In this thesis we explore the possibility of creating continuousvariable \nquantum systems that are capable of supporting universal \nquantum computation. We begin by examining the measurement-based \nmodel, which employs sequences of measurements on highly entangled \nresource states, known as a cluster states. We suggest a method for \nthe construction of Gaussian cluster states based on ensembles of atoms \nand quantum non-demolition interactions. We then go on to expand \nour model to allow for the inclusion of light modes as part of the cluster. \nThis yields a new class of states, the composite cluster states. \nThis leads us to propose a new architecture for the measurement-based \nmodel that uses these composite clusters to increase resource e ciency \nand reduce computational errors. \nThe second part of this thesis concerns topological quantum computation. \nIn states exhibiting topological degrees of freedom, quantum \ninformation can be stored as a non-local property of the physical system \nand manipulated by braiding quasiparticles known as anyons. Here \nwe show how these ideas can be extended to continuous variables. We \nestablish a continuous variable analogue of the Kitaev toric code, show \nthat excitations correspond to continuous versions of Abelian anyons \nand investigate their behaviour under the condition of nite squeezing \nof the resource state. \nFinally, we expand our continuous variable topological model to \ninclude non-abelian excitations by constructing superpositions of CV \ntoric code anyons. We derive the fusion and braiding behaviour of \nthese non-abelian excitations and nd that they correspond to a CV \nanalog of Ising anyons. Using these resources, we go on to suggest \na computational scheme that encodes qubits within the fusion spaces \nof the CV Ising anyons and derive one- and two-qubit quantum gates \noperations that are implemented in a topological manner.
In this thesis we explore the possibility of creating continuousvariable \nquantum systems that are capable of supporting universal \nquantum computation. We begin by examining the measurement-based \nmodel, which employs sequences of measurements on highly entangled \nresource states, known as a cluster states. We suggest a method for \nthe construction of Gaussian cluster states based on ensembles of atoms \nand quantum non-demolition interactions. We then go on to expand \nour model to allow for the inclusion of light modes as part of the cluster. \nThis yields a new class of states, the composite cluster states. \nThis leads us to propose a new architecture for the measurement-based \nmodel that uses these composite clusters to increase resource e ciency \nand reduce computational errors. \nThe second part of this thesis concerns topological quantum computation. \nIn states exhibiting topological degrees of freedom, quantum \ninformation can be stored as a non-local property of the physical system \nand manipulated by braiding quasiparticles known as anyons. Here \nwe show how these ideas can be extended to continuous variables. We \nestablish a continuous variable analogue of the Kitaev toric code, show \nthat excitations correspond to continuous versions of Abelian anyons \nand investigate their behaviour under the condition of nite squeezing \nof the resource state. \nFinally, we expand our continuous variable topological model to \ninclude non-abelian excitations by constructing superpositions of CV \ntoric code anyons. We derive the fusion and braiding behaviour of \nthese non-abelian excitations and nd that they correspond to a CV \nanalog of Ising anyons. Using these resources, we go on to suggest \na computational scheme that encodes qubits within the fusion spaces \nof the CV Ising anyons and derive one- and two-qubit quantum gates \noperations that are implemented in a topological manner.
Topological quantum computation encodes quantum information in the internal fusion space of non-Abelian anyonic quasiparticles, whose braiding implements logical gates. This goes beyond Abelian topological order (TO) such as the toric code, as its anyons lack internal structure. However, the simplest non-Abelian generalizations of the toric code do not support universality via braiding alone. Here we demonstrate that such minimally non-Abelian TOs can be made universal by treating anyon fusion as a computational primitive. We prepare a 54-qubit TO wavefunction associated with the smallest non-Abelian group, S3, on Quantinuum's H2 quantum processor. This phase of matter exhibits cyclic anyon fusion rules, known to underpin universality, which we evidence by trapping a single non-Abelian anyon on the torus. We encode logical qutrits in the nonlocal fusion space of non-Abelian fluxes and, by combining an entangling braiding operation with anyon charge measurements, realize a universal topological gate set and read-out, which we further demonstrate by topologically preparing a magic state. This work establishes S3 TO as simple enough to be prepared efficiently, yet rich enough to enable universal topological quantum computation.
We describe how continuous-variable Abelian anyons, created on the surface of a continuous-variable analog of Kitaev's lattice model can be utilized for quantum computation. In particular, we derive protocols for the implementation of quantum gates using topological operations. We find that the topological operations alone are insufficient for universal quantum computation, which leads us to study additional nontopological operations such as offline squeezing and single-mode measurements. It is shown that these in conjunction with a non-Gaussian element allow for universal quantum computation using continuous-variable Abelian anyons.
The quantum fusion technique enables the bonding of two cluster states together, and therefore provides a promising method to grow large-scale cluster states. Quantum fusion was first proposed in a discrete-variable regime. In terms of the resource usage, quantum fusion can be divided into two categories, which are type-I and type-II fusions. Type-I fusion consumes one qubit in the fusion procedure, while type-II consumes two qubits. However, for continuous-variable (CV) analogs, concrete schemes for realizing quantum fusions of cluster states with arbitrary topologies have not been shown yet. In this paper, we present the CV analog of quantum fusion schemes. We show that CV type-I fusion can be realized by an ancilla and two quantum nondemolition interactions, and CV type-II fusion can be realized by an entanglement swapping scheme. With finite squeezing considered, we analyze the extra errors added by the fusion procedures in CV cluster-state quantum computation. Our schemes are universal and can be generalized to fuse cluster states with arbitrary topologies.
