Abstract:
Quantum computer includes optical systems arranged in series each of the plurality of optical systems includes first half-wave plate, first polarizing beam splitter, first switching mirror, first photodetector, first polarization rotator, optical cavity which contains atom, second switching mirror, second photodetector, second polarization rotator, and high reflection mirror, first polarization beam splitter outputting third light beam received from first switching mirror or second switching mirror to adjacent one of optical systems, third switching mirrors each provided between adjacent two optical systems, each of third switching mirrors reflecting or transmitting light beam output from one of two optical systems, light sources each providing light beam to corresponding optical system, and measurement system which measures polarization of incoming light beam.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application is based upon and claims the benefit of priority from prior Japanese Patent Application No. 2005-026654, filed Feb. 2, 2005, the entire contents of which are incorporated herein by reference. 
     BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention relates to a quantum computer and quantum computation method utilizing coupling of an optical cavity with an atom. 
     2. Description of the Related Art 
     L.-M. Duan et al. have proposed a new method for realizing a controlled phase flip gate (see, for example, L.-M. Duan and H. J. Kimble, Phys. Rev. Lett. 92, 127902 (2004)). In this proposal, quantum bits are expressed by polarization of photons. However, in quantum computers, it is preferable to express quantum bits using atomic states that are more stable and easily usable as memories. 
     In light of this, Y.-F. Xiao et al. have proposed a controlled phase flip gate in which a change in light beam intensity in a cavity due to strong coupling between the cavity and an atom is utilized like the method of Duan, but the quantum bits are expressed by ground states of the atoms (see, for example, Y.-F. Xiao, X.-M. Lin., J. Gao, Y. Yang, Z.-F. Han, and C.-C. Guo, Phys. Rev. A 70, 042314 (2004); and Y.-F. Xiao, Z.-F. Han, Y. Yang, and C.-C. Guo, Phys. Lett. A 330, 137 (2004)). 
     Xiao et al. state in these papers that the methods are scalable since they exhibit a low error rate. 
     However, Xiao et al. suggest nothing about the specific structure of a quantum computer containing three or more quantum bits. Moreover, even if a researcher in this technical field has mastered the methods proposed by Xiao et al., it is still not obvious for them to contrive any specific structure of the quantum computer containing three or more quantum bits. 
     BRIEF SUMMARY OF THE INVENTION 
     In accordance with a first aspect of the invention, there is provided a quantum computer comprising: 
     a plurality of optical systems arranged in series each of the plurality of optical systems comprising:
         a first half-wave plate which receives a light beam;   a first polarizing beam splitter which receives a light beam passing through the first half-wave plate;   a first switching mirror which reflects or transmits a first light beam transmitted by the first polarizing beam splitter;   a first photodetector which detects the first light beam transmitted by the first switching mirror;   a first polarization rotator which receives the first light beam reflected by the first switching mirror;   an optical cavity which receives the first light beam passing through the first polarization rotator and contains an atom;   a second switching mirror which reflects or transmits a second light beam reflected by the first polarizing beam splitter;   a second photodetector which detects the second light beam transmitted by the second switching mirror;   a second polarization rotator which receives the second light beam reflected by the second switching mirror; and   a high reflection mirror which receives the second light beam passing through the second polarization rotator and reflects the received light beam in a direction opposite to an incident direction of the received light beam, the first polarization beam splitter outputting a third light beam received from the first switching mirror or the second switching mirror to adjacent one of the optical systems;       

     a plurality of third switching mirrors each provided between adjacent two optical systems, each of the third switching mirrors reflecting or transmitting a light beam output from one of the two optical systems; 
     a plurality of light sources each providing the light beam to the corresponding optical system; and 
     a measurement system which measures polarization of an incoming light beam, the measurement system comprising:
         a second half-wave plate which receives the incoming light beam output from the last-stage optical system, the last-stage optical system arranged at an down end of the plurality of optical systems:   a second polarizing beam splitter which receives the incoming light beam passing through the second half-wave plate; and   a pair of third and fourth photodetectors, the third photodetector detecting the incoming light beam reflected by the second polarizing beam splitter, the fourth photodetector detecting the incoming light beam transmitted by the second polarizing beam splitter.       

     In accordance with a second aspect of the invention, there is provided a quantum computer comprising: 
     a plurality of optical systems arranged in series each of the plurality of optical systems comprising:
         a first half-wave plate which receives the light beam;   a first polarizing beam splitter which receives a light beam passing through the first half-wave plate;   a first switching mirror which reflects or transmits a first light beam transmitted by the first polarizing beam splitter;   a first photodetector which detects the first light beam transmitted by the first switching mirror;   a first polarization rotator which receives the first light beam reflected by the first switching mirror;   an optical cavity which receives the first light beam passing through the first polarization rotator and contains an atom;   a second switching mirror which reflects or transmits a second light beam reflected by the first polarizing beam splitter;   a second photodetector which detects the second light beam transmitted by the second switching mirror;   a second polarization rotator which receives the second light beam reflected by the second switching mirror; and   a high reflection mirror which receives the second light beam passing through the second polarization rotator and reflects the received light beam in a direction opposite to an incident direction of the received light beam, the first polarization beam splitter outputting a third light beam received from the first switching mirror or the second switching mirror to adjacent one of the optical systems; and       

     a plurality of third switching mirrors each provided between adjacent two optical systems, each of the third switching mirrors reflecting or transmitting a light beam output from one of the two optical systems. 
    
