Physics’ Evolution Toward Computing

Mike Stannett

Takashi Mihara

A computation is an evolution from one physical state to another, and so are Today’s computers, i.e., the computations of them are changes of electric signals. What are differences of quantum computers from classical computers? The computational principles of classical computers are based on classical physics. In principle, we can construct them from devices based on classical physics

Pierre-Louis Giscard, Omid Faizy Namarvar, Christian Bonhomme

We present novel, exotic types of frame changes for the calculation of quantum evolution operators. We detail in particular the biframe, in which a physical system's evolution is seen in an equal mixture of two different standard frames at once. We prove that, in the biframe, convergence of all series expansions of the solution is quadratically faster than in `conventional' frames. That is, if in laboratory frame or after a standard frame change the error at order $n$ of some perturbative series expansion of the evolution operator is on the order of $ε^n$, $0<ε<1$, for a computational cost $C(n)$ then it is on the order of $ε^{2n+1}$ in the biframe for the same computational cost. We demonstrate that biframe is one of an infinite family of novel frames, some of which lead to higher accelerations but require more computations to set up initially, leading to a trade-off between acceleration and computational burden.

Michael J. Dinneen, Nies, André, Khoussainov, Bakhadyr

Smith, Isaac David

Any computational process is built from a sequence of components. In this context, there are at least two ways in which “component” can be interpreted: physically and logically. The former can be understood as referring to the hardware for the computation, i.e., the physical substrate where the computation “takes place”, while the latter can be understood as referring to the software, i.e., the mathematical mapping of inputs to outputs. These two interpretations are by no means distinct: each logical component is implemented as the evolution of a physical component in some state. Accordingly, the logical requirements of a computation also specify the minimal physical requirements of the computational device. In quantum computational processes, the physical components are constructed from materials whose states and evolutions are described by quantum mechanics. At the logical level, this permits the use of quantum information theory: states are represented by vectors in a Hilbert space and evolutions are specified via unitary dynamics and measurements. In measurement-based quantum computational processes, the latter form of evolution, i.e. that driven by measurements, is most prominent. In this context, certain quantum states taken on the quality of a computational resource. That is to say, it is the combination of state and measurement together that produce the desired evolution for a given logical component. This thesis treats several topics related to the preparation and use of such resource states. In particular, much of the thesis will be concerned with states that support universal quantum computation; that is, for which any quantum computation is possible within the framework of MBQC. The first contribution in this regard consists in establishing a connection between a different framework of universal computation, namely that of parity quantum computing, and MBQC. This connection relates to both preparation and use since the former was developed in the context of quantum optimisation (a use) and utilises a specific type of encoding of quantum states (a preparation). The next sections consider the requirements under which specific sets of logical components suffice for universal quantum computation. These sets commonly arise in quantum information theory, and as state-measurement combinations in MBQC and parity quantum computing in particular. In addition to establishing results on the sizes of such sets, this thesis investigates the consequences for parity quantum computing and MBQC, including, e.g., considerations for the design of physical devices and the compilation of computations. Finally, several limits to the preparation and use of states for MBQC are investigated. In one case, an information-theoretic quantity is considered which can be used to quantify uncertainty regarding some piece of information. In the context of MBQC, the piece of information can pertain to, e.g., the choice of computation to perform on a given state, such as in the context of blind quantum computing. Thereafter, a method is developed capable of witnessing the inability to prepare various states under certain constraints that are relevant in, e.g., the context of quantum networks.