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数理解析研究所講究録

On Beam Splittings and Quantum Logical Gate on Fock Space (New Development of Infinite-Dimensional Analysis and Quantum Probability)

Wolfgang Freudenberg, Masanori Ohya, Noboru Watanabe

Published Apr 1, 2000• 0 citations
OpenAlex

Abstract

In usual computer, there is an upper bound of computational speed because of irreversibility of logical gate.In order to avoid this demerit, Fredkin and Tof- foli [3] proposed a conservative logical gate.After that, Milburn [4] constructed a physical model of reversible quantum gate with beam splittings and a Kerr medium.This model is called FTM (Fredkin $-$ Toffoli $-$ Milburn gate) in this paper.This FTM gate was described by the quantum channel and the efficiency of information transmission of the FTM gate was discussed in [10].FTM gate is using a photon number state as an input state for control gate.The photon number state might be difficult to realize physically.In this paper, we introduced a new device on symmetric Fock space in order to avoid this difficulty.In Section 1, we briefly review noisy quantum channels and generalized beam splittings.In Section 2, we explain the quantum channel for FTM gate based on the notation of Section 1.In Section 3, we intruduced a new operator on symmetric Fock space and discuss the truth table for our new gate.

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The theory of logical gates in quantum computation has suggested new forms of quantum logic, called quantum computational logics. The basic semantic idea is the following: the meaning of a sentence α is identified with a quantum information quantity, represented by a density operator of a Hilbert space, whose dimension depends on the logical complexity of α. At the same time, the logical connectives of the language are interpreted as operations defined in terms of quantum logical gates. Standard quantum computational models can be described as special cases of Fock space models, where the meaning of any sentence is localized in a precise sector of a Fock space ℱ. From an intuitive point of view, the increasing number of particles described in the different sectors of ℱ can be interpreted as increasing information.

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In classical computer, there exist inevitable demerits for discussing log- ical gates.One of the demerits is an irreversibility of logical gates, that is the AND and the OR gates.This property causes to the restriction of computtional speed for the classical computer.$\mathrm{T}\mathrm{h}\dot{\mathrm{e}}$ re are several kind of ap- proaches for avoiding these demetits.One of these approaches is proposed by Feynman [1].He proved that every logical gates can be contructed by combining with only two reversible gates, i.e., the NOT and the Controlled NOT (CNOT) gates.By the way, there are several approaches for realizing quantum logical gates.One of those approaches is the study by means of a nuclear magnetic 1

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This thesis covers several topics in optical quantum computation. A quantum computer is a computational device which is able to manipulate information by performing unitary operations on some physical system whose state can be described as a vector (or mixture of vectors) in a Hilbert space. The basic unit of information, called the qubit, is considered to be a system with two orthogonal states, which are assigned logical values of 0 and 1. Photons make excellent candidates to serve as qubits. They have little interactions with the environment. Many operations can be performed using very simple linear optical devices such as beam splitters and phase shifters. Photons can easily be processed through circuit-like networks. Operations can be performed in very short times. Photons are ideally suited for the long-distance communication of quantum information. The great difficulty in constructing an optical quantum computer is that photons naturally interact weakly with one another. This thesis first gives a brief review of two early approaches to optical quantum computation. It will describe how any discrete unitary operation can be performed using a single photon and a network of beam splitters, and how the Kerr effect can be used to construct a two photon logic gate. Second, this work provides a thorough introduction to the linear optical quantum computer developed by Knill, Laflamme, and Milburn. It then presents this author's results on the reliability of this scheme when implemented using imperfect photon detectors. This author finds that quantum computers of this sort cannot be built using current technology. Third, this dissertation describes a method for constructing a linear optical quantum computer using nearly orthogonal coherent states of light as the qubits. It shows how a universal set of logic operations can be performed, including calculations of the fidelity with which these operations may be accomplished. It discusses methods for reducing and correcting errors and recovering from failed operations. Lastly it describes an analysis of the long distance transmission of the coherent state qubits and shows how transmission errors can be corrected.

