JP2009271624A - Quantum circuit, quantum arithmetic unit, and quantum arithmetic method - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To provide a technology for calculating a sum a+b of binary numbers a, b with a linear size and linear depth by using only a neighborhood quantum bit arithmetic operation without using any auxiliary quantum bit. <P>SOLUTION: Each digit of a binary number b is associated with the initial state of quantum bits Q<SB>2i+1</SB>(i=0, through n-1) in the order of small one, and each digit of a binary number a is associated with the initial state of quantum bits Q<SB>2i+2</SB>(i=0 through n-1) in the order of small one. A quantum arithmetic operation performs a quantum operation to the quantum bits Q1 through Q<SB>2n+1</SB>. Each digit of the sum a+b is associated with the state after operation of the quantum bits Q<SB>2i+1</SB>(i=0 through n) in the order of small one. <P>COPYRIGHT: (C)2010,JPO&INPIT

Description

  The present invention relates to a technique for performing a quantum operation using a quantum circuit.

  2. Description of the Related Art In recent years, attention has been focused on a technique for solving a problem that is difficult to solve by conventional information theory by applying the principle of quantum mechanics, which is a physical law applied to a fine world. For example, a quantum computer can execute a plurality of operations in parallel using a superposition state that is a phenomenon unique to the world of quantum mechanics (quantum parallelism). And it is suggested that the quantum computer can calculate much faster than the current computer with respect to problems such as factorization by the quantum parallelism and the degeneracy of the state by observation.

For example, in Non-Patent Document 1, when two binary numbers a = a _n−1 ... A ₀ and b = b _n−1 ... B ₀ and z of 0 or 1 are given as inputs in a quantum computer. , A + b = s _n ... S ₀ and a quantum circuit that outputs z (+) s _n , s _n−1, s ₀ and a is disclosed. Where n is a natural number, a _i ∈ {0,1} (i = 0,..., N−1), b _i ∈ {0,1} (i = 0,..., N−1), As α, βε {0,1}, α (+) β represents addition (exclusive OR) of α and β in mod2.

  FIG. 15 is a diagram showing the quantum circuit disclosed in Non-Patent Document 1. In FIG.

  As shown in this figure, the quantum circuit of FIG. 15 includes a MAJ (Majorrity) gate, a UMA (UnMajorrity and Add) gate, and a CNOT gate. Here, the quantum circuit means a combination of quantum basic gates such as CNOT gates.

  The MAJ gate is a quantum circuit that calculates x, y, and z majorities xy (+) yz (+) zx indicated by three qubits, respectively, and is defined as shown in FIG. As shown in this figure, the MAJ gate is composed of a CNOT gate and a Toffoli gate defined in FIGS. 6 and 7, respectively.

  The UMA gate is defined as shown in FIG. As shown in this figure, the UMA gate is also composed of a CNOT gate and a Toffoli gate. In the drawing

は、前述した（＋）と同じ排他的論理和を示す。

Indicates the same exclusive OR as (+) described above.

In the quantum circuit of Non-Patent Document 1, 2n qubits representing two binary numbers a = a _n-1 ... A ₀ and b = b _n−1 ... B ₀ | a ₀ > _{,. −1} >, | b ₀ >,..., | B _n−1 >, one qubit | z> indicating z of 0 or 1, and one qubit | 0> which is an auxiliary qubit. Used as input. FIG. 15 shows an example in which a and b are each represented by 5 qubits (n = 5), but it is easy to extend this to an adder circuit of n qubit bits.

In the case of this quantum circuit, the number of three basic gates (CNOT gate and Toffoli gate in this example) necessary for executing the arithmetic processing increases in proportion to the number of digits of a and b. Furthermore, the number of calculation steps necessary for executing the arithmetic processing also increases in proportion to the number of digits a and b. Here, the number of basic operations proportional to the number of input digits is called “linear size”, and the number of calculation steps proportional to the number of input digits is called “linear depth”. That is, the quantum circuit of Non-Patent Document 1 is a quantum circuit that performs an operation with a linear size and a linear depth.
SA Cuccaro, TG Draper, SA Kutin, and DP Moulton, “A new quantum ripple-carry addition circuit”, [online], [searched May 1, 2008], Internet <http://arxiv.org/abs / quant-ph / 0410184v1>, 2004

  Since it is physically difficult to realize a large number of qubits and operate them cooperatively, it is desirable not to use auxiliary qubits. In addition, in order to facilitate control of the qubit, it is desirable to configure the quantum circuit only by quantum operations on neighboring qubits. Here, the neighborhood qubit is a constant c qubits that are consecutively arranged and do not depend on the number of digits n of the binary numbers a and b.

  However, the quantum circuit described in Non-Patent Document 1 requires an auxiliary qubit and has a problem that it includes a quantum operation for a qubit that is not a neighboring qubit.

  In view of the above problems, the present invention provides a quantum circuit, a quantum operation device, and a quantum operation method that calculate the sum a + b of two binary numbers a and b by using only a quantum operation on neighboring qubits without using auxiliary qubits. The purpose is to provide.

