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Polariton Graph Simulator (Optimizer): an analog Hamiltonian simulator

Popular description

The search for an optimal solution is analogous to looking for the lowest point in a mountainous terrain with many valleys, trenches, and drops. A hiker may go downhill and think that they have reached the lowest point of the entire landscape, but there may be a deeper drop just behind the next mountain. Such a search may seem daunting in natural terrain, but imagine its complexity in high-dimensional space! This is exactly the problem to tackle when the objective function to minimise represents a real-life problem with many unknowns, parameters, and constraints.

Modern supercomputers can only deal with a small subset of such problems when the dimension of the function to be minimised is small or when the underlying structure of the problem allows it to find the optimal solution quickly even for a function of large dimensionality. Even a hypothetical quantum computer, if realised, offers at best the quadratic speed-up for the “brute-force” search for the global minimum.

What if instead of moving along the mountainous terrain in search of the lowest point, one fills the landscape with a magical dust that only shines at the deepest level, becoming an easily detectible marker of the solution?

Our "magic dust" is created by shining a laser at stacked layers of selected atoms such as gallium, arsenic, indium, and aluminium. The electrons in these layers absorb and emit light of a specific colour. Polaritons are ten thousand times lighter than electrons and may achieve sufficient densities to form a new state of matter known as a Bose-Einstein condensate, where the quantum phases of polaritons synchronise and create a single macroscopic quantum object that can be detected through photoluminescence measurements.

But how to create a potential landscape that corresponds to the function to be minimised and to force polaritons to condense at its lowest point? To do this, we focused on a particular type of optimisation problem, but a type that is general enough so that any other hard problem can be related to it, namely minimisation of the XY model which is one of the most fundamental models of statistical mechanics. We have shown that we can create polaritons at vertices of an arbitrary graph: as polaritons condense, the quantum phases of polaritons arrange themselves in a configuration that correspond to the absolute minimum of the objective function.

XY Model is a universal classical spin model alongside other universal spin models such as an Ising and Heisenberg models. They are characterised by the given degrees of freedom, ``spins", by their interactions, ``couplings," and by the associated cost function, ``Hamiltonian". Various physical platforms have been proposed to simulate such models using superconducting qubits, optical lattices, coupled lasers etc. We introduced polariton graphs as a new platform for finding the global minimum of classical XY Hamiltonians in a variety of geometries and coupling strengths. This systems is based on well-established semiconductor and optical control technologies and benefit from flexible tunability and easy readability. Polariton condensates can be imprinted into any two-dimensional graph by spatial modulation of the pumping laser offering straightforward scalability. Polariton simulators have the potential to reach the global minimum of the XY Hamiltonian in a bottom-up approach by gradually increasing excitation density to threshold. This is an advantage over classical or quantum annealing techniques, where the global ground state is reached through transitions over metastable excited states with an increase of the cost of the search with the size of the system.

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Relevant publications

K.P.Kalinin and N.G.Berloff "Gain-dissipative simulators for large-scale hard cla ssical optimisation," arXiv:1805.01371 (2018)

We show that various gain-dissipative platforms based on the networks of optical parametric oscillators, lasers and various non-equilibrium Bose-Einstein condensates suffer from the same problem: the parameters of the hard optimization problems they intend to solve depend on the node occupancies that are not a priory known, which limits the applicability of the gain-dissipative simulators to the classes of problems easily solvable by classical computations. We show how to overcome this difficulty and formulate the principles of operation of such simulators for solving the NP-hard large-scale optimisation problems such as constant modulus continuous quadratic optimisation and quadratic binary optimisation for any general matrix. To solve such problems any gain-dissipative simulator has to implement a feedback mechanism for the dynamical adjustment of the gain and coupling strengths, so that occupancy of each node is the same. The estimates of the time operation of the physical implementation of the gain-dissipative simulators for large matrices show the speed-up of the several orders of magnitude in comparison with classical computations.

K.P.Kalinin and N.G.Berloff, "Blockchain platform with proof-of-work based on analog Hamiltonian optimisers," arXiv:1802.10091 (2018)

We reconsider the basis of blockchain encryption and suggest to move from currently used proof-of-work schemes to the proof-of-work performed by analog Hamiltonian optimisers. This approach has a potential to significantly increase decentralisation of the existing blockchains and to help achieve faster transaction times, therefore, removing the main obstacles for blockchain implementation.