    
     
       BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWING 
         FIG. 1  is a view illustrating an optical system for performing a CZ gate operation on atomic and photonic quantum bits; 
         FIG. 2  is a view illustrating the energy level structure of an atom having three energy levels; 
         FIG. 3  is a view illustrating a quantum circuit equivalent to a CZ gate acting on quantum bit  1  and quantum bit  2 , which uses additional quantum bit  3 ; 
         FIG. 4  is a view illustrating an optical system for performing a CZ gate operation on two atomic quantum bits; 
         FIG. 5  is a view illustrating an optical system giving extensibility to a quantum computer according to an embodiment; 
         FIG. 6  is a view illustrating the quantum computer of the embodiment; 
         FIG. 7  is a view useful in explaining how to realize a CZ gate by the quantum computer of  FIG. 6 ; 
         FIG. 8  is a view illustrating how to realize a switching mirror using a ring cavity; 
         FIG. 9  is a view illustrating how to realize a switching mirror using an etalon; 
         FIG. 10  is a view useful in explaining a method for reading a quantum bit in the quantum computer of  FIG. 6 ; 
         FIG. 11  is a view illustrating a quantum-circuit expression for  FIG. 10 ; 
         FIG. 12  is a view illustrating the energy level structure of an atom having four energy levels; 
         FIG. 13  is a view illustrating a quantum computer for performing a CZ gate operation on atomic and photonic quantum bits, according to example 1; 
         FIG. 14  is a view illustrating the energy level of a Pr +3  ion doped in Y 2 SiO 5  crystal; 
         FIG. 15  is a view illustrating light beam frequencies for performing the Raman transition; 
         FIG. 16  is a view illustrating a quantum computer, according to example 2, for performing a CZ gate operation on atomic and photonic quantum bits, utilizing a single-photon pulse; 
         FIG. 17  is a view illustrating a quantum computer for performing a CZ gate operation on two atomic quantum bits, according to example 3; 
         FIG. 18  is a view illustrating a quantum computer, according to example 4, in which etalons are used as the switching mirrors; and 
         FIG. 19  is a view illustrating a quantum computer, according to example 5, for performing a CZ gate operation on quantum bits corresponding to two atoms contained in cavities that are not adjacent to each other. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     the embodiments of the present invention have been developed in light of the above problems, and aim to provide a quantum computer and quantum computation method that are extensible in the number of quantum bits. 
     Quantum computers and methods according to embodiments of the invention will be described in detail, referring to the accompanying drawings. 
     Firstly, an explanation will be given of fundamental matters related to the quantum computers and methods of the embodiments. 
     If an atom is contained in an optical cavity and is strongly coupled with the cavity, when a light beam having a resonant frequency of the cavity enters the cavity, the intensity of the intracavity light is quite different from that in the case where such an atom does not exist. 
     In general, the strong coupling between a cavity and an atom means the coupling constant g is greater than the damping rate κ of the cavity and the relaxation rate γ of the atom (g&gt;κ, γ). 
     Explaining this phenomenon in more detail, an incident light cannot enter the cavity when an atom strongly coupled with the cavity exists in the cavity, while the incident light enters the cavity when such an atom does not exist. The intensity of the incident light beam is limited. If the intensity is too high, the intensity change of the intracavity light is not caused by such an atom. Therefore, the intensity of the incident light beam must be set to as a low level as enables the intensity change to be caused. As described later, by utilizing this phenomenon, a controlled phase flip gate acting on a quantum bit expressed by polarization of a photon and a quantum bit expressed by the ground states of an atom can be realized. 
     The controlled phase flip gate (also called a CZ gate) is a universal gate, and is used together with one-quantum-bit gates to perform an arbitrary quantum computation. The CZ gate performs the following transformation.
 
α 00 |0&gt;|0&gt;+α 01 |0&gt;|1&gt;+α 10 |1&gt;|0&gt;+α 11 |1&gt;|1&gt;→α 00 |0&gt;|0&gt;+α 01 |0&gt;|1&gt;+α 10 |1&gt;|0&gt;−α 11 |1&gt;|1&gt;  (1)
 
