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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Programming non-trivial algorithms in the measurement-based quantum computation model

Paul M. Alsing, A. Matthew Smith, Michael L. Fanto, Christopher C. Tison, Gordon E. Lott

Published Oct 13, 2014• 1 citation
Publisher

Abstract

We provide a set of prescriptions for implementing a quantum circuit model algorithm as measurement based quantum computing (MBQC) algorithm via a large cluster state. As means of illustration we draw upon our numerical modeling experience to describe a large graph state capable of searching a logical 8 element list (a non-trivial version of Grover's algorithm with feedforward). We develop several prescriptions based on analytic evaluation of cluster states and graph state equations which can be generalized into any circuit model operations. Such a resulting cluster state will be able to carry out the desired operation with appropriate measurements and feed forward error correction. We also discuss the physical implementation and the analysis of the principal 3-qubit entangling gate (Tooli) required for a non-trivial feedforward realization of an 8-element Grover search algorithm.

DOI
10.1117/12.2067490

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We provide a set of prescriptions for implementing a quantum circuit model algorithm as measurement based quantum computing (MBQC) algorithm via a large cluster state. As means of illustration we draw upon our numerical modeling experience to describe a large graph state capable of searching a logical 8 element list (a non-trivial version of Grover's algorithm with feedforward). We develop several prescriptions based on analytic evaluation of cluster states and graph state equations which can be generalized into any circuit model operations. Such a resulting cluster state will be able to carry out the desired operation with appropriate measurements and feed forward error correction. We also discuss the physical implementation and the analysis of the principal 3-qubit entangling gate (Tooli) required for a non-trivial feedforward realization of an 8-element Grover search algorithm.
We provide a set of prescriptions for implementing a quantum circuit model algorithm as measurement based quantum computing (MBQC) algorithm1, 2 via a large cluster state. As means of illustration we draw upon our numerical modeling experience to describe a large graph state capable of searching a logical 8 element list (a non-trivial version of Grover's algorithm3 with feedforward). We develop several prescriptions based on analytic evaluation of cluster states and graph state equations which can be generalized into any circuit model operations. Such a resulting cluster state will be able to carry out the desired operation with appropriate measurements and feed forward error correction. We also discuss the physical implementation and the analysis of the principal 3-qubit entangling gate (Toffoli) required for a non-trivial feedforward realization of an 8-element Grover search algorithm.
We provide a set of prescriptions for implementing a circuit model algorithm as measurement-based quantum computing algorithm via a large discrete cluster state constructed sequentially, from qubits implemented as single photons. We describe a large optical discrete graph state capable of searching logical 4 and 8 element lists as an example. To do so we have developed several prescriptions based on analytic evaluation of the evolution of discrete cluster states and graph state equations. We describe the cluster state as a sequence of repeated entanglement and measurement steps using a small number of single photons for each step. These prescriptions can be generalized to implement any logical circuit model operation with appropriate single-photon measurements and feed forward error corrections. Such a cluster state is not guaranteed to be optimal (i.e. minimum number of photons, measurements, run time).
We provide a set of prescriptions for implementing a circuit model algorithm as measurement based quantum computing via a large discrete cluster state constructed sequentially, from qubits implemented as single photons. We describe a large optical discrete graph state capable of searching logical 4 and 8 element lists as an example. To do so we have developed several prescriptions based on analytic evaluation of the evolution of discrete cluster states and graph state equations. We describe the cluster state as a sequence of repeated entanglement and measurement steps using a small number of single photons for each step. These prescriptions can be generalized to implement any logical circuit model operation with appropriate single photon measurements and feed forward error corrections. Such a cluster state is not guaranteed to be optimal (i.e. minimum number of photons, measurements, run time).
Measurement-based quantum computing (MBQC), an alternate paradigm for formulating quantum algorithms, can lead to potentially more flexible and efficient implementations as well as to theoretical insights on the role of entanglement in a quantum algorithm. Using the graph-theoretical ZX-calculus, we describe and apply a general scheme for reformulating quantum circuits as MBQC implementations. After illustrating the method using the two-qubit Deutsch–Jozsa algorithm, we derive a ZX graph-diagram that encodes a general MBQC implementation for the three-qubit Deutsch–Jozsa algorithm. This graph describes an 11-qubit cluster state on which single-qubit measurements are used to execute the algorithm. Particular sets of choices of the axes for the measurements can be used to implement any realization of the oracle. In addition, we derive an equivalent lattice cluster state for the algorithm.
Quantum subroutines, like their classical counterparts, serve as essential components of larger algorithms. They can be realized using gate based architectures or measurement based quantum computation (MBQC), where computation proceeds through adaptive measurements on highly entangled cluster states. In this work, we implement a phase oracle as a reusable subroutine within the MBQC framework using experimentally feasible cluster states. Unlike prior demonstrations of Grover's algorithm in MBQC that omit explicit oracle construction, our approach provides a transparent formulation of the oracle subroutine, including its measurement patterns, byproduct correlations, density-matrix simulations, and corresponding MBQC measurement circuits.
The Measurement Based Quantum Computation (MBQC) model achieves universal quantum computation by employing projective single qubit measurements with classical feedforward on a highly entangled multipartite cluster state. Rapid advances in improving scalability of quantum computing systems have enabled the generation of large cluster states for implementing MBQC on various platforms. This review focuses on three such efforts, each utilizing a different quantum computing technology viz., superconducting qubits, trapped ion qubits and squeezed photon states. MBQC is being increasingly employed on optical platforms which can generate large entangled resource states but lack the ability to perform deterministic entangling gates.