Quantum computers offer impressive computational speed-ups over their present-day (classical) counterparts. In the measurement-based model, quantum computation is driven by single-site measurements on a large entangled quantum state known as a cluster state. This thesis explores extensions of the measurement-based model for quantum computation in qubit and continuous-variable systems. Within the qubit setting, we consider the task of characterizing how well a small-scale measurement-based quantum device can perform logic gates. We adapt a pre-existing scheme known as randomized benchmarking into the setting of measurement-based quantum computation on a one-dimensional cluster state. A key feature of randomized benchmarking is that it uses random sequences of gates. We show how the intrinsic randomness of measurement-based quantum computation can be harnessed when implementing them. Within the continuous-variable setting, we consider optical cluster states that can be generated with current technology. We propose a compact method for generating universal cluster states based on optical-parametric-oscillator technology. We consider how finite squeezing effects manifest in computation and show that pre-existing measurement-based protocols are suboptimal. We propose new measurement-based protocols that have better noise properties, compactness, and circuit flexibility. As an application, we introduce a measurement-based method for implementing interferometry. In this model, the finite squeezing noise can be dealt with as a photon-loss process. Building further on this work, we investigate the resource requirements of a measurement-based boson-sampling device, proving simultaneous efficiency in time, space, and squeezing (energy) resources. These results offer new insights into how to build, use, and characterize a measurement-based quantum computer.
Quantum simulations are an invaluable tool in condensed matter physics, enabling the precise generation and control of physical phenomena abstracted from the complexity of the underlying many-body system. At the heart of such complexity lies entanglement, the intrinsically quantum correlations that have attracted renewed interest as a fundamental resource in quantum computation and information. In this thesis, we explore the use of one- and two-dimensional lattice models to simulate and harness such correlations for next-generation quantum technologies. We begin in one dimension with a spin-chain model that allows complete control over bipartite correlation structures. Specifically, we introduce a model of nearest-neighbour couplings and transverse field terms, whose exact solution yields a q-deformed rainbow ground state formed from a tensor product of q-deformed singlets, each with directly tunable entanglement. This framework acts as a universal simulator of free-fermion entanglement spectra, with numerical studies confirming high-fidelity realisations across a range of examples. We then turn to strong correlations in two dimensions, where topologically ordered systems host exotic quasiparticles known as anyons. Braiding non-Abelian anyons enacts unitary transformations on a logically encoded fusion space, offering a natural pathway to fault-tolerant quantum computation. Using Kitaev’s quantum double model D(G), we focus on the smallest non-Abelian instance, D(S3), realised on a lattice of d=6 qudits. We develop a minimal protocol in which braiding and fusion are simulated solely through operators that create and measure anyons, demonstrating that D(S3) anyons generate magic states, a vital resource for universal quantum computation. Finally, a dense encoding scheme is presented, rendering the minimal realisation of this protocol accessible with current quantum technologies and a circuit model for realisation within the framework of linear optics is presented. Notably, error analysis of this encoding scheme reveals an inherent passive protection provided by the topological encoding, extending beyond conventional notions of topological fault tolerance.
Quantum measurements often destroy the “quantumness” of a state. Combining measurements and entanglement, however, unlocks new opportunities in quantum computing, as it transcends the boundaries between quantum phases. Measurement-based quantum computation is a perfect example: from an array of locally entangled qubits, individual measurements teleport and modify some logical information. Since quantum operations are prone to errors, this information is encoded in the topological Toric code, thus enabling fault-tolerant computation. Since the Toric code has a 2-dimensional structure, the resource state needed to perform measurement-based quantum computation is necessarily 3-dimensional and is called the 3D cluster state. In this thesis, we provide new tools to enlarge the capabilities of topological measurement-based quantum computation and reduce its operational cost. First, we introduce a lattice defect in the underlying cluster state that improves the expressivity of the computation. Second, we develop general ideas towards circuit compaction and quantify the resources used by the new methods. Furthermore, here, the Toric code is created thanks to local measurements, while it cannot be created with local unitary operations only. We investigate the mathematical roots of this intriguing property, and we show that long-range entanglement arises from locally measuring some general short-range entangled state. Specifically, we assume that the initial short-range entangled state is in a non-trivial symmetry protected topological phase for an abelian group on an infinite spin chain. That is, there is a global operation generated by local unitary operators labeled by the element of the group that leaves the state invariant. On infinite quantum spin chains, these symmetry protected topological phases have been completely characterized. We then perform on-site measurements on increasing intervals of the chain. This creates a sequence of measured states that are eigenstates of the local unitaries associated with elements of a subgroup of the symmetry group. Using the algebraic properties of the initial state, we show that this sequence of measured states has increasing long-range correlations. The entanglement bounds necessarily deteriorate with the size of the measured intervals; thus, the measured states cannot be uniformly short-range entangled.