     The CZ gate employed in this description is similar in principle to that proposed by the above-mentioned Xiao et al., but different therefrom in structure. 
     The CZ gate employed in the embodiments is substantially the CZ gate proposed by the above-mentioned Duan et al., which is performed on a quantum bit expressed by polarization of a photon and a quantum bit expressed by the ground states of an atom. Referring first to  FIG. 1 , an optical system for realizing this CZ gate will be described. 
     As shown in  FIG. 1 , this optical system comprises a polarizing beam splitter (PBS)  101 , two polarization rotators (in  FIG. 1 , quarter-wave plates (QWPs)  201  and  202 ), a high reflection mirror  301  and a one-sided optical cavity  401  containing an atom with three energy levels such as shown in  FIG. 2 . 
     The PBS  101  reflects a vertically polarized light beam, and passes therethrough a horizontally polarized light beam. 
     The QWPs  201  and  202  invert vertical and horizontal polarizations of the light passing through the QWPs twice. The PBS  101  and QWPs  201  and  202 , which have the above properties, are used to separate an incident light beam and a reflected light beam from each other. In this case, a circularly polarized light beam enters the one-sided optical cavity  401 . If a Faraday rotator and a half-wave plate (HWP) are used instead of the QWPs, an incident light beam and a reflected light beam can be separated from each other, and a linearly polarized light beam enters the one-sided optical cavity  401 . In the embodiment, the QWPs are used. 
     The high reflection mirror  301  reflects a light beam in a direction opposite to the direction of the incident. 
     The frequency of an incident photon is set equal to the resonant frequency of the cavity. The one-sided optical cavity  401  is, for example, a Fabry-Perot cavity composed of a partially transmitting mirror and a high reflection mirror. 
     Referring then to  FIG. 2 , the energy levels of the atom contained in the one-sided optical cavity  401  will be described.  FIG. 2  is a view illustrating the energy level of the three-level atom. As shown in  FIG. 2 , only the atomic transition between |1&gt; and |2&gt; is coupled with an incident light beam (cavity mode). 
     In the embodiment, the stable ground states |0&gt; and |1&gt; are used to express quantum bits. The transition between |1&gt; and |2&gt; is strongly coupled with an incident light beam (cavity mode). The expression that the transition between |1&gt; and |2&gt; is strongly coupled with an incident light beam (cavity mode) means that the following three conditions are satisfied: i) a coupling constant between the cavity mode and the |1&gt;–|2&gt; transition is greater than the decay rates of both the cavity and the atom; ii) the transition frequency corresponding to the |1&gt;–|2&gt; transition is equal to the frequency (i.e., equal to the resonant frequency of the cavity) of the incident light beam; iii) owing to a rule of selection, the |1&gt;–|2&gt; transition is coupled with the circularly polarized light beam of the incident light beam, and is not coupled with a light whose polarization is opposite to that of the light mentioned above. 
     On the other hand, the |0&gt;–|2&gt; transition does not interact with the incident light beam (cavity mode) because of a large detuning. For simplicity, the incident light beam is assumed to be a single-photon pulse. In the embodiment, a coherent light beam may also be used as the incident light beam. In this case, if the emission of a light beam is stopped when a single-photon is detected, the same result as in the case of using a single-photon pulse can be acquired. Assume here that the coherent light beam is so weak that the intensity change of the intracavity light depending on the existence of the atom strongly coupled with the cavity can be observed. 
     Assume that the initial state is given by
 
|Ψ 0 &gt;=α 00 |0&gt;| V&gt;+α   01 |0&gt;| H&gt;+α   10 |1&gt;| V&gt;+α   11 |1&gt;| H&gt;,   (2)
 
where the first ket vectors indicate the states of the atom, and the second ket vectors indicate the polarized states of the incident photon. V and H represent vertical polarization (hereinafter referred to as “V-polarization”) and horizontal polarization (hereinafter referred to as “H-polarization”), respectively. Further, assume that V and H correspond to bit “0” and bit “1”, respectively.
 
     Referring again to  FIG. 1 , the V-polarized photon is reflected by the PBS  101  and guided to the high reflection mirror  301 . The photon is then reflected by the high reflection mirror  301  and returned to the PBS  101 . The returned photon is an H-polarized one since it has passed through the QWP  202  twice. Therefore, at this time, it passes through the PBS  101 . 
     The H-polarized photon passing through the PBS  101  is guided to the one-sided optical cavity  401 . If the atomic state in the cavity  401  is |0&gt;, the photon enters the cavity  401  and then reflects therefrom, since the cavity  401  is equivalent to a vacuum cavity in this case. In contrast, if the atomic state in the cavity  401  is |1&gt;, the photon reflects therefrom without entering it. 
     As found from a simple calculation based on classical optics, the phase of a photon which enters the cavity and reflects from it differs by 180 degrees from that of a photon which reflects from the cavity without entering it. In light of this, the phase flip which occur only when the polarization of a light beam is H-polarization and the atomic state is |1&gt; can be realized. This is equivalent to the realization of a CZ gate in which the ground states |0&gt; and |1&gt; of an atom are used as a control bit, and the polarized states |V&gt; and |H&gt; of a photon are used as a target bit. This CZ gate performs a transformation given by
 
|Ψ 0 &gt;=α 00 |0&gt;| V&gt;+α   01 |0&gt;| H&gt;+α   10 |1&gt;| V&gt;+α   11 |1&gt;| H&gt;→|Ψ   1 &gt;=α 00 |0&gt;| H&gt;+α   01 |0&gt;| V&gt;+α   10 |1&gt;| H&gt;−α   11 |1&gt;| V&gt;   (3)
 
     In the above expression, the replacement of |V&gt; and |H&gt; is caused by the QWPs  201  and  202 . 
     As described above, a CZ gate acting on atomic and photonic quantum bits is realized simply by applying the photon to a cavity strongly coupled with the atom, apart from the replacement of the V and H polarizations. 
     Referring then to  FIG. 3 , a description will be given of the fact that a CZ gate between two atoms can be realized using the CZ gate between the atom and the photon.  FIG. 3  is a view illustrating a quantum circuit equivalent to a CZ gate acting on quantum bit  1  and quantum bit  2 , which uses additional quantum bit  3 . 
     In  FIG. 3 , M indicates bit reading, and Z M  indicates that if the result of M is 0, nothing is performed, whereas if the result of M is 1, a phase flip gate (hereinafter referred to as “a Z gate”) operation is performed. The Z gate operation is defined by
 
|0&gt;→|0&gt;, |1&gt;→−|1&gt;  (4)
 
