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Journal of Physics Conference Series

Casting Rubik’s Group into a Unitary Representation for Reinforcement Learning

Sebastiano Corli, Lorenzo Del Moro, Davide Emilio Galli, Enrico Prati

Published Jun 1, 2023• 5 citations
Publisher

Abstract

Abstract Rubik’s Cube is one of the most famous combinatorial puzzles involving nearly 4.3 × 1019 possible configurations. However, only a single configuration matches the solved one. Its mathematical description is expressed by the Rubik’s group, whose elements define how its layers rotate. We develop a unitary representation of the Rubik’s group and a quantum formalism to describe the Cube based on its geometrical constraints. Using single particle quantum states, we describe the cubies as bosons for corners and fermions for edges. By introducing a set of four Ising-like Hamiltonians, we managed to set the solved configuration of the Cube as the global ground state for all the Hamiltonians. To reach the ground state of all the Hamiltonian operators, we made use of a Deep Reinforcement Learning algorithm based on a Hamiltonian reward. The Rubik’s Cube is successfully solved through four phases, each phase driven by a corresponding Hamiltonian reward based on its energy spectrum. We call our algorithm QUBE, as it employs quantum mechanics to tackle the combinatorial problem of solving the Rubik’s Cube. Embedding combinatorial problems into the quantum mechanics formalism suggests new possible algorithms and future implementations on quantum hardware.

