Quantum Computing. Melanie Swan

computation, the idea is to package the quantum mechanical matrix manipulations such that they run quantum states that are executed with a set of gates that offer the same kind of Boolean logic as in classical computing.

3.2.2 Bit and qubit

      In classical computing, the bit is the fundamental computational unit. The bit is an abstract mathematical entity that is either a 0 or a 1. Computations are constructed as a series of manipulations of 0s and 1s. In the physical world, a bit might be represented in terms of a voltage inside a computer, a magnetic domain on a hard disk, or light in an optical fiber. The qubit (quantum bit) is the equivalent system in quantum mechanics. The qubit is likewise an abstract mathematical entity (a logical qubit), existing in a superposition state of being both a 0 and a 1, until collapsed in the measurement at the end of the computation into being a classical 0 or 1. The qubit can be instantiated in different ways in the physical world. There are realizations of qubits in atoms, photons, electrons, and other kinds of physical systems. The quantum state of a qubit is a vector in a 2D space. This is a linear combination of the 1 and the 0 (the trajectory or probability that it is in the 1 or the 0 state). A model of computation can be built up by assigning states closer to the 0 as being 0 and states closer to the 1 as being 1 (when measured).

      A bit is always in a state of either 1 or 0. A qubit exists in a state of being both 1 and 0 until it is collapsed into a 1 or a 0 at the end of the computation. A bit is a classical object that exists in an electronic circuit register. A qubit is a quantum object (an atom, photon, or electron) that bounces around in a 3D space with a different probability of being at any particular place in the 3D sphere called a Hilbert space (and has vector coordinates in the X, Y, and Z directions). Figure 3.1 shows the physical space of the states of the bit and the qubit.

      The interpretation is that whereas a classical bit is either on or off (in the state of 1 or 0), a qubit can be on and off (1 and 0) at the same time, a property called superposition. One example of this is the spin of the electron in which the two levels can be understood as spin-up and spin-down. Another example is the polarization of a single photon in which the two states can be taken to be the vertical polarization and the horizontal polarization (single photons are often transmitted in communications networks on the basis of polarization). In a classical system, a bit needs to be in one state or the other. However, in a quantum mechanical system, the qubit can be in a coherent superposition of both states or levels of the system simultaneously, a property which is fundamental to quantum mechanics and indicates the greater potential range of computation in quantum systems.

      Figure 3.1. Potential states of bit and qubit.

      Compared to classical states, quantum states are much richer and have more depth. Superposition means that quantum states can have weight in all possible classical states. Each step in the execution of a quantum algorithm mixes the states into more complex superpositions. For example, starting with the qubit in a position of 0–0–0 leads to a superposition of 1–0–0, 1–0–1, and 1–1–1. Then each of the three parts of the superposition state branches out into even more states. This indicates the extensibility of quantum computers that could allow faster problem solving than is available in classical computers.

3.2.3 Creating qubits

      A qubit can be created in any quantum system which has two levels of energy that can be manipulated (Steane, 1997). Qubits can be conceived as being similar to harmonic oscillators at the macroscale. Physical systems that vibrate in a wave-like form between two levels of energy are called harmonic oscillators. Some examples include electrical circuits with oscillating current, sound waves in gas, and pendulums. Harmonic oscillators can be modeled as a wave function that cycles between the peak and trough energy levels. The same wave function concept is true at the quantum scale. In this sense, whenever there is a quantum system with two levels of energy, it can be said to be a qubit and possibly engaged as a two-state quantum device. This implies that there can be many different ways of building qubits. Hence, the method for creating qubits might be an engineering choice similar to the way that different methods have been used in classical computing for the physical implementation of logic gates (methods have ranged over time and included vacuum tubes, relays, and most recently integrated circuits).

3.3Quantum Hardware Approaches

3.3.1 The DiVincenzo criteria

      The DiVincenzo criteria have been proposed as standards that constitute the five elements of producing a well-formed quantum computer (DiVincenzo, 2000). The criteria are having (1) a scalable system of well-characterized qubits, (2) qubits that can be initialized with fidelity (typically to the zero state), (3) qubits that have a long-enough coherence time for the calculation (with low error rates), (4) a universal set of quantum gates (that can be implemented in any system), and (5) the capability of measuring any specific qubit in the ending result.

      There are several approaches to quantum computing (Table 3.1) (McMahon, 2018). Those with the most near-term focus are superconducting circuits, ion trapping, topological matter, and quantum photonics. Irrespective of the method, the objective is to produce quantum computing chips that perform computations with qubits, using a series of quantum logic gates that are built into quantum circuits, whose operation is programmed with quantum algorithms. Quantum systems may be accessed locally or as a cloud service. As of June 2019, one method is commercially available, which is superconducting circuits. Verification of computational claims is a considerable concern. External parties such as academic scientists are engaged to confirm, verify, and benchmark the results of different quantum systems, for example, for Google (Villalonga et al., 2019) and for IonQ (Murali et al., 2019).

Table 3.1. Quantum computing hardware platforms.

3.3.2 Superconducting circuits: Standard gate model

      The most prominent approach to quantum computing is superconducting circuits. Qubits are formed by an electrical circuit with oscillating current and controlled by electromagnetic fields. Superconductors are materials which have zero electrical resistance when cooled below a certain temperature. (In fact, it is estimated that more than half of the basic elements in the periodic table become superconducting if they are cooled to sufficiently low temperatures.) Mastering superconducting materials could be quite useful since as a general rule, about 20% of electricity is lost due to resistance. The benefit of zero electrical resistance for quantum computing is that electrons can travel completely unimpeded without any energy dissipation. When the temperature drops below the critical level, two electrons (which usually repel each another) form a weak bond and become a so-called Cooper pair that experiences no resistance when going through metal (tunneling) and which can be manipulated in quantum computing.

      Superconducting materials are used in quantum computing to produce superconducting circuits that look architecturally similar to classical computing circuits, but are made from qubits. There is an electrical circuit with oscillating current in the shape of a superconducting loop that has the circulating current and a corresponding magnetic field that can hold the qubits in place. Current is passed through the superconducting loop in both directions to create the two states of the qubit. More technically, the superconducting loop is a superconducting quantum interference device (SQUID) magnetometer (a device for measuring magnetic fields), which has two superconductors separated by thin insulating layers to form two parallel Josephson junctions. Josephson junctions are key to quantum computing because they are nonlinear superconducting inductors that create the energy levels needed to make a distinct qubit.

      Specifically, the nonlinearity of the Josephson inductance breaks the degeneracy of the energy-level