CN103414477A - Method for structuring state transition graph and grid graph of quantum convolutional codes - Google Patents
Method for structuring state transition graph and grid graph of quantum convolutional codes Download PDFInfo
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Abstract
The invention belongs to the field of error correction of quantum encoding, and particularly relates to a method for structuring a state transition graph and a grid graph of quantum convolutional codes. The structured state transition graph and the grid graph of the quantum convolutional codes are the most powerful tools for analyzing a Viterbi decoding algorithm of the quantum convolutional codes. In the encoding scheme of the quantum convolutional codes, each encoding circuit corresponds to only one encoding correction operation, and k-bit information is encoded into n-bit digits through the encoding operations. Corresponding mathematical manipulation is performed on the encoding operations, therefore, a corresponding encoding matrix is obtained, and the encoding matrix gives the one-to-one correspondence relation of quantum bits and encoded operators. By analyzing the state transition process of the encoded operators acting on convolution bits, the state transition graph of the quantum convolutional codes is obtained. The state transition graph of the quantum convolutional codes only describes the transition processes among different states, and if the relation between the changes of the states and the time is considered, the corresponding grid graph can be drawn.
Description
Technical field
The present invention generally is applied in the quantum Coding Theory, specifically is applied in the Viterbi decoding algorithm of quantum convolution code.
Background technology
Along with constantly carrying out of quantum information science research, the superiority of quantum communications and quantum calculation shows especially out more and more, large several Factorization for example, if with present supercomputer, decompose the numeral of a hundreds of position, the suitable time (hundreds of 100000000 years) in life-span in needs use and universe, within the time of a few minutes, just can complete with the quantum computer with same clock frequency.In a single day quantum theory is applied to practice, and a lot of fields essence will occur and change.Yet in actual environment, the quantum bit in quantum communications and quantum calculation does not isolate, it interacts with external environment condition constantly, thereby the quantum bit coherence is destroyed, and causes quantum decoherence.In quantum communications, quantum message to be transmitted also can be subject to the impact of quantum noise in channel, causes quantum state that inevitable mistake occurs.Therefore, quantum computer is become a reality, realize reliable quantum communications, a key problem is exactly to overcome the quantum noise brought by eliminating coherence.Research shows, the quantum channel coding techniques is to overcome a kind of effective ways of eliminating coherence with the correct amount sub-error, it can not only make quantum computer effectively calculate in noisy environment, also can make quantum information on the quantum channel with noise, realize communication reliably.And quantum convolution code quantum channel coding techniques a kind of just.
In existing classical channel coding technology, convolution code is a kind of and the diverse code of block code, each yard bit is not only relevant with current input, also relevant with adjacent bit, owing between bit, having the coherence, in the situation that encoder complexity is identical, the performance of convolution code is better than block code.The Viterbi algorithm is the optimum decoding algorithm of convolution code, because it realizes simply being widely used in deep space communication, satellite communication and mobile communication.And classical state transition diagram and grid chart are to analyze the most capable instrument of Viterbi algorithm.
It is many in classical channel coding theorem that by quantum, to be stablized the basic thought that subcode uses for reference the same with describing mode, we wish that the state transition diagram played a significant role in classical coding theory and grid chart also can be extended to quantum and stablize subcode, make it become the computing basis of analyzing quantum Viterbi decoding algorithm.
Summary of the invention
Main purpose of the present invention is the state transition diagram of explanation structure quantum convolution code and the method for grid chart, for the optimum decoding algorithm of analyzing the quantum convolution code lays the foundation.
The invention describes a kind of method for drafting of state transition diagram of quantum convolution code.The known code parameter is the quantum convolution code of [[n, k, m]], k position information exchange is crossed encoding operation and is encoded into the long code word in n position, and m refers to code storage, and defining its encoding operation is U, its corresponding encoder matrix is V, and we are called the state transitions process (M occurred for the coding operator on convolution position, m position
J-1→ M
j) be the corresponding state transition diagram of this encoding operation U.Likely, each may be represented as a state node in the drawings in the institute that this state transition diagram can travel through coding operator on the convolution position, and such node has 4
mIndividual.Every two nodes connect with a directed edge, represent the transfer process of coding operator on the interior convolution of adjacent encoder time quantum position, one group of mark is arranged on every limit, and in mark, left data represents the k position coding operator of current time input, and right data represents the n position coding operator of current time output; Each node stretches out 4
k* 2
(n-k)The bar limit enters other nodes, and the limit that enters simultaneously each node has 4
k* 2
(n-k)Bar.
