May 1713

We introduce two quantitative measures of the strength of causal relations. These two measures capture the maximum and minimum changes in a quantum system induced by changes in another system. We show that both measures possess important properties, such as continuity and faithfulness, and can be evaluated through optimization over orthogonal input states. For the maximum causal effect, we provide numerical lower bounds based on a variational algorithm, which can be used to estimate the strength of causal relations without performing a full quantum process tomography. To illustrate the application of our algorithm, we analyze two paradigmatic examples, the first involving a coherent superposition of direct cause and common cause and the second involving communication through a coherent superposition of two completely depolarizing channels.

This is the second in a series of “graphical grokking” papers in which we study how stabiliser codes can be understood using the ZX calculus. In this paper we show that certain complex rules involving ZX diagrams, called spider nest identities, can be captured succinctly using the scalable ZX calculus, and all such identities can be proved inductively from a single new rule using the Clifford ZX calculus. This can be combined with the ZX picture of CSS codes, developed in the first “grokking” paper, to give a simple characterisation of the set of all transversal diagonal gates at the third level of the Clifford hierarchy implementable in an arbitrary CSS code.

We study non-terminating graph rewriting models, whose local rules are applied non-deterministically — and yet enjoy a strong form of determinism, namely space-time determinism. Of course in the case of terminating computation it is well-known that the mess introduced by asynchronous rule applications may not matter to the end result, as confluence conspires to produce a unique normal form. In the context of non-terminating computation however, confluence is a very weak property, and (almost) synchronous rule applications is always preferred e.g. when it comes to simulating dynamical systems. Here we provide sufficient conditions so that asynchronous local rule applications conspire to produce well-determined events in the space-time unfolding of the graph, regardless of their application orders. Our first example is an asynchronous simulation of a dynamical system. Our second example features time dilation, in the spirit of general relativity.