power of our universe. Table 2. Effective field theory operators fall

! In 1+1 classical simulations do the job. / µ 2 R,c 66.2 66.0 65.8 65.6 65.4 65.2 65.0 64.8 64.6 64.4 0.17 0.16 0.15 0.14 0.13 0.12 0.

Approximate values for the lattice critical parameter / µ 2 R,c( ) (top)andtheh i critical exponent ( ) (bottom) obtained from linear

18 ! In 1+1 classical simulations do the job. / µ 2 R,c Ẽ fit h i Ẽ h i Figure 16. Approximate values for the lattice critical parameter / µ 2 R,c( ) (top)andtheh i critical exponent ( ) (bottom) obtained from linear fits to the lowest-lying excitation energies Ẽ and from power-law fits to the order-parameter h i for values of approaching the continuum limit! 0. The line in (a) corresponds to the fourth fit of Table I.

Quantum computing hardware

Quantum computing hardware aka Experimental Aspects of Quantum Computation PHYS 576 Class format 1 st hour: introduction by BB 2 nd and 3 rd hour: two student presentations, about 40 minutes each followed