According to one aspect of the invention, to calculate the sum a + b of two binary numbers a and b by operating on _{2n + 1} qubits Q ₁ ,..., Q _{2n + 1} ,
Each digit of the binary number b corresponds to the initial state of the qubit Q _{2i + 1} (i = 0,..., N−1) from the smaller one, and each digit of the binary number a corresponds to the qubit Q _{2i + 2} ( i = 0,..., n−1) respectively, and each digit of the sum a + b corresponds to the state after the operation of the quantum bit Q _{2i + 1} (i = 0 _,.
Applying CNOT with qubit Q _{2n-k + 1} as target bit, qubit Q _2n-k as control bit, and applying CNOT with qubit Q _{2n-k + 1} as control bit and qubit Q _2n-k as target bit Are sequentially repeated by increasing k by 1 from k = 0 to k = 2n−3,
Apply Toffoli with qubits Q ₁ and Q ₂ as control bits and qubit Q ₃ as a target bit,
Apply MAJ2 to qubits Q _{2k + 3} , Q _{2k + 4} , Q _{2k + 5} , apply SWAP to qubits Q _{2k + 2} , Q _{2k + 3} , apply SWAP to qubits Q _{2k + 3} , Q _{2k + 4} , and apply qubits Q _{2k + 2} , Q _{2k + 3} The operation of applying Toffoli using the control bit and qubit Q _{2k + 4} as the target bit is sequentially repeated by increasing k by 1 from k = 0 to k = n−3,
Apply MAJ2 to the qubits Q _2n−1 , Q _2n , Q _{2n + 1} ,
Applying SWAP to the qubits Q _2n-2 and Q _2n−1 , applying CNOT with the qubit Q _2n−1 as the control bit and the qubit Q _2n as the target bit, the qubits Q _2n−2 and Q _2n− Apply SWAP to ₁
The operation of applying UMA2 to qubits Q _2n-2k-4 , Q _2n-2k-3 , Q _2n-2k-2 is sequentially repeated by increasing k by 1 from k = 0 to k = n-4,
Apply Toffoli with qubits Q ₂ and Q ₃ as control bits and qubit Q ₄ as target bits,
Apply the NOT to qubit _{Q 1,} to apply the SWAP the qubit _Q 2, _{Q 3,} to apply the SWAP the qubit _Q 4, _{Q 5,} applying the SWAP the qubit _Q 3, _{Q 4,} quantum Apply Toffoli with bits Q ₅ and Q ₆ as control bits and qubit Q ₇ as a target bit,
SW = 0 is applied to the qubits Q _{2k + 4} , Q _{2k + 5} , Q _{2k + 6} , Q _{2k + 7,} and the operation of applying Toffoli with the qubits Q _{2k + 6} , Q _{2k + 7} as the control bits and the qubits Q _{2k + 8 as the} target bits is k = 0 to k = n -4 by increasing k by 1 up to -4
The operation of applying SWAP2 to the qubits Q _2n-2k-4 , Q _2n-2k-3 , Q _2n-2k-2 , Q _2n-2k-1 is performed by ₁ from k = 0 to k = n-3. Increase and repeat sequentially,
Apply the NOT to qubit _{Q 2,} control bits qubit _Q 2, _{Q 3,} applying the Toffoli targeting bits qubit _{Q 4,} to apply the SWAP the qubit _Q 1, _{Q 2,} qubit Q apply the SWAP to _{2, Q 3,}
The operation of applying CNOT using the qubit Q _{2k + 1} as the target bit and the qubit Q _{2k + 2} as the control bit is sequentially repeated by increasing k by 1 from k = 0 to k = n−1.
Perform an operation that is substantially equivalent to the operation.

  Without using the auxiliary qubit, the sum a + b of the two binary numbers a and b can be calculated only by the operation on the neighboring qubit.

  Hereinafter, an embodiment of a quantum circuit, a quantum operation device, and a quantum operation method of the present invention will be described. The same parts are denoted by the same reference numerals, and redundant description is omitted.

The quantum circuit, the quantum operation device, and the quantum operation method according to this embodiment have two n (n is a natural number) binary numbers a = a _n−1 ... A ₀ and b = b _n−1 ... B ₀ and 0. or when 1 is a z is given as input, the sum _{a + b = s n ... s} 0 is calculated, and z (+) _{s n} and _{s n-1} ... _{s 0} and _{a = a n-1 ... a} 0 Is output. As i = 0,..., n−1, a _i ∈ {0,1}, b _i ∈ {0,1}, s _i ∈ {0,1}. (+) Is addition in mod2.

In the quantum circuit, the quantum circuit, and the quantum operation method according to this embodiment, a quantum operation is performed on 2n + 1 quantum bits Q ₁ ,..., Q _{2n + 1} (see FIG. 20) in order to calculate the sum a + b. Unlike the quantum circuit of Non-Patent Document 1, no auxiliary qubit is used. Here, performing a quantum operation on a qubit may be abbreviated as performing an operation on the qubit.

Each digit of the binary number b corresponds to the initial state of the qubit Q _{2i + 1} (i = 0,..., N−1) from the smallest one, and each digit of the binary number a has the qubit Q _{2i + 2} ( This corresponds to the initial state of i = 0,..., n−1). That is, assuming that i = 0,..., N−1, the digit of b _i corresponds to the initial state of the qubit Q _{2i + 1} , and the digit of a _i corresponds to the initial state of the qubit Q _{2i + 2} .

The sum a + b of the two n-digit binary numbers a and b may be n + 1 digits due to carry. The qubit Q _{2n + 1} is used to store the most significant digit which is the (n + 1) th digit of the sum a + b.

  The quantum circuit, quantum operation device, and quantum operation method according to this embodiment operate on neighboring qubits. Here, the neighborhood qubit is a constant c qubits that are consecutively arranged and do not depend on the number of digits n of the binary numbers a and b. In this embodiment, c = 3. By configuring the quantum circuit and the quantum operation method only with operations on neighboring qubits, it is easy to physically implement them, and the accuracy of operations on qubits increases.

  The quantum circuit, quantum operation device, and quantum operation method according to this embodiment have a linear size and a linear depth. The linear size means that the basic operation number is proportional to the number of digits n of the binary numbers a and b. The linear depth means that the number of calculation steps is proportional to the number of digits n of the binary numbers a and b.

