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Requirements on Nonlinear Optical Quantum Gates

Mingyin Patrick Leung

Published Jul 1, 2009• 0 citations
OpenAlex

Abstract

Quantum information science has shown that computers which exploit the quantum nature of particles, namely quantum computers, can outperform contemporary computers in some computational tasks. The fundamental building blocks of a quantum computer are quantum logical gates and quantum bits (qubits). Previous research has shown that the optical approach to quantum computing is promising. However, linear optical quantum computing (LOQC) schemes require a huge amount of resource, which makes large scale LOQC impractical, and hence there have been renewed interests in nonlinear optical quantum computing schemes, where less resource is required. The performance of these quantum gates depends on the properties of the nonlinear media. However, requirements on some of the properties for high performance quantum gates are not fully known. This thesis intends to bridge this gap of knowledge and examines the necessary conditions on several types optical nonlinearities that are common in two-qubit quantum gates schemes. These types of nonlinearities are, namely two-photon absorption, $\chi^{(2)}$ nonlinearity and $\chi^{(3)}$ cross-Kerr nonlinearity. The two-photon absorption based quantum Zeno gate is modeled in this thesis. It is shown that for practical absorbers, the photon loss significantly lowers the quantum fidelity of the Zeno gate. Nevertheless, this thesis proposes to use the Zeno gate for fusing optical cluster states. With the best theoretical estimate of single photon loss in the absorbers, the Zeno gate can outperform linear optical schemes. This thesis also proposes to embed the Zeno gate in the teleportation-type of two-qubit gate, namely GC-Zeno gate, such that the success rate of the gate can be traded off for higher gate fidelity. The effect of some mode matching error and detector inefficiency on the GC-Zeno gate are also considered here. It is shown that the photon loss requirement as well as the mode matching requirement are both stringent for having a fault tolerant GC-Zeno gate. This thesis models some of the properties of a $\chi^{(3)}$ optical medium and explores how they affect the fidelity of the cross-Kerr nonlinearity based quantum gate. This thesis shows that for a cross-Kerr medium with fast time response but negligible wave dispersion, the medium would induce spectral entanglement between the input photons and this significantly lowers the fidelity of the quantum gate. Nevertheless, when the dispersion has a stronger effect than the time response, and if phase noise is negligible, it is possible to achieve a quantum gate with high fidelity. However, the noise is actually significant, and this thesis suggests that spectral filtering can be applied to prohibit the occurrence of the noise. The requirements on employing optical $\chi^{(2)}$ nonlinearity for quantum computing are also examined. This study models the spectral effects of a $\chi^{(2)}$ medium on its efficiency. It is shown in this thesis that since the Hamiltonian of the medium does not commute at different times, the unitary operation should be modeled by a Dyson series, which leads to undesired spectral entanglement that lowers the efficiency of the medium. However, in the case of periodical poling, the unitary operation can be modeled by a Taylor series, where under some phase matching conditions, the medium can have a high efficiency. Furthermore, this thesis proposes a Bell measurement scheme and a quantum gate scheme based on $\chi^{(2)}$ nonlinearity that can always outperform linear optics even when the nonlinearity strength is weak. In the case of sufficiently strong nonlinearity, a quantum gate with high success rate can be achieved. In summary, this thesis models some of the properties of two-photon absorbers, $\chi^{(2)}$ nonlinearity and $\chi^{(3)}$ nonlinearity, and shows that it is possible to achieve the conditions required for high performance quantum gates, however these conditions are experimentally challenging to meet.

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According to idealized models, a strong Kerr non-linearity may be used to build optical quantum gates for optical quantum information processing by inducing conditional phase shifts on quantum states. Recently, Shapiro (PRA 73, 062305 (2006)) argued that for a Kerr medium with non-instantaneous but fast response, essentially no phase shift is induced on two-single-photon input states, and thus a quantum gate build from such a medium cannot work. Here we show that a fast response Kerr medium induces some but very little phase shifts on a two-single-photon input state, and it is insufficient for high fidelity quantum computation. We point out that this is caused by the medium imparting spectral entanglement to the input photons. We further show that a way to circumvent this problem and achieve a high fidelity gate, is to engineer the dispersion properties of the medium to give a dominant spectral effect over the non-instantaneous response, in addition to satisfying a phase matching condition.