The invention describes a kind of method for drafting of grid chart of quantum convolution code.The known code parameter is the quantum convolution code of [[n, k, m]], and k position information exchange is crossed encoding operation and is encoded into, the code word that the n position is long, m refers to code storage, if carry out altogether coding N+t time, wherein front N time for input message, for coding circuit, make zero for latter t time, set of node is divided into N+t+1 subset D
J, wherein | D
0|=1, | D
J|=2
m, 1≤j≤N+t; Every two nodes connect with a directed edge, all from node D
J-1Set out and arrive node D
JThe directed edge set be called E
j, E
jBe called the j joint of grid chart, one group of mark is arranged on every limit, in mark, left data represents the k position coding operator of current time input, and right data represents the n position coding operator of current time output; In the moment, each node stretches out 4 at each coding
k* 2
(n-k)The bar limit enters other nodes, and the limit that enters simultaneously each node has 4
k* 2
(n-k)Bar.
When coding circuit was determined, described encoding operation U is unique to be determined.
The pass of described encoder matrix V and encoding operation U is
Wherein
Referring to pauli group's operator, can calculate encoder matrix V one to one by encoding operation U, is one 2 rank, (n+m) * 2 (n+m) matrixes.
By encoding operation U, calculate encoder matrix V, the focus that we pay close attention to has been transferred to the coding operator acted on bit to be encoded from quantum bit to be encoded.
What state transition diagram was described is to act on the state transitions process (M that the coding operator on the convolution position occurs
J-1→ M
J), M
JWhile referring to the j time coding, act on the coding operator state on the convolution position.
If need transmission N section information to be encoded, the cataloged procedure of quantum convolution code is divided into N+t scramble time unit, and front N section is for input message, and rear t section is got back to full the output of coding circuit | 0 > bit.
The accompanying drawing explanation
The stabistor of Fig. 1 [[n, k, m]] quantum convolution code means.
N step convolution coding circuit before Fig. 2.
T step convolution coding circuit after Fig. 3.
Fig. 4 coding operator transition diagram.
Fig. 5 [[2,1,1]] quantum convolution coding circuit.
Fig. 6 [[2,1,1]] quantum convolution code state transition diagram.
Fig. 7 [[2,1,1]] quantum convolution code grid chart.
Embodiment
In Fig. 1, M
j,iFor the Pauli operator of n+m position, j represents the current time unit, 1≤i≤n-k.The stabistor of quantum convolution code is in the overlapped m of adjacent moment position, and this m position is called the convolution position.
In Fig. 2,
Be called logical bit, be used to inputting the k position information in the current time unit, after encoding operation U, become the long code word in n position
While remaining m position | P
jFor next, constantly encode, initial condition | P
0Be the complete of m position | 0>state.
In Fig. 3, input is complete on logical bit | and 0 > bit, remainder is constant.Its effect is that the output of coding circuit is got back to entirely | 0 > bit.
In Fig. 4, M
J-1Mean to act in j-1 time quantum the state of coding operator on convolution position, m position, M
JThe state that means coding operator on convolution position, the interior m position of j time quantum, the definition initial condition
The state that means coding operator on the logical bit of the interior k of j time quantum position;
Mean to act in j time quantum the state of coding operator on convolution position, m position.