•Sep 15, 2022•0 citations

Implementation of a quantum Fredkin gate using an entanglement resource

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Summary form only given. Photonic qubit logic provides flexible, low noise scheme for quantum information processing. However, implementation of increasing scale requires numbers of resources beyond what is readily available. A recent demonstration [1] has shown how to add a control operation to existing unitary transformations, providing a practical method for moving to medium-scale quantum circuits. Here we use the entanglement-based version of this scheme to implement a complex quantum gate - the Fredkin, or 3-qubit swap gate - with significantly less resources than its decomposition into a 2-qubit gate array.In the technique proposed in [1], the Hilbert space is expended in order to have a four-level system for each qubit. Each system is composed of a qubit sub-system on which the target operation acts, and a second qubit sub-system, which remains unaffected. The control is added by gates that switch between the different subsystems according to the control qubit. Despite its conceptual simplicity, this technique still uses too many gates for practical implementations with current photonic technology. A more feasible way is a version that replaces the gates with pre-existing entanglement between the gates with qubit system and the second subsystem. The experiment is based on four possible photon pairs -called x, o, s and e - with photons either only in x and e, leading to the swapping, or only in o and s, leading to no swapping (Fig. 1). Each qubit is encoded twice and the two last modes are only used for triggering. The coincidences and the polarization beam splitter on two first mode ensure that the possibilities for the origin of the photons are linked to the state of the control qubit.The four possible photon pairs are encoding in the polarization mode coming from two polarization entangling down-conversion crystals. One crystal give a photon pair either in modes x or o and the second one a pair in mode s or e . We then separate spatially the polarization mode using beam displacer or Sagnac interferometers to preserve the phase stability (Fig . 2) . From the simulation we expect a fidelity around 85% for the truth table.

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A quantum Fredkin gate (Conference Presentation)

Raj B. Patel, Joseph Ho, Franck Ferreyrol +2 more

One of the greatest challenges in modern science is the realisation of quantum computers which, as their scale increases, will allow enhanced performance of tasks across many areas of quantum information processing. Quantum logic gates play a vital role in realising these applications by carrying out the elementary operations on the qubits; a key aim is minimising the resources needed to build these gates into useful circuits. While the salient features of a quantum computer have been shown in proof-of-principle experiments, e.g., single- and two-qubit gates, difficulties in scaling quantum systems to encode and manipulate multiple qubits has hindered demonstrations of more complex operations. This is exemplified by the classical Fredkin (or controlled-SWAP) gate [1] for which, despite many theoretical proposals [2,3] relying on concatenating multiple two-qubit gates, a quantum analogue has yet to be realised. Here, by directly adding control to a two-qubit SWAP unitary [4], we use photonic qubit logic to report the first experimental demonstration of a quantum Fredkin gate [5]. Our scheme uses linear optics and improves on the overall probability of success by an order of magnitude over previous proposals [2,3]. This optical approach allows us to add control an arbitrary black-box unitary which is otherwise forbidden in the standard circuit model [6]. Additionally, the action of our gate exhibits quantum coherence allowing the generation of the highest fidelity three-photon GHZ states to date. The quantum Fredkin gate has many applications in quantum computing, quantum measurements [7] and cryptography [8,9]. Using our scheme, we apply the Fredkin gate to the task of direct measurements of the purity and state overlap of a quantum system [7] without recourse to quantum state tomography.