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Quantum information science has shown that computers which exploit the quantum nature of particles, namely quantum computers, can outperform contemporary computers in some computational tasks. The fundamental building blocks of a quantum computer are quantum logical gates and quantum bits (qubits). Previous research has shown that the optical approach to quantum computing is promising. However, linear optical quantum computing (LOQC) schemes require a huge amount of resource, which makes large scale LOQC impractical, and hence there have been renewed interests in nonlinear optical quantum computing schemes, where less resource is required. The performance of these quantum gates depends on the properties of the nonlinear media. However, requirements on some of the properties for high performance quantum gates are not fully known. This thesis intends to bridge this gap of knowledge and examines the necessary conditions on several types optical nonlinearities that are common in two-qubit quantum gates schemes. These types of nonlinearities are, namely two-photon absorption, χ(2)\chi^{(2)}χ(2) nonlinearity and χ(3)\chi^{(3)}χ(3) cross-Kerr nonlinearity. The two-photon absorption based quantum Zeno gate is modeled in this thesis. It is shown that for practical absorbers, the photon loss significantly lowers the quantum fidelity of the Zeno gate. Nevertheless, this thesis proposes to use the Zeno gate for fusing optical cluster states. With the best theoretical estimate of single photon loss in the absorbers, the Zeno gate can outperform linear optical schemes. This thesis also proposes to embed the Zeno gate in the teleportation-type of two-qubit gate, namely GC-Zeno gate, such that the success rate of the gate can be traded off for higher gate fidelity. The effect of some mode matching error and detector inefficiency on the GC-Zeno gate are also considered here. It is shown that the photon loss requirement as well as the mode matching requirement are both stringent for having a fault tolerant GC-Zeno gate. This thesis models some of the properties of a χ(3)\chi^{(3)}χ(3) optical medium and explores how they affect the fidelity of the cross-Kerr nonlinearity based quantum gate. This thesis shows that for a cross-Kerr medium with fast time response but negligible wave dispersion, the medium would induce spectral entanglement between the input photons and this significantly lowers the fidelity of the quantum gate. Nevertheless, when the dispersion has a stronger effect than the time response, and if phase noise is negligible, it is possible to achieve a quantum gate with high fidelity. However, the noise is actually significant, and this thesis suggests that spectral filtering can be applied to prohibit the occurrence of the noise. The requirements on employing optical χ(2)\chi^{(2)}χ(2) nonlinearity for quantum computing are also examined. This study models the spectral effects of a χ(2)\chi^{(2)}χ(2) medium on its efficiency. It is shown in this thesis that since the Hamiltonian of the medium does not commute at different times, the unitary operation should be modeled by a Dyson series, which leads to undesired spectral entanglement that lowers the efficiency of the medium. However, in the case of periodical poling, the unitary operation can be modeled by a Taylor series, where under some phase matching conditions, the medium can have a high efficiency. Furthermore, this thesis proposes a Bell measurement scheme and a quantum gate scheme based on χ(2)\chi^{(2)}χ(2) nonlinearity that can always outperform linear optics even when the nonlinearity strength is weak. In the case of sufficiently strong nonlinearity, a quantum gate with high success rate can be achieved. In summary, this thesis models some of the properties of two-photon absorbers, χ(2)\chi^{(2)}χ(2) nonlinearity and χ(3)\chi^{(3)}χ(3) nonlinearity, and shows that it is possible to achieve the conditions required for high performance quantum gates, however these conditions are experimentally challenging to meet.
The use of photons as qubits is a promising implementation for quantum computation. The inability of photons to interact, especially with the environment, makes them an ideal physical candidate. However, this also makes them a difficult system to perform two qubit gates on. Recent breakthroughs in photonic quantum computing have shown methods around the requirement of direct photon-photon interaction. In this thesis we study three recently discovered schemes for optical quantum computation. \n \n We first investigate the so called linear optical quantum computing (LOQC) scheme, exploring a method to improve the original proposal by constructing a photon-number QND detector that succeeds with a high probability. In doing this we present a new type of LOQC teleporter, one that can detect the presence of a single photon in an arbitrary polarisation state when the input state is a sum of vacuum and multi-photon terms. This new type of teleporter is an improvement on the original scheme in that the entangled states required can be made offline with fewer entangling operations. \n \n We next investigate the so called quantum bus (qubus) scheme for photonic quantum computing. We show a scheme to measure the party of n qubit states by using a single qubus mode, controlled rotations and displacements. This allows for the syndrome measurements of any stabilizer quantum error correcting code. We extend these results to a fault tolerant scheme to measure an arbitrary Pauli operator of weight n, incorporating so called single bit teleportations. We investigate the construction of a Toffoli gate by using a single qubus mode, controlled rotations and displacements that works with a success probability of at least 25%. We also investigate the use of single bit teleportations to construct a universal set of gates on coherent state type logic and in the construction of cluster states. \n \nWe finally investigate the optical Zeno gate, a gate that uses the Zeno effect in the form of two photon absorbers to induce a CSIGN. We model realistic two photon absorption and include realistic single photon loss for this gate, examining the use of encoding to overcome the single photon loss, showing that our results are competitive with the leading equivalent LOQC scheme.
