JP4295679B2 - Quantum calculation method, quantum computer, and quantum calculation program - Google Patents

Description

  The present invention relates to a quantum computation method, a quantum computer, and a quantum computation program for performing quantum computation, and more particularly, to a quantum computation method, a quantum computer, and a quantum computation program for performing an exchange operation between qubits.

<Quantum computer>
Research on quantum computers that perform calculations based on the operating principles of quantum mechanics has been actively conducted around the world. As an extension of the Turing machine, which is a mathematical model of the current computer, Deutsche, UK, proposed a quantum Turing machine in 1985, which laid the foundation for quantum computer research. After that, a quantum circuit, which is a computational model different from the quantum Turing machine, was proposed, and it was theoretically proved that the quantum Turing machine is equivalent in performance to the quantum circuit. However, the fundamental differences in performance between quantum Turing machines and ordinary Turing machines were not theoretically elucidated, and quantum circuits could not be physically realized at that time, so the idea of quantum computers is general Has not been widely recognized. The reason why quantum computers attracted worldwide attention was that in 1994, Shower of AT & T (at that time) announced a quantum algorithm that solved factorization in polynomial time.

  If a quantum computer can be implemented, it can break most public key cryptography, including RSA cryptography, using the Shore algorithm. Since the discovery of Shore, a great deal of research has been published in both software and hardware. Realization of a quantum computer requires hardware that expresses a qubit that is a basic unit of information in quantum computation, and a rotation operation of 1 qubit and a CNOT (control NOT) operation of 2 qubits in the hardware. . For example, in 2001, IBM succeeded in factoring 15 using an NMR quantum computer with 7 qubits. In this experiment, qubits were represented by atomic nuclear spins, and one qubit rotation operation and CNOT The operation was realized by controlling the electromagnetic field.

A lot of hardware has been proposed to realize quantum computers, but quantum computers using spin are one of the promising candidates. This type of quantum computer includes a method using an electron spin in a quantum dot (Non-Patent Document 1) and a method using an atomic nuclear spin (Non-Patent Documents 2 and 3) as qubits.
<Problems of quantum computers using spin>
Thus, quantum computers using spin are promising, but some problems are pointed out in the principle. For example, in Non-Patent Document 4, the CNOT operation changes the total spin amount, which is the basic conserved amount of quantum physics. Therefore, the quantum computer using spins has a limit in calculation accuracy, and ensures sufficient accuracy. It is pointed out that an ancillary qubit (auxiliary qubit) that is 100 times the required amount is necessary for the above.

<Solution of Non-Patent Document 5>
Techniques for solving such problems have also been proposed. For example, Non-Patent Document 5 proposes a method for realizing a quantum computer by introducing the concept of an encoded qubit (embedded qubit) and expressing a logical qubit with a plurality of qubits. In this method, an arbitrary rotation operation and CNOT operation on a logical qubit is realized only by performing an exchange operation on the qubit. When the qubit is realized by spin, the exchange operation is an operation that does not change the total spin amount, and solves the difficulty in Non-Patent Document 4.

The method proposed in Non-Patent Document 5 will be described more specifically. In this method, a logical qubit is expressed by 3 qubits. Specifically, the following states in the three qubit space are set to logical 0 and 1 (based on | 0 _L > and | 1 _L >).
| 0 _L > = 1 / √2 (| 010) − | 100))
| 1 _L > = 1 / √6 (2 | 001) − | 010) − | 100))
This | 0 _L > and | 1 _L > are orthogonal, and even if these quantum states are transformed using the spin exchange operation, | 0 _L > and | 1 _L > are in a two-dimensional space created by | 0 _L > and | 1 _L >. Closed. Conversely, an arbitrary point on the two-dimensional space created by | 0 _L > and | 1 _L > includes the first qubit and second qubit exchange operation (including partial exchange) and the second qubit. It can be shown that it can be made from | 0 _L > or | 1 _L > by the exchange operation of the third qubit (including partial exchange). Therefore, it can be seen that an arbitrary rotation operation on the logical qubit can be realized by a spin exchange operation.

を生成する操作を意味する。なお、原子の核スピンを量子ビットとして用いる場合、この操作は、交換対象の原子に磁場を与えることによって実現できる。 Here, the exchange operation (including partial exchange) is a quantum state of 2 qubits

Means an operation to generate When the atomic nuclear spin is used as a qubit, this operation can be realized by applying a magnetic field to the exchange target atom.

T is a parameter indicating the degree of exchange (how much quantum state is exchanged between two qubits). Specifically, for example, when the nuclear spin of an atom is used as a qubit, a value obtained by multiplying the time for applying the magnetic field by a constant is t. In addition, a complete exchange of quantum states is performed in the case of t = 1/2 + kπ (k is an integer), and an exchange operation other than t = 1/2 + kπ is referred to as a partial exchange.
In Non-Patent Document 5, a CNOT operation on two logical qubits is realized by performing an exchange operation as shown in FIG.