•Dec 7, 2016•0 citations
In usual computer, there is an upper bound of computational speed because of irreversibility of logical gate.In order to avoid this demerit, Fredkin and Tof- foli [3] proposed a conservative logical gate.After that, Milburn [4] constructed a physical model of reversible quantum gate with beam splittings and a Kerr medium.This model is called FTM (Fredkin −-− Toffoli −-− Milburn gate) in this paper.This FTM gate was described by the quantum channel and the efficiency of information transmission of the FTM gate was discussed in [10].FTM gate is using a photon number state as an input state for control gate.The photon number state might be difficult to realize physically.In this paper, we introduced a new device on symmetric Fock space in order to avoid this difficulty.In Section 1, we briefly review noisy quantum channels and generalized beam splittings.In Section 2, we explain the quantum channel for FTM gate based on the notation of Section 1.In Section 3, we intruduced a new operator on symmetric Fock space and discuss the truth table for our new gate.
The theory of logical gates in quantum computation has suggested new forms of quantum logic, called quantum computational logics. The basic semantic idea is the following: the meaning of a sentence α is identified with a quantum information quantity, represented by a density operator of a Hilbert space, whose dimension depends on the logical complexity of α. At the same time, the logical connectives of the language are interpreted as operations defined in terms of quantum logical gates. Standard quantum computational models can be described as special cases of Fock space models, where the meaning of any sentence is localized in a precise sector of a Fock space ℱ. From an intuitive point of view, the increasing number of particles described in the different sectors of ℱ can be interpreted as increasing information.
In classical computer, there exist inevitable demerits for discussing log- ical gates.One of the demerits is an irreversibility of logical gates, that is the AND and the OR gates.This property causes to the restriction of computtional speed for the classical computer.The˙\mathrm{T}\mathrm{h}\dot{\mathrm{e}}The˙ re are several kind of ap- proaches for avoiding these demetits.One of these approaches is proposed by Feynman [1].He proved that every logical gates can be contructed by combining with only two reversible gates, i.e., the NOT and the Controlled NOT (CNOT) gates.By the way, there are several approaches for realizing quantum logical gates.One of those approaches is the study by means of a nuclear magnetic 1
This thesis covers several topics in optical quantum computation. A quantum computer is a computational device which is able to manipulate information by performing unitary operations on some physical system whose state can be described as a vector (or mixture of vectors) in a Hilbert space. The basic unit of information, called the qubit, is considered to be a system with two orthogonal states, which are assigned logical values of 0 and 1. Photons make excellent candidates to serve as qubits. They have little interactions with the environment. Many operations can be performed using very simple linear optical devices such as beam splitters and phase shifters. Photons can easily be processed through circuit-like networks. Operations can be performed in very short times. Photons are ideally suited for the long-distance communication of quantum information. The great difficulty in constructing an optical quantum computer is that photons naturally interact weakly with one another. This thesis first gives a brief review of two early approaches to optical quantum computation. It will describe how any discrete unitary operation can be performed using a single photon and a network of beam splitters, and how the Kerr effect can be used to construct a two photon logic gate. Second, this work provides a thorough introduction to the linear optical quantum computer developed by Knill, Laflamme, and Milburn. It then presents this author's results on the reliability of this scheme when implemented using imperfect photon detectors. This author finds that quantum computers of this sort cannot be built using current technology. Third, this dissertation describes a method for constructing a linear optical quantum computer using nearly orthogonal coherent states of light as the qubits. It shows how a universal set of logic operations can be performed, including calculations of the fidelity with which these operations may be accomplished. It discusses methods for reducing and correcting errors and recovering from failed operations. Lastly it describes an analysis of the long distance transmission of the coherent state qubits and shows how transmission errors can be corrected.
Summary form only given. Photonic qubit logic provides flexible, low noise scheme for quantum information processing. However, implementation of increasing scale requires numbers of resources beyond what is readily available. A recent demonstration [1] has shown how to add a control operation to existing unitary transformations, providing a practical method for moving to medium-scale quantum circuits. Here we use the entanglement-based version of this scheme to implement a complex quantum gate - the Fredkin, or 3-qubit swap gate - with significantly less resources than its decomposition into a 2-qubit gate array.In the technique proposed in [1], the Hilbert space is expended in order to have a four-level system for each qubit. Each system is composed of a qubit sub-system on which the target operation acts, and a second qubit sub-system, which remains unaffected. The control is added by gates that switch between the different subsystems according to the control qubit. Despite its conceptual simplicity, this technique still uses too many gates for practical implementations with current photonic technology. A more feasible way is a version that replaces the gates with pre-existing entanglement between the gates with qubit system and the second subsystem. The experiment is based on four possible photon pairs -called x, o, s and e - with photons either only in x and e, leading to the swapping, or only in o and s, leading to no swapping (Fig. 1). Each qubit is encoded twice and the two last modes are only used for triggering. The coincidences and the polarization beam splitter on two first mode ensure that the possibilities for the origin of the photons are linked to the state of the control qubit.The four possible photon pairs are encoding in the polarization mode coming from two polarization entangling down-conversion crystals. One crystal give a photon pair either in modes x or o and the second one a pair in mode s or e . We then separate spatially the polarization mode using beam displacer or Sagnac interferometers to preserve the phase stability (Fig . 2) . From the simulation we expect a fidelity around 85% for the truth table.
One of the greatest challenges in modern science is the realisation of quantum computers which, as their scale increases, will allow enhanced performance of tasks across many areas of quantum information processing. Quantum logic gates play a vital role in realising these applications by carrying out the elementary operations on the qubits; a key aim is minimising the resources needed to build these gates into useful circuits. While the salient features of a quantum computer have been shown in proof-of-principle experiments, e.g., single- and two-qubit gates, difficulties in scaling quantum systems to encode and manipulate multiple qubits has hindered demonstrations of more complex operations. This is exemplified by the classical Fredkin (or controlled-SWAP) gate [1] for which, despite many theoretical proposals [2,3] relying on concatenating multiple two-qubit gates, a quantum analogue has yet to be realised. Here, by directly adding control to a two-qubit SWAP unitary [4], we use photonic qubit logic to report the first experimental demonstration of a quantum Fredkin gate [5]. Our scheme uses linear optics and improves on the overall probability of success by an order of magnitude over previous proposals [2,3]. This optical approach allows us to add control an arbitrary black-box unitary which is otherwise forbidden in the standard circuit model [6]. Additionally, the action of our gate exhibits quantum coherence allowing the generation of the highest fidelity three-photon GHZ states to date. The quantum Fredkin gate has many applications in quantum computing, quantum measurements [7] and cryptography [8,9]. Using our scheme, we apply the Fredkin gate to the task of direct measurements of the purity and state overlap of a quantum system [7] without recourse to quantum state tomography.