Patent Publication Number: US-7583438-B2

Title: Quantum circuit and quantum computer

Description:
TECHNICAL FIELD 
     The present invention relates to quantum circuits and quantum computers and, more particularly, to a quantum circuit and a quantum computer using a controlled-unitary transform. 
     BACKGROUND ART 
     In order to realize computational capacity extending beyond conventional computer limits, research on quantum computation has become active in recent years. For example, the use of the algorithm disclosed by P. W. Shor at the 35th Annual Symposium of the Foundations of Computer Science in 1995 achieves the factorization of a large integer into prime factors in time expressed by a polynomial indicating the number of bits. 
     In conventional computers, execution time increases as an exponential function of the number of bits. In the conventional computers, execution time proportional to the number of bits is required to retrieve an N-bit database. Using the algorithm proposed by L. Grover, Physical Review Letters, Vol. 79, p. 325, 1997, however, the database can be retrieved in time proportional to the square root of N. 
     The above quantum computation is realized using a fundamental unit called a quantum bit (qubit) substituting for a classical bit. The qubit is characterized by including a linear superposition of a state |0&gt; corresponding to a logical value 0 and a state |1&gt; corresponding to a logical value 1, i.e., |Ψ&gt;=a|0&gt;+b|1&gt;(a 2 +b 2 =1). Further, quantum gates for performing a unitary transform of |Ψ&gt;→U|Ψ&gt; to one qubit, controlled-unitary transform gates for performing the unitary transform to another qubit depending on the state of the one qubit, and a quantum circuit composed of the above gates are needed. 
     The above-mentioned content will now be described using Shor&#39;s algorithm to resolve an integer N into prime factors as an example. 
     This algorithm is implemented using the combination of conventional computation (classical computation) and quantum computation. 
     1. When N is an even number, 2 is output as a factor. 
     2. Whether p and q satisfying conditions that p≧1, q≧2, and N=p q  exist is determined by classical computation. If they exist, p is output as a factor. 
     3. Select an integer a (1&lt;a&lt;N) at random. The greatest common divisor of a and N is calculated using Euclidean algorithm. When the greatest common divisor is larger than 1, the greatest common divisor is output. 
     4. Order r of a modN is computed using a quantum computer for finding order. 
     5. When r is an odd number or an even number, which satisfies that a r/2 =−1(modN), the routine is returned to step 3. In other cases, the greatest common divisor of a r/2 −1 and N and the greatest common divisor of a r/2 +1 and N are computed. When these greatest common divisors are not self-evident factors (1 or N), they are output. 
     The quantum computer used in step 4 is realized by a quantum circuit shown in  FIG. 1 . In this case, the number of bits is 4. The quantum circuit shown in  FIG. 1  includes a circuit  21  for applying a phase depending on order and an inverse Fourier transform circuit  22  for obtaining the order from the phase. A controlled-unitary transform gate U a    23  in the circuit  21  for applying the phase depending on the order is given by
 
 cU   a |0&gt;| x&gt;=| 0&gt;| x&gt;,  
 
 cU   a |1&gt;| x&gt;=| 1 &gt;|ax  modN&gt;,  (1)
 
where c denotes a constant. In this circuit, a qubit is transformed as
 
                               cU   a     ⁡     (          0   〉     +        1   〉       )       ⁢        1   〉       =       ∑     j   =   0       ⁢       ∑     s   =   0       r   -   1       ⁢       exp   ⁡     [       2   n     ⁢   πⅈ   ⁢           ⁢     sj   /   r       ]       ⁢   j           〉     ⁢          u   s     〉       ,           (   2   )               
where |u s &gt; is defined by
 
                          u   s     〉     =       1     r       ⁢       ∑     k   =   0       r   -   1       ⁢       exp   ⁡     [       -   2     ⁢   πⅈ   ⁢           ⁢     sk   /   r       ]       ⁢              a   k     ⁢           ⁢   mod   ⁢           ⁢   N     〉     .                   (   3   )               
When a first qubit is subjected to an inverse Fourier transform, the following expression is obtained.
 
