CN113052319A - Cramer-schmitt orthogonalization method for quantum computation - Google Patents

Cramer-schmitt orthogonalization method for quantum computation Download PDF

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CN113052319A
CN113052319A CN202110311886.1A CN202110311886A CN113052319A CN 113052319 A CN113052319 A CN 113052319A CN 202110311886 A CN202110311886 A CN 202110311886A CN 113052319 A CN113052319 A CN 113052319A
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余旭涛
孟凡旭
张在琛
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Southeast University
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Abstract

The invention discloses a Cramer-Schmidt orthogonalization method based on quantum computation, which comprises the following steps of: s1: preparing a traditional linear independent vector set into a corresponding quantum state set in a quantum mode; s2: and (4) implementing Cramer-Schmitt orthogonalization according to a quantum block coding technology, and carrying out an orthogonalization process on the quantum state sets in the step S1 to obtain a group of orthogonal quantum state sets. Aiming at the traditional Cramer-Schmidt orthogonalization method, the invention adopts a quantum block coding technology and a unitary matrix linear combination technology to reduce the complexity of the traditional Cramer-Schmidt orthogonalization, thereby obtaining more stable performance.

Description

Cramer-schmitt orthogonalization method for quantum computation
Technical Field
The invention relates to the field of classical linear algebra and quantum computation, in particular to a Cramer-Schmitt orthogonalization method based on quantum computation.
Background
In 1982, Feynman described the great potential of quantum computing, and suggested to construct quantum computers on the basis of the principles of quantum mechanics, to exploit the potential of quantum computing. Further, Shor in 1994 proposes a polynomial time quantum algorithm of prime factorization and discrete logarithm problems; in 1995, Grover proposed a quantum algorithm that searches over a search space without structures. The quantum computing algorithms show special potential of quantum computing, and provide acceleration for traditional algorithms. Because the traditional Cramer-Schmitt orthogonalization method is widely applied to the fields of wireless communication and artificial intelligence, the traditional method often relates to the operation of a large-scale high-latitude vector set, so that the traditional Cramer-Schmitt orthogonalization method relates to the problems of high computational complexity, and a better processing mode is not available at present. The best current classical clement-schmitt orthogonalization method is able to construct a set of orthogonal vectors with a complexity of O (poly (nt)). At present, quantum computation has not been applied to precedent of the Cramer-Schmitt orthogonalization method.
Disclosure of Invention
The technical problem is as follows: the invention aims to provide a Cramer-Schmitt orthogonalization method based on quantum computation, which reduces the complexity of the traditional Cramer-Schmitt orthogonalization method to be lower than that of the traditional Cramer-Schmitt orthogonalization method
Figure BDA0002990109320000011
Therefore, the method is more suitable for a large-scale high-dimensional Cramer-Schmidt orthogonalization method to obtain a faster operation speed.
The technical scheme is as follows: to achieve the purpose, the Cramer-Schmitt orthogonalization method based on quantum computation comprises the following steps:
s1: constructing the traditional linear independent vector set into a corresponding quantum state set through a quantum mode;
s2: and (4) realizing quantum Cramer-Schmitt orthogonalization according to a quantum block coding technology, and carrying out an orthogonalization process on the quantum state set in the step S1 to obtain a group of orthogonal quantum state sets.
Wherein the content of the first and second substances,
in step S1, a set of linearly independent vectors
Figure BDA0002990109320000012
Being an n-dimensional real vector, the corresponding quantum state form is obtained by equation (1):
Figure BDA0002990109320000021
in the formula (1), the reaction mixture is,
Figure BDA0002990109320000022
is a vector
Figure BDA0002990109320000023
Is a two norm, i.e.
Figure BDA0002990109320000024
In step S1, the process of orthogonalizing the set of quantum states by claime-schmitt orthogonalization is expressed by equation (2):
Figure BDA0002990109320000025
in the formula (2, | yiT is an intermediate state generated by a quantum claime-schmitt orthogonalization method.
In step S2, y isiT is an intermediate state generated by a quantum gram-smith orthogonalization method, | y according to a quantum state unclonable principle i1,2, t is very difficult to prepare directly and can be prepared efficiently without the presence of a direct unitary transform, | yi>I 1,2, t is passed through
Figure BDA0002990109320000026
The linear combination structure of (3) is as shown in formula (3):
Figure BDA0002990109320000027
the quantum claimer-schmitt orthogonalization is obtained by the following method:
s2.1: coefficient w obtained from equation (3)ik K 1,2,., t, i 1,2,, t, is a unitary matrix of formula (4):
Figure BDA0002990109320000028
s2.2: the quantum state of formula (5) is constructed according to the linear combination technique of quantum unitary matrix
Figure BDA00029901093200000211
Figure BDA0002990109320000029
In the formula (5, | t⊥>Is an unnormalized quantum state and
Figure BDA00029901093200000210
|φ>is a quantum state of an arbitrary logt bit;
Figure BDA0002990109320000037
is a proprietary representation within a quantum, which is a state requiring logt quantums 0;
s2.3: constructing a controlled marker unitary mapping to operate on the quantum state in equation (5) to obtain a quantum state as equation (6):
Figure BDA0002990109320000031