Here, the horizontal axis in FIG. 12 represents the time axis, and each horizontal bar represents each qubit. The arrow represents a partial exchange. Furthermore, the numbers indicated by the alphabets next to it represent parameters indicating the degree of exchange. Two logical qubits are represented by qubit sets 1 to 3 and 4 to 6, respectively.
As described above, an arbitrary rotation operation and CNOT operation, that is, quantum computation can be performed by expressing one logical qubit with 3 qubits and performing only an exchange operation between qubits.

<Problem of the method of Non-Patent Document 5>
However, there are problems with this method. In general, when one logical qubit is expressed by three qubits, the space generated from the two logical qubits and the exchange operation has nine dimensions (see, for example, Non-Patent Document 6). However, if it is limited to the CNOT operation, the operation becomes a four-dimensional space operation, and the remaining five dimensions become a useless space. In other words, if the parameter t described above can be strictly controlled in the exchange operation for realizing the above CNOT operation, the result of the CNOT operation to the quantum state in the four-dimensional space should be a mapping to the same four-dimensional space. is there. However, in actual physical operation, it is impossible to strictly operate the parameter t of the exchange operation. Therefore, when the exchange operation of FIG. 12 is actually performed on the quantum state of the four-dimensional space, “Leakage” to the remaining five-dimensional space occurs.

That is, when one logical qubit is represented by three qubits, the exchange operation for six qubits represented by two logical qubits is a linear operation in the 9-dimensional space “W” because of its physical requirements. . A necessary and sufficient condition for this exchange operation to be some kind of quantum computation on two logical qubits is that the points in the four-dimensional space “V ₁ ” are mapped onto the same space by this exchange operation (however, this The four-dimensional space “V ₁ ” is a direct product space of the two-dimensional space of each of the two logical qubits. However, when the exchange operation of FIG. 12 is actually performed, the point v ₁ in V ₁ is not mapped to the point v ₁ ′ in V ₁ due to the error of the parameter t, and the point v ₁ ”+ v in W ₁ is not mapped. _"would be mapped to (although, v _{2" 2} is a point in V _2, V ₂ is the orthogonal complement space V ₁ in W.). this is the leakage problem. it should be noted that generally this Such v ₂ ″ is called a leakage component, and when v ₂ ″ is not a zero vector, leakage is caused by physical operation. However, “leakage” treated below refers to an error in parameter t, which is an artificial operation. This is a case where the error is limited to the above.
The occurrence of such “leakage” is one of the biggest problems for quantum computers that are realized by expressing logical qubits in a high-dimensional space as described in Non-Patent Document 5. In a more general description, reducing errors is important in quantum computers.

<Conventional solution>
As a conventional method for solving this problem, there is a method described in Non-Patent Document 7. In this method, “leakage” caused by quantum computation performed by expressing one logical qubit with 3 qubits is converted into an error in quantum computation and corrected. Hereinafter, this method will be described.

First, bases of three qubits are taken as | 0>, | 1>, | 2>,. However, | 0> and | 1> correspond to 0 and 1 of one logical qubit, respectively. That is, | 2>,..., | 7> corresponds to the “leak” component.
Also, in order to correct one logical qubit, three ancilla qubits that are 0 when interpreted as logical qubits are facilitated.

Then, the following mapping is realized by an exchange operation.
| 0,0> → | 0,0>
| 1,0> → | 1,0>
| Other bases, 0> → || 0> and | 1> points on the space (V ₁ ) spanned by the bases, arbitrary points>
Here, i of | i, j> indicates the state base of the logical qubit (3 qubits) to be corrected using the above-mentioned base, and j indicates the state base of the auxiliary bit. It is shown using. Moreover, this operation is leaking (the state | 2>, ..., | 7 > component) and a point on the V ₁ of the as the state error (there should on quantum computing a point on the V ₁ of the as the current state This is an operation for conversion to (difference).

Thereafter, this quantum calculation error is corrected by a known method (for example, Non-Patent Documents 8, 9, and 10).
D. Loss and DP DiVincenzo, Quantum computation with quantum dots, Physical Review A 57, pp.120-126, 1998. Gershenfield, Chuang, Bulk spin resonance quantum computation, Science 275, 350, 1997. BE Kane, A silicon-based nuclear spin quantum computer, Nature 393, 133, 1998. M. Ozawa, Conservative quantum computing, Physical Review Letters 89, No. 5 (2000) 057902. DP DiVincenzo, D. Bacon, J. Kempe, S. Burkard, KB Whaley, Universal quantum computation with the exchange interaction, Nature 408, pp.339--342 (2000). J. Kempe, D. Bacon, DA Lidar, KB Whaley, Theory of Decoherence-free fault-tolerant universal quantum computation, Physical Review A 63, 042307 (2001). J. Kempe, D. Bacon, DP DiVincenzo, KB Whaley, Encoded Universality from a Single Physical Interaction, Quantum Information and Computation 1, 241 (2001). PW Shor. Fault-tolerant quantum computation.IN Preceedings, 37th Annual Symposium on Fundamentals of Computer Science, pages 55-65, IEEE Press, Los Alamitos, CA, 1996. AM Steane.Error Correcting Codes in Quantum Theory.Phys. Rev. Lett., 77: 793, 1996. AM Steane. Efficient fault-tolerant quantum computing. Nature, 399: 124, May 1993.