  The quantum circuit, quantum operation device, and quantum operation method according to this embodiment are based on CNOT, Toffoli, NOT, and SWAP shown in FIGS. 6 to 9, respectively.

Hereinafter, assuming that i = 0,..., N−1, the initial state of the qubit Q _{2i + 1} is | b _i >, the initial state of the qubit Q _{2i + 2} is | a _i >, and the initial state of the qubit Q _{2n + 1} A case where | z> will be described as an example.

  The quantum circuit and the quantum operation method according to this embodiment include five steps shown in FIGS. 16 to 19, and the processing of these steps is realized by the quantum operation device. A functional block diagram of the quantum arithmetic device is illustrated in FIG. FIG. 1 to FIG. 5 illustrate quantum circuits when n = 5, that is, binary numbers a and b are 5 digits.

<Step S1 (see FIG. 16)>
The CNOT operating unit 11 (see FIG. 20) applies CNOT using the qubit Q _{2n-k + 1} as the target bit and the qubit Q _2n-k as the control bit, and controls the qubit Q _{2n-k + 1} as the control bit and qubit Q _{2n. The} operation of applying CNOT with _-k as a target bit is sequentially repeated by increasing k by 1 from k = 0 to k = 2n-3 (step S1).

Specifically, step S1 includes steps S1a to s1e. The control unit 9 sets the variable k stored in the storage unit 10 to k = 0 (step S1a). The CNOT operating unit 11 applies CNOT to these quantum bits Q _{2n-k + 1} and Q _{2n-k using} the quantum bit Q _{2n-k + 1} as the target bit and the quantum bit Q _2n-k as the control bit (step S1b). Subsequently, the CNOT operating unit 11 applies CNOT to these qubits Q _{2n-k + 1} and Q _{2n-k using} the qubit Q _{2n-k + 1} as a control bit and the qubit Q _2n-k as a target bit (step) S1c). The controller 9 determines whether k = 2n−3 (step S1d). If k = 2n−3 is not satisfied, the control unit 9 increments k by 1 (step S1e) and proceeds to step S1b. If k = 2n-3, the process proceeds to step S2.

By the operation of step S1, the value of z is the second and subsequent digits _b n ... _{b 1} each dispensing the second and subsequent digit _a n ... _{a 1} and binary b a binary a.

<Step S2 (see FIG. 16)>
Step S2 includes steps S2a to S2c.

First, the Toffoli operation unit 13 applies Toffoli using the qubits Q ₁ and Q ₂ as control bits and the qubit Q ₃ as a target bit (step S2a).

Subsequently, MAJ2 is applied to the qubits Q _{2k + 3} , Q _{2k + 4} , Q _{2k + 5} , SWAP is applied to the qubits Q _{2k + 2} , Q _{2k + 3,} and SWAP is applied to the qubits Q _{2k + 3} , Q _{2k + 4} . The operation of applying Toffoli using bits Q _{2k + 2} and Q _{2k + 3} as control bits and quantum bit Q _{2k + 4} as a target bit is sequentially repeated by increasing k by 1 from k = 0 to k = n−3 (step S2b).

  Here, MAJ2 is a quantum circuit that outputs the largest value among the three input values, and is defined in FIG. For example, if two or more input values are | 0>, | 0> is output, and if two or more input values are | 1>, | 1> is output.

Step S2b specifically includes steps S2b1 to S2b5. The control unit 9 sets the variable k stored in the storage unit 10 to k = 0 (step S2b1). The MAJ2 operation unit 14 applies MAJ2 to the qubits Q _{2k + 3} , Q _{2k + 4} , and Q _{2k + 5} (step S2b2). The SWAP operation unit 12 applies SWAP to the qubits Q _{2k + 2} and Q _{2k + 3} (step S2b3). Subsequently, the SWAP operation unit 12 applies SWAP to the qubits Q _{2k + 3} and Q _{2k + 4} (step S2b4). The Toffoli operating unit 13 uses the qubits Q _{2k + 2} , Q _{2k + 3} as control bits and the qubit Q _{2k + 4} as a target bit, and applies Toffoli to these qubits Q _{2k + 2} , Q _{2k + 3} , Q _{2k + 4} (step S2b5). The controller 9 determines whether k = n−3 (step S2b6). If k = n−3 is not true, the controller 9 increments k by 1 (step S2b7), and proceeds to step S2b2. If k = n−3, the process proceeds to step S2c.

At the end of step S2, the MAJ2 operation unit 14 applies MAJ2 to the qubits Q _2n−1 , Q _2n , Q _{2n + 1} (step S2c).
By the operation of step S2, the carry | c _i > (i = 1,..., N) of each digit is calculated and expressed in the form | a _i (+) c _i >. Here, | c ₀ > = | 0> and | c _{i + 1} > = | a _i b _i (+) b _i c _i (+) c _i a _i >.

Incidentally, appearing in the lower right of FIG. _{2 | z (+) s 5} > of | _{s 5>} is, | equal to _{c 5>.} This is because | a ₅ > = | b ₅ > = 0.

<Step S3 (see FIG. 17)>
Step S3 includes steps S3a to S3e.

First, the SWAP operation unit 12 applies SWAP to the qubits Q _2n−2 and Q _2n−1 (step S3a). The CNOT operating unit 11 applies the CNOT to these quantum bits Q _2n-1 and Q _{2n using} the quantum bit Q _2n-1 as a control bit and the quantum bit Q _2n as a target bit (step S3b). The SWAP operation unit 12 applies SWAP to the qubits Q _2n−2 and Q _2n−1 (step S3c).