s2.4: constructing unitary operation T of formula (7) from quantum states of formula (6)i：
Figure BDA0002990109320000032
S2.5: for quantum state
Figure BDA0002990109320000033
Performing unitary operation to obtain the quantum state as shown in equation (8)
Figure BDA0002990109320000034
In the formula (8), the reaction mixture is,
Figure BDA0002990109320000035
is an unnormalized quantum state and is orthogonal to the first term in the formula;
s2.6: measurement of formula (8) yields a quantum state of formula (9):
Figure BDA0002990109320000036
in the formula (9, | yk+1>Is an orthogonal quantum state, quantum state y, generated by quantum claimer-Schmitt orthogonalizationiT can be obtained in the same manner as described above.
Has the advantages that: the invention discloses a Cramer-Schmitt orthogonalization method based on quantum computation, which aims at solving the problem of processing large-scale high-latitude vector Cramer-Schmitt orthogonalization, adopts a quantum method to realize the Cramer-Schmitt orthogonalization method, and reduces the complexity of vector set operation in the traditional Cramer-Schmitt orthogonalization method, thereby obtaining a group of orthogonal quantum state sets by a more efficient method, and being better suitable for application scenes such as artificial intelligence, mode recognition, big data processing and the like.
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FIG. 1 is a flow chart of a method in accordance with an embodiment of the present invention;
FIG. 2 is a flow chart of quantum state construction of the vector set of step S1 according to the embodiment of the present invention;
fig. 3 is a flowchart of step S2 according to the embodiment of the present invention.
Detailed Description
The technical solution of the present invention will be further described with reference to the following embodiments.
The specific embodiment discloses a claime-schmitt orthogonalization method based on quantum computation, which comprises the following steps as shown in fig. 1:
s1: constructing the traditional linear independent vector set into a corresponding quantum state set through a quantum mode;
s2: and (4) implementing Cramer-Schmitt orthogonalization according to a quantum block coding technology, and carrying out an orthogonalization process on the quantum state sets in the step S1 to obtain a group of orthogonal quantum state sets.
FIG. 2 is a flow chart of quantum state preparation in step S1, wherein in step S1, the vectors are linearly independent
Figure BDA0002990109320000041
The corresponding quantum state form can be obtained by (1):
Figure BDA0002990109320000042
in the formula (1), the reaction mixture is,
Figure BDA0002990109320000043
is a vector
Figure BDA0002990109320000044
Is a two norm, i.e.
Figure BDA0002990109320000045
Fig. 3 is a flowchart of step S2, and the clehm-schmitt orthogonalization process of the quantum state set in S1 in step S2 is expressed according to equation (2):
Figure BDA0002990109320000046
in the formula (2, | yi>I 1,2, t is a set of orthogonal quantum states.
In step S2, y isi>I 1,2, t is an intermediate state generated by a quantum gram-smith orthogonalization method, and y is an intermediate state generated according to a quantum state unclonable principlei>I 1,2, t is very difficult to prepare directly and can be prepared efficiently without the presence of a direct unitary transform. However, because of the collection of quantum states in step S1
Figure BDA0002990109320000047
Can be effectively prepared, | yi>I 1,2, t may be determined by
Figure BDA0002990109320000048
The linear combination structure of (3) is as shown in formula (3):
Figure BDA0002990109320000049
the quantum claimer schmitt orthogonalization method can be obtained by:
s2.1: coefficient w obtained from equation (3)ik K 1,2,., t, i 1,2,, t, is a unitary matrix of formula (4):
Figure BDA0002990109320000051
s2.2: preparing quantum state shown as formula (5) according to quantum unitary matrix linear combination technology
Figure BDA00029901093200000511
Figure BDA0002990109320000052
In the formula (5, | t⊥>Is an unnormalized quantum state and
Figure BDA0002990109320000053
|φ>is a quantum state of an arbitrary logt bit;
s2.3: constructing a controlled marker unitary mapping to operate on the quantum state in equation (5) to obtain a quantum state as equation (6):
Figure BDA0002990109320000054
s2.4: according to formula (6)The quantum state structure is unitary operation T of formula (7)i：
Figure BDA0002990109320000055
S2.5: for quantum state
Figure BDA0002990109320000056
Performing unitary operation to obtain the quantum state as shown in equation (8)
Figure BDA0002990109320000057
In the formula (8), the reaction mixture is,
Figure BDA0002990109320000058
is an unnormalized quantum state and is orthogonal to the first term in the formula;
s2.6: measurement of formula (8) yields a quantum state of formula (9):
Figure BDA0002990109320000059
Figure BDA00029901093200000510
in the formula (9, | yk+1>Is an orthogonal quantum state, quantum state y, generated by quantum claimer-Schmitt orthogonalizationi>I 1,2, t can be obtained in the same manner as described above.
The complexity of the method is as follows: quantum state
Figure BDA0002990109320000061
Can be in the complexity of O [ poly (logn)]Is efficiently prepared, | yi>Can be obtained from the obtained coefficient wikK 1,2, t is constructed by linear combination of quantum unitary operations, the complexity of which can be approximated as
Figure BDA0002990109320000062
The quantum state form of the spatial covariance matrix can be in complexity. Unitary operation TiCan be determined by complexity
Figure BDA0002990109320000063
Is effectively constructed; by pairs
Figure BDA0002990109320000064
Measurement is performed to obtain the final orthogonal quantum state set yi>I 1,2, t, whose complexity can be approximated as
Figure BDA0002990109320000065
Wherein A is represented by
Figure BDA0002990109320000066
A matrix of columns.

