Abstract
We introduce a new family of models for measurement-based quantum computation which are deterministic and approximately universal. The resource states which play the role of graph states are prepared via 2-qubit gates of the form exp(−iπ2nZ⊗Z). When n=2, these are equivalent, up to local Clifford unitaries, to graph states. However, when n>2, their behaviour diverges in two important ways. First, multiple applications of the entangling gate to a single pair of qubits produces non-trivial entanglement, and hence multiple parallel edges between nodes play an important role in these generalised graph states. Second, such a state can be used to realise deterministic, approximately universal computation using only Pauli Z and X measurements and feed-forward. Even though, for n>2, the relevant resource states are no longer stabiliser states, they admit a straightforward, graphical representation using the ZX-calculus. Using this representation, we are able to provide a simple, graphical proof of universality. We furthermore show that for every n>2 this family is capable of producing all Clifford gates and all diagonal gates in the n-th level of the Clifford hierarchy.
Highlights
Measurement-based quantum computation (MBQC) describes a family of alternatives to the quantum circuit model, where one starts with a highly entangled resource state and performs all computation via measurements
While individual measurements introduce non-determinism, a deterministic quantum computation can be recovered by allowing future operations to depend on past measurement outcomes, a concept known as feedforward
Graph states are a family of stabiliser states that can be described by an undirected graph, where each edge represents a pair of qubits entangled via a controlled-Z operation
Summary
Measurement-based quantum computation (MBQC) describes a family of alternatives to the quantum circuit model, where one starts with a highly entangled resource state and performs all computation via measurements. One can consider hypergraph states [13], a generalisation of graph states produced by multi-qubit n-controlled-Z operations, represented graphically as hyper-edges These were recently shown to admit a universal model of computation using Pauli measurements and feed-forward [31]. We introduce a new family of generalisations of graph states which admit universal deterministic computation using only Pauli X and Z measurements and feed-forward. The diagrams keep track of the extra (Clifford) errors introduced by propagating errors forward, and it enables us to derive a technique for performing Pauli and Clifford corrections purely by means of single-qubit measurement choices in the bases {X, Z} This yields a measurement-based model which is very flexible. It is a topic of active research to extend these techniques to hypergraph-based models, where the role of the ZX-calculus is played by the recently-introduced ZH-calculus [3]
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