Subsequently, the UMA2 operation unit 15 performs an operation of applying UMA2 to the qubits Q _2n-2k-4 , Q _2n-2k-3 , Q _2n-2k-2 from k = 0 to k = n-4. k is incremented by 1 and repeated in sequence (step S3d). UMA2 is a quantum gate defined in FIG.

Step S3d specifically includes steps S3d1 to s3d4. The control unit 9 sets the variable k stored in the storage unit 10 to k = 0 (step S3d1). The UMA2 operation unit 15 applies UMA2 to the qubits Q2n _-2k-4 , Q2n _-2k-3 , and Q2n _-2k-2 (step S3d2). The controller 9 determines whether k = n−4 (step S3d3). If k = n−4 is not satisfied, the control unit 9 increments k by 1 (step S3d4) and proceeds to step S3d2. If k = n-4, the process proceeds to step S3e.

At the end of step S3, the Toffoli operation unit 13 uses the qubits Q ₂ , Q ₃ as control bits and the qubit Q ₄ as a target bit, and applies Toffoli to these qubits Q ₂ , Q ₃ , Q ₄ ( Step S3e).

As a result of the operation in step S3, assuming that i = 2,..., N, | a _i (+) c _i > is | a _i (+) c ₁ (+) b ₀ >, where c _i does not depend on i. It is expressed by

<Step S4 (see FIG. 18)>
Step S4 includes steps S4a to S4e.

NOT operation unit 16 applies a NOT to qubit _{Q 1} (step S4a). The SWAP operation unit 12 applies SWAP to the qubits Q ₂ and Q ₃ (step S4b). The SWAP operation unit 12 applies SWAP to the qubits Q ₄ and Q ₅ (step S4c). The SWAP operation unit 12 applies SWAP to the qubits Q ₃ and Q ₄ . The Toffoli operating unit 14 uses the qubits Q ₅ , Q ₆ as control bits and the qubit Q ₇ as a target bit, and applies Toffoli to these qubits Q ₅ , Q ₆ , Q ₇ (step S4d).

Subsequently, the operation of applying SWAP2 to the qubits Q _{2k + 4} , Q _{2k + 5} , Q _{2k + 6} , Q _{2k + 7} and applying Toffoli to the qubits Q _{2k + 6} , Q _{2k + 7} , Q _{2k + 8} from k = 0 to k = n−4 Is incremented by 1 and repeated sequentially (step S4e).

  SWAP2 is a quantum circuit defined by FIG.

Specifically, step S4e includes steps S4e1 to S4e5. The control unit 9 sets the variable k stored in the storage unit 10 to k = 0 (step S4e1). The SWAP2 operation unit 17 applies SWAP2 to the qubits Q _{2k + 4} , Q _{2k + 5} , Q _{2k + 6} , and Q _{2k + 7} (step S4e2). The Toffoli operation unit 14 uses the qubits Q _{2k + 6} , Q _{2k + 7} as control bits, and the qubit Q _{2k + 8} as a target bit, and applies Toffoli to these qubits Q _{2k + 6} , Q _{2k + 7} , Q _{2k + 8} (step S4e3). The control unit 9 determines whether k = n−4 (step S4e4). If k = n−4 is not satisfied, the control unit 9 increments k by 1 (step S4e5) and proceeds to step S4e2. If k = n−4, go to step 5.

As a result of the operation in step s4, | a _i > is obtained from | a _i (+) c ₁ (+) b ₀ > with i = 2,.

<Step S5 (see FIG. 19)>
Step S5 includes steps S5a to S5f.

The SWAP2 operation unit 17 performs an operation of applying SWAP2 to the qubits Q _2n-2k-4 , Q _2n-2k-3 , Q _2n-2k-2 , Q _2n-2k-1 from k = 0 to k = The k is incremented by 1 up to n-3 and repeated sequentially (step S5a). SWAP2 is a quantum gate defined in FIG.

Specifically, step S5a includes steps S5a1 to S5a4. The control unit 9 sets the variable k stored in the storage unit 10 to k = 0 (step S5a1). The SWAP2 operation unit 17 applies SWAP2 to the qubits Q2n _-2k-4 , Q2n _-2k-3 , Q2n _-2k-2 , and Q2n _-2k-1 (step S5a2). The controller 9 determines whether k = n−3 (step S5a3). If k = n−3 is not true, the controller 9 increments k by 1 (step S5a4), and proceeds to step S5a2. If k = n−3, the process proceeds to step S5b.

NOT operation unit 16 applies a NOT to qubit _{Q 2} (step S5b). The Toffoli operation unit 13 applies Toffoli to the qubits Q ₂ , Q ₃ , and Q ₄ (Step S5c). The SWAP operation unit 12 applies SWAP to the qubits Q ₁ and Q ₂ (step S5d). Subsequently, the SWAP operation unit 12 applies SWAP to the qubits Q ₂ and Q ₃ (step S5e).

The CNOT operation unit 11 sequentially repeats the operation of applying CNOT using the qubit Q _{2k + 1} as the target bit and the qubit Q _{2k + 2} as the control bit, increasing k by 1 from k = 0 to k = n−1 (step S5f). ).

Specifically, step S5f includes steps S5f1 to S5f4. The control unit 9 sets the variable k stored in the storage unit 10 to k = 0 (step S5f1). The CNOT operating unit 11 applies CNOT to these quantum bits Q _{2k + 1} and Q _{2k + 2 using} the quantum bit Q _{2k + 1} as a target bit and the quantum bit Q _{2k + 2} as a control bit (step S5f2). The controller 9 determines whether k = n−1 (step S5f3). If k = n−1 is not satisfied, the controller 9 increments k by 1 (step S5f4), and proceeds to step S5f2. If k = n−1, the process ends.