We model an optical implementation of a controlled-SIGN gate that makes use of the quantum Zeno effect [Franson et al., Phys. Rev. A 70, 062302 (2004)] in the presence of photon loss. Gate operation is severely affected. However, we show that by using photon loss codes proposed for linear optical quantum computation (LOQC), performance is greatly enhanced, outperforming LOQC equivalents. The technique can be applied to other types of nonlinearities, making the implementation of nonlinear optical gates much more attractive.
Quantum computing is an exciting field that promises great improvements to our ability to solve certain intractable classes of problem. Implementing a quantum computer will require a physical system in which quantum effects are strongly evident. Many different physical systems have been proposed for this task, including ion traps, quantum dots, neutral atoms and photons. This thesis focuses on optical implementations of a quantum computer, and specifically on single-photon designs, as opposed to those utilising coherent states. Photonic systems have shown a great deal of promise for quantum computing tasks since it was shown by Knill, Laflamme and Milburn (KLM) that scalable efficient quantum computing could be performed using only linear optical elements and measurement. This was made viable through the use of error-correcting codes. The work in this thesis covers the use of error-correcting codes in single-photon linear optics quantum computing (LOQC). The central feature of these schemes is the parity code introduced by KLM and used to overcome the gate errors that are inherent in this type of LOQC. First is a discussion of an improved scheme that incorporates the fusion gates used in Browne and Rudolph's cluster state proposal in order to simplify the encoding and gate procedures. This results in high-probability gates with a resource usage comparable to the cluster state schemes. The remainder of the thesis examines the application of further error correcting codes to handle other errors that may arise in any practical implementation of an optical quantum computer. The most significant type of error in any optical system is photon loss, and hence an extension of the parity code is presented that allows loss errors to be detected and corrected. Procedures for performing universal quantum computation whilst employing this encoding are also described. These enable the establishment of loss thresholds for parity-state based LOQC, revealing this code is capable of handling up to 10\% loss in all elements during general computation, or 17\% if it is used solely as an active memory. Although loss is the most common type of optical error, other types of error will also become an issue when attempting to produce large quantum computing systems. An analysis of the error thresholds and resource usage which can be expected when applying a general error correcting code such as the Steane code to this type of LOQC system is also presented. The error thresholds and resource requirements established are greater than those found for cat state optical quantum computing, but fall below those calculated for cluster states. However, the resource requirements of the scheme are significantly smaller than the cluster state requirements for an equivalent circuit. Finally, a new experiment is proposed, achievable with today's technology, for testing the loss-tolerant encoding described earlier. Expansions of the experiment that would demonstrate tolerance of actual experimental error are also discussed.
According to idealized models, a strong Kerr non-linearity may be used to build optical quantum gates for optical quantum information processing by inducing conditional phase shifts on quantum states. Recently, Shapiro (PRA 73, 062305 (2006)) argued that for a Kerr medium with non-instantaneous but fast response, essentially no phase shift is induced on two-single-photon input states, and thus a quantum gate build from such a medium cannot work. Here we show that a fast response Kerr medium induces some but very little phase shifts on a two-single-photon input state, and it is insufficient for high fidelity quantum computation. We point out that this is caused by the medium imparting spectral entanglement to the input photons. We further show that a way to circumvent this problem and achieve a high fidelity gate, is to engineer the dispersion properties of the medium to give a dominant spectral effect over the non-instantaneous response, in addition to satisfying a phase matching condition.
In principle the Zeno effect controlled-sign gate of Franson et al (2004 Phys. Rev. A 70 062302) is a deterministic two-qubit optical gate. However, when realistic values of photon loss are considered its fidelity is significantly reduced. Here we consider the use of measurement based quantum processing techniques to enhance the operation of the Zeno gate. With the help of quantum teleportation, we show that it is possible to achieve a Zeno CNOT gate (GC-Zeno gate) that gives (near) unit fidelity and moderate probability of success of 0.8 with a two-photon to one-photon absorption ratio κ = 10 4 . We include some mode-mismatch effects and estimate the bounds on the mode overlap and κ for which fault tolerant operation would be possible.
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.