K.Kalinin, P.G.Lagoudakis, and N.G.Berloff, "Matter wave coupling of spatially separated and unequally pumped polariton condensates," Phys. Rev. B, 97, 094512 (2018)

The introduction of non-equal pumping increases the complexity of the type of the problems that can be solved by polariton condensates arranged in a graph configuration. If equally pumped polaritons condensates arrange their phases to solve the constrained quadratic minimisation problem with a real symmetric matrix, the non-equally pumped condensates solve that problem for a general Hermitian matrix. i

K.Kalinin, P.G. Lagoudakis, and N.G.Berloff, "Simulating the spectral gap with polariton graphs,", in review by Phys. Rev. Letts. arXiv:1709.04683 (2017)

Theoretical validation of the polariton graph simulator/optimiser as a global minimizer of the XY model. It is shown that polariton graphs/optimizers also reproduce the lower excited states of the XY model and therefore can be used to simulate the spectral gap.

K.Kalinin, P.G. Lagoudakis, and N.G.Berloff, "Exotic states of matter with polariton chains," Phys. Rev. B (Rapid Comm) 97, 161101(R) (2018)

It is shown theoretically that the low energy states of polaritonic linear chains demonstrate various classical regimes: ferromagnetic, antiferromagnetic and frustrated spiral phases, where quantum or thermal fluctuations are expected to give rise to a spin liquid state. At the same time nonlinear interactions at higher pumping intensities bring about phase chaos and novel exotic phases. This provides the route to study novel exitic states with polariton graphsimulator/optimizer.

P.G.Lagoudakis and N.G. Berloff, "A Polariton Graph Simulator" New Journal of Physics 19, 125008 (2017)

A polaiton graph simulator/optimizer is discussed theoretically establishing (1) an analytical solution for a single condensate -- "polariton qubit" -- that is used as a building block of the polariton simulator and (2) connection with the laser networks and Kuramoto model.

N. G. Berloff, M. Silva, K. Kalinin, A. Askitopoulos, J. D. Töpfe, P. Cillibrizzi, W. Langbein and P. G. Lagoudakis "Realizing the classical XY Hamiltonian in polariton simulators," Nature Materials 16(11) 1120 (2017)

It is established theoretically and confirmed experimentally that polariton condensates arranged in a graph realise the absolute minimum of the XY Model and therefore polariton graphs form a platform for quantum simulatotions for solving NP-hard problems. Maximum of 45 "qubits" is realised.

H. Ohadi, R. L. Gregory, T. Freegarde, Y. G. Rubo, A. V. Kavokin, N. G. Berloff, and P. G. Lagoudakis, "Nontrivial phase coupling in polariton multiplets," Phys. Rev. X 6, 031032 (2016)

Non-resonant excitations of two and three spots is considered in theory and experiment. The dependence of the interactions between the condensates is shown to depend on the polariton outflow wavevector and the distance between the condensates. Non-trivial coupling of three condensates is established.

G Christmann, G Tosi, N G Berloff, P Tsotsis, P S Eldrige, Z Hatzopoulos, P G Savvidis, J J Baumberg "Oscillatory solitons and time-resolved phase locking of two polariton condensates," New Journal of Physics, 16 (10), 103039 (2014)

Non-resonant pumped excitations are considered showing the phase locking of two condensates which is linked to the Josephson coupling.

G. Tosi, G. Christmann, N. G. Berloff, P. Tsotsis, T. Gao, Z. Hatzopoulos, P. G. Savvidis, J. J. Baumberg "Geometrically locked vortex lattices in semiconductor quantum fluids," Nature Communications, 3 , 1243 (2013)

Experimental realisation of the theroretical proposal of Keeling-Berloff ", to non-resonantly pump polaritons in the corners of an equilaterial triangle and a square. For the triangle only in-phase configurations were observed, for the square two possibilities were observed: ferro- and anti-ferromagnetic configurations.

G. Tosi, G. Christmann, N.G. Berloff, P. Tsotsis, T. Gao, Z. Hatzopoulos, P.G. Savvidis, J.J. Baumberg,   "Sculpting oscillators with light within a nonlinear quantum fluid", Nature Physics, 8, 190-194 (2012)

The fisrt experimental realization of two non-resonantly pumped condensates. The exitation is well above the threshold, so many states are excited at the same time.

J. Keeling, N. G. Berloff "Controllable half-vortex lattices in an incoherently pumped polariton condensate", arXiv:1102.5302 (2011)

The first proposal to non-resonantly pump polaritons in a graph -- in the corners of an equilaterial triangle -- to observe nontrivial states due to polaritons flowing from the pump sites.