By the processing in step S5, each digit of the sum a + b = s _n ... S ₀ corresponds to the state after the operation of the qubit Q _{2i + 1} (i = 0,..., N) from the smaller one. Specifically, as exemplified in the case of n = 5 in FIG. 5, for each digit of s _n-1 ... S ₀ , qubits Q _{2i + 1} = | s _i > (i = 0,..., N−1) ) And s _n digits are quantum bits Q _{2n + 1} = | z (+) S _n >.
Note that the state after the operation of the qubit Q _{2i + 2} (i = 0,..., N−1) corresponds to the smaller of the digits of the binary number b.

  Thus, the sum a + b of two binary numbers a and b can be calculated without using auxiliary qubits. By not using auxiliary qubits, addition can be performed with fewer qubits than in the past, and control of qubits is facilitated.

[Quantum arithmetic unit]
Next, the quantum arithmetic device 1 that performs operations on qubits according to the quantum circuit of the present invention will be described. As illustrated in FIG. 20, the quantum computation device 1 of the present invention includes a control unit 9, a storage unit 10, a CNOT operation unit 11, a SWAP operation unit 12, a Toffoli operation unit 13, a MAJ2 operation unit 14, a UMA2 operation unit 15, A NOT operation unit 16, a SWAP2 operation unit 17, and qubits Q ₁ ,..., Q _{2n + 1} are included.

  The quantum arithmetic device 1 can be realized by a single quantum computer. The control unit 9 and the storage unit 10 may be realized by a classical computer as will be described later. As physical systems realized by quantum computers, for example, methods using ion traps (JI Cirac and P. Zoller, Quantum computations with cold trapped ions, Physical Review Letter 74; 4091, 1995), photon polarization and optical path as qubits, etc. (Y. Nakamura, M. Kitagawa, K. Igeta, In 3-rd Proc. Asia-Pacific Phys. Comf., World Scientific, Singapore, 1988), a method using each spin in a liquid (Gershenfield, Chuang , Bulk spin resonance quantum computation, Science, 275; 350, 1997), a method using nuclear spins in silicon crystals (BE Kane, A silicon-based nuclear spin quantum computer, Nature 393, 133, 1998), in quantum dots Method using electron spin (D. Loss and DP DiVincenzo, Quantum computation with quantum dots, Physical Review A 57, 120-126, 1998), method using superconducting element (Y. Nakamura, Yu. A. Pashkin and JS Tsai , Coherent control of macroscopic quantum states in a single- cooper pair box, Nature 393, 786-788, 1999). For details on how to implement quantum computers for each physical system, see `` http://www.ipa.go.jp/security/fy11/report/contents/crypto/crypto/report/QuantumComputers/contents/doc/qc_survey. pdf "and" MA Nielsen and IL Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Chapter 7 Physical Realization ".

<Quantum bit>
In an ion trap quantum computer, for example, a quantum bit is realized by utilizing the ground state and excited state of ions. When nuclear spins are used as qubits, for example, “TD Ladd, et al.,“ All-Silicon quantum computer, ”Phys. Rev. Lett., Vol. 89, no. 1, 017901-1‐ [017901-4, July 1, 2002.], each qubit is generated on a Si (111) substrate or the like. The initial quantum state of the qubit may be, for example, one obtained by an operation by a quantum circuit for other operations, or the substrate on which each qubit is generated is cooled to the order of mK (millikelvin) or less. Then, after aligning the spin direction, a predetermined electromagnetic wave pulse may be applied and generated. When the polarization of a photon is used as a qubit, for example, parametric down conversion (PDC) (for example, “PG Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, AV Sergienko, and Y Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett., 75: 4337-4341, 1995. “PG Kwiat, E. Waks, AG White, I. Appelbaum, and PH Eberhard, “Ultrabright source of polarization-entangled photons,” Phys. Rev. A, 60: R773-R776, 1999. ”) is used. In this case, as the initial quantum state of each qubit, for example, the one obtained by an operation by a quantum circuit of another operation is used. In addition, by performing operations such as Walsh Hadamard transform, CNOT, rotation, etc. realized by a beam splitter, a polarization rotation element, etc. on a single photon generated by parametric down conversion, etc., the above initial quantum state is generated. Also good.
In addition, it is good also as preparing a qubit by the method described in said literature.

  In addition, when it is necessary to preserve the quantum state of the qubit before and after the operation, for example, the amount of electrons in the quantum dot, the nuclear spin, or the charge (Cooper pair) in the superconductor is quantized. Quantum memories that store data using bits may be used (A. Barenco, D. Deutsch, and A. Ekert, Phys. Rev. Lett. 74, 4083 (1995), Hideaki Matsueda, Journal of the Institute of Electronics, Information and Communication Engineers) A Vol.J81-A No.12 (1998) 1978, THOosterkamp et.al., Nature 395,873 (1998), D. Loss and DP DiVincenzo, Phys. Rev. A57 (1998) 120. T. Oshima, quant- ph / 0002004, http://arxiv.org/abs/quant-ph/0002004, BEKane, A silicon-based nuclear spin quantum computer, Nature, 393, 133 (1998), http: //www.snf.unsw edu.au/, Y. Nakamura, Yu. A. Pashkin and JSTsai, Nature 398 (1999) 768).