     Further, H in  FIG. 3  represents an Hadamard gate (hereinafter referred to “an H gate”), which is defined by 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
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     The H gate can be realized using an HWP in the case of polarization of a photon. 
     The quantum circuit shown in  FIG. 3  is equivalent to a CZ gate acting on quantum bits  1  and  2 . Explaining in more detail, the CZ gate acting on quantum bits  1  and  2  can be realized by an H gate acting on additional quantum bit  3 , CZ gates acting on quantum bits  1  and  3  and on quantum bits  2  and  3 , measurement of quantum bit  3 , and a Z gate operation acting on quantum bit  1  which is performed or not performed depending on the measurement result. This will now be described briefly. Assume that the initial state is given by
 
|Ψ 0 &gt;=α 00 |00&gt;+α 01 |01&gt;+α 10 |10&gt;+α 11 |11&gt;)|0&gt;  (6)
 
     In this case, the state of the quantum circuit immediately before bit reading M is given by 
     
       
         
           
             
               
                 
                   
                     
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     In accordance with the result of bit reading M performed thereafter, quantum bits  1  and  2  are varied in the following manners:
 
|Ψ 2 &gt;=α 00 |00&gt;+α 01 |01&gt;+α 10 | 10 &gt;−α 11 &gt;
 
|Ψ 2 &gt;=α 00 |00&gt;+α 01 |01&gt;−α 10 | 10 &gt;+α 11 &gt;  (8)
 
     Accordingly, if nothing is done when quantum bit  3  is 0, and a Z gate operation is performed when quantum bit  3  is 1, the state becomes
 
|Ψ 3 &gt;=α 00 |00&gt;+α 01 |01&gt;+α 10 | 10 &gt;−α 11 &gt;  (9)
 