In Fig. 5, | a initial condition be | 0 >, | b the input message position, | c > each input | 0 > state, according to coding circuit, obtain encoding operation U and be
a，b，c∈{0，1}。
1 quantum convolution code
In the quantum information theory, the two-dimentional Hilbert space on the corresponding complex field of 1 quantum bit, establish the quantum state that single quantum bit is corresponding and be
Wherein α and β are plural number, and meet | α |
2+ | β |
2=1, | 0>and | 1>represent one group of orthogonal basis in this Hilbert space.We use
The Hilbert sky of single quantum bit is asked.By that analogy, the tensor product in n corresponding n the two-dimentional Hilbert space of quantum bit
Corresponding space representation is
The Quantum Error Correcting Codes that code parameters is [[n, k]] is 2
nDimension Hilbert space
In one 2
kN-dimensional subspace n, this subspace is expressed as C
n, its cataloged procedure can be described as
The k bit information
Carry out after encoding operation U being encoded as the code word of n bit
Encoding operation U meets direct transform.In the present invention, we only consider to meet the encoding operation of clifford conversion.
The quantum convolution code: code parameters is the quantum convolution code of [[n, k, m]], can mean its stabistor generator with Fig. 1.
The state transition diagram of 2 quantum convolution codes
Suppose to need transmission N section information to be encoded, the cataloged procedure of quantum convolution code is divided into the N+t step to carry out, and front N step cataloged procedure is by shown in Figure 2, and rear t step cataloged procedure is by shown in Figure 3.For each quantum convolution code, if its coding circuit is definite, its encoding operation U is also unique determines.Pass through formula
Can calculate the corresponding encoder matrix V of this quantum convolution code, we consider that how encoder matrix operates the coding operator of convolution code, obtains required state transition diagram thus now.
Code parameters is the quantum convolution code of [[n, k, m]], and in N+t scramble time unit, by encoder matrix V, the coding operator acted on each coded-bit has following transfer process:
1≤j≤N+t, specific to each scramble time unit, available Fig. 4 means.
The state transition diagram of quantum convolution code: the known code parameter is the quantum convolution code of [[n, k, m]], and its encoding operation is U, and its corresponding encoder matrix is V, and we are called the state transitions process (M occurred for coding operator on convolution position, m position
J-1→ M
j) be the corresponding state transition diagram of this encoding operation U, and meet:
Likely, each may be represented as a state node in the drawings in the institute that 1 this state transition diagram can travel through coding operator on the convolution position, and such node has 4
mIndividual;
2 every two nodes connect with a directed edge, represent the transfer process of coding operator on the interior convolution of adjacent encoder time quantum position, one group of mark is arranged on every limit, in mark, left data represents the k position coding operator of current time input, right data represents the n position coding operator of current time output, limit mark (Z, XY) representative (Y: Z: I) V=XY: I from the Y state to the I state;
3 each node stretch out 4
k* 2
(n-k)The bar limit enters other nodes, and the limit that enters simultaneously each node has 4
k* 2
(n-k)Bar.
For a n=2, k=1, m=1 quantum convolution code, its coding circuit as shown in Figure 5, calculates the encoder matrix V of 6 * 6 by encoding operation U:
The state transition diagram of this quantum convolution code as shown in Figure 6.
The grid chart of 3 quantum convolution codes
Although state transition diagram can be illustrated under the information sequence of different inputs, the state transitions process that on convolution position, m position, coding operator occurs, but can not express the relation of this state transition diagram and time, in order to mean the relation of each state and time, we can mean with grid chart.
The grid chart of quantum convolution code: the known code parameter is the quantum convolution code of [[n, k, m]], and total N+t scramble time unit, according to its state transition diagram, can obtain corresponding grid chart, and this grid chart is a directed graph that meets following condition:
1 set of node can be divided into N+t+1 subset D
J, wherein | D
0|=1, | D
J|=2
m, 1≤j≤N+t;
2 every two nodes connect with a directed edge, all from node D
J-1Set out and arrive node D
JThe directed edge set be called E
J, E
jBe called the j joint of grid chart, one group of mark is arranged on every limit, in mark, left data represents the k position coding operator of current time input, and right data represents the n position coding operator of current time output;
3 in each encodes constantly, and each node stretches out 4
k* 2 (
n-k) the bar limit enters other nodes, the limit that enters simultaneously each node has 4
k* 2
(n-k)Bar.