<CNOT operation unit 11>
In an ion trap quantum computer, for example, ions are arranged on a straight line, and CNOT is realized by laser beam irradiation aiming at each ion. In addition, when using photon polarization as a qubit, for example, a polarization beam splitter or the like is used, and TB Pittman, MJ Fitch, BC Jacobs, JD Franson: “Experimental Controlled-NOT Logic Gate for Single Photons in the Coincidence Basis , “Quant-ph / 0303095, http://arxiv.org/abs/quant-ph/0303095”, the CNOT operation is realized by the Pittman et al. Method. When nuclear spin is used as a qubit, for example, the CNOT operation unit 11 can be realized by applying a predetermined electromagnetic wave pulse to the qubit.
In addition, you may implement | achieve CNOT operation part 11 by the method described in said literature.

<SWAP operation unit 12, Toffoli operation unit 13, MAJ2 operation unit, UMA2 operation unit, NOT operation unit 16, SWAP2 operation unit 17>
When the polarization of photons is used as a qubit, each calculation can be realized using the above-described CNOT configuration, polarization rotation element, or the like. When the nuclear spin is used as a qubit or ion trap, each process can be realized by the above-described electromagnetic wave pulse or laser beam irradiation.

<Control unit 9, storage unit 10>
The control unit 9 may be realized, for example, by causing a known computer to execute a predetermined program, or may be realized by the quantum computer described above. The storage unit 10 may be a readable / writable semiconductor memory such as DRAM, SRAM, flash memory, NV (Nonvolatile) RAM, or the above-described quantum memory.

<Single qubit operation>
In the ion trap quantum computer, for example, ions are arranged on a straight line, and the operation of a single qubit is realized by laser beam irradiation aimed at each ion. When nuclear spins are used as qubits, each processing is realized by electromagnetic pulse or laser beam irradiation. Moreover, when using the polarization of a photon as a qubit, it implement | achieves by a polarization rotation element etc., for example.

[Modifications, etc.]
All or part of the qubits Q ₁ ,..., Q _{2n + 1} may be an overlap of | 0> and | 1>. By applying the quantum circuit and the quantum operation method according to the present invention to the qubits Q ₁ ,..., Q _{2n + 1 in} which | 0> and | 1> are superposed, parallel processing becomes possible. For example, if each of the qubits Q ₁ ,..., Q _2n corresponding to each digit of the n-digit binary numbers a and b is an overlap of | 0> and | 1>, 2 ²ⁿ A sum a + b for all combinations of binary numbers a and b can be calculated in parallel.

  Instead of the MAJ2 operation unit 14 performing the MAJ2 operation on the qubit, the CNOT operation unit 11 may perform an operation substantially equivalent to the MAJ2 operation defined in FIG. Similarly, instead of the UMA2 operation unit 15 performing the UMA2 operation on the qubit, the CNOT operation unit 11, the SWAP operation unit 12, and the Toffoli operation unit 13 perform operations substantially equivalent to the UMA2 operation defined in FIG. You may go.

  Here, substantially the same operation means an operation with the same output. To describe UMA2 as an example, the UMA2 operation unit 15 performs an operation of performing UMA2 defined in FIG. 13 by one quantum operation, and an operation of sequentially performing each operation constituting UMA2 (that is, the first and second operations). And Toffoli operation on the third qubit, SWAP operation on the second and third qubits, CNOT operation on the second and third qubits, and SWAP on the first and second qubits The operation for performing the operation is an operation that is substantially equal because the outputs are the same.

  Further, it is known that the SWAP defined in FIG. 9 can be composed of three CNOTs as shown in FIG. In other words, the SWAP operation defined in FIG. 9 is substantially equal to the three CNOT operations shown in FIG. Therefore, the UMA2 operation defined in FIG. 13 is substantially equivalent to the UMA3 operation of FIG. 23 in which the second SWAP in the UMA2 is replaced with three CNOTs.

  Also, as shown in FIG. 22, it is known that when the CNOT is performed twice without changing the control bit and the target bit, the original quantum state is restored. That is, the operation of performing CNOT twice without changing the control bit and the target bit is substantially the same as the operation of performing nothing. Therefore, the UMA3 operation defined in FIG. 23 is substantially equal to the UMA4 operation defined in FIG. 24 in which the fourth CNOT operation and the fifth CNOT operation of the UMA3 are not performed. Operations substantially equivalent to the UMA2 operation exist in addition to the UMA3 operation and the UMA4 operation exemplified here.

  Just as there are a plurality of operations that are substantially equal to the UMA2 operation, there are operations that are substantially equal to the operations by the quantum circuit, quantum operation device, and quantum operation method of the present invention. Instead of the operations in steps S1 to S5 described above, an operation substantially equivalent to this operation may be performed.

  In addition, the various processes described above are not only executed in time series according to the description, but may be executed in parallel or individually according to the processing capability of the apparatus that executes the processes or as necessary.

  Needless to say, other modifications are possible without departing from the spirit of the present invention.

The figure which illustrates step S1 of the quantum circuit in case of n = 5. The figure which illustrates step S2 of the quantum circuit in case of n = 5. The figure which illustrates step S3 of the quantum circuit in case of n = 5. The figure which illustrates step S4 of the quantum circuit in case of n = 5. The figure which illustrates step S5 of the quantum circuit in case of n = 5. The figure which shows CNOT. The figure which shows Toffoli. The figure which shows NOT. The figure which shows SWAP. The figure which shows MAJ. The figure which shows UMA. The figure which shows MAJ2. The figure which shows UMA2. The figure which shows SWAP2. A quantum circuit for calculating the sum of two binary numbers according to the prior art. The flowchart showing the process of step S1 and step S2. The flowchart showing the process of step S3. The flowchart showing the process of step S4. The flowchart showing the process of step S5. The figure which illustrates the functional block of a quantum arithmetic unit The figure which shows SWAP '. The figure which shows the quantum circuit which applies CNOT twice. The figure which shows UMA3. The figure which shows UMA4.

