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Chapter 4
Advanced Quantum Computing: Interference and Entanglement
Abstract
The special properties of quantum objects (atoms, ions, and photons) are superposition, interference, and entanglement. Superposition refers to particles existing across all possible states simultaneously. Interference is the situation where intervention from noise in the environment damages the quantum object, and also the possibility that the wave functions of particles can either reinforce or diminish each other. Entanglement means that groups of particles are connected and can interact in ways such that the quantum state of each particle cannot be described independently of the state of the others even when the particles are separated by a large distance. One of the most important implications of entanglement is that qubits can be error-corrected, which will likely be necessary for the advent of universal quantum computing. An application of quantum computing that is already available is certifiably random bits, a proven source of randomness, which is used in secure cryptography.
4.1Introduction
One surprise is that there may be many more useful short-term applications of quantum computing with currently available NISQ devices than has been thought possible without full-blown universal quantum computers. NISQ devices are noisy intermediate-scale quantum devices (Preskill, 2018). For example, even near-term quantum computing devices may allow computations as elaborate as the simulation of quantum field theories (Jordan et al., 2012).
4.1.1 Quantum statistics
Quantum superposition, entanglement, and interference (SEI) properties come together in the discipline of quantum statistics. Quantum phenomena have a signature. They produce certain kinds of recognizable quantum statistical distributions that could only have come from quantum mechanical systems. This includes patterns from interference (through amplitudes), superposition (through qubit spins, such as in quantum annealing), and entanglement (through Bell pairs and otherwise). These quantum signatures are unique and identifiable. This is not surprising, given that quantum statistics means studying how wave functions behave in a quantum mechanical system, in a statistical format (i.e. distribution-based). The key point is that only a quantum system could have produced such output, and thus it can be used as a source of provable randomness.
As an indication of the unique signifiers of quantum phenomena, one relevant interpretation of quantum statistics is Porter–Thomas distribution. These are distributions in which the probabilities themselves are exponentially distributed random variables. The quantum statistics are known and have been developed elsewhere in physics to model quantum many-body systems. The practical application is that quantum statistical distributions can be set up to generate either predictable patterns or randomness. In particular, many applications require a guaranteed source of randomness. True randomness instills trust in believing that events have been fairly determined. Some of the immediate applications for randomness are cryptography (setting the parameters for a system that cannot be back-doored or otherwise breached), and blockchains more generally, both in cryptography and in facilitating the creation of next-generation consensus algorithms (PBFT) based on entropy. Other uses for randomness include running lotteries (picking numbers fairly) and auditing election results (selecting precincts to review at random).
At present, most randomness is not guaranteed to be random, and a potential trend would be the widespread use of quantum computers to generate guaranteed randomness for use in various security applications.
4.2Interference
In physics, wave interference is a phenomenon in which two waves in a system have either a reinforcing or canceling effect upon one another. There can be positive coherence if the two waves are in the same phase, reinforcing each other in a stronger way, as in the ocean when multiple big waves come into shore at once. Alternatively, there can be negative coherence if waves are in phases that counterpose or cancel each other out, such as when there is noise from the environment.
Interference is used in building quantum circuits and calculating with vectors in quantum computing. A quantum circuit harnesses the qubit wave action with matrix multiplications (linear algebra). Each time a vector (corresponding to qubit position) is multiplied by a matrix (the computational movement through the quantum gate system), the matrix combines numbers in the vector, and the combination either reinforces the numbers or cancels them out. In this real physical sense, coherent wave behavior is calculated as a vector passing through quantum gates. This is an important factor that is competing against the fact that coherent wave behavior is the environmental noise of the system. The coherent action of the waves is fragile, and can be easily destroyed if the system has too much noise or other interference.
This is a challenge in quantum computing because irrespective of the qubit-generation method (superconducting circuits, trapped ions, topological matter, etc.), there is always going to be noise in the system, and if the noise overwhelms the coherent wave activity, the quantum computer is not going to work. Hence, quantum error correction becomes important for mitigating the noise.
4.2.1 Interference and amplitude
The wave behavior of qubits and interference is seen in modeling coherent wave action through quantum gates (protecting against noise in quantum circuit design), and also in another property of the quantum mechanical domain, amplitude. Whereas probabilities are assigned to the different possible states of the world in classical systems, amplitudes are the analog in quantum systems. Amplitudes are more complicated than probabilities, in that they can interfere destructively and cancel each other out, be complex numbers, and not sum to one. A quantum computer is a device that maintains a state that is a superposition of every configuration of qubits measured in amplitude. For practical computation, the amplitudes are converted into probabilities (probability is the squared absolute value of its amplitude). A key challenge is figuring out how to obtain a quantum speed advantage by exploiting amplitudes. This is not as straightforward as using the superposition property of qubits to model a greater number of possibilities, since simply measuring random configurations will not