                     1     r       ⁢          s   /   r     〉     ⁢          u   s     〉             (   4   )               
When the first qubit is measured, s/r is obtained, so that the estimate of r can be obtained. Each Hadamard gate  24  performs the transform expressed as
 
     
       
         
           
             
               
                 
                   
                     
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     In addition, the inverse Fourier transform uses controlled-rotation gates  25  each performing a kind of controlled-unitary transform. Each controlled-rotation gate  25  is given by 
     
       
         
           
             
               
                 
                   
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     To realize Shor&#39;s algorithm, a nuclear magnetic resonance (NMR) technology using spin nuclei in an organic molecule as quantum bits has been utilized. 
     Recently, an experiment in factoring 15=3×5 into prime factors has been reported by L. M. K. Vandersypen, M. Steffen, G. Breyta, C. S. Yannonl, M. H. Sherwood, and I. L. Chuang, Nature, Vol. 414, p. 883, 2001. 
     DISCLOSURE OF INVENTION 
     In the above-mentioned conventional system, however, in principle, it is difficult to realize computation with 10 qubits or more and it is impossible to implement practically meaningful computation. Studies on the reduction of the number of qubits necessary for quantum computation have been conducted. According to a report by S. Beauregard, Los Alamos Preprint Archive, quant-ph/0205095 (2002), the number of qubits necessary for the factorization of an n-bit integer into prime factors can be reduced to 2n+3. In this case, to factorize, e.g., a 40-bit integer into prime factors, 83 bits are required. Accordingly, it is difficult to implement computation by a method using spin nuclei in an organic molecule as qubits. 
     Even using other methods for realizing computation with qubits, qubits lose coherence due to the influence of an environment. Thus, it is difficult to perform multi-bit quantum computation. 
     In consideration of the above circumstances, an object of the present invention is to provide a quantum circuit and a quantum computer capable of performing multi-bit quantum computation. 
     To accomplish the above object, the present invention provides: 
     [1] A quantum circuit in which a quantum bit is represented by the polarization directions of light, a sequence of polarized light pulses indicating a quantum bit string is sequentially supplied, and the amount of polarization rotation on a certain light pulse and the amount of phase difference are determined on the basis of a result of the measurement of the preceding input light pulse sequence, thus realizing a controlled-unitary transform. 
     [2] The quantum circuit according to the above [1], characterized in that the controlled-unitary transform causes the phase difference between polarization indicating a |0&gt; state and polarization indicating a |1&gt; state. 
     [3] The quantum circuit according to the above [2], characterized in that outputs of a polarization beam splitter are coupled through a polarization maintaining fiber, and a phase modulator is arranged in a position deviated from the middle point of the polarization maintaining fiber, thus causing the phase difference between the polarization indicating the |0&gt; state and that indicating the |1&gt; state. 
     [4] The quantum circuit according to the above [1], [2], or [3], characterized in that in the light pulses representing the quantum bits, the number of photons included in one pulse is larger than 1. 
     [5] A quantum computer including the quantum circuit according to the above [1], [2], [3], or [4]. 
     The quantum circuit according to the present invention is characterized in that a quantum bit is represented by the polarization directions of light, a sequence of polarized light pulses indicating a quantum bit string is sequentially supplied, and the amount of polarization rotation on a certain light pulse and the amount of phase difference are determined on the basis of a result of the measurement of the preceding input light pulse sequence, thus realizing a controlled-unitary transform. 
     In the quantum circuit, the controlled-unitary transform is realized by measuring the state of a control quantum bit and supplying a result of the measurement as an electric signal to a modulator. Thus, computation for multiple quantum bits can be sequentially performed. 
     Advantageously, therefore, an arbitrary number of quantum bits can be transformed by the controlled-unitary transform for one quantum bit. 
     In the quantum circuit according to the present invention characterized in that the controlled-unitary transform causes the phase difference between polarization indicating a |0&gt; state and polarization indicating a |1&gt; state, outputs of a polarization beam splitter are coupled through a polarization maintaining fiber, and a phase modulator is arranged in a position deviated from the middle point of the polarization maintaining fiber, thus causing the phase difference between the polarization indicating the |0&gt; state and that indicating the |1&gt; state, polarized light in the |0&gt; state and polarized light in the |1&gt; state propagate through the same fiber. Thus, it is possible to prevent a reduction in coherence caused by fluctuations in the index of refraction in the fiber. 
     Further, in the quantum circuit of the present invention, the number of photons included in each of light pulses representing quantum bits can be larger than 1. Consequently, even when a large error is included in the input state or the unitary transform, an error in the measurement result can be reduced. 
     Therefore, in the quantum computer including the quantum circuit of the present invention, advantageously, the accuracy of the quantum circuit at the preceding stage is not strongly demanded, so that multi-bit computation can be easily performed. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a diagram of a circuit in a quantum computer for realizing Shor&#39;s algorithm. 
         FIG. 2  is a diagram showing a quantum inverse Fourier transform circuit according to an embodiment of the present invention. 
     