Frequently Asked Questions

1. Why do you refer to a polariton graph simulator as “quantum”?

As we discussed in NJP (2017) paper “the word ”quantum” could be attached to our proposal for a simulator to reflect the statistical nature of polariton condensates. The process of Bose-Einstein condensation is inherent to quantum statistics where a large fraction of bosons occupies the lowest quantum state, at which point macroscopic quantum phenomena become apparent. The use of the classical mean-field equations to describe the kinetics of the condensate does not negate the quantum statistic nature of its existence. At the same time, the proposed simulator has a quantum speed-up which is associated with the stimulated process of condensation i.e. an accelerated relaxation to the global ground quantum state.”

2. Polariton Graph Simulator simulates the classical XY Model. Does this mean that it can not address various aspects of quantum systems?

There is an understandable misconception that classical systems cannot be used to address various aspects of quantum systems. In fact there are many classical tools that are routinely used to address quantum systems, such as numerical simulations, tensor network theory, density functional theory, and quantum Monte Carlo algorithms. The theoretical threshold between classical and quantum simulations underlines the fundamental but non-trivial and often contentious questions about quantum versus classical simulators. This discussion has recently been held within the community. We would follow the account given in several recent publications on this topic quoting specifically that of Johnson, Clark and Jaksch “What is a quantum simulator?” EPJ Quantum Technology 2014, 1:10. According to their account a device is classified as “quantum” if preparing the state relies on quantum effects: “our assignment of the quantum in quantum simulator based only on the device avoids the assumption that only simulating quantum models is hard enough to potentially benefit from a quantum device. This is not so: finding the ground state of even a classical Ising model is NP-hard and thus thought to be inefficient on both a classical and quantum device [Barahona, F.: On the computational complexity of ising spin glass models. Journal of Physics A: Mathematical and General 15, 3241 (1982) and Bernstein, E., Vazirani, U.: Quantum complexity theory. SIAM J on Computing 26, 1411–1473 (1997)].” According to this definition our polariton simulator can be classified as “quantum”, since the process of the condensation that guarantees the achievement of the global minimum of the classical XY Hamiltonian is the property of quantum statistics, namely Bose-Einstein condensation.

3. Can a classical XY model give any information about the quantum nature of the considered system?

The power and universality of the classical XY model follows from its appearance in the general framework of critical phenomena. In Chapter 3 of the book “Superfluid States of Matter” by B. Svistunov, E. Babaev and N. Prokof’ev (2015) the authors concentrate on the use of the classical XY model to analyse quantum phase transitions and state: “The concept of the universality class is extremely powerful for continuous phase transitions since it is sufficient to study in detail just one representative model to be able to make solid predictions about all other systems featuring a similar broken-symmetry transition. This is so no matter whether the system is quantum or classical, continuous or on a lattice, and regardless of the microscopic details of interactions.” In that chapter (and references therein) the authors discuss the use of the classical XY model to study various quantum phase transitions. The transition points are associated with the development of long-range correlations in the statistical classical XY model. The use of the XY model for topological quantum information could come from the topological projection. For instance, superfluidity is about topological order in the phase field; “whether the phase gradient is related to the current flow or not is a secondary consideration that does not change the essence of the transition point and emergence of order at low temperature. One can apply the same notions of topological protection and quantization of closed-contour integrals, introduce vortices, and consider the free energy increase in response to phase gradients.” Notably, the 2016 Nobel Prize in Physics was awarded to Kosterlitz and Thouless for their research into the KT (or BKT) quantum phase transition between superfluid and normal states in two dimensions. In the seminal paper Kosterlitz used the classical XY model with nearest-neighbour interactions on a two-dimensional square lattice [Kosterlitz, Journal of Physics C: Solid State Physics, 1974]. Furthermore, the envisioned advantage of analog simulators lies in how the computational time scales with the size of the problem. Among others Dieter Jaksch and co-workers write “The benefit of a quantum device is that the size of problems that can be tackled in a reasonable time grows significantly more quickly with the size of the simulating device than it does for a classical device, thus it is envisaged that quantum devices will one day be able to solve larger problems than their classical counterparts.” [from Johnson, Clark and Jaksch “What is a quantum simulator?” EPJ Quantum Technology 2014, 1:10]. This is precisely the benefit of the polariton platform since the time it takes for bosonic stimulation to create a condensate at the global minimum of the XY Hamiltonian scales linearly with the number of vertices.

4. What is the efficiency, trustworthiness, limitations etc. of your platform?

As Johnson, Clark and Jaksch succinctly put it: “In this sense trustworthiness is not a clear-cut topic that is established upon the initial development of a simulator. Instead, it is the result of a complex, time-consuming process in the period that follows. It is the responsibility of critics not to be overly harsh and unfairly demanding of new simulators to provide immediate proof of their trustworthiness, but it is also the responsibility of proponents not to declare trustworthiness before their simulator has earned it.”