Claims (5)

1. A Cramer-Schmidt orthogonalization method based on quantum computation is characterized in that: the method comprises the following steps:
s1: constructing the traditional linear independent vector set into a corresponding quantum state set through a quantum mode;
s2: and (4) realizing quantum Cramer-Schmitt orthogonalization according to a quantum block coding technology, and carrying out an orthogonalization process on the quantum state set in the step S1 to obtain a group of orthogonal quantum state sets.
2. The method of claime-schmitt orthogonalization based on quantum computation of claim 1, characterized in that: in step S1, a set of linearly independent vectors
Figure FDA0002990109310000011
Being an n-dimensional real vector, the corresponding quantum state form is obtained by equation (1):
Figure FDA0002990109310000012
in the formula (1), the reaction mixture is,
Figure FDA0002990109310000013
is a vector
Figure FDA0002990109310000014
Is a two norm, i.e.
Figure FDA0002990109310000015
3. The method of claime-schmitt orthogonalization based on quantum computation of claim 1, characterized in that: in step S1, the process of orthogonalizing the set of quantum states by claime-schmitt orthogonalization is expressed by equation (2):
Figure FDA0002990109310000016
in the formula (2, | yi>I 1,2, t is an intermediate state generated by a quantum gram-smith orthogonalization method.
4. The method of claime-schmitt orthogonalization based on quantum computation of claim 3, characterized in that: in step S2, y isi>I 1,2, t is an intermediate state generated by a quantum gram-smith orthogonalization method, and y is an intermediate state generated according to a quantum state unclonable principlei1,2, t is very difficult to prepare directly and can be prepared efficiently without the presence of a direct unitary transform, | yi1,2, t is passed through
Figure FDA0002990109310000017
The linear combination structure of (3) is as shown in formula (3):
Figure FDA0002990109310000018
5. the method of claime-schmitt orthogonalization based on quantum computation of claim 3, characterized in that: the quantum claimer-schmitt orthogonalization is obtained by the following method:
s2.1: coefficient w obtained from equation (3)ikK 1,2,., t, i 1,2,, t, is a unitary matrix of formula (4):
Figure FDA0002990109310000021
s2.2: the quantum state of formula (5) is constructed according to the linear combination technique of quantum unitary matrix
Figure FDA00029901093100000211
Figure FDA0002990109310000022
In the formula (5, | t⊥>Is an unnormalized quantum state and
Figure FDA0002990109310000023
|φ>is a quantum state of an arbitrary logt bit;
Figure FDA0002990109310000024
is a proprietary representation within a quantum, which is a state requiring logt quantums 0;
s2.3: constructing a controlled marker unitary mapping to operate on the quantum state in equation (5) to obtain a quantum state as equation (6):
Figure FDA0002990109310000025
s2.4: the quantum state structure according to formula (6) is as in formula (7)) Unitary operation Ti：
Figure FDA0002990109310000026
S2.5: for quantum state
Figure FDA0002990109310000027
Performing unitary operation to obtain the quantum state as shown in equation (8)
Figure FDA0002990109310000028
In the formula (8), the reaction mixture is,
Figure FDA0002990109310000029
is an unnormalized quantum state and is orthogonal to the first term in the formula;
s2.6: measurement of formula (8) yields a quantum state of formula (9):
Figure FDA00029901093100000210
in the formula (9, | yk+1>Is an orthogonal quantum state, quantum state y, generated by quantum claimer-Schmitt orthogonalizationiT can be obtained in the same manner as described above.
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Citations (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN106126187A (en) * 2016-06-20 2016-11-16 符建 A kind of light field parallel computation unit based on orthogonal pseudo-random phase code and method
CN111582491A (en) * 2020-04-30 2020-08-25 合肥本源量子计算科技有限责任公司 Construction method and device of quantum line

Patent Citations (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN106126187A (en) * 2016-06-20 2016-11-16 符建 A kind of light field parallel computation unit based on orthogonal pseudo-random phase code and method
CN111582491A (en) * 2020-04-30 2020-08-25 合肥本源量子计算科技有限责任公司 Construction method and device of quantum line

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