Quantum Physics

From Feynman diagrams via Penrose graphical notation to quantum circuits, graphical languages are widely used in quantum theory and other areas of theoretical physics. The category-theoretical approach to quantum mechanics yields a new set of graphical languages, which allow rigorous pictorial reasoning about quantum systems and processes. One such language is the ZX-calculus, which is built up of elements corresponding to maps in the computational and the Hadamard basis. This calculus is universal for pure state qubit quantum mechanics, meaning any pure state, unitary operation, and post-selected pure projective measurement can be represented. It is also sound, meaning any graphical rewrite corresponds to a valid equality when translated into matrices.

While the calculus is not complete for general quantum mechanics, I show that it is complete for stabilizer quantum mechanics and for the single-qubit Clifford+T group. This means that within those subtheories, any equality that can be derived using matrices can also be derived graphically.

The ZX-calculus can thus be applied to a wide range of problems in quantum information and quantum foundations, from the analysis of quantum non-locality to the verification of measurement-based quantum computation and error-correcting codes. I also show how to construct a ZX-like graphical calculus for Spekkens' toy bit theory and give its associated completeness proof.

Quantum Adiabatic Optimization proposes to solve discrete optimization problems by mapping them onto quantum spin systems in such a way that the optimal solution corresponds to the ground state of the quantum system. The standard method of preparing these ground states is using the adiabatic theorem, which tells us that quantum systems tend to remain in the ground state of a time-dependent Hamiltonian which transforms sufficiently slowly. In this talk I'll show that alternative strategies using non-adiabatic effects can in some cases be dramatically faster for instances which are hard for the traditional adiabatic method.

I will also discuss Simulated Quantum Annealing (SQA), which is a classical simulation of adiabatic optimization at non-zero temperature based on Path-Integral Quantum Monte Carlo. SQA is widely used in practice to study adiabatic optimization, but relatively little is known about the rate of convergence of the markov chain that underlies the algorithm. By focusing on a family of instances which adiabatic optimization solves in polynomial time, but require exponential time to solve using classical (thermal) simulated annealing, I will present numerical evidence as well as a work-in-progress proof that SQA can be exponentially faster than classical simulated annealing.

The Church-Turing thesis is one of the pillars of computer science; it postulates that every classical system has equivalent computability power to the so-called Turing machine. While this thesis is crucial for our understanding of computing devices, its implications in other scientific fields have hardly been explored. What if we consider the Church-Turing thesis as a law of nature? In this talk I will present our first results in connection with quantum information theory [1] by showing that computer science laws have profound implications for some of the most fundamental results of quantum theory.  First I will show how they question our knowledge on what a mixed quantum state is, as we identified situations in which ensembles of quantum states defining the same mixed state, indistinguishable according to the quantum postulates, do become distinguishable when prepared by a computer (or any classical system). Then I will introduce a new loophole for Bell-like experiments: if some of the parties in a Bell-like experiment use a computer to decide which measurements to make, then the computational resources of an eavesdropper have to be limited in order to have a proper observation of non-locality.