Explanation of symbols

DESCRIPTION OF SYMBOLS 1 Quantum arithmetic unit 9 Control part 10 Memory | storage part 11 CNOT operation part 12 SWAP operation part 13 Toffoli operation part 14 MAJ2 operation part 15 UMA2 operation part 16 NOT operation part 17 SWAP2 operation part

Claims (3)

A quantum circuit that calculates the sum a + b of two binary numbers a and b by operating on _{2n + 1} qubits Q ₁ ,..., Q _{2n + 1} ,
Each digit of the binary number b corresponds to the initial state of the qubit Q _{2i + 1} (i = 0,..., N−1) from the smaller one, and each digit of the binary number a corresponds to the qubit Q _{2i + 2} ( i = 0,..., n−1) respectively, and each digit of the sum a + b corresponds to the state after the operation of the quantum bit Q _{2i + 1} (i = 0 _,.
Applying CNOT with qubit Q _{2n-k + 1} as target bit, qubit Q _2n-k as control bit, and applying CNOT with qubit Q _{2n-k + 1} as control bit and qubit Q _2n-k as target bit Are sequentially repeated by increasing k by 1 from k = 0 to k = 2n−3,
Apply Toffoli with qubits Q ₁ and Q ₂ as control bits and qubit Q ₃ as a target bit,
Apply MAJ2 to qubits Q _{2k + 3} , Q _{2k + 4} , Q _{2k + 5} , apply SWAP to qubits Q _{2k + 2} , Q _{2k + 3} , apply SWAP to qubits Q _{2k + 3} , Q _{2k + 4} , and apply qubits Q _{2k + 2} , Q _{2k + 3} The operation of applying Toffoli using the control bit and qubit Q _{2k + 4} as the target bit is sequentially repeated by increasing k by 1 from k = 0 to k = n−3,
Apply MAJ2 to the qubits Q _2n−1 , Q _2n , Q _{2n + 1} ,
Applying SWAP to the qubits Q _2n-2 and Q _2n−1 , applying CNOT with the qubit Q _2n−1 as the control bit and the qubit Q _2n as the target bit, the qubits Q _2n−2 and Q _2n− Apply SWAP to ₁
The operation of applying UMA2 to qubits Q _2n-2k-4 , Q _2n-2k-3 , Q _2n-2k-2 is sequentially repeated by increasing k by 1 from k = 0 to k = n-4,
Apply Toffoli with qubits Q ₂ and Q ₃ as control bits and qubit Q ₄ as target bits,
Apply the NOT to qubit _{Q 1,} to apply the SWAP the qubit _Q 2, _{Q 3,} to apply the SWAP the qubit _Q 4, _{Q 5,} applying the SWAP the qubit _Q 3, _{Q 4,} quantum Apply Toffoli with bits Q ₅ and Q ₆ as control bits and qubit Q ₇ as a target bit,
SW = 0 is applied to the qubits Q _{2k + 4} , Q _{2k + 5} , Q _{2k + 6} , Q _{2k + 7,} and the operation of applying Toffoli with the qubits Q _{2k + 6} , Q _{2k + 7} as the control bits and the qubits Q _{2k + 8 as the} target bits is k = 0 to k = n -4 by increasing k by 1 up to -4
The operation of applying SWAP2 to the qubits Q _2n-2k-4 , Q _2n-2k-3 , Q _2n-2k-2 , Q _2n-2k-1 is performed by ₁ from k = 0 to k = n-3. Increase and repeat sequentially,
Apply the NOT to qubit _{Q 2,} control bits qubit _Q 2, _{Q 3,} applying the Toffoli targeting bits qubit _{Q 4,} to apply the SWAP the qubit _Q 1, _{Q 2,} qubit Q apply the SWAP to _{2, Q 3,}
The operation of applying CNOT using the qubit Q _{2k + 1} as the target bit and the qubit Q _{2k + 2} as the control bit is sequentially repeated by increasing k by 1 from k = 0 to k = n−1.
A quantum circuit that performs operations that are substantially equivalent to operations. A quantum arithmetic device that calculates a sum a + b of two binary numbers a and b by operating on _{2n + 1} qubits Q ₁ ,..., Q _{2n + 1} ,
Each digit of the binary number b corresponds to the initial state of the qubit Q _{2i + 1} (i = 0,..., N−1) from the smaller one, and each digit of the binary number a corresponds to the qubit Q _{2i + 2} ( i = 0,..., n−1) respectively, and each digit of the sum a + b corresponds to the state after the operation of the quantum bit Q _{2i + 1} (i = 0 _,.
Applying CNOT with qubit Q _{2n-k + 1} as target bit, qubit Q _2n-k as control bit, and applying CNOT with qubit Q _{2n-k + 1} as control bit and qubit Q _2n-k as target bit Are sequentially repeated by increasing k by 1 from k = 0 to k = 2n−3,
Apply Toffoli with qubits Q ₁ and Q ₂ as control bits and qubit Q ₃ as a target bit,
Apply MAJ2 to qubits Q _{2k + 3} , Q _{2k + 4} , Q _{2k + 5} , apply SWAP to qubits Q _{2k + 2} , Q _{2k + 3} , apply SWAP to qubits Q _{2k + 3} , Q _{2k + 4} , and apply qubits Q _{2k + 2} , Q _{2k + 3} The operation of applying Toffoli using the control bit and qubit Q _{2k + 4} as the target bit is sequentially repeated by increasing k by 1 from k = 0 to k = n−3,
Apply MAJ2 to the qubits Q _2n−1 , Q _2n , Q _{2n + 1} ,
