Computational Phase Transition Signature in Gibbs Sampling

06/25/2019
by   H. Philathong, et al.
0

Gibbs sampling is fundamental to a wide range of computer algorithms. Such algorithms are set to be replaced by physics based processors-be it quantum or stochastic annealing devices-which embed problem instances and evolve a physical system into an ensemble to recover a probability distribution. At a critical constraint to variable ratio, decision problems-such as propositional satisfiability-appear to statistically exhibit an abrupt transition in required computational resources. This so called, algorithmic or computational phase transition signature, has yet-to-be observed in contemporary physics based processors. We found that the computational phase transition admits a signature in Gibbs' distributions and hence we predict and prescribe the physical observation of this effect. We simulate such an experiment, that when realized experimentally, we believe would represent a milestone in the physical theory of computation.

READ FULL TEXT VIEW PDF
POST COMMENT

Comments

There are no comments yet.

Authors

page 5

12/16/2020

Programmable Quantum Annealers as Noisy Gibbs Samplers

Drawing independent samples from high-dimensional probability distributi...
02/13/2019

Assessing Solution Quality of 3SAT on a Quantum Annealing Platform

When solving propositional logic satisfiability (specifically 3SAT) usin...
06/25/2020

Between-Domain Instance Transition Via the Process of Gibbs Sampling in RBM

In this paper, we present a new idea for Transfer Learning (TL) based on...
07/13/2020

Benchmarking 16-element quantum search algorithms on IBM quantum processors

We present experimental results on running 4-qubit unstructured search o...
10/01/2018

Joint Activity Detection and Channel Estimation for IoT Networks: Phase Transition and Computation-Estimation Tradeoff

Massive device connectivity is a crucial communication challenge for Int...
02/20/2018

Memcomputing: Leveraging memory and physics to compute efficiently

It is well known that physical phenomena may be of great help in computi...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.

References

  • (1) David Deutsch. Quantum theory, the Church–Turing principle and the universal quantum computer. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 400(1818):97–117, 1985.
  • (2) Alonzo Church. An unsolvable problem of elementary number theory. American journal of mathematics, 58(2):345–363, 1936.
  • (3) Alan Mathison Turing. On computable numbers, with an application to the Entscheidungsproblem. Proceedings of the London mathematical society, 2(1):230–265, 1937.
  • (4) James M Crawford and Larry D Auton. Experimental results on the crossover point in random 3-SAT. In Artificial Intelligence, volume 81, pages 31–57, 1996.
  • (5) Ehud Friedgut, Jean Bourgain, et al. Sharp thresholds of graph properties, and the k-SAT problem. Journal of the American mathematical Society, 12(4):1017–1054, 1999.
  • (6) Bart Selman and Scott Kirkpatrick. Critical behavior in the computational cost of satisfiability testing. Artificial Intelligence, 81(1-2):273–295, 1996.
  • (7) Andrew Lucas. Ising formulations of many np problems. Frontiers in Physics, 2:5, 2014.
  • (8) JD Biamonte. Nonperturbative k-body to two-body commuting conversion Hamiltonians and embedding problem instances into Ising spins. Physical Review A, 77(5):052331, 2008.
  • (9) James Daniel Whitfield, Mauro Faccin, and JD Biamonte. Ground-state spin logic. EPL (Europhysics Letters), 99(5):57004, 2012.
  • (10) Scott Kirkpatrick, Daniel C Gelatt, and Mario P Vecchi. Optimization by simulated annealing. Science, 220(4598):671–680, 1983.
  • (11) Shoko Utsunomiya, Kenta Takata, and Yoshihisa Yamamoto. Mapping of Ising models onto injection-locked laser systems. Optics express, 19(19):18091–18108, 2011.
  • (12) Takahiro Inagaki, Yoshitaka Haribara, Koji Igarashi, Tomohiro Sonobe, Shuhei Tamate, Toshimori Honjo, Alireza Marandi, Peter L McMahon, Takeshi Umeki, Koji Enbutsu, et al. A coherent Ising machine for 2000-node optimization problems. Science, 354(6312):603–606, 2016.
  • (13) D Pierangeli, G Marcucci, and C Conti. Large-scale photonic ising machine by spatial light modulation. Physical Review Letters, 122(21):213902, 2019.
  • (14) Alireza Marandi, Zhe Wang, Kenta Takata, Robert L Byer, and Yoshihisa Yamamoto. Network of time-multiplexed optical parametric oscillators as a coherent Ising machine. Nature Photonics, 8(12):937, 2014.
  • (15) Micha Nixon, Eitan Ronen, Asher A Friesem, and Nir Davidson. Observing geometric frustration with thousands of coupled lasers. Physical review letters, 110(18):184102, 2013.
  • (16) Natalia G Berloff, Matteo Silva, Kirill Kalinin, Alexis Askitopoulos, Julian D Töpfer, Pasquale Cilibrizzi, Wolfgang Langbein, and Pavlos G Lagoudakis. Realizing the classical XY Hamiltonian in polariton simulators. Nature materials, 16(11):1120, 2017.
  • (17) David Dung, Christian Kurtscheid, Tobias Damm, Julian Schmitt, Frank Vewinger, Martin Weitz, and Jan Klaers. Variable potentials for thermalized light and coupled condensates. Nature Photonics, 11(9):565, 2017.
  • (18) Kirill Kalinin and Natalia G Berloff. Global optimization of spin Hamiltonians with gain-dissipative systems. Scientific reports, 8(1):17791, 2018.
  • (19) Mark W Johnson, Mohammad HS Amin, Suzanne Gildert, Trevor Lanting, Firas Hamze, Neil Dickson, R Harris, Andrew J Berkley, Jan Johansson, Paul Bunyk, et al. Quantum annealing with manufactured spins. Nature, 473(7346):194, 2011.
  • (20) Rami Barends, Alireza Shabani, Lucas Lamata, Julian Kelly, Antonio Mezzacapo, Urtzi Las Heras, Ryan Babbush, Austin G Fowler, Brooks Campbell, Yu Chen, et al. Digitized adiabatic quantum computing with a superconducting circuit. Nature, 534(7606):222, 2016.
  • (21) R Harris, J Johansson, AJ Berkley, MW Johnson, T Lanting, Siyuan Han, P Bunyk, E Ladizinsky, T Oh, I Perminov, et al.