     Thus, it is found that a CZ gate acting on quantum bits  1  and  2  can be realized by the quantum circuit shown in  FIG. 3 . 
     Here, a Z gate acting on the atomic quantum bit can be realized by the Raman transition. The Raman transition indicates a phenomenon in which Rabi oscillation between |0&gt; and |1&gt; is caused by a light beam of two frequencies the difference of which is equal to the |0&gt;–|1&gt; transition frequency, and which do not equal the |0&gt;–|2&gt; and |1&gt;–|2&gt; transition frequencies. 
     To perform a CZ gate operation on two atomic quantum bits, in the quantum circuit of  FIG. 3 , quantum bits  1  and  2  are expressed by the respective atomic states, and quantum bit  3  is expressed by polarization of the photon. Further, CZ gates acting on quantum bits  1  and  3  and on quantum bits  2  and  3  are performed using the scheme shown in  FIG. 1 . 
     Referring to  FIG. 4 , it will be described how the quantum circuit of  FIG. 3  is realized using the scheme shown in  FIG. 1 .  FIG. 4  shows an optical system for performing a CZ gate operation on two atomic quantum bits. 
     As shown in  FIG. 4 , the optical system comprises PBSs  101 ,  102  and  103 , QWPs  201 ,  202 ,  203  and  204 , high reflection mirrors  301 ,  302 ,  303  and  304 , one-sided optical cavities  401  and  402 , HWPs  501 ,  502  and  503 , and photodetectors  601  and  602 . 
     The HWPs  501  to  503  provide H gates acting on photonic quantum bits expressed by polarization as mentioned above. 
     The photodetectors  601  and  602  detect whether photons come or not. 
     The other device components are similar to those shown in  FIG. 1 . 
     Quantum bits  1 ,  2  and  3  shown in  FIG. 3  are expressed by the ground states of an atom in the one-sided optical cavity  401  ( FIG. 4 ), by those of an atom in the one-sided optical cavity  402  ( FIG. 4 ), and by the polarization of a photon, respectively. As mentioned above, |0&gt; and |1&gt; of the photonic quantum bit correspond to V polarization and H polarization, respectively. 
     In the CZ gate shown in  FIG. 1 , polarization replacement occurs as indicated in the expression (3). In light of the replacement, to realize the quantum circuit of  FIG. 3  by the optical system shown in  FIG. 4 , it is sufficient if the HWPs  502  and  503  perform a gate operation (hereinafter referred to “the H′ gate operation”) given by the following expression, which differs only in sign from the H gate operation given by the expression (5). 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
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     The operation of the H′ gate is equivalent to that of the H gate after a NOT gate. 
     Further, if the HWP  501  also performs the H′ gate operation, an H-polarized light beam is used as an incident light beam instead of a V-polarized light beam. Since it is simple if all the HWPs have the same function, it is hereinafter assumed that all the HWPs are used to perform the H′ gate operation, and an H-polarized light beam is used as an incident light beam. 
     From the above, it is found that the optical system of  FIG. 4  can realize a CZ gate on two atomic quantum bits using a photon. However, it is not obvious how to construct the optical system in order to increase the number of quantum bits. The embodiment of the invention proposes a method that enables a quantum computer using the CZ gate of  FIG. 4  to employ as many quantum bits as possible in principle. This method will now be described. 
     The first two of the three PBSs shown in  FIG. 4 , i.e., the PBSs  101  and  102 , are used to perform CZ gates on the atomic and photonic quantum bits, while the last PBS  103  is used to measure polarization of a photon. To make the quantum computer extensible, these two functions are made to be executed by a single PBS. As described later, to this end, it is sufficient to use a special mirror (hereinafter referred to as “the switching mirror) which can switch its reflectivity between low (for high transmission) and high (for high reflection). 
     Referring to  FIG. 5 , an optical system for imparting extensibility to a quantum computer will be described.  FIG. 5  shows an optical system in which a single polarizing beam splitter functions for both a quantum gate and a polarization measuring unit, by using a switching mirror that can switch its reflectivity between low (for high transmission) and high (for high reflection). 
     The optical system of  FIG. 5  is acquired by adding, to the optical system of  FIG. 1 , an HWP  501 , photodetectors  601  and  602 , and switching mirrors  701  and  702 . When the switching mirror has high reflection, the optical system of  FIG. 5  executes a CZ gate operation on atomic and photonic quantum bits after an H′ gate operation on the photonic quantum bit. In contrast, when the switching mirror performs high transmission, the optical system functions as a polarization-measuring unit for measuring polarization of a photon after an H′ gate operation on the photonic quantum bit. If optical systems similar to the above are prepared and connected to each other via switching mirrors, a CZ gate operation can be performed on quantum bits corresponding to atoms included in any adjacent two of the optical systems. Further, when the optical systems are connected using the switching mirrors, a photon can be directly guided from the outside to any one of the optical systems via the corresponding switching mirror. 
     Referring to  FIG. 6 , a description will be given of how to connect the optical systems.  FIG. 6  shows a quantum computer according to the embodiment. 
     As shown in  FIG. 6 , optical systems, i.e., an optical system  1 , optical system  2 , . . . , optical system N (N indicates the total number of the optical systems) starting from the left, are connected by switching mirrors. The switching mirror k connects the optical system k to the optical system (k+1) (k=1, 2, . . . , N−1). The optical system  2 , . . . , optical system N are similar to the optical system shown in  FIG. 5 . However, the optical system  1  and the optical system after the optical system N, shown in  FIG. 6 , are not necessarily similar to that of  FIG. 5 . Since the optical system  1  does not need to measure polarization of a photon, it may not include photodetectors and switching mirrors. Further, since the optical system after the optical system N merely measures polarization, it only includes a PBS, photodetectors, and an HWP for an H′ gate operation. 
     When optical systems are connected as shown in  FIG. 6 , a CZ gate operation can be performed on quantum bits corresponding to atoms included in any adjacent two of the optical systems. As a result, a quantum computer in which as many quantum bits as possible are employed can be constructed. 
     Referring then to  FIG. 7 , a description will be given of how to execute a CZ gate operation on quantum bits k and (k+1) in  FIG. 6 , where the quantum bit expressed by the ground states of the atom in the optical system k in  FIG. 6  is called quantum bit k. 
