SmallTalks "Equivalence of continuous- and discrete-variable gate-based quantum computers with finite energy"

Abstract 

Quantum computers are computational devices that are expected to solve certain problems much faster than classical computers. The speedup comes from the fact that quantum computers can utilize phenomena from quantum physics like superposition, entanglement, and interference. This new form of information processing is expected to enable the quantum computer to display huge speedups for tasks such as integer factorization and simulation of chemical reactions.

For a quantum system to be a suitable basis in quantum computer, it must be able to store information and be manipulated to perform calculations. The quantum system that most resembles a classical bit, which is the smallest unit of information in a classical computer, is a two-level quantum system, which is known as a qubit. A classical bit can only take the value 0 or 1, but a qubit can exist in a superposition between 0 and 1. This means the qubit can exist in both states simultaneously until it is measured. Higher-dimensional systems can also be used to encode information, which is usually called a qudit. These approaches fall under the umbrella of discrete-variable quantum computing.

Quantum systems, however, offer more features than just a discrete number of levels. Other physical properties, such as the amplitude and phase (or position and momentum–like variables) of electromagnetic fields, can also be used to encode information. These degrees of freedom form a continuous, effectively infinite, basis. Quantum computers that utilize these types of quantum systems are known as continuous-variable quantum computers.

Although quantum computers are expected to outperform classical computers on certain tasks, a natural question to ask in light of the discrete- and continuous-variable approaches is: Is there a difference in the computational powers of discrete-variable versus continuous-variable quantum computers? It is straightforward to see that continuous-variable quantum computers are at least as powerful as discrete-variable ones since a continuous system can contain any finite-dimensional system as a subset. In other words, a discrete-variable encoding can always be embedded inside a continuous-variable system. In this work, we investigate the converse case: how well can continuous-variable quantum computers be simulated by discrete-variable quantum computers?

We prove that discrete-variable quantum computers can simulate realistic continuous-variable devices. We do so by creating a method to convert general continuous-variable states to discrete-variable ones. Similarly, we describe how to convert continuous-variable operations and measurements to their discrete-variable counterparts. By applying this method and examining the error contributed by each approximation, we find that the total error can always be made arbitrarily small. This means that using a realistic continuous-variable quantum computer will not give an advantage over using a discrete-variable device.