    
    
     BEST MODE FOR CARRYING OUT THE INVENTION 
     An embodiment of the present invention will now be described below with reference to the drawings. 
       FIG. 2  is a diagram showing a quantum inverse Fourier transform circuit according to the embodiment of the present invention. 
     In this diagram, a sequence of input light pulses serving as a quantum bit (qubit) string is incident on a terminal  101  of a circulator  1  and is then emitted from a terminal  102 . Emitted light is incident on a terminal  201  of a polarization beam splitter  2 . The polarization beam splitter  2  splits the light into vertically polarized light to be emitted from a terminal  202  and horizontally polarized light to be emitted from a terminal  203 . An output of the terminal  202  is connected to a polarization maintaining fiber segment  12  such that polarization is aligned to the slow axis of the polarization maintaining fiber segment  12 . Further, the output is coupled through another polarization maintaining fiber segment  12  to the terminal  203  of the polarization beam splitter  2  via a phase modulator  3 . In this instance, the slow axis of the polarization maintaining fiber segment  12  is set so as to obtain horizontal polarization in agreement with the terminal  203 . The polarization maintaining fiber segment  12  is longer than the other one connecting the terminal  202  of the polarization beam splitter  2  to the phase modulator  3  by 4 m. Thus, a fiber loop, including the polarization maintaining fiber segments  12 , connecting the terminals  202  and  203  can be constructed. In this fiber loop, the vertically polarized light travels clockwise and the horizontally polarized light proceeds counterclockwise. Both the polarizations are again combined into light by the polarization beam splitter  2 . The combined light is emitted from the terminal  201  and is then returned to the terminal  102  of the circulator  1 . An output is generated from a terminal  103  of the circulator  1 . The output from the terminal  103  of the circulator  1  is incident on a terminal  501  of a polarization beam splitter  5  via a polarization controller  4 . The polarization controller  4  compensates the difference between polarization states in the fiber segments and the respective components and also acts as the Hadamard gate  24  in  FIG. 1 . The polarization beam splitter  5  splits incident light into vertically polarized light to be emitted from a terminal  502  and horizontally polarized light to be emitted from a terminal  503 . Detectors  6  and  7  are connected to the terminals  502  and  503  of the polarization beam splitter  5 , respectively. The difference between outputs of the detectors  6  and  7  is obtained by a subtractor  8 . After that, the difference therebetween is supplied to a discriminator  9 . An output of the discriminator  9  is supplied to a control circuit  10 . The control circuit  10  determines an output voltage of a pulse generator  11 . An output of the pulse generator  11  is supplied to the phase modulator  3 . 
     According to the present embodiment, the phase modulator  3  is arranged at a distance of 2 m from the middle point of the fiber loop. The distance D is not limited to this value. The length L of the fiber loop, the repetition time T of the light pulse sequence, the pulse width t of the pulse generator  11  can be determined so as to satisfy the following relationship. When v denotes light velocity in the fiber,
 
 T&gt;L/v,   (7)
 
 t&lt;D/v   (8)
 
     Expression (7) denotes a condition to prevent the interference between different qubits in the polarization beam splitter  2 . Properly speaking, the condition may not necessarily be satisfied. It is desirable that time during which different pulses collide be longer than the coherence time of light pulses. Expression (8) represents a condition that only one polarized light is subjected to phase modulation. 
     The structure according to the embodiment has been described in detail above. The control circuit  10  shown in  FIG. 2  can include a general shift register or the like. Those skilled in the art can easily construct the control circuit  10  on the basis of the operation, which will be explained below. 
     Subsequently, each qubit shown in  FIG. 2  is represented by the polarization directions of a light pulse. Horizontally polarized light corresponds to the |0&gt; state and vertically polarized light corresponds to the |1&gt; state. Assuming that a phase φ is expressed in binary decimal notation, i.e., φ=0.φ 1 φ 2  . . . φ n  (φ 1 , φ 2 , . . . , φ n =0, 1), an input state for an inverse Fourier transform can be expressed by 
     
       
         
           
             
               
                 
                   
                     