However, in the case of the method described in Non-Patent Document 7, there is a problem that a new cost is generated with respect to qubits and calculation time.
That is, in the case of the method described in Non-Patent Document 7, a new auxiliary bit is required every time the logical qubit is corrected. This means that the number of qubits required for quantum computation is greatly increased.
In addition, when a logical qubit is corrected, a new operation using an auxiliary bit and a new operation for correcting an error in quantum calculation are required. This means that this greatly increases the computational cost required for quantum computation.

  The present invention has been made in view of such a point, and in a quantum computation technique that executes quantum computation using only an exchange operation, it is possible to perform “leakage” without generating new costs related to qubits and computation time. It aims at providing the technology which improves the problem of "."

In the present invention, in order to solve the above-described problem, in a plurality of physical qubits, logical qubits (n (n ≧ 2) physical qubits) are used as coding units, and the first logical qubit is set to the first The second logical qubit (≠ first logical qubit) is encoded with the second base (≠ first base). Then, an exchange operation between the encoded first logical qubit and the second logical qubit is performed. The “physical qubit” means a physical qubit unit such as a spin 1/2 particle.
Here, even when logical qubits are encoded using a plurality of types of bases, an arbitrary rotation operation or CNOT search can be implemented by an exchange operation as in the conventional example. However, when a plurality of types of bases are used, the dimension of the space generated by the exchange operation is different from that when a single base is used. In some cases, the “leakage” problem can be fundamentally improved by changing the dimension of the generated space.

For example, n = 3, and the first basis and the second basis are
1 / √2 (| 010> − | 100>),
1 / √6 (2 | 001> − | 010> − | 100>),
1 / √2 (| 101>-| 011>),
1 / √6 (2 | 110 > − | 101 > − | 011 > ),
The space generated by the exchange operation of two quantum logic bits encoded by different bases is not 9-dimensional, but linear between a 5-dimensional subspace and a 9-dimensional subspace. It becomes a bond. Specifically, the exchange operation result between two logical qubits encoded with these different bases becomes the following quantum state (details will be described later). Note that | ψ ₀ > is a five-dimensional space, and | ψ ₁ > is a nine-dimensional space.
| Ψ> = (1 / √2) · | ψ ₀ > + (1 / √2) · | ψ ₁ > (1)

となる。 Here, an error accumulated in the five-dimensional space | ψ ₀ > in the above formula (1) is denoted by Δ _0, and an error accumulated in the nine-dimensional space | ψ ₁ > is denoted by Δ ₁ . In this case, the error Δ of the entire expression (1) is

It becomes.

  In general, when a physical operation is performed on a qubit in a certain subspace, the accumulated amount of error caused by the physical operation is smaller as the dimension of the subspace is lower. That is, such an error has any dimensional component in the subspace, and the component value may be positive or negative. When a positive error component and a negative error component are accumulated on the same dimension, these error components are canceled out. Further, the probability that the two generated errors have the same dimension component is higher as the dimension of the subspace is lower. As a result, the lower the subspace dimension, the smaller the amount of error that is ultimately accumulated.

Therefore, placing the error accumulated by the 9-dimensional space created by the conventional replacement operation of the logical qubit consisting of three physical qubits and Δ _2, Δ _{0 <Δ 2} and delta ₁ ≒ delta ₂ relationship is established . The relationship of Δ <Δ ₂ is established by these relationships and the above-described equation (2). This means that the general error accumulation amount when encoded with the first basis and the second basis described above is smaller than that when encoded with the conventional single basis. . And the fact that a general error accumulation amount can be reduced also means that the amount of “leakage” can be reduced.

  In the present invention, the first logical qubit is encoded with the first basis, the second logical qubit is encoded with the second basis, and the first logical qubit encoded with the different basis and the second logical qubit Therefore, it is possible to generate a space having a dimension different from that of the exchange operation using only a single basis. As a result, the amount of “leakage” can be reduced depending on how n and the base are selected.

Embodiments of the present invention will be described below with reference to the drawings.
<Overview>
In order to substantially reduce “leakage”, it is necessary to make the space generated from the exchange operation of two logical qubits as small as possible and to reduce unnecessary space dimensions. In this embodiment, by using a plurality of types of bases (binary representation method) for encoding each logical qubit, unnecessary spatial dimensions are reduced, and “leakage” is essentially reduced.

