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IEEE Spectrum

Quantum computers scale up: Constructing a universal quantum computer with a large number of qubits will be hard but not impossible

R. Versluis, Chad Hagen

Published Apr 1, 2020• 10 citations
Publisher

Abstract

The classic Rubik's Cube has 43,252,003,274,489,856,000 different states. You might well wonder how people are able to take a scrambled cube and bring it back to its original configuration, with just one color showing on each side. Some people are even able to do this blindfolded after viewing the scrambled cube once. Such feats are possible because there's a basic set of rules that always allow someone to restore the cube to its original state in 20 moves or less.

DOI
10.1109/mspec.2020.9055969

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The classic Rubik's Cube has 43,252,003,274,489,856,000 different states. You might well wonder how people are able to take a scrambled cube and bring it back to its original configuration, with just one color showing on each side. Some people are even able to do this blindfolded after viewing the scrambled cube once. Such feats are possible because there's a basic set of rules that always allow someone to restore the cube to its original state in 20 moves or less.
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.
Permutation puzzles, such as the Rubik's Cube and the 15 puzzle, are enjoyed by the general public and mathematicians alike. Here we introduce quantum versions of permutation puzzles where the pieces of the puzzles are replaced with indistinguishable quantum particles. The moves in the puzzle are achieved by swapping or permuting the particles. We show that simply permuting the particles can be mapped to a classical permutation puzzle, even though the identical particles are entangled. However, we obtain a genuine quantum puzzle by adding a quantum move: the square root of swap. The resulting puzzle cannot be mapped to a classical permutation puzzle. We focus predominately on the quantization of the $2\ifmmode\times\else\texttimes\fi{}2$ slide puzzle and briefly treat the $2\ifmmode\times\else\texttimes\fi{}2\ifmmode\times\else\texttimes\fi{}1$ Rubik's Cube.
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.
If quantum computers are ever built, they might need only minutes to perform tasks that would take today's fastest supercomputer millions of years. Quantum bits, or qubits, which store and process information, can assume the values of 1 and 0 simultaneously. That means 100 qubits can calculate in just one step a computation that ordinary bits would need to repeat 2100 times. Confused or intrigued? Check out the University of Oxford's Centre for Quantum Computation Web site (http://www.qubit.org/). Besides serving experts, “we try to maintain information which will interest people who simply wonder what quantum computing is all about,” says one of the Web masters, Benjamin Simon.
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.
Introduction In an ordinary computer, information is stored in a collection of tiny circuits each of which is designed to have two stable and easily distinguishable configurations: each represents a bit . In our study of quantum cryptography, we have seen how it can be useful to express information not in ordinary bits but in qubits . Whereas a bit can have only two values, say 0 and 1, a qubit can be in any quantum superposition of ∣0〉 and ∣1〉. Moreover, a qubit can be entangled with other qubits. Thus one might wonder whether a quantum computer, in which the basic elements for storing and processing information are qubits, can outperform an ordinary (classical) computer in certain ways. This question was addressed by researchers starting in the 1980s. In terms of practical consequences, perhaps the most dramatic answer has been given by Peter Shor in his 1994 factoring algorithm for a quantum computer, an algorithm that is exponentially faster than any known classical algorithm. As we have seen in Chapter 1, the difficulty of factoring a product of two large primes is the basis of the security of the RSA cryptosystem. So if one could build a large enough quantum computer – and there is no reason in principle why this could not be done – the RSA system would be rendered ineffective. In this chapter we present the basics of quantum computation and then focus on Shor's factoring algorithm.