DOI
10.1088/1742-6596/2533/1/012006

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Abstract Rubik’s Cube is one of the most famous combinatorial puzzles involving nearly 4.3 × 1019 possible configurations. However, only a single configuration matches the solved one. Its mathematical description is expressed by the Rubik’s group, whose elements define how its layers rotate. We develop a unitary representation of the Rubik’s group and a quantum formalism to describe the Cube based on its geometrical constraints. Using single particle quantum states, we describe the cubies as bosons for corners and fermions for edges. By introducing a set of four Ising-like Hamiltonians, we managed to set the solved configuration of the Cube as the global ground state for all the Hamiltonians. To reach the ground state of all the Hamiltonian operators, we made use of a Deep Reinforcement Learning algorithm based on a Hamiltonian reward. The Rubik’s Cube is successfully solved through four phases, each phase driven by a corresponding Hamiltonian reward based on its energy spectrum. We call our algorithm QUBE, as it employs quantum mechanics to tackle the combinatorial problem of solving the Rubik’s Cube. Embedding combinatorial problems into the quantum mechanics formalism suggests new possible algorithms and future implementations on quantum hardware.
Abstract Rubik’s cube is one of the most famous combinatorial puzzles involving nearly 4.3 × 10 19 possible configurations. Its mathematical description is expressed by the Rubik’s group, whose elements define how its layers rotate. We develop a unitary representation of such group and a quantum formalism to describe the cube from its geometrical constraints. Cubies are described by single particle states which turn out to behave like bosons for corners and fermions for edges, respectively. When in its solved configuration, the cube, as a geometrical object, shows symmetries which are broken when driven away from this configuration. For each of such symmetries, we build a Hamiltonian operator. When a Hamiltonian lies in its ground state, the respective symmetry of the cube is preserved. When all such symmetries are preserved, the configuration of the cube matches the solution of the game. To reach the ground state of all the Hamiltonian operators, we make use of a deep reinforcement learning algorithm based on a Hamiltonian reward. The cube is solved in four phases, all based on a respective Hamiltonian reward based on its spectrum, inspired by the Ising model. Embedding combinatorial problems into the quantum mechanics formalism suggests new possible algorithms and future implementations on quantum hardware.
We introduce a reinforcement learning algorithm designed to identify the fixed points of a given quantum operation. The method iteratively constructs the unitary transformation that maps the computational basis onto the basis of fixed points through a reward-penalty scheme based on quantum measurements. In cases where the operation corresponds to a Hamiltonian evolution, this task reduces to determining the Hamiltonian eigenstates. The algorithm is first benchmarked on random Hamiltonians acting on two and three qubits and then applied to many-body systems of up to six qubits, including the transverse-field Ising model and the all-to-all pairing Hamiltonian. In both cases, the algorithm is demonstrated to perform successfully; in the pairing model, it can also reveal hidden symmetries, which can be exploited to restrict learning to specific symmetry sectors. Finally, we discuss the possibility of post-selecting high-fidelity states even when full convergence has not been reached.
Quantum hardware and quantum-inspired algorithms are becoming increasingly popular for combinatorial optimization. However, these algorithms may require careful hyperparameter tuning for each problem instance. We use a reinforcement learning agent in conjunction with a quantum-inspired algorithm to solve the Ising energy minimization problem, which is equivalent to the Maximum Cut problem. The agent controls the algorithm by tuning one of its parameters with the goal of improving recently seen solutions. We propose a new Rescaled Ranked Reward (R3) method that enables a stable single-player version of self-play training and helps the agent escape local optima. The training on any problem instance can be accelerated by applying transfer learning from an agent trained on randomly generated problems. Our approach allows sampling high quality solutions to the Ising problem with high probability and outperforms both baseline heuristics and a black-box hyperparameter optimization approach.
The advantage of quantum computers over classical computers fuels the recent trend of developing machine learning algorithms on quantum computers, which can potentially lead to breakthroughs and new learning models in this area. The aim of our study is to explore deep quantum reinforcement learning (RL) on photonic quantum computers, which can process information stored in the quantum states of light. These quantum computers can naturally represent continuous variables, making them an ideal platform to create quantum versions of neural networks. Using quantum photonic circuits, we implement Q learning and actor-critic algorithms with multilayer quantum neural networks and test them in the grid world environment. Our experiments show that 1) these quantum algorithms can solve the RL problem and 2) compared to one layer, using three layer quantum networks improves the learning of both algorithms in terms of rewards collected. In summary, our findings suggest that having more layers in deep quantum RL can enhance the learning outcome.
Abstract We propose and evaluate a quantum-inspired algorithm for solving Quadratic Unconstrained Binary Optimization (QUBO) problems, which are mathematically equivalent to finding ground states of Ising spin-glass Hamiltonians. The algorithm employs Matrix Product States (MPS) to compactly represent large superpositions of spin configurations and utilizes a discrete driving schedule to guide the MPS toward the ground state. At each step, a driver Hamiltonian—incorporating a transverse magnetic field—is combined with the problem Hamiltonian to enable spin flips and facilitate quantum tunneling. The MPS is updated using the standard Density Matrix Renormalization Group (DMRG) method, which iteratively minimizes the system’s energy via multiple sweeps across the spin chain. Despite its heuristic nature, the algorithm reliably identifies global minima, not merely near-optimal solutions, across diverse QUBO instances. We first demonstrate its effectiveness on intermediate-level Sudoku puzzles from publicly available sources, involving over 200 Ising spins with long-range couplings dictated by constraint satisfaction. We then apply the algorithm to MaxCut problems from the Biq Mac library, successfully solving instances with up to 251 nodes and 3,265 edges. We discuss the advantages of this quantum-inspired approach, including its scalability, generalizability, and suitability for industrial-scale QUBO applications.
Abstract The quantum Rubik's cube exhibits distinctive dual characteristics of high dimensionality and multi‐projection. Its multi‐projection surfaces can present quantum system information from diverse angles, whereas its high‐dimensional nature can be harnessed to probe complex system architectures. In this study, a quantum Rubik's cube is integrated with the Heisenberg model to formulate a novel Hamiltonian form. A fourth‐order quantum Rubik's cube is used to investigate the quantum phase transition, and a fifth‐order Rubik's cube is used to simulate the jumps of electrons across different energy levels, demonstrating the effectiveness of this approach. This holds significant potential for enabling physicists to delve more profoundly into the characteristics and laws of quantum systems, thereby providing substantial support for the theoretical advancement of quantum physics.