<Outline configuration>
FIG. 2 is a conceptual diagram illustrating a schematic configuration of the quantum computer 1 in this embodiment.
As illustrated in FIG. 2, the quantum computer 1 includes a qubit substrate 10 in which a plurality of physical qubits 11a to 11c, 12a to 12c, and 13a to 13c are arranged, and physical qubits 11a to 11c, 12a to 12c, and 13a to An initialization unit 20 for initializing the quantum state of 13c, arranged on a qubit substrate with logical qubits 11 to 13 (n (n ≧ 2) physical qubits / n = 3 in this example) as encoding units An encoding unit 30 that encodes each of the generated logical qubits with any of m (m ≧ 2) types of bases, and an exchange operation unit 40 that performs an exchange operation between logical qubits encoded with different bases have.

Here, the physical quantum bits 11a to 11c, 12a to 12c, and 13a to 13c in this example are spin ½ particles such as atomic nuclear spins and electron spins in quantum dots. In addition, each logical qubit 11 to 13 in this example is configured by n adjacent (n = 3 in this example) physical qubits. Here, “adjacent” is a concept including a case where physical qubits are sequentially adjacent, and is not limited to a case where all n physical qubits are adjacent.
In FIG. 2, three logical qubits 11 to 13 are configured by nine physical qubits 11a to 11c, 12a to 12c, and 13a to 13c arranged in a line. The number and arrangement of are not limited to this.

<Specific configuration example>
FIG. 3 is a block diagram illustrating a specific configuration of the quantum computer 1 of the present embodiment.
As illustrated in FIG. 3, the quantum computer 1 of this example includes a qubit board 10, a cooling unit 21 that constitutes an “initialization unit 20”, a cooling control unit 22, and a magnet 31 that constitutes an “encoding unit 30”. And the magnetic field control unit 32, the laser light emitting unit 41, the laser control unit 42, the mirror 43, the mirror driving unit 44, the cantilever 50, the CPU (Central Processing Unit) 60, and the quantum computer 1 that constitute the “exchange operation unit 40”. Is a program memory 70 that stores a quantum computation program for causing the CPU 60 to execute, an input value memory 80 that stores classical information of an input value to be operated, and an operation content that stores information indicating the physical operation content of a logical qubit A memory 90 is included.

  Here, the qubit substrate 10 of this example is a Si (111) substrate or the like, for example, “J. -L. Lim, et al., JAP84, 255 (1998)” or “Phys. Rev. Lett. The physical qubit as illustrated in FIG. 2 is formed by the method described in “2002”. The cooling unit 21 is, for example, a laser cooling device or a dilution cooler that can cool the qubit substrate 10 to the [mK (millikelvin)] order or less.

  Further, in this example, the qubit substrate 10 is fixed to one side of the cantilever 50, and the magnet 31 is disposed on the side opposite to the fixed surface. The magnetic field control unit 32 is arranged and configured so as to control the magnetic field of the magnet 31. The cooling unit 21 is arranged and configured so that the qubit board 10 can be cooled, and the cooling control unit 22 is electrically connected to the cooling unit 21 so that instructions to the cooling unit 21 are possible. The laser beam emitting unit 41 is provided at a position where the emitted laser beam is incident on the mirror 43, and the laser control unit 42 is electrically connected to the laser beam emitting unit 41 so that the laser beam emitting unit 41 can be instructed. Connected. The mirror 43 is arranged and configured so that the laser beam emitted from the laser beam emitting unit 41 can be reflected to an arbitrary place on the qubit substrate 10, and the mirror driving unit 44 can control the reflection angle of the mirror 43. Configured as follows. The CPU 60 is electrically connected to the cooling control unit 22, the magnetic field control unit 32, the laser control unit 42, and the mirror driving unit 44 so as to control them, and further includes a program memory 70, an input value memory 80, and The operation content memory 90 is configured to be able to read data.

<Processing>
Next, the quantum calculation method of this embodiment will be described.
FIG. 1 is a flowchart for explaining a quantum calculation method in this embodiment. Hereinafter, the processing procedure of this embodiment will be described with reference to this figure.
[Initialization]
First, the initialization unit 20 (FIG. 2) initializes the physical qubits 11a to 11c, 12a to 12c, and 13a to 13c of the qubit substrate 10 (step S1). Specifically, for example, the CPU 60 (FIG. 3) gives an instruction to the cooling control unit 22 according to the quantum calculation program read from the program memory 70, and the cooling control unit 22 that receives this instruction drives the cooling unit 21. . Thereby, the qubit substrate 10 is cooled to several mK or less, and the spin directions of the physical qubits 11a to 11c, 12a to 12c, and 13a to 13c are aligned (see, for example, “Phys. Rev. Lett. 2002”). . 4A is a conceptual diagram showing the initialized physical qubits 11a to 11c, 12a to 12c, and 13a to 13c. This state corresponds to “0” of each physical quantum bit. Furthermore, the vertical axis and the horizontal axis in FIG.