For n=2, k=1, the quantum convolution code of m=1, its grid chart is as shown in Figure 7.
Claims (5)
1. method of constructing quantum convolution code state transition diagram, it is characterized in that: code parameters is [[n, k, m]] the quantum convolution code, k position information exchange is crossed encoding operation and is encoded into the long code word in n position, m refers to code storage, its encoding operation meets unitary transformation, its encoding operation is carried out to 2 rank, (n+m) * 2 (n+m) encoder matrixs that the clifford conversion can calculate a correspondence, under the effect of this matrix, the state change process of dissection coding operator on convolution position, m position, and summarize with figure, this figure is by state node, directed edge, input coding operator and output encoder operator form, and the institute that can travel through coding operator on the convolution position likely, each may be represented as a state node in the drawings, such node has 4
mindividual, every two nodes connect with a directed edge, represent the transfer process of coding operator on the interior convolution of adjacent encoder time quantum position, one group of mark is arranged on every limit, and in mark, left data represents the k position coding operator of current time input, and right data represents the n position coding operator of current time output, each node stretches out 4
k* 2
(n-k)the bar limit enters other nodes, and the limit that enters simultaneously each node has 4
k* 2
(n-k)bar.
2. a kind of method of constructing quantum convolution code state transition diagram as claimed in claim 1 is further characterized in that: according to known coding circuit, draw state transition diagram, for the drafting of grid chart lays the foundation.
3. method of constructing quantum convolution code grid chart, it is characterized in that: can describe the state transition diagram in each scramble time unit, code parameters is [[n, k, m]] the quantum convolution code, k position information exchange is crossed encoding operation and is encoded into the long code word in n position, m refers to code storage, if carry out altogether coding N+t time, wherein before N time for input message, for coding circuit, make zero for latter t time, this figure is by state node collection, directed edge collection, input coding operator and output encoder operator form, and wherein set of node can be divided into N+t+1 subset, and each subset definition is D
j, | D
0|=1, | D
J|=2
m, 1≤j≤N+t; Every two nodes connect with a directed edge, all from node D
J-1Set out and arrive node D
jThe directed edge set be called E
J, one group of mark is arranged on every limit, in mark, left data represents the k position coding operator of current time input, right data represents the n position coding operator of current time output; In the moment, each node stretches out 4 at each coding
k* 2
(n-k)The bar limit enters other nodes, and the limit that enters simultaneously each node has 4
k* 2 (
n-k) bar.
4. a kind of method of constructing quantum convolution code grid chart as claimed in claim 3, be further characterized in that: can express state transition diagram and the relation between the scramble time, and clearly describe each state changes within the scramble time process.
5. a kind of method of constructing quantum convolution code grid chart as claimed in claim 3, be further characterized in that: for the optimum decoding algorithm of analyzing the quantum convolution code provides strong instrument.
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Cited By (4)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN103873074A (en) * | 2013-11-28 | 2014-06-18 | 西安电子科技大学 | Quantum Viterbi decoding algorithm based on grid chart at decoding end |
| CN109740758A (en) * | 2019-01-09 | 2019-05-10 | 电子科技大学 | A kind of kernel method based on quantum calculation |
| CN110210073A (en) * | 2019-05-10 | 2019-09-06 | 腾讯科技（深圳）有限公司 | Quantum noise process analysis method, device, equipment and storage medium |
| CN114091363A (en) * | 2020-08-04 | 2022-02-25 | 合肥本源量子计算科技有限责任公司 | Computational fluid dynamics simulation method, device and equipment based on quantum algorithm |
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| US20080201631A1 (en) * | 2004-05-28 | 2008-08-21 | France Telecom | Method for Error Correction Coding Comprising Local Error Detection Codes, Corresponding Decoding Method, Transmitting, Receiving and Storage Device and Program |
| CN101409564A (en) * | 2008-11-25 | 2009-04-15 | 南京邮电大学 | Construction method for quantum low-density parity check code base on stabilizing subcode |
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| US20080201631A1 (en) * | 2004-05-28 | 2008-08-21 | France Telecom | Method for Error Correction Coding Comprising Local Error