Simulated Quantum Annealing For General Ising Models

Transcription

1 Simulated Quantum Annealing For General Ising Models Thomas Neuhaus Jülich Supercomputing Centre, JSC Forschungszentrum Jülich Jülich, Germany t.neuhaus@fz-juelich.de November 23

2 On the Talk Quantum Annealing DWAVE Simulated Quantum Properties T. Neuhaus, JSC Simulated Quantum Annealing For General Ising Models 2/2

3 Quantum Annealing Board in a Box supposed to do Quantum Annealing: H Ising = M J ij σi z σj z + N h i σi z Γ N σ x i <ij> i i Ising model J ij + transverse field Γ number of clauses M non-planar graph h i + number of spins N T. Neuhaus, JSC Simulated Quantum Annealing For General Ising Models 3/2

4 Quantum Annealing switch to adiabatic control: λ M H Ising = λ J ij σi z σj z <ij> + λ N h i σi z ( λ) i N i σ x i 6 4 Landau Zener crossing gap correlation length E 2-2 ξ GAP = diabatic probability (E E ) min QPT LAMBDA P DIABATIC = e +π 2 h ξ GAP 2 τ λ P ADIABITC = P DIABATIC T. Neuhaus, JSC Simulated Quantum Annealing For General Ising Models 4/2

5 Chain Tooth Sampling Schedules short trajectories of length τ O() repetitions dynamics: on the chip : Schrödinger equation : simul. quantum annealing : Newtons equation of motion : simulated annealing named QA,SQA,CA,SA 2 Histogram: N[H Ising (λ = )] λ(τ).5 τ success probability: P to find a ground state run time at median success: τ τ = ln(.5) ln( P ) τ T. Neuhaus, JSC Simulated Quantum Annealing For General Ising Models 5/2

6 Landau Zener and the Quantum Monte Carlo problems in HARD 2SAT Metropolis updates Trotter regularization Multi Spin Coding run time in MCS ln(τ/ < τ >) vs. 2ln(ξ GAP / < ξ GAP >): diabatic probabilty P LZ vs. ξgap : N=5.8 N= P LZ /2 ln[τ/<τ>] ln[ξ/<ξ>] T. Neuhaus, JSC Simulated Quantum Annealing For General Ising Models 6/2

7 Residual Energy in Annealing SA : Huse, D. Fisher PRL 986: SQA : Santoro, Martonak, Tosatti, Car, Sience 22 2D Ising glass on J ij +2 flat distribution finding: ξ SA < ξ SQA < 6 ɛ(τ) ln(τ) ξ ξ = min[2, 2(d θ)/(d s /2+2Ψ θ)] droplet model T. Neuhaus, JSC Simulated Quantum Annealing For General Ising Models 7/2

8 DWAVE 22: USC-Lockheed Martin Quantum Computation Center N=28+N=52 23: Nasa, QUAIL, Quantum Artificial Intelligence Laboratory N=52 in the near future apply for computertime at: superconducting fluxes chip T. Neuhaus, JSC Simulated Quantum Annealing For General Ising Models 8/2

9 DWAVE chip mount chip mount T. Neuhaus, JSC Simulated Quantum Annealing For General Ising Models 9/2

10 DWAVE fridge box T. Neuhaus, JSC Simulated Quantum Annealing For General Ising Models /2

11 Connectivity Graph 2D arrangement of K 4,4 K 4,4 : completely connected bipartite graph of 4+4 spins N=28 spins(similar for N=52) the DWAVE graph is bipartite the DWAVE graph is non-planar regular lattice manifolds are somewhat orthogonal T. Neuhaus, JSC Simulated Quantum Annealing For General Ising Models /2

12 Ensemble Correlations P SQA vs. P SA EA spin glass non-planar: EA spin glass planar: P SQA P SQA P SA P SA d random magnet: random magnet on full graph: P SQA P SQA P SA P SA T. Neuhaus, JSC Simulated Quantum Annealing For General Ising Models 2/2

13 Ensemble Histograms H[P SQA ]:[EA-BIMODAL] EA spin glass non-planar: EA spin glass planar: 5 SQA 5 SQA H(P) H(P) P d random magnet: P random magnet on full graph: N=64 SQA 2 SQA H(P) H(P) P P T. Neuhaus, JSC Simulated Quantum Annealing For General Ising Models 3/2

