Measurement based quantum computation requires the generation of a cluster state (quantum resource) prior to starting a computation. Generation of this entangled state can be difficult with many schemes already proposed. We present an abstract scheme which can create 2D cluster states as a universal resource for quantum computing. We find a linear scaling of grid size with cluster depth. The scheme is also capable of creating more exotic topologies including 3D structures and the unit cell for topological error correction. We note its relevance to the cavity QED scheme in [30] although it could be applied to various architectures.
Measurement based quantum computation requires the generation of a cluster state (quantum resource) prior to starting a computation. Generation of this entangled state can be difficult with many schemes already proposed. We present an abstract scheme which can create 2D cluster states as a universal resource for quantum computing. We find a linear scaling of grid size with cluster depth. The scheme is also capable of creating more exotic topologies including 3D structures and the unit cell for topological error correction. We note its relevance to the cavity QED scheme in [30] although it could be applied to various architectures.
Measurement-based quantum computation offers exponential computational speed-up through simple measurements on a large entangled cluster state. We propose and demonstrate a scalable scheme for the generation of photonic cluster states suitable for universal measurement-based quantum computation. We exploit temporal multiplexing of squeezed light modes, delay loops, and beam-splitter transformations to deterministically generate a cylindrical cluster state with a two-dimensional (2D) topological structure as required for universal quantum information processing. The generated state consists of more than 30,000 entangled modes arranged in a cylindrical lattice with 24 modes on the circumference, defining the input register, and a length of 1250 modes, defining the computation depth. Our demonstrated source of two-dimensional cluster states can be combined with quantum error correction to enable fault-tolerant quantum computation.
Entanglement is one of the most mysterious and fleeting properties of nature. It is also a powerful resource for quantum information processing. Efforts to design and build a quantum computer are severely constrained by the difficulties of preserving useful qubit entanglement against the environment. Concatenated fault-tolerant error correction provides a framework to perform scalable computation, but is extremely expensive in terms of resources. Direct topological implementations promise robust entanglement and control, but require physical systems with as yet unobserved controllable topological phases. Here we present the first practical scheme for constructing a topological quantum computer based on a 3D cluster state lattice of photons on chip. In contrast to the resource requirements of other schemes, a target error rate of 10^(-16) requires approximately 3000 components per logical qubit. This constitutes a significant advance towards practical quantum computing and leads the way for topological architectures in other implementations.
Entanglement is the key resource for measurement-based quantum computing. It is stored in quantum states known as cluster states, which are prepared offline and enable quantum computing by means of purely local measurements. Universal quantum computing requires cluster states that are both large and possess (at least) a two-dimensional topology. Continuous-variable cluster states-based on bosonic modes rather than qubits-have previously been generated on a scale exceeding one million modes, but only in one dimension. Here, we report generation of a large-scale two-dimensional continuous-variable cluster state. Its structure consists of a 5- by 1240-site square lattice that was tailored to our highly scalable time-multiplexed experimental platform. It is compatible with Bosonic error-correcting codes that, with higher squeezing, enable fault-tolerant quantum computation.
Quantum computation is a new computational paradigm which can provide fundamentally faster computation than in the classical regime. This is dependent on finding efficient quantum algorithms for problems of practical interest. One of the most successful tools in developing new quantum algorithms is the quantum walk. In this thesis, we explore two applications of the discrete time quantum walk. In addition, we introduce an experimental scheme for generating cluster states, a universal resource for quantum computation. We give an explicit construction which provides a link between the circuit model of quantum computation, and a graph structure on which the discrete time quantum walk traverses, performing the same computation. We implement a universal gate set, proving the discrete time quantum walk is universal for quantum computation, thus confirming any quantum algorithm can be recast as a quantum walk algorithm. In addition, we study factors affecting the efficiency of the quantum walk search algorithm. Although there is a strong dependence on the spatial dimension of the structure being searched, we find secondary dependencies on other factors including the connectivity and disorder (symmetry). Fairly intuitively, as the connectivity increases, the efficiency of the algorithm increases, as the walker can coalesce on the marked state with higher probability in a quicker time. In addition, we find as disorder in the system increases, the algorithm can maintain the quantum speed up for a certain level of disorder before gradually reverting to the classical run time. Finally, we give an abstract scheme for generating cluster states. We see a linear scaling, better than many schemes, as doubling the size of the generating grid in our scheme produces a cluster state which is double the depth. Our scheme is able to create other interesting topologies of entangled states, including the unit cell for topological error correcting schemes.
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.
Topological measurement-based quantum computation (MBQC) enables one to carry out universal fault-tolerant quantum computation via single-qubit measurements with a family of large entangled states called cluster states as resources. Raussendorf's three-dimensional cluster states (RTCSs) based on the surface codes are mainly considered for topological MBQC. In such schemes, however, the logical Hadamard, phase (Z1/2), and T(Z1/4) gates which are essential for building up arbitrary logical gates are not implemented natively without using state distillation or lattice dislocations, to the best of our knowledge. In particular, state distillation generally consumes many ancillary logical qubits; thus it is a severe obstacle against practical quantum computing. To solve this problem, we suggest an MBQC scheme via a family of cluster states called color-code-based cluster states (CCCSs) based on the two-dimensional color codes instead of the surface codes. We define logical qubits, construct elementary logical gates, and describe error correction schemes. We show that all logical Clifford gates generated by the cnot, Hadamard, and phase gates can be implemented natively in a fault-tolerant manner, although the T gate still requires state distillation to be fault-tolerant. The error thresholds of MBQC via CCCSs for logical-Z errors are calculated to be 2.7%--2.8%, which are comparable to the values for RTCSs, assuming a simple error model where physical qubits have nontrivial errors independently with the same probability. We analyze and compare the resource overheads of both the schemes. In particular, we show that the number of physical qubits required for implementing a phase gate with CCCSs is at least about 26 times smaller than with RTCSs using state distillation, for the same code distance.