Quantum Turing Machines and Quantum Prover-Verifier Interactions by Abel Molina Prieto A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Doctor of Philosophy in Computer Science (Quantum Information) Waterloo, Ontario, Canada, 2020 c Abel Molina Prieto 2020 Examining Committee Membership The following served on the Examining Committee for this thesis. The decision of the Examining Committee is by majority vote. External Examiner: Simon Perdrix Charg´ede Recherche, Institut des Sciences de l'Information et de Leurs Interactions, Centre National de la Recherche Scientifique Supervisor: John Watrous Professor, Cheriton School of Computer Science, University of Waterloo Internal Member: Richard Cleve Professor and IQC Chair, Cheriton School of Computer Science, University of Waterloo Internal-External Member: David Gosset Associate Professor, Department of Combinatorics & Optimization, University of Waterloo Other Member(s): Ashwin Nayak Professor, Department of Combinatorics & Optimization, Cross-Appointed Faculty, Cheriton School of Computer Science, University of Waterloo ii Author's Declaration I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. iii Abstract We present results on quantum Turing machines and on prover-verifier interactions. In our work on quantum Turing machines, we continue the line of research opened by Yao (1993), who proved that quantum Turing machines and quantum circuits are poly- nomially equivalent computational models: t ≥ n steps of a quantum Turing machine running on an input of length n can be simulated by a uniformly generated family of quan- tum circuits with size quadratic in t, and a polynomial-time uniformly generated family of quantum circuits can be simulated by a quantum Turing machine running in polynomial time. We then first revisit the simulation of quantum Turing machines with uniformly gen- erated quantum circuits, and present a variation on the simulation method employed by Yao together with an analysis of it. This analysis reveals that the simulation of quantum Turing machines can be performed by quantum circuits having depth linear in t, rather than quadratic depth, and can be extended easily to many variants of quantum Turing ma- chines, such as ones having multi-dimensional tapes. Our analysis is based on an extension of a method of Arrighi, Nesme, and Werner (2011) that allows for the localization of causal unitary evolutions, involving abstract lemmas that might be of independent interest. We also consider the more complex extension of our variant to the circuit simulation of multi-tape quantum Turing machines, where our variant provides a circuit with O(tk) size and O(tk−1) depth for the simulation of t steps of a machine with k tapes. This can be contrasted with the O(tk) depth corresponding to the generalization of Yao's simulation by Nishimura and Ozawa (2002). Our usage of abstract techniques regarding the localization of causal unitary evolutions allows again for a simplification of the algebraic manipulation aspects of the construction. We also discuss the further extension to the case of oracle quantum Turing machines. In our work on prover-verifier interactions, we first consider a protocol under the name of perfect/conclusive quantum state exclusion. This means to be able to discard with certainty at least one out of n possible quantum state preparations by performing a measurement of the resulting state. When all the preparations correspond to pure states and there are no more of them than their common dimension, it is an open problem whether POVMs give any additional power for this task with respect to projective measurements. This is the case even for the simple case of three states in three dimensions, which is discussed by Caves, Fuchs and Schack (2002) as unsuccessfully tackled. In our work, we give an analytical proof that in this case POVMs do indeed not give any additional power with respect to projective measurements. We also discuss possible generalizations of our work, including an application of Quadratically Constrained Quadratic Programming that might be of special interest. iv We additionally consider the problem of quantum hedging, a particular kind of quantum correlation that arises between parallel instances of prover-verifier interactions. M. and Watrous (2012) studied a protocol that exhibited a perfect form of quantum hedging, where the risk for the prover of losing a first game can completely offset the corresponding risk for a second game. We take a step towards a better understanding of this hedging phenomenon by giving a characterization of the prover's optimal behavior for a natural generalization of this