[Encoding on the first basis]
Next, the encoding unit 30 uses the logical qubits as encoding units, and each of the first logical qubits 11, 11 a to 11 c and 13 a to 13 c (white circles in FIG. 4B), 13 is encoded with the first basis (step S2). Specifically, for example, according to the quantum calculation program read from the program memory 70 by the CPU 60 (FIG. 3), the value to be calculated is read from the input value memory 80, and an instruction according to the value is given to the magnetic field control unit 32. The magnetic field control unit 32 changes the magnetic field of the magnet 31 according to this instruction, and encodes the first logical qubits 11 and 13 with the first base (for example, see “Ref. Quant-ph / 9905096v2 (1999)”). .)

Hereinafter, this encoding process will be described in detail. As the first base, for example,
1 / √2 (| 010> − | 100>),
1 / √6 (2 | 001> − | 010> − | 100>),
1 / √2 (| 101>-| 011>),
1 / √6 (2 | 110 > − | 101 > − | 011 > ),
A pair selected from can be used, but in this example,
_{| 0 L> = 1 / √2} (| 010> - | 100>) ... (3)
| 1 _L > = 1 / √6 (2 | 001 > − | 010 > − | 100 > ) (4)
Is the first basis. Further, | 0 _L > and | 1 _L > correspond to logical qubits “0” and “1”, respectively.

を施す（ステップＳ１２）。次に、その物理量子ビット１１ａを反転制御ビットとし、物理量子ビット１１ｂを目標ビットとした反転制御ＮＯＴ演算（反転制御ビットが０のときに目標ビットを反転させる）を行う（ステップＳ１３）。そして最後に、物理量子ビット１１ａを反転させる（ステップＳ１４）。 FIG. 5 is a diagram illustrating a process of encoding the logical qubit 11 into the state (“0”) of Expression (3). In addition, the arrow in this figure has shown progress of time (FIG. 6-8 is also the same).
As shown in this figure, when the logical qubit 11 is encoded to “0” in the first base of this example, first, the physical qubit 11a is inverted (step S11), and then Hadamard transform is performed on it.

(Step S12). Next, an inversion control NOT operation (inverting the target bit when the inversion control bit is 0) using the physical qubit 11a as the inversion control bit and the physical qubit 11b as the target bit is performed (step S13). Finally, the physical qubit 11a is inverted (step S14).

を施す（ステップＳ２４）。そして、物理量子ビット１１ａ，１１ｂを制御ビットとし、物理量子ビット１１ｃを目標ビットとした制御ＮＯＴ演算を行い（ステップＳ２５）、さらに、物理量子ビット１１ａ，１１ｂを反転制御ビットとし、物理量子ビット１１ｃを目標ビットとした反転制御ＮＯＴ演算を行う（ステップＳ２６）。その後、物理量子ビット１１ｂ，１１ｃを制御ビットとし、物理量子ビット１１ａを目標ビットとした制御ＮＯＴ演算を行い（ステップＳ２７）、さらにその後、物理量子ビット１１ｃを制御ビットとし、物理量子ビット１１ｂを目標ビットとした制御アダマール変換Ｈを施す（ステップＳ２８）。そして、最後に物理量子ビット１１ｃを制御ビットとし、物理量子ビット１１ｂを目標ビットとした制御ＮＯＴ演算を行う（ステップＳ２９） FIG. 6 is a diagram showing a process of encoding the logical qubit 11 into the state (“1”) of the equation (4).
As shown in this figure, when the logical qubit 11 is encoded to “1” in the first base of this example, first, the physical qubit 11a is inverted (step S21), and the Hadamard transform H is applied to it (step S21). Step S22). Next, an inversion control NOT operation is performed using the physical qubit 11a as an inversion control bit and the physical qubit 11b as a target bit (step S23). Next, unitary transformation is applied to the physical qubit 11a.

(Step S24). Then, a control NOT operation using the physical qubits 11a and 11b as control bits and the physical qubit 11c as a target bit is performed (step S25), and the physical qubits 11a and 11b are used as inversion control bits, and the physical qubit 11c. An inversion control NOT operation is performed using as a target bit (step S26). Thereafter, a control NOT operation is performed using the physical qubits 11b and 11c as control bits and the physical qubit 11a as a target bit (step S27). Thereafter, the physical qubit 11c is used as a control bit and the physical qubit 11b is set as a target. A control Hadamard transform H, which is a bit, is applied (step S28). Finally, a control NOT operation is performed using the physical qubit 11c as a control bit and the physical qubit 11b as a target bit (step S29).

