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Conference on Lasers and Electro-Optics

Photonic quantum computing: Shor’s algorithm and the road to fault-tolerance

B. P. Lanyon, Till J. Weinhold, Nathan K. Langford, Marco Barbieri, Marcelo Pereira de Almeida, Alexei Gilchrist, Daniel F. V. James, Andrew G. White

Published May 4, 2008• 5 citations
Publisher

Abstract

This work describes the implementation of a compiled version of Shor's algorithm in a photonic system by utilizing single photons and by employing the nonlinearity induced by measurement. This study also demonstrates the core processes, coherent control, and resultant entangled states that are required in a full-scale implementation of Shor's algorithm. Results show that the performance of quantum algorithm is not the same as the performance of the underlying quantum circuit, and stress the importance of developing techniques for characterising quantum algorithms.

DOI
10.1109/qels.2008.4553035

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This work describes the implementation of a compiled version of Shor's algorithm in a photonic system by utilizing single photons and by employing the nonlinearity induced by measurement. This study also demonstrates the core processes, coherent control, and resultant entangled states that are required in a full-scale implementation of Shor's algorithm. Results show that the performance of quantum algorithm is not the same as the performance of the underlying quantum circuit, and stress the importance of developing techniques for characterising quantum algorithms.
Shor's powerful quantum algorithm for factoring represents a major challenge in quantum computation. Here, we implement a compiled version in a photonic system. For the first time, we demonstrate the core processes, coherent control, and resultant entangled states required in a full-scale implementation. These are necessary steps on the path towards scalable quantum computing. Our results highlight that the algorithm performance is not the same as that of the underlying quantum circuit and stress the importance of developing techniques for characterizing quantum algorithms.
The experiments described in this thesis form an investigation into the path towards establishing the requirements of quantum computing in a linear optical system. Our qubits are polarisation encoded photons for which the basic operations of quantum computing, single qubit rotations, are a well understood problem. The difficulty lies in the interaction of photons. To achieve these we use measurement induced non-linearities. The first experiment in this thesis describes the thorough characterisation of a controlled-sign gate based on such non-linearities. The photons are provided as pairs generated through parametric down-conversion, and as such share correlations unlikely to carry over into large scale implementations of the future. En route to such larger circuits, a characterisation of the actions of the controlled-sign gate is conducted, when the input qubits have been generated independently from each other, revealing a large drop in process fidelity. To explore the cause of this degradation of the gate performance a thorough and highly accurate model of the gate is derived including the realistic description of faulty circuitry, photon loss and multi-photon emission by the source. By simulating the effects of the various noise sources individually, the heretofore largely ignored multi-photon emission is identified as the prime cause of the degraded gate performance, causing a drop in fidelity nearly three times as large as any other error source. I further draw the first comparison between the performance of an experimental gate to the error probabilities per gate derived as thresholds for fault-tolerant quantum computing. In the absence of a single vigourous threshold value, I compare the gate performance to the models that yielded the highest threshold to date as an upper bound and to the threshold of the Gremlin-model, which allows for the most general errors. Unsurprisingly this comparison reveals that the implemented gate is clearly insufficient, however just remedying the multi-photon emission error will allow this architecture to move to within striking distance of the boundary for fault-tolerant quantum computing. The utilised methodology can be applied to any gate in any architecture and can, combined with a suitable model of the noise sources, become an important guide for developments required to achieve fault tolerant quantum computing. The final experiment on the path towards linear optical quantum computing is the demonstration of a pair of basic versions of Shor's algorithm which display the essential entanglement for the algorithm. The results again highlight the need for extensive measurements to reveal the fundamental quality of the implemented algorithm, which is not accessible with limited indicative measurements. In the second part of the thesis, I describe two experiments on other forms of entanglement by extending the actions of a Fock-State filter, a filter that is capable of attenuating single photon states stronger than multi-photon states, to produce entangled states. Furthermore this device can be used in conjunction with standard wave-plates to extend the range of operations possible on the bi-photonic qutrit space, showing that this setup suffices to produce any desired qutrit state, thereby giving access to new measurement capabilities and in the process creating and proving the first entanglement between a qubit and a qutrit.
We have developed an integrated waveguide approach to photonic quantum circuits for high performance, miniaturization and scalability. We have begun to address the challenges of scaling up quantum circuits using new insights into how controlled operations can be efficiently realised, and demonstrated Shor's algorithm with consecutive CNOT gates and the iterative phase estimation algorithm. We have shown how quantum circuits can be reconfigured, using thermo-optic phase shifters to realise a highly reconfigurable quantum circuit able to perform almost any function on two photonic qubits, and electro-optic phase shifters in lithium niobate to rapidly manipulate the path and polarisation of telecomm wavelength single photons. We have addressed miniaturisation using multimode interference coupler architectures to directly implement NxN Hadamard operations and the `Boson sampling problem', and by using high refractive index contrast materials such as SiOxNy, in which we have implemented quantum walks of correlated photons, and Si, in which we have demonstrated generation of orbital angular momentum states of light. We have incorporated microfluidic channels for the delivery of samples to measure the concentration of a blood protein with entangled states of light. We have begun to address the integration of superconducting single photon detectors and diamond and non-linear single photon sources. Finally, we give an overview of recent work on fundamental aspects of quantum measurement, including a quantum version of Wheeler's delayed choice experiment.
Quantum information science has shown that harnessing quantum mechanical effects can dramatically improve performance for certain tasks in communication, computation and measurement. Of the various physical systems being pursued, single particles of light - photons - are often the logical choice [1]. In addition to single photon sources and detectors, photonic quantum technologies will rely on sophisticated optical circuits [2]. Recently we reported the implementation of quantum optic integrated circuits, which not only dramatically reduces the footprint of quantum circuits, but allows unprecedented stability and higher performance. We demonstrated silica on silicon circuits that implement key components for quantum information, including CNOT gates [3] and single-qubit operations [4]. These components show promising progresses toward fault tolerance operation [5]. We also used integrated waveguides to implement a circuit that performs a compiled version of Shor's quantum algorithm [6] to factorize 15. Here we report the demonstration of circuits that extend the capabilities of the components already demonstrated, to take full advantage of the integrated optics architecture.
We report an experimental demonstration of a complied version of Shor's algorithm using four photonic qubits. We choose the simplest instance of this algorithm, that is, factorization of N=15 in the case that the period r=2 and exploit a simplified linear optical network to coherently implement the quantum circuits of the modular exponential execution and semiclassical quantum Fourier transformation. During this computation, genuine multiparticle entanglement is observed which well supports its quantum nature. This experiment represents an essential step toward full realization of Shor's algorithm and scalable linear optics quantum computation.
Photonics has been a promising platform for implementing quantum technologies owing to its scalability and robustness. In this paper, we demonstrate the encoding of information in 32 time bins or dimensions of a single photon. A practical scheme for manipulating the single photon in high dimensions is experimentally realized to implement a compiled version of Shor's algorithm on a single photon. Our work demonstrates the powerful information-processing capacity of a high-dimensional quantum system for complex quantum information tasks.