Quantum Computing. Melanie Swan

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 time speed-ups

      The normal speed-up in quantum computing compared to classical computing is due to the superposition of 0s and 1s, in that the quantum circuit can process 0s and 1s at the same time. This provides massive parallelism by being able to process all of the problem inputs at the same time. Photonics allows an additional speed-up to the regular speed-up of quantum computing. In photonic quantum computing, superposition can be used not only for problem inputs but also for processing gates (Procopio et al., 2015). Time can be accelerated by superpositioning the processing gates. Standard quantum architectures have fixed gate arrangements, whereas photonic quantum architectures allow the gate order to be superimposed as well. This means that when computations are executed, they run through circuits that are themselves superpositioned. The potential computational benefit of the superposition of optical quantum circuits is an exponential advantage over classical algorithms and a linear advantage over regular quantum algorithms.

3.3.7 Neutral atoms, diamond defects, quantum dots, and nuclear magnetic resonance

      Overall, there are many methods for generating qubits and computing with them (Table 3.3). In addition to the four main approaches (superconducting circuits, ion traps, Majorana fermions, and photonics), four additional approaches are discussed briefly. These include neutral atoms, diamond defects (nitrogen-vacancy defect centers), quantum dots, and nuclear magnetic resonance (NMR).

      3.3.7.1Neutral atoms

      An early-stage approach to quantum computing is neutral atoms. Neutral atoms are regular uncharged atoms with balanced numbers of protons and electrons, as opposed to ions that are charged because they have had an electron stripped away from them or added to them. Qubits are produced by exciting neutral atoms trapped in optical lattices or optical arrays, and qubits are controlled in computation by another set of lasers. The neutral atoms are trapped in space with lasers. An optical lattice is made with interfering laser beams from multiple directions to hold the atoms in wells (an egg carton-shaped structure). Another method is holding the atoms in an array with optical tweezers. Unlike ions (which have strong interactions and repel each other), neutral atoms can be held in close confinement with each other and manipulated in computation. Atoms such as cesium and rubidium are excited into Rydberg states from which they can be manipulated to perform computation (Saffman, 2016). Researchers have been able to accurately program a two-rubidium atom logic gate 97% of the time with the neutral atoms approach (Levine et al., 2018), as compared to 99% fidelity with superconducting qubits. A 3D array of 72 neutral atoms has also been demonstrated (Barredo et al., 2018).

Table 3.3. Qubit types by formation and control parameters.

Qubit type	Qubit formation (DiVincenzo criterion #1)	Qubit control for computation (DiVincenzo criteria #2–5)
1.Superconducting circuits	Electrical circuit with oscillating current	Electromagnetic fields and microwave pulses
2.Trapped ions	Ion (atom stripped of one electron)	Ions stored in electromagnetic traps and manipulated with lasers
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