Applying SWAP to the qubits Q _2n-2 and Q _2n−1 , applying CNOT with the qubit Q _2n−1 as the control bit and the qubit Q _2n as the target bit, the qubits Q _2n−2 and Q _2n− Apply SWAP to ₁
The operation of applying UMA2 to qubits Q _2n-2k-4 , Q _2n-2k-3 , Q _2n-2k-2 is sequentially repeated by increasing k by 1 from k = 0 to k = n-4,
Apply Toffoli with qubits Q ₂ and Q ₃ as control bits and qubit Q ₄ as target bits,
Apply the NOT to qubit _{Q 1,} to apply the SWAP the qubit _Q 2, _{Q 3,} to apply the SWAP the qubit _Q 4, _{Q 5,} applying the SWAP the qubit _Q 3, _{Q 4,} quantum Apply Toffoli with bits Q ₅ and Q ₆ as control bits and qubit Q ₇ as a target bit,
SW = 0 is applied to the qubits Q _{2k + 4} , Q _{2k + 5} , Q _{2k + 6} , Q _{2k + 7,} and the operation of applying Toffoli with the qubits Q _{2k + 6} , Q _{2k + 7} as the control bits and the qubits Q _{2k + 8 as the} target bits is k = 0 to k = n -4 by increasing k by 1 up to -4
The operation of applying SWAP2 to the qubits Q _2n-2k-4 , Q _2n-2k-3 , Q _2n-2k-2 , Q _2n-2k-1 is performed by ₁ from k = 0 to k = n-3. Increase and repeat sequentially,
Apply the NOT to qubit _{Q 2,} control bits qubit _Q 2, _{Q 3,} applying the Toffoli targeting bits qubit _{Q 4,} to apply the SWAP the qubit _Q 1, _{Q 2,} qubit Q apply the SWAP to _{2, Q 3,}
The operation of applying CNOT using the qubit Q _{2k + 1} as the target bit and the qubit Q _{2k + 2} as the control bit is sequentially repeated by increasing k by 1 from k = 0 to k = n−1.
A quantum arithmetic device that performs an operation substantially equivalent to the operation. A quantum operation method for calculating a sum a + b of two binary numbers a and b by operating on _{2n + 1} qubits Q ₁ ,..., Q _{2n + 1} ,
Each digit of the binary number b corresponds to the initial state of the qubit Q _{2i + 1} (i = 0,..., N−1) from the smaller one, and each digit of the binary number a corresponds to the qubit Q _{2i + 2} ( i = 0,..., n−1) respectively, and each digit of the sum a + b corresponds to the state after the operation of the quantum bit Q _{2i + 1} (i = 0 _,.
Applying CNOT with qubit Q _{2n-k + 1} as target bit, qubit Q _2n-k as control bit, and applying CNOT with qubit Q _{2n-k + 1} as control bit and qubit Q _2n-k as target bit Are sequentially repeated by increasing k by 1 from k = 0 to k = 2n−3,
Apply Toffoli with qubits Q ₁ and Q ₂ as control bits and qubit Q ₃ as a target bit,
Apply MAJ2 to qubits Q _{2k + 3} , Q _{2k + 4} , Q _{2k + 5} , apply SWAP to qubits Q _{2k + 2} , Q _{2k + 3} , apply SWAP to qubits Q _{2k + 3} , Q _{2k + 4} , and apply qubits Q _{2k + 2} , Q _{2k + 3} The operation of applying Toffoli using the control bit and qubit Q _{2k + 4} as the target bit is sequentially repeated by increasing k by 1 from k = 0 to k = n−3,
Apply MAJ2 to the qubits Q _2n−1 , Q _2n , Q _{2n + 1} ,
Applying SWAP to the qubits Q _2n-2 and Q _2n−1 , applying CNOT with the qubit Q _2n−1 as the control bit and the qubit Q _2n as the target bit, the qubits Q _2n−2 and Q _2n− Apply SWAP to ₁
The operation of applying UMA2 to qubits Q _2n-2k-4 , Q _2n-2k-3 , Q _2n-2k-2 is sequentially repeated by increasing k by 1 from k = 0 to k = n-4,
Apply Toffoli with qubits Q ₂ and Q ₃ as control bits and qubit Q ₄ as target bits,
Apply the NOT to qubit _{Q 1,} to apply the SWAP the qubit _Q 2, _{Q 3,} to apply the SWAP the qubit _Q 4, _{Q 5,} applying the SWAP the qubit _Q 3, _{Q 4,} quantum Apply Toffoli with bits Q ₅ and Q ₆ as control bits and qubit Q ₇ as a target bit,
SW = 0 is applied to the qubits Q _{2k + 4} , Q _{2k + 5} , Q _{2k + 6} , Q _{2k + 7,} and the operation of applying Toffoli with the qubits Q _{2k + 6} , Q _{2k + 7} as the control bits and the qubits Q _{2k + 8 as the} target bits is k = 0 to k = n -4 by increasing k by 1 up to -4
The operation of applying SWAP2 to the qubits Q _2n-2k-4 , Q _2n-2k-3 , Q _2n-2k-2 , Q _2n-2k-1 is performed by ₁ from k = 0 to k = n-3. Increase and repeat sequentially,
Apply the NOT to qubit _{Q 2,} control bits qubit _Q 2, _{Q 3,} applying the Toffoli targeting bits qubit _{Q 4,} to apply the SWAP the qubit _Q 1, _{Q 2,} qubit Q apply the SWAP to _{2, Q 3,}
The operation of applying CNOT using the qubit Q _{2k + 1} as the target bit and the qubit Q _{2k + 2} as the control bit is sequentially repeated by increasing k by 1 from k = 0 to k = n−1.
A quantum computation method that performs an operation substantially equivalent to the operation.

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