[Encoding on second basis]
Next, in the encoding unit 30, a second logical qubit (≠ first logical qubit) 12 composed of three physical qubits 12a to 12c (black circles in FIG. 4B) is converted into a second basis ( ≠ 1st basis) (step S3). Specifically, for example, similarly to the encoding on the first basis, the magnetic field control unit 32 changes the magnetic field of the magnet 31 under the control of the CPU 60 (FIG. 3), and the second logical qubit 12 is changed to the second logical qubit 12. Encode on the basis of. In this embodiment, encoding is performed so that the first logical qubit and the second logical qubit are alternately arranged. That is, the encoding unit 20 (FIG. 2) encodes adjacent logical qubits with different bases.

Hereinafter, this encoding process will be described in detail. As the second base, for example,
1 / √2 (| 010> − | 100>),
1 / √6 (2 | 001> − | 010> − | 100>),
1 / √2 (| 101>-| 011>),
1 / √6 (2 | 110 > − | 101 > − | 011 > ),
A pair selected from (a pair different from the first basis) can be used, but in this example,
_{| 0 L> = 1 / √2} (| 101> - | 011>) ... (6)
_{| 1 L> = 1 / √6} (2 | 110> - | 101> - | 011>) ... (7)
Is the second basis. Also, | 0 _L > and | 1 _L > correspond to “0” and “1” of the logical qubits, respectively.

FIG. 7 is a diagram illustrating a process of encoding the logical qubit 12 into the state (“0”) of Expression (6).
As shown in this figure, when the logical qubit 12 is encoded to “0” in the second base of this example, first, the physical qubit 12a is inverted (step S31), and Hadamard transform H is performed on it (step S31). Step S32). Next, a control NOT operation is performed using the physical qubit 12a as a control bit and the physical qubit 12b as a target bit (step S33). Finally, the physical qubit 12c is inverted (step S34).

FIG. 8 is a diagram showing a process of encoding the logical qubit 12 into the state (“1”) of Expression (7).
As shown in this figure, when the logical qubit 12 is encoded to “1” by the second base in this example, first, the physical qubit 12a is inverted (step S41), and the Hadamard transform H is applied to it (step S41). Step S42). Next, an inversion control NOT operation is performed using the physical qubit 12a as an inversion control bit and the physical qubit 12b as a target bit (step S43). Next, the unitary transformation A of Expression (5) is performed on the physical qubit 12a (step S44). Then, a control NOT operation is performed using the physical qubits 12a and 12b as control bits and the physical qubit 12c as a target bit (step S45). Further, the physical qubits 12a and 12b are used as inversion control bits, and the physical qubit 12c. An inversion control NOT operation is performed using as a target bit (step S46). Thereafter, a control NOT operation is performed using the physical qubits 12b and 12c as control bits and the physical qubit 12a as a target bit (step S47). Thereafter, the physical qubit 12c is used as a control bit and the physical qubit 12b is set as a target. A control Hadamard transform H, which is a bit, is applied (step S48). Next, the physical qubit 12a is inverted (step S49), and an inversion control NOT operation is performed using the physical qubit 12c as an inversion control bit and the physical qubit 12b as a target bit (step S50). 12c is inverted (step S51).

[Exchange operation]
Next, the exchange operation unit 40 performs an exchange operation between the encoded first logical qubits 11 and 13 and the encoded second logical qubit 12 (step S4). This exchange operation is, for example, an exchange operation between adjacent logical qubits, and is an exchange operation (including partial exchange) for the control NOT shown in FIG.
FIG. 9 is a diagram conceptually showing this exchange operation. As shown in FIG. 9, in this step, the first logical qubit 11 (physical qubits 11a to 11c) and the second logical qubit 12 (physical qubits 12a to 12c) encoded with different bases are used. Is subjected to a quantum operation consisting of a plurality of consecutive exchange operations.

  Specifically, for example, the CPU 60 (FIG. 3) reads information indicating the operation content from the operation content memory 90 according to the quantum calculation program read from the program memory 70, and the laser control unit 42 and the mirror drive unit 44 according to this information. Give instructions to. In accordance with these instructions, the laser control unit 42 and the mirror driving unit 44 control the laser beam emitting unit 41 and the mirror 43, respectively, irradiate the logical qubits that perform the exchange operation with laser light, and perform the exchange operation.

<Features of this embodiment>
As described above, in this embodiment, since a plurality of types of bases are used for encoding each logical qubit, it is possible to generate a space of a different dimension from the case of performing an exchange operation using only a single base. It becomes possible. Therefore, depending on the number of physical qubits constituting the logical qubit and the selection of the basis, the dimension of the generated space can be limited compared to the case of using only a single basis, and the occurrence of “leakage” is fundamentally reduced. can do.

