hot enough so that the atoms settle in the lowest energy landscape, which makes the strongest material. Similarly, quantum annealing is based on the idea of finding the lowest energy configuration of a system.
Quantum annealing is deployed as a method for solving optimization problems by using quantum adiabatic evolution to find the ground state of a system (adiabatic means heat does not enter or leave the system). Run on a quantum computer, the quantum annealing process starts from a ground state which is the quantum mechanical superposition of all possible system states with equal weights. The system then evolves per the time-dependent Schrödinger equation in a natural physical evolution to settle in a low-energy state. The computational problem to be solved is framed in terms of an energy optimization problem in which the low-energy state signals the answer. (The quantum annealing process is described in more detail in Chapter 10.)
Overall, the quantum annealing process allows the system of spins (spinning atoms of qubits) to find a low-energy state. Superconducting circuits in the quantum annealing model can be thought of as programmable annealing engines (Kaminsky & Lloyd, 2004). Optimization problems are framed such that they can be instantiated in the form of an energy landscape minimization. Although annealing machines are not general-purpose quantum computers, one advantage is that since annealing systems constantly attempt to reach the lowest energy state, they are more tolerant and resistant to noise than gate model systems and may require much less error correction at large scales.
3.3.4 Ion trapping
Another prominent approach to quantum computing is trapped ions. In these quantum chips, ions are stored in electromagnetic traps and manipulated by lasers and electromagnetic fields. Ions are atoms which have been stripped of or received electrons, which leaves them positively or negatively charged and therefore more easily manipulatable. The advantage of ion trap qubits is that they have a long coherence time (making calculations easier) and (like annealing machines) may require less error correction at large scales. A single qubit trap may accommodate 30–100 qubits, and 23 qubits have been demonstrated in a research context (Murali et al., 2019).
The IonQ quantum chip uses ytterbium ions, which unlike superconducting qubits, do not need to be supercooled to operate. Bulky cryogenic equipment is not required, and the entire system occupies about one cubic meter, as opposed to a much larger footprint for superconducting circuit machines. The chip is a few millimeters across. It is fabricated with silicon and contains 100 electrodes that confine and control the ions in an ultrahigh-vacuum environment.
To operate, the ion trap quantum computer holds the ions in a geometrical array (a linear array for IonQ). Laser beams encode and read information to and from individual ions by causing transitions between the electronic states of the ion. The ions influence each other through electrostatic interactions and their coupling can be controlled. More specifically, the IonQ ions form a crystal structure because they repel each other (since they are all of the same isotope of the same element (ytterbium-171)). The electrodes underneath the ions hold the charged particles together in a linear array by applying electrical potentials. The lasers initialize the qubits, entangle them through coupling, and produce quantum logic gates to execute the computation. At the end of the computation, another laser causes ions to fluoresce if they are in a certain qubit state. The fluorescence is collected to measure each qubit and compute the result of the computation. One design principle is already becoming clear in such ion trap systems, that the number of qubits scales as the square root of the gates.
3.3.5 Majorana fermions and topological quantum computing
An interesting and somewhat exotic approach for building a universal quantum computer is Majorana fermions. Qubits are made from particles in topological superconductors and electrically controlled in a computational model based on their movement trajectories (called “braiding”). One of the main benefits of topological quantum computing is physical error correction (error correction performed in the hardware, not later by software). The method indicates very low initial error rates as compared with other approaches (Freedman et al., 2002).
Topological superconductors are novel classes of quantum phases that arise in condensed matter, characterized by structures of Cooper pairing states (i.e. quantum computable states) that appear on the topology (the edge and core) of the superconductor (hence the name topological superconductors). The Cooper pairing states are a special class of matter called Majorana fermions (particles identified with their own antiparticles). Topological invariants constrained by the symmetries of the systems produce the Majorana fermions and ensure their stability.
As the Majorana fermions bounce around, their movement trajectories resemble a braid made out of different strands. The braids are wave functions that are used to develop the logic gates in the computation model (Wang, 2010). Majorana fermions appear in particle–antiparticle pairs and are assigned to quantum states or modes. The computation model is built up around the exchange of the so-called Majorana zero modes in a sequential process. The sequentiality of the process is relevant as changing the order of the exchange operations of the particles changes the final result of the computation. This feature is called non-Abelian, denoting that the steps in the process are non-commuting (non-exchangeable with one another). Majorana zero modes obey a new class of quantum statistics, called non-Abelian statistics, in which the exchange operations of particles are non-commutative.
The Majorana zero modes (modes indicate a specific state of a quantum object related to spin, charge, polarization, or other parameter) are an important and unique state of the Majorana fermionic system (unlike other known bosonic and fermionic matter phases). The benefit of the non-Abelian quantum statistics of the Majorana zero modes is that they can be employed for wave function calculations, namely to average over the particle wave functions in sequential order. The sequential processing of particle wave function behavior is important for constructing efficient logic gates for quantum computation. Researchers indicate that well-separated Majorana zero modes should be able to manifest non-Abelian braiding statistics suitable for unitary gate operations for topological quantum computation (Sarma et al., 2015).
Majorana fermions have only been realized in the specialized conditions of temperatures close to 1 K (−272°C) under high magnetic fields. However, there are recent proposals for more reliable platforms for producing Majorana zero modes (Robinson et al., 2019) and generating more robust Majorana fermions in general (Jack et al., 2019).
3.3.6 Quantum photonics
Qubits are formed from either matter (atoms or ions) or light (photons). Quantum photonics is an important approach to quantum computing given its potential linkage to optical networks, in the fact that global communications networks are based on photonic transfer. In quantum photonics, single photons or squeezed states of light in silicon waveguides are used to represent qubits, and they are controlled in a computational model in cluster states (entangled states of multiple photons). Quantum photonics can be realized in computing chips or in free space. Single photons or squeezed states of light are sent through the chip or the free space for the computation and then measured with photon detectors at the other end.
For photonic quantum computing, a cluster state of entangled photons must be produced. The cluster state is a resource state of multidimensional highly entangled qubits. There are various ways of generating and using the cluster state (Rudolph, 2016). The general process is to produce photons, entangle them, compute with them, and measure the result. One way of generating cluster states is in lattices of qubits with Ising-type interactions (phase transitions). Lattices translate well into computation. Cluster states are represented as graph states, in which the underlying graph is a connected subset of a d-dimensional lattice. The graph states are then instantiated as a computation graph with directed operations to perform the computation.
3.3.6.1Photonic
