title: Sequential measurement in quantum learning
Creators: name: Spencer-Wood, Hector
Publisher: name: University of Glasgow
Year: 2024

description: In an increasingly quantum world with more and more quantum technologies nearing practical use, the importance of interacting directly with quantum data is becoming clear. Although doing so often leads to advantages, it also presents us with some uniquely quantum challenges: for example, information about a quantum system cannot, in general, be extracted without disturbing the state of the system. In this thesis, we primarily focus on how performing a learning task on quantum data disturbs it, and affects one’s ability to learn about it again in the future. In particular, we focus on the learning task of unsupervised binary classification, and how it affects quantum data when it is performed on a subset of it. In such a binary classification task, we are given a dataset that is made up of qubits that are each in one of two unknown pure states, and our aim is to cluster, with optimal probability of success, the data points into two groups based on their state. To investigate how well we can perform this task sequentially, we first consider a base case of a three-qubit dataset, made up of qubits that are each in one of two unknown states, and investigate how an intermediate classification on a two-qubit subset affects our ability to subsequently classify the whole dataset. We analytically derive and plot the tradeoff between the success rates of the two classifications and find that, although the intermediate classification does indeed affect the subsequent one in a non-trivial way, there is a remarkably large region where the first classification does not force the second away from its optimal probability of success. We then describe this scenario as a quantum circuit and simulate the tradeoff using Qiskit’s AerSimulator. Following on from this, we go on to investigate whether an intermediate measurement can leave a subsequent one unaffected in the more general setting of an n-qubit dataset, again made up of qubits that are each in one of two unknown states. We see that numerics hint that nothing about the order of the qubits in a (n − 1)-qubit dataset can be learnt without affecting a subsequent classification on the full dataset. We make steps to prove that this is indeed the case and show that an immediate consequence of this is that, for some m &gt; 1, a non-trivial intermediate classification on n−m qubits will always negatively affect a subsequent one on all n qubits. We conclude this line of work by deriving two bounds to how successful an intermediate classification of n − 1 qubits can be without affecting the following n-qubit one, hypothesising that one of these is optimal.
 
We then shift our focus to the field of indefinite causal order (ICO). Motivated by ICO’s connection to non-commutivity, we explore the idea of implementing quantum key distribution (QKD) in an indefinite causal regime. After showing that it is possible to share a key in an ICO, we find that, unlike other QKD protocols in the literature, eavesdroppers can be detected without publicly discussing a subset of the shared key. Indeed, we show that this is true for any individual attack in which the eavesdroppers abide by the causal structure chosen by the sharing parties. Further, we prove the security of this protocol for a subclass of these individual attacks. We then ask whether this “private detection” is a truly consequence of ICO and show that there is a definite causal ordered strategy that appears to yield the same phenomenon. Although we note that there are hints of some more subtle differences between the definite and indefinite causal cases, we conclude that carrying out QKD in an ICO is unlikely to offer any advantage, at least when considered in the form that we did. Finally, we close this thesis by summarising what we have found and noting some possible directions for future study.