For example, a logical qubit is configured using three physical qubits,
1 / √2 (| 010> − | 100>),
1 / √6 (2 | 001> − | 010> − | 100>),
1 / √2 (| 101>-| 011>),
1 / √6 (2 | 110 > − | 101 > − | 011 > ),
When encoding is performed using two bases determined by a pair selected from the above, a space generated by the base is a 9-dimensional subspace corresponding to the total angular momentum S = 1 and the angular momentum S _z = 0 of the Z component. And a linear combination (superposition) with a five-dimensional subspace corresponding to S = 0 and S _z = 0 (for example, “Sun Fu Tuan, Akio Sakurai,“ Modern Quantum Mechanics (above) ”, Yoshioka (See Bookstore, issued on January 15, 2003, pages 276-297).

Also, exchange operation is a linear operation performed without changing the angular momentum S _z of total angular momentum S and Z components, quantum states generated by the exchange operation is also as described above in the formula (1) 5 It is a linear combination of a three-dimensional subspace and a nine-dimensional subspace.
In this case, the general error Δ of the entire space is expressed by the above-described equation (2), and is accumulated in the 9-dimensional space created by the conventional logical qubit exchange operation including three physical qubits. It is smaller than the typical error delta ₂ that. As a result of reducing the general error, the amount of “leakage” can also be reduced.

<Application examples>
The present invention is not limited to the embodiment described above. For example, in this embodiment, the physical qubits are generated in a line on the qubit board (FIG. 4). However, as shown in FIG. The logical qubits 220 may be formed by three adjacent physical qubits 210 that are formed in parallel (two-dimensional square lattice qubit array). Also in this example, white circles and black circles indicate differences in bases to be encoded (the same applies to the examples of FIGS. 10B and 11). Furthermore, the vertical axis and the horizontal axis in FIG. 10 each indicate a space (the same applies to FIG. 11 below).

Further, as in the qubit substrate 300 of FIG. 10B, the three physical qubits 310 constituting each logical qubit 320 are arranged at the vertices of an isosceles triangle, and the respective logical qubits 320 are alternately arranged. (Two-dimensional kagome lattice qubit array).
Further, as in the qubit board 400 of FIG. 11, the three physical qubits 410 constituting each logical qubit 420 are arranged in a straight line in different directions for each base, and the respective logical qubits 420 are arranged alternately. (Two-dimensional Kagome lattice qubit array).

Also, logical qubits and physical qubits may be arranged in three dimensions.
Furthermore, in this embodiment, an example in which logical qubits are encoded using two types of bases has been described, but logical qubits may be encoded using three or more types of bases.
In this embodiment, the logical qubits are encoded so that the expressions of physically adjacent logical qubits are always different. However, at least a part of the logical qubits encoded on the same basis is continuous. It is good also as arranging.

Furthermore, in this embodiment, an example in which spin 1/2 particles are used as physical qubits is shown, but physical qubits may be configured using other physical systems.
In this embodiment, the description of observation is omitted, but the measurement of quantum data is performed by, for example, generating a sequence of physical qubits for measurement that have the same quantum state as the physical qubit to be operated on a qubit substrate. For this, a nuclear magnetic resonance probe method (MRFM) (see, for example, “Phys. Rev. Lett. 2002”) or the like is applied.

Furthermore, the method of this embodiment is a completely independent method from the conventional solution. That is, it is a method that can be used in combination with the conventional solution. Therefore, “leakage” can be reduced more effectively by adding the method of this embodiment to the conventional solution.
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.

As described above, the quantum calculation program stored in the program memory 70 is a program for causing a computer to execute processing necessary for controlling the quantum computer 1. The program describing the processing content can be recorded on a computer-readable recording medium. The computer-readable recording medium may be any medium such as a magnetic recording device, an optical disk, a magneto-optical recording medium, or a semiconductor memory. Specifically, for example, the magnetic recording device may be a hard disk device or a flexible Discs, magnetic tapes, etc. as optical disks, DVD (Digital Versatile Disc), DVD-RAM (Random Access Memory), CD-ROM (Compact Disc Read Only Memory), CD-R (Recordable) / RW (ReWritable), etc. As the magneto-optical recording medium, MO (Magneto-Optical disc) or the like can be used, and as the semiconductor memory, EEP-ROM (Electronically Erasable and Programmable-Read Only Memory) or the like can be used.
Also, such a program is distributed by selling, transferring, or lending a portable recording medium such as a DVD or CD-ROM in which the program is recorded. Furthermore, such a program may be stored in a storage device of a server computer, and the program may be distributed by transferring the program from the server computer to another computer via a network.

