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arXiv:1003.1545 · Lecture notes in computer science

Simple Sets of Measurements for Universal Quantum Computation and Graph State Preparation

Yasuhiro Takahashi

Published Jan 1, 2011• 2 citations
arXivHTMLPDF
DOI
10.1007/978-3-642-18073-6_3

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We consider the problem of minimizing resources required for universal quantum computation using only projective measurements. The resources we focus on are observables, which describe projective measurements, and ancillary qubits. We show that the set of observables {Z ⊗ X, ( cos θ)X + ( sin θ)Y all θ ∈ [0, 2π)} with one ancillary qubit is universal for quantum computation. The set is simpler than a previous one in the sense that one-qubit projective measurements described by the observables in the set are ones only in the (X, Y) plane of the Bloch sphere. The proof of the universality immediately implies a simple set of observables that is approximately universal for quantum computation. Moreover, the proof implies a simple set of observables for preparing graph states efficiently.
The direct compilation of algorithm-specific graph states in measurement-based quantum computation (MBQC) can lead to resource reductions in terms of circuit depth, entangling gates, and even the number of physical qubits. In this work, we extend previous studies on algorithm-tailored graph states to periodic sequences of generalized Pauli rotations, which commonly appear in, e.g., Trotterized Hamiltonian simulation. We first implement an enhanced simulated-annealing-based algorithm to find optimal periodic graph states within local-Clifford equivalent MBQC resources. In addition, we derive a novel scheme for the preparation of resource states based on a graph state and a ladder of CNOT gates, which we term anticommutation-based MBQC, since it uncovers a direct relationship between the graph state and the anticommutation matrix for the set of Hamiltonians generating the computation. We also deploy our two approaches to derive universal resource states from minimal universal sets of generating Hamiltonians. Finally, we demonstrate and compare both of our methods based on various examples from condensed matter physics and universal quantum computation.
Measurement based (MB) quantum computation allows for universal quantum computing by measuring individual qubits prepared in entangled multipartite states, known as graph states. Unless corrected for, the randomness of the measurements leads to the generation of ensembles of random unitaries, where each random unitary is identified with a string of possible measurement results. We show that repeating an MB scheme an efficient number of times, on a simple graph state, with measurements at fixed angles and no feedforward corrections, produces a random unitary ensemble that is an $\ensuremath{\varepsilon}$-approximate ttt design on nnn qubits. Unlike previous constructions, the graph is regular and is also a universal resource for measurement based quantum computing, closely related to the brickwork state.