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Journal of Modern Optics

Translating non-trivial algorithms from the circuit model to the measurement-based quantum computing model

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

Published Mar 24, 2015• 1 citation
Publisher

Abstract

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).

DOI
10.1080/09500340.2015.1014437

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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).
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.
Photons are a natural resource in quantum information, and the last decade showed significant progress in high-quality single photon generation and detection. Furthermore, photonic qubits are easy to manipulate and do not require particularly strongly sealed environments, making them an appealing platform for quantum computing. With the one-way model, the vision of a universal and largescale quantum computer based on photonics becomes feasible. In one-way computing, the input state is not an initial product state∣0⟩⊗n\vert\boldsymbol{0}\rangle^{\otimes n}∣0⟩⊗n, but a socalled cluster state. A series of measurements on the cluster state's individual qubits and their temporal order, together with a feed-forward procedure, determine the quantum circuit to be executed. We propose a pipeline to convert a QASM circuit into a graph representation named measurement-graph (m-graph), that can be directly translated to hardware instructions on an optical one-way quantum computer. In addition, we optimize the graph using ZX-Calculus before evaluating the execution on an experimental discrete variable photonic platform.
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 computers promise to be more efficient and powerful than their classical counterparts. In the one-way quantum computer model [1], a sequence of measurements processes qubits, which are initially prepared in a highly entangled cluster state. We present here an optical implementation of a 4-qubit cluster state and we show how different algorithms can be experimentally realized by performing the corresponding sequence of measurements. The key advantage of this scheme over the standard network approach of quantum computing is that inherent, randomly induced measurement errors can classically be fed-forward and corrected by adapting the basis of subsequent measurement.