  The quantum computer 1 that executes the program distributed in this way, for example, first stores the program recorded on the portable recording medium or the program transferred from the server computer once in its own program memory 70. When executing the process, the CPU 60 reads the program stored in its own program memory 70 and executes the process according to the read program. As another execution form of such a program, the CPU 60 may directly read the program from a portable recording medium and execute processing according to the program. Further, the program is transferred to the CPU 60 from the server computer. Each time, the processing according to the received program may be executed.

  The present invention can be used directly for quantum computation of any quantum computer using spin.

The flowchart for demonstrating the quantum calculation method in 1st Embodiment. The conceptual diagram which illustrated schematic structure of the quantum computer in 1st Embodiment. The block diagram which illustrated the concrete composition of the quantum computer of the 1st implementation. (A) is the conceptual diagram which showed the initialized physical qubit, (b) is the conceptual diagram which showed the mode that it encoded for every logical qubit. The figure which showed the process which encodes a logic qubit to the state of Formula (3). The figure which showed the process which encodes a logic qubit to the state of Formula (4). The figure which showed the process which encodes a logic qubit to the state of Formula (6). The figure which showed the process which encodes a logic qubit to the state of Formula (7). The figure which showed exchange operation notionally. The figure which showed the modification of the qubit board | substrate. The figure which showed the modification of the qubit board | substrate. The figure which illustrated exchange operation for realizing CNOT operation.

Explanation of symbols

DESCRIPTION OF SYMBOLS 1 Quantum computer 10,200,300,400 qubit board 20 Initialization part 30 Encoding part 40 Exchange operation part

Claims (15)

A quantum computation method for performing quantum computation using a logical qubit composed of n (n ≧ 2) physical qubits as a coding unit,
An encoding unit encoding the first logical qubit with a first basis;
The encoding unit encoding a second logical qubit different from the first logical qubit with a second base different from the first base;
An exchange operation unit performing an exchange operation between the encoded first logical qubit and the encoded second logical qubit;
A quantum computation method characterized by comprising: The quantum calculation method according to claim 1,
The physical qubit is
Spin 1/2 particles,
A quantum computation method characterized by that. The quantum calculation method according to claim 1,
Each logical qubit above is
Composed of adjacent physical qubits,
A quantum computation method characterized by that. The quantum calculation method according to claim 1,
The first logical qubit and the second logical qubit are alternately arranged.
A quantum computation method characterized by that. The quantum calculation method according to claim 1,
N is 3;
Each of the above bases is
1 / √2 (| 010> − | 100>),
1 / √6 (2 | 001> − | 010> − | 100>),
1 / √2 (| 101>-| 011>),
1 / √6 (2 | 110> − | 101> − | 011>),
Determined by the pair selected from
A quantum computation method characterized by that. The quantum calculation method according to claim 5,
The first basis is
{1 / √2 (| 010> − | 100>), 1 / √6 (2 | 001> − | 010> − | 100>)},
The second basis is
{1 / √2 (| 101> − | 011>), 1 / √6 (2 | 110> − | 101> − | 011>)}.
A quantum computation method characterized by that. The quantum calculation method according to claim 1,
The above exchange operation
It is an exchange operation for control NOT operation.
A quantum computation method characterized by that. A quantum computer that performs quantum computation using a logical qubit composed of n (n ≧ 2) physical qubits as an encoding unit,
A qubit substrate on which a plurality of physical qubits are arranged;
An encoding unit that encodes each logical qubit arranged on the qubit board with any of m (m ≧ 2) types of bases;
An exchange operation unit for performing an exchange operation between the logical qubits encoded in different bases;
A quantum computer characterized by comprising: The quantum computer according to claim 8, wherein
The physical qubit is
Spin 1/2 particles,
A quantum computer characterized by that. The quantum computer according to claim 8, wherein
Each logical qubit above is
Composed of adjacent physical qubits,
A quantum computer characterized by that. The quantum computer according to claim 8, wherein
The encoding unit is
Encode adjacent logical qubits with different basis,
The exchange operation part
Exchange operation between adjacent logical qubits,
A quantum computer characterized by that. The quantum computer according to claim 8, wherein
N is 3;
Each of the above bases is
1 / √2 (| 010> − | 100>),
1 / √6 (2 | 001> − | 010> − | 100>),
1 / √2 (| 101>-| 011>),
1 / √6 (2 | 110> − | 101> − | 011>),
Determined by the pair selected from
A quantum computer characterized by that. The quantum computer according to claim 12, wherein
M is 2;
The first basis is
{1 / √2 (| 010> − | 100>), 1 / √6 (2 | 001> − | 010> − | 100>)},
The second basis is
{1 / √2 (| 101> − | 011>), 1 / √6 (2 | 110> − | 101> − | 011>)}.
A quantum computer characterized by that. The quantum computer according to claim 8, wherein
The exchange operation part
Perform exchange operation for control NOT calculation,
A quantum computer characterized by that.   15. A quantum computation program for causing a computer to execute processing necessary for controlling the quantum computer according to claim 8.

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