     As shown in  FIG. 7 , a single H-polarized photon pulse enters the optical system k through a switching mirror (k−1)  707  set for high transmission. Using the switching mirrors  701  and  702  in the optical system k for high reflection, a CZ gate operation is performed on the quantum bit k and the photonic quantum bit. Using the switching mirror k  708  for high reflection, the photon is guided to the optical system (k+1). Using the switching mirrors  703  and  704  in the optical system (k+1) for high reflection, a CZ gate operation is performed on the quantum bit (k+1) and the photonic quantum bit. Using the switching mirror (k+1)  709  for high reflection, the photon is guided to the optical system (k+1). Using the switching mirrors  705  and  706  in the optical system (k+2) for high reflection, polarization of the photon is measured. In accordance with the measurement result, a Z gate operation is performed on the quantum bit k. 
     By the above operation, a CZ gate acting on the quantum bits k and (k+1) is realized. At the same time, a CZ gate acting on the quantum bits m and (m+1) (m&lt;k−2, m&gt;k+2) can be also realized. Thus, the quantum computer of the embodiment can employ as many quantum bit as possible, and can perform two-quantum-bit gate operations in a parallel manner. 
     Referring to  FIGS. 8 and 9 , implementation of a switching mirror will be described.  FIG. 8  shows a case where a ring cavity is used as a switching mirror, while  FIG. 9  shows a case where an etalon is used as a switching mirror. 
     The ring cavity can adjust the cavity length. As shown in  FIG. 8 , the ring cavity comprises input/output mirrors  801  and  802 , high-reflection mirrors  301  and  302 , and cavity length adjuster  901 . The input/output mirrors  801  and  802  have the same transmission factor. The cavity length adjuster  901  adjusts the optical cavity length. The adjuster  901  can be formed of an electrooptic modulator if the adjustment is based on a change in refractive index, or formed of a piezoelectric transducer if the adjustment is based on a physical distance. The electrooptic modulator changes the optical distance by changing the refractive index, which depends on an applied voltage. Since the electrooptic modulator exhibits a quicker response than the piezoelectric transducer, if the cavity length adjuster  901  utilizes the electrooptic modulator, the cavity length can be adjusted quickly. 
     In the above structure, if the cavity length adjuster  901  sets the cavity length to a value that causes the cavity not to resonate with an incident light beam, the incident light beam is reflected therefrom, whereas if the adjuster  901  sets the cavity length to a value that causes the cavity to resonate with an incident light beam, the incident light beam is transmitted therethrough. Thus, the ring cavity can be used as a switching mirror. 
     The etalon  1701  shown in  FIG. 9  transmits or reflects a light beam depending upon the incident angle of the light beam. In other words, the etalon  1701  can be used as a switching mirror by adjusting the incident angle of the light beam. The incident angle is adjusted by, for example, rotating the etalon  1701  about an axis. If a high reflection mirror  301  is prepared as shown in  FIG. 9 , the direction of the light beam reflected from the etalon  1701  can be adjusted. 
     Referring to  FIG. 10 , the method for reading a quantum bit will be described. 
     In this case, the quantum bit k is read. Using the switching mirror (k−1)  705  for high transmission, a single H-polarized photon pulse is guided to the optical system k. Using the switching mirrors  701  and  702  in the optical system k for high reflection, a CZ gate operation is performed on the quantum bit k and the photonic quantum bit. Using the switching mirror k  706  for high reflection, the photon is guided to the optical system (k+1). Using the switching mirrors  703  and  704  in the optical system (k+1) for high reflection, polarization of the photon is measured. As a result, reading of the quantum bit k can be realized, as will be described. Thus, the quantum computer of the embodiment can also perform reading of a quantum bit efficiently. 
     Referring to the quantum circuit shown in  FIG. 11 , the principle of reading the quantum bit k will be described.  FIG. 11  is a quantum circuit diagram useful in explaining why the method shown in  FIG. 10  enables to read the quantum bit k. 
     The quantum circuit of  FIG. 11  is a quantum-circuit expression for  FIG. 10 . In the quantum circuit of  FIG. 11 , the input of the photonic quantum bit is |0&gt;, and is subjected to an H gate operation. This quantum circuit is a circuit for reading the quantum bit k. Further, this reading method can be applied to both the quantum bit k and quantum bit m (m&lt;k−2, m&gt;k+2) at the same time. Thus, in the quantum computer of the embodiment, quantum bit reading can also be performed in a parallel manner. 
     Further, in the quantum computer of the embodiment, if four energy levels (including three lower level states |0&gt;, |1&gt; and |3&gt;, and one upper level |2&gt;) are used instead of three energy levels, a CZ gate operation can also be performed on quantum bits corresponding to atoms contained in cavities which are not adjacent to each other. Referring to  FIG. 12 , the case of using the four-level atoms will be described. 
     To perform a CZ gate operation on quantum bits k and (k+n) (n is an integer not less than 2), |1&gt; of the quantum bits (k+1) to (k+n−1) are shifted to |3&gt;, and the CZ gates are performed on the quantum bits k and (k+n) in a similar manner to that on the quantum bits k and (k+1), with the switching mirrors k to (k+n−1) set for high reflection. This is because the cavities (k+1) to (k+n−1) are equivalent to a vacant cavity, and hence a photon enters the cavity (k+n) in the same state as that in which it enters the optical system (k+1). Thus, a CZ gate operation can also be performed on quantum bits corresponding to atoms contained in cavities that are not adjacent to each other. 
     As described above, a special mirror having its reflectivity changeable between low and high, which can be implemented by a ring cavity having an adjustable cavity length or by an etalon having an adjustable incident angle, enables a single polarizing beam splitter to serve for both a CZ gate and a polarization measuring unit. As a result, a controlled phase flip gate can be performed on any two quantum bits with connected optical systems of the same structure. 
     As explained above, in the embodiment, atomic stable states are used as quantum bits, and the phenomenon is utilized in which the intensity of a light in an optical cavity varies because of strong coupling of the cavity with the atom when a light beam enters the cavity. As a result, the embodiment of the invention can provide a quantum computer and quantum computation method in which two quantum bit gate operations can be performed in a parallel manner, and the number of quantum bits can be easily increased. 
     Examples according to the embodiment of the invention will be described. 
     EXAMPLE 1 
     Referring to  FIG. 13 , a description will be given of an example of the CZ gate on atomic and photonic quantum bits shown in  FIG. 1 . 