14 SQA + SA Run-Times :[NO-SQA-SPEEDUP] EA spin glass non-planar: EA spin glass planar: lnτ SQA SA lnτ N / d random magnet: N lnτ SQA SA lnτ N / random magnet on full graph: N 3 28 SQA SA SQA SA lnτ lnτ N / lnτ N N T. Neuhaus, JSC Simulated Quantum Annealing For General Ising Models 4/2

15 SQA vs. QA at N=8 in EA:[chip is quantum] P SQA P QA P SA P SA.2 5 SQA QA H(P) 2 H(P) P P T. Neuhaus, JSC Simulated Quantum Annealing For General Ising Models 5/2

16 Troyer et. al. 23 arxiv: v2 [quant-ph] EA J = ± on full graph N=8 machine finds groundstates N=8 machine operates quantum no QA speedup over SA N=8 quantum annealing at.5 success is 5µsec N=8 8-core Xeon SA at.5 success is 4µsec Troyer, Talk, N=52: T. Neuhaus, JSC Simulated Quantum Annealing For General Ising Models 6/2

17 23 DWAVE Benchmark e+8 e+6 SQA SA DWAVE random EA on full Dwave Graph MCS=.5 nano-sec November 23 τ/[µsec] N.5 T. Neuhaus, JSC Simulated Quantum Annealing For General Ising Models 7/2

18 Conlusion some understanding on annealing boards no known non-trivial problem class where SQA effectivly wins over SA the simulated data are not asymptotic no one knows what happens at N = 248 T. Neuhaus, JSC Simulated Quantum Annealing For General Ising Models 8/2

19 appendix I arxiv: v2 [quant-ph]; Quantum annealing with more than one hundred qubits ; Sergio Boixo, Troels F. Rannow, Sergei V. Isakov, Zhihui Wang, David Wecker, Daniel A. Lidar, John M. Martinis, Matthias Troyer: Quantum technology is maturing to the point where quantum devices, such as quantum communication systems, quantum random number generators and quantum simulators, may be built with capabilities exceeding classical computers. A quantum annealer, in particular, solves hard optimisation problems by evolving a known initial configuration at non-zero temperature towards the ground state of a Hamiltonian encoding a given problem. Here, we present results from experiments on a 8 qubit D-Wave One device based on superconducting flux qubits. The strong correlations between the device and a simulated quantum annealer, in contrast with weak correlations between the device and classical annealing or classical spin dynamics, demonstrate that the device performs quantum annealing. We find additional evidence for quantum annealing in the form of small-gap avoided level crossings characterizing the hard problems. To assess the computational power of the device we compare it to optimised classical algorithms. T. Neuhaus, JSC Simulated Quantum Annealing For General Ising Models 9/2

20 appendix II arxiv:35.494v [quant-ph]; Classical signature of quantum annealing ; John A. Smolin, Graeme Smith: A pair of recent articles concluded that the D-Wave One machine actually operates in the quantum regime, rather than performing some classical evolution. Here we give a classical model that leads to the same behaviors used in those works to infer quantum effects. Thus, the evidence presented does not demonstrate the presence of quantum effects. T. Neuhaus, JSC Simulated Quantum Annealing For General Ising Models 2/2

21 appendix III arxiv: v [quant-ph]; Comment on: Classical signature of quantum annealing ; Lei Wang, Troels F. Rannow, Sergio Boixo, Sergei V. Isakov, Zhihui Wang, David Wecker, Daniel A. Lidar, John M. Martinis, Matthias Troyer: In a recent preprint (arxiv:35.494) entitled "Classical signature of quantum annealing" Smolin and Smith point out that a bimodal distribution presented in (arxiv: ) for the success probability in the D-Wave device does not in itself provide sufficient evidence for quantum annealing, by presenting a classical model that also exhibits bimodality. Here we analyze their model and in addition present a similar model derived from the semi-classical limit of quantum spin dynamics, which also exhibits a bimodal distribution. We find that in both cases the correlations between the success probabilities of these classical models and the D-Wave device are weak compared to the correlations between a simulated quantum annealer and the D-Wave device. Indeed, the evidence for quantum annealing presented in arxiv: is not limited to the bimodality, but relies in addition on the success probability correlations between the D-Wave device and the simulated quantum annealer. The Smolin-Smith model and our semi-classical spin model both fail this correlation test. T. Neuhaus, JSC Simulated Quantum Annealing For General Ising Models 2/2