    Experimental demonstration of a robust and scalable flux qubit.

    Physical Review B, 81(13):134510, 2010.
  • (22) R Harris, Y Sato, AJ Berkley, M Reis, F Altomare, MH Amin, K Boothby, P Bunyk, C Deng, C Enderud, et al. Phase transitions in a programmable quantum spin glass simulator. Science, 361(6398):162–165, 2018.
  • (23) Andrew D King, Juan Carrasquilla, Jack Raymond, Isil Ozfidan, Evgeny Andriyash, Andrew Berkley, Mauricio Reis, Trevor Lanting, Richard Harris, Fabio Altomare, et al. Observation of topological phenomena in a programmable lattice of 1,800 qubits. Nature, 560(7719):456, 2018.
  • (24) Stephen A Cook. The complexity of theorem-proving procedures. In Proceedings of the third annual ACM symposium on Theory of computing, pages 151–158. ACM, 1971.
  • (25) Bart Selman, David G Mitchell, and Hector J Levesque. Generating hard satisfiability problems. Artificial intelligence, 81(1-2):17–29, 1996.
  • (26) Melven R Krom. The decision problem for a class of first-order formulas in which all disjunctions are binary. Mathematical Logic Quarterly, 13(1-2):15–20, 1967.
  • (27) Béla Bollobás, Christian Borgs, Jennifer T Chayes, Jeong Han Kim, and David B Wilson. The scaling window of the 2-SAT transition. Random Structures & Algorithms, 18(3):201–256, 2001.
  • (28) Vašek Chvátal and Bruce Reed.

    Mick gets some (the odds are on his side)(satisfiability).

    In Proceedings., 33rd Annual Symposium on Foundations of Computer Science, pages 620–627. IEEE, 1992.
  • (29) Andreas Goerdt. A threshold for unsatisfiability. Journal of Computer and System Sciences, 53(3):469–486, 1996.
  • (30) Armin Biere. PicoSAT essentials. Journal on Satisfiability, Boolean Modeling and Computation, 4:75–97, 2008.
  • (31) Mohammad Taghi Hajiaghayi and Gregory B Sorkin. The satisfiability threshold of random 3-SAT is at least 3.52. arXiv preprint math/0310193, 2003.
  • (32) Elitza Maneva and Alistair Sinclair. On the satisfiability threshold and clustering of solutions of random 3-SAT formulas. Theoretical Computer Science, 407(1-3):359–369, 2008.
  • (33) I. Zacharov, R. Arslanov, M. Gunin, D. Stefonishin, S. Pavlov, O. Panarin, A. Maliutin, S. Rykovanov, and M. Fedorov.

    ‘Zhores’ – Petaflops Supercomputer for Data-Driven Modeling, Machine Learning and Artificial Intelligence, feb 2019.

  • (34) Francisco Barahona. On the computational complexity of Ising spin glass models. Journal of Physics A: Mathematical and General, 15(10):3241, 1982.

Appendix A Ising Spin Embedding

Finding ground energy and ground state configurations of physical systems such as spin glasses is NP-hard barahona1982computational . 2-SAT and 3-SAT instances can be directly mapped onto the Hamiltonian minimization problem, see e.g. lucas2014ising ; biamonte2008nonperturbative . The standard procedure is as follows.

We convert logical bits to spins

(7)

The penalty Hamiltonian is constructed by real linear extension of by means of the following invertible mapping between Boolean variables and projectors,

(8)

and

(9)

where and acting on the spin.

By this construction, clauses in a SAT instance are mapped onto projectors . Hence, the Hamiltonian can be constructed by summing over all the clauses in an instance,

(10)

where assigns the value of corresponding to the clause. The ground state space is spanned by solutions of the SAT problem, and the ground energy is equal to minimal number of unsatisfiable clauses.

Substituting the projectors in Eq. (10), where is the Pauli matrix along the quantization axis, the Hamiltonian can be written in the form of the generalized Ising Hamiltonian with k-body interaction. In case of MAX 2-SAT, the Hamiltonian is a sum over projectors onto -SAT clauses

(12)

The coefficient indicates the local field of the spin and the couplings encode the 2-body interaction between the spin and the spin.

This method provides a way to physically realize SAT instances as spin Hamiltonians. If such a physical system is built, cooling the system into its ground state is equivalent to solving the SAT problems.