     To confirm whether a CZ gate therebetween is realized or not, bit reading previously described with reference to  FIG. 10  is performed. Explaining in more detail, first an atom is preset in a certain known state, then the process shown in  FIG. 10  is performed on the atom to measure the state of the atom, and finally it is verified that the measured state corresponds to the preset certain state. When the CZ gate and the bit reading have succeeded, a V-polarized light beam is observed if the preset certain state of the atom is |0&gt;. In contrast, if the preset certain state of the atom is |1&gt;, an H-polarized light beam has to be observed. Accordingly, this example can also be regarded as an example of the reading method. 
     In example 1, Pr 3+  ions contained in Y 2 SiO 5  crystal are used as atoms having such three levels as shown in  FIG. 2 . The Pr 3+  ions have the energy level structure shown in  FIG. 14 . Further, the Pr 3+  ions have the property that they absorb only a linearly polarized light beam. In light of this, in example 1, a Faraday rotator  1501  and HWP  502  are provided, instead of a QWP, in front of an optical cavity  1301  so that a linearly polarized light beam enters the cavity  1301 . The optical cavity  1301  is formed by mirror-polishing the surface of the crystal. The orientation of the crystal is set so that maximum absorption of a V-polarized light beam can be realized. Accordingly, the HWP  502  in front of the optical cavity  1301  is adjusted to apply a V-polarized light beam to the cavity  1301 . 
     In this example, the cavity length is 10 mm, the waist radius of the cavity mode is 5 μm, and the transmittance of the input mirror is 10 −6 . In this case, the coupling constant g between the cavity mode and the atom at the waist is 30 kHz, and the damping rate κ of a photon in the cavity mode is 4 kHz. Further, the relaxation rate γ of the exited state of the ion is about 6 kHz. Since in this case, the condition g&gt;κ, γ is satisfied, the coupling of the cavity with the atom in this example is strong. 
     The optical cavity  1301  is placed in a cryostat  1401  and kept at 1.4 K by liquid helium therein. A ring dye laser  1101  having a stabilized frequency is used as a light source. The output of the ring dye laser  1101  is an H-polarized light beam. An HWP  504  is provided for converting the output of the laser  1101  into a V-polarized light beam. Acoustooptic modulators  1201  to  1204  perform frequency adjustment. 
     In example 1, firstly, an initial state in which only one ion is related to a gate operation is prepared. To this end, firstly, a light beam that resonates with the optical cavity  1301  is applied thereto from the input mirror (i.e., from the left of the optical cavity  1301  in  FIG. 13 ) for a while (for about 1 second). As a result, the ions contained in the cavity mode and having an energy level coupled with the cavity mode are made to be in the states which are not coupled with the mode. 
     Subsequently, a light beam, whose direction is vertical to the cavity mode, of a frequency higher by 17.30 MHz than the resonant frequency of the cavity is applied to the waist. As a result, the ions positioned near the waist can be returned to the energy level that is coupled with the cavity mode. After that, a light beam is guided to the high-reflection mirror of the cavity  1301  (i.e., from the right of the optical cavity  1301  in  FIG. 13 ). While scanning the frequency of the light beam, the intensity of a light beam output through the input mirror is measured. As a result of the measurement, peaks away from each other by 9.5 kHz were observed. This result indicates that one ion is coupled with the cavity mode since the coupling constant g is 30 kHz (9.5=2×30/2π). Thus, the state in which only one ion related to the gate operation exists in the cavity mode can be prepared. Moreover, the state of the ion is |1&gt; in  FIG. 14 , which means that the ion is prepared in a known initial state. 
     After thus preparing one ion of |1&gt;, a weak H-polarized coherent light beam with an intensity 1 fW (hereinafter, a “weak coherent light beam” indicates a coherent light beam having an intensity of 1 fW) was applied to the cavity via the HWP  501 , and a light beam reflected therefrom was measured by photodetectors  601  and  602 , with the result that a V-polarized photon was acquired. Further, after preparing one ion of |1&gt;, two light beams having two frequencies as shown in  FIG. 15  was applied to the ion so that the ion was shifted to |0&gt; by the Raman transition, and the same experiment as the above was performed. As a result, a V-polarized photon was observed. These results indicate that a CZ gate operation on atomic and photonic quantum bits succeeded. 
     EXAMPLE 2 
     In example 2, a single-photon pulse is guided into a cavity, unlike example 1. In example 1, a weak coherent light beam is guided to the cavity. Referring to  FIG. 16 , a description will be given of a quantum computer, according to example 2, in which a CZ gate operation is performed on an atom and a photon using a single-photon pulse. 
     The right side system of  FIG. 16  (i.e., the optical systems located rightward with respect to the PBS  101 , acoustooptic modulator  1204  and beam splitter  1008 ) are similar to those employed in example 1 of  FIG. 13  except that the incident light beam is a single-photon pulse. The left side system of  FIG. 16  (including the PBS  101 , acoustooptic modulator  1204  and beam splitter  1008 ) are used to generate a single-photon pulse. An optical cavity  1301  included in the left side system is similar to an optical cavity  1302  for a CZ gate. 
     A method for generating a single-photon pulse using the left side system will be described. Firstly, a state in which only one ion is in |0&gt; is prepared in the left cavity  1301  by the same method as employed in example 1. Subsequently, a light beam of a frequency higher by 17.30 MHz than the resonant frequency of the optical cavity  1301  is gradually applied to the ion, thereby shifting the state of the ion (atom) to |1&gt; and generating a single photon in the cavity mode via adiabatic passage based on the principle of quantum mechanics. After a while, a single photon in the cavity mode is ejected from the right hand mirror of the optical cavity  1301 . The ejected photon, which is V-polarized, is converted into an H-polarized photon by the HWP  501  and the Faraday rotator  1501 . The H-polarized photon is guided to the right side optical system through the PBS  101 . Thus, a CZ gate operation can be realized using a single-photon pulse. 
     Also in example 2, it was found as in example 1 that when the state of the atom in the right hand optical cavity  1301  was initially set to |0&gt;, a V-polarized photon was observed, while when the state of the atom was initially set to |1&gt;, an H-polarized photon was observed. From these results, it is verified that a CZ gate operation between the atom and the photon succeeded. 
     EXAMPLE 3 
     Referring to  FIG. 17 , a description will be given of example 3 related to a CZ gate between two atoms. 
     Since it is necessary to guide a light beam to a cavity  1302  to lastly read the state of an atom in the cavity  1302 , an optical system including a cavity  1301  is coupled with an optical system including the cavity  1302  via a ring cavity. As shown in  FIG. 17 , the ring cavity comprises high reflection mirrors  305  and  306 , input mirrors  801  and  802 , and cavity length adjuster  901 . 