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     The polarization states of input light pulses sequentially indicate |0&gt;+exp[2πi0.φ n ]|1&gt;, |0&gt;+exp[2πi0.φ n−1 φ n ]|1&gt;, . . . , and |0&gt;+exp[2πi0.φ 1 φ 2  . . . φ n ]|1&gt;. In other words, the polarization is given at a point on the great circle of y 2 +z 2 =1 on the yz plane on the Poincare sphere including 45-degree linear polarization to circular polarization. 
     A first light pulse passes through the circulator  1  and is then divided into segments in two directions toward the terminals  202  and  203  by the polarization beam splitter  2 . The first light pulse is not subjected to phase modulation by the phase modulator  3 . After the segments pass through the fiber loop, the polarization state combined by the polarization beam splitter  2  is the same as the input polarization state. After the light pulse passes through the circulator  1 , the polarization state is transformed by the polarization controller  4  as shown in Expression (5). In other words, when φ n =0, only the detector  7  detects light obtained by polarization split through the polarization beam splitter  5 . When φ n =1, only the detector  6  detects the light. Therefore, the value of φ n  can be found by comparing the outputs of the detectors  6  and  7 . 
     Regarding a second light pulse, an output voltage of the pulse generator  11  is adjusted so that the second light pulse is subjected to phase modulation by −φ n π/2. Pulse generation time is set so that when vertically polarized light passes through the phase modulator  3 , a pulse is applied to the phase modulator  3 , and when horizontally polarized light passes therethrough, a pulse is not applied thereto. According to the present embodiment, the vertically polarized light arrives at the phase modulator  3  earlier than the horizontally polarized light by about 20 ns corresponding to the 4-m fiber. In other words, the vertically polarized light arrives at the phase modulator  3  earlier than the horizontally polarized light by about 20 ns corresponding to a voltage fiber of the pulse generator  11 . The width of voltage pulse of the pulse generator  11  is set such that it is shorter than the difference in arrival time between the vertically polarized light and the horizontally polarized light and is longer than the time width of light pulse. According to the present embodiment, the light pulse width is 100 ps and the voltage pulse width of the pulse generator  11  is 10 ns. The respective polarization states pass through the fiber loop and are then combined by the polarization beam splitter  2 . The combined polarization state is |0&gt;+exp[2πi0.φ n−1 ]|1&gt;. Regarding light passing through the polarization controller  4  and being subjected to polarization split through the polarization beam splitter  5 , therefore, when φ n−1 =0, only the detector  7  detects the light. When φ n−1 =1, only the detector  6  detects the light. The outputs of the detectors  6  and  7  are compared to each other, so that the value of φ n−1  can be found. 
     Regarding a third light pulse, the output voltage of the pulse generator  11  is adjusted so that only the polarization in the |1&gt; state is subjected to phase modulation by −φ n−1 π/2−φ n π/4. Due to the phase modulation, the state of light incident on the polarization beam splitter  5  is |0&gt;+exp[2πi0.φ n−2 ]|1&gt;. The outputs of the detectors  6  and  7  are compared to obtain the value of φ n−2 . 
     In a manner similar to the above, the output voltage of the pulse generator  11  is adjusted so that only the polarization in the |1&gt; state of a k-th light pulse is subjected to phase modulation by an amount expressed as follows. 
                   -       ∑     j   =   1       k   -   1       ⁢       πϕ     n   -   j   +   1       /     2     k   -   j                   (   10   )               
Due to the phase modulation, light incident on the polarization beam splitter  5  has a state expressed by |0&gt;+exp[2πi0.φ n−k+1 ]|1&gt;. The outputs of the detectors  6  and  7  are compared to each other, so that the value of φ n−k+1  can be obtained.
 
     As mentioned above, the operation for performing phase modulation to the next qubit by an amount determined based on a result of the measurement of the preceding input qubit is repeated, so that the inverse Fourier transform can be performed to an arbitrary number of qubits. 
     According to the present embodiment, the amplitude of phase modulation is determined using the results of all qubits measured so far. The number of photons per pulse is larger than 1 and the difference in output between the detectors  6  and  7  is large. Therefore, if the amount of phase modulation is not accurate, the measurement result includes no error. As a result of the experiment, it was found that the inverse Fourier transform can be performed with accuracy using the measurement results of up to the fourth previous bit. 
     The quantum circuit described in the present embodiment is combined with a circuit for providing a phase depending on the order, so that Shor&#39;s algorithm can be realized. In this case, the number of photons is set larger than 1 in the circuit for providing a phase depending on the order. Therefore, even if an error in phase is large, quantum computation with a small error can be performed. 
     The present invention is not limited to the foregoing embodiment but many modifications and variations are possible within the spirit of the present invention. The modifications and variations are not excluded from the scope of the present invention. 
     As described in detail above, the present invention has the following advantages. 
     (A) First, a controlled-unitary transform can be performed to an arbitrary number of qubits using the controlled-unitary transform for one qubit. 
     The reason is that the operation of applying phase modulation with an amplitude, determined based on a result of the measurement of a qubit, to the next qubit is repeated. 
     (B) Second, an error, which is caused by disturbance in the environment and is included in the measurement result, is small. 
     The reason is as follows: Since a fiber loop is used and polarized light in the |0&gt; state and that in the |1&gt; state propagate through the same fiber, the common disturbance is applied, so that the disturbance does not affect the phase difference between the polarized light in the |0&gt; state and that in the |1&gt; state. 
     In addition, the number of photons to be used can be increased. Consequently, it is possible to easily distinguish between the |0&gt; state and the |1&gt; state. Thus, even if the phase difference is large, measurement with a small error can be performed. 
     INDUSTRIAL APPLICABILITY 
     According to the present invention, the quantum circuit and the quantum computer can realize multi-bit quantum computation. The quantum circuit can be used as one of essential modules of the quantum computer.