protocol. Furthermore, we discuss how the usage of the logarithmic utility principle to analyze prover-verifier interactions could justify further study of quantum hedging. v Acknowledgements The research presented in this thesis was conducted under the supervision and with the collaboration of John Watrous. Thanks to him are due for numerous insightful con- versations, without which this research would not have been possible. I thank as well the home members of the committee Richard Cleve, David Gosset and Ashwin Nayak for their time and feedback during my completion of the PhD program. Thanks for his time and willingness to participate in the defense procedure are due as well to the external examiner Simon Perdrix. This research did also benefit from feedback and conversations with Juani Bermejo Vega, Jonathan Buss, Andrea Coladangelo, Alessandro Cosentino, Ronald de Wolf, Philippe Faist, Alex B. Grilo, Nicol´asGuar´ın-Zapata,Stacey Jeffery, Nathaniel Johnston, Artem Kaznatcheev, George Knee, Robin Kothari, Debbie Leung, Sanketh Menda, Alexandre Nolin, Christopher Perry, Jitendra Prakash, Daniel Puzzuoli, Burak S¸ahino˘glu, Luke Scha- effer, Jamie Sikora and Jon Tyson, as well as coauthors Srinivasan Arunachalam and Vincent Russo, and anonymous referees. The work presented here was partially conducted during visits hosted by the Center for Quantum Technologies in Singapore and the Institut de Recherche en Informatique Fondamentale in Paris. Thanks are due to hosts Rahul Jain and Fr´ed´ericMagniez, as well as to Ashwin Nayak and the rest of the team handling the regular collaboration between the Institute for Quantum Computing and these host institutions. This research was funded through the Natural Sciences and Engineering Research Coun- cil of Canada, the Mike and Ophelia Lazaridis Graduate Fellowship program, the David R. Cheriton Graduate Scholarship program and the University of Waterloo President's Graduate Scholarship program. vi Table of Contents List of Figures viii 1 Introduction1 1.1 Quantum Turing machines...........................1 1.2 Quantum prover-verifier interactions.....................3 1.3 Summary of results...............................4 1.3.1 Quantum Turing machines.......................4 1.3.2 Prover-verifier interactions.......................5 1.4 Notation.....................................5 I Quantum Turing machines7 2 Causality and locality8 2.1 Setting......................................9 2.2 Results...................................... 10 3 Single-tape quantum Turing machines 16 3.1 Deterministic Turing Machines......................... 16 3.1.1 Definition................................ 17 3.1.2 The classic Boolean circuit simulation of deterministic Turing machines 17 3.2 Quantum Turing machines........................... 22 vii 3.2.1 Definition................................ 22 3.2.2 Looped-tape quantum Turing machines................ 25 3.2.3 Other variants of quantum Turing machines............. 27 3.3 A variant of the simulation of quantum Turing machines by a quantum circuit 27 3.3.1 Registers in the simulation....................... 28 3.3.2 Operators in the simulation...................... 31 3.3.3 Locality and parallelism........................ 32 3.3.4 Behaviour of the local gate G ..................... 35 3.3.5 Recapitulation of the simulation procedure.............. 37 3.3.6 Complexity analysis.......................... 37 3.3.7 Differences with Yao's original simulation............... 41 3.3.8 Sensitivity to model choice....................... 43 3.4 Equivalence between unitarity and isometricity for QTM evolution operators 46 3.4.1 Setting.................................. 46 3.4.2 Result and proof............................ 47 3.4.3 Generalizations............................. 48 4 Multi-tape quantum Turing machines 49 4.1 Setting...................................... 49 4.2 Extension to standard multi-tape quantum Turing machines of our variant for the simulation of quantum Turing machines............... 51 4.2.1 Setup for extension........................... 51 4.2.2 Obstacle to naive proof for the extension............... 55 4.2.3 Making the extension work....................... 56 4.2.4 Parallelism and complexity....................... 58 4.3 Oracle quantum Turing machines....................... 62 4.3.1 Definition................................ 62 4.3.2 A first circuit simulation........................ 64 viii 4.3.3 More complex circuit simulations with more standard oracle gate models.................................. 67 4.3.4 Other models and further work.................... 76 II Quantum prover-verifier interactions 78 5 Quantum state exclusion