     Further, the quantum computer shown in  FIG. 17  is fundamentally similar in structure to the CZ gate with extensibility shown in  FIG. 7 , therefore example 3 can also be regarded as an example of the CZ gate with extensibility. 
     In example 3, to confirm whether a CZ gate operation between two atoms has succeeded, the fact is noticed that a control NOT gate (CNOT) operation can be performed by three gate operations—(an H gate operation on a target bit)→a CZ gate operation→(an H gate operation on a target bit)—. Resulting from the CNOT gate operation, 
     |0&gt; |0&gt;, |0&gt; |1&gt;, |1&gt; |0&gt;, |1&gt; |1&gt; are respectively changed to 
     |0&gt; |0&gt;, |0&gt; |1&gt;, |0&gt; |1&gt;, |1&gt; |1&gt; 
     By conforming whether these changes have occurred or not, it is confirmed whether the CZ gate operation between two atoms has succeeded. 
     Firstly, one ion in each cavity mode is prepared in |1&gt; by the same method as employed in example 1. Subsequently, two light beams having two frequencies as shown in  FIG. 15  is applied to the ion in the cavity  1302 , and then an H gate operation is performed by the Raman transition. After that, a weak coherent light beam is applied to the HWP  501 . Cavity length adjusters  901 ,  902  and  903  are all adjusted to cause all ring cavities to serve as high reflection mirrors. Polarization of a light beam is measured by photodetectors  603  and  604 . If polarization of a light beam is V-polarization, a control unit  1601  performs nothing, whereas if polarization of a light beam is H-polarization, the control unit  1601  performs a Z gate operation on the ion in the cavity  1301 . Lastly, an H gate operation is performed on the ion in the cavity  1302  by the Raman transition. 
     If the CZ gate operation between two atoms has succeeded, the state of the ion in the cavity  1301  has to be kept at |1&gt;, and the state of the ion in the cavity  1302  has to be shifted to |0&gt;. To confirm this, the states of the ions in the cavities  1301  and  1302  are checked in this order. 
     To check the state of the ion in the cavity  1301 , the cavity length adjuster  901  is adjusted to cause the ring cavity including it to serve as a high reflection mirror, and the cavity length adjusters  902  and  903  are adjusted to cause the ring cavities including them to serve as high transmission mirrors. After that, a weak coherent light beam is guided to the cavity  1301  through the HWP  501 , and polarization of the light beam is measured by the photodetectors  601  and  602 . 
     To check the state of the ion in the cavity  1302 , the cavity length adjuster  901  is adjusted to cause the ring cavity including it to serve as a high transmission mirror, and the cavity length adjusters  902  and  903  are adjusted to cause the ring cavities including them to serve as high reflection mirrors. After that, a weak coherent light beam is guided from the input mirror  801  to the cavity  1302  through the HWP  503 , and polarization of the light beam is measured by the photodetectors  603  and  604 . 
     As a result of the measurement, an H-polarized light beam was observed in the case of the ion in the cavity  1301 , and a V-polarized light beam was observed in the case of the ion in the cavity  1302 . From this, it was confirmed that the states of the ions in the cavities  1301  and  1302  were |1&gt; and |0&gt;, respectively. 
     Similarly, if the initial states of the ions are |1&gt; and |0&gt;, |0&gt; and |1&gt;, and |0&gt; and |0&gt;, they have to be changed, after the above operation is performed thereon, to |1&gt; and |1&gt;, |0&gt; and |1&gt;, and |0&gt; and |0&gt;, respectively. 
     It was confirmed that these changes occurred. The results of the measurements indicate that CZ gate operations were realized between the two atoms (ions). 
     EXAMPLE 4 
     Referring then to  FIG. 18 , a description will be given of example 4 in which an etalon is used as a switching mirror. Example 4 differs from example 3 shown in  FIG. 17  only in that in the former, etalons with a finesse of about 100 is used instead of the ring cavities. The other structures are similar to those of example 3. 
     In example 4, an incident light beam enters each etalon at an incident angle of about 5 degrees. After CZ gate operations were performed on atoms as in example 3, correct results were acquired. 
     EXAMPLE 5 
     Referring to  FIG. 19 , a description will be given of example 5 in which a CZ gate operation is performed on quantum bits corresponding to atoms contained in cavities that are not adjacent to each other as shown in  FIG. 12 . 
     In example 5, a Pr 3+  ion is used as an ion in each cavity as in example 1. Since the Pr 3+  ion has three ground states as shown in  FIG. 14 , the ground state other than |0&gt; and |1&gt;, i.e., |3&gt;, is utilized. 
     Firstly, the state of an ion in a cavity  1302  in  FIG. 19  was set to |3&gt;. Cavity length adjusters  901  to  906  were all adjusted to cause all ring cavities to serve as high reflection mirrors. The states of the ions in cavities  1301  and  1303  were both set to |1&gt;. Next, a CNOT gate operation was performed on the ions in the cavities  1301  and  1303  in the same manner as in example 3. 
     Thereafter, the cavity length adjusters  902  and  903  were adjusted so that the corresponding ring cavities serve as high transmission mirrors, whereby a weak coherent light beam is guided to an HWP  501 , and the state of the ion in the cavity  1301  was read by photodetectors  601  and  602 . Subsequently, the cavity length adjuster  906  was adjusted so that the corresponding ring served as a high transmission mirror, and the cavity length adjusters  904  and  905  were adjusted so that the corresponding ring cavities served as high reflection mirrors. After that, a weak coherent light beam is guided to the HWP  505 , and a CZ gate operation on the ion in the cavity  1303  and the photon was performed, whereby the state of the ion in the cavity  1303  was read by photodetectors  605  and  606 . As a result, it was found that the states of the ions in the cavities  1301  and  1303  were |1&gt; and |0&gt;, respectively. 
     Similar operations were performed with the initial states of the ions in the cavities  1301  and  1303  set to |0&gt; and |0&gt;, |0&gt; and |1&gt;, and |1&gt; and |0&gt;, respectively. As a result, their states were changed to |0&gt; and |0&gt;, |0&gt; and |1&gt;, and |1&gt; and |1&gt;, respectively. These results indicate that CZ gate operations on the atoms in the cavities  1301  and  1303  that are not adjacent to each other succeeded. 
     The quantum computer and quantum computation method of the embodiments are extensible in the number of quantum bits. 
     Additional advantages and modifications will readily occur to those skilled in the art. Therefore, the invention in its broader aspects is not limited to the specific details and representative embodiments shown and described herein. Accordingly, various modifications may be made without departing from the spirit or scope of the general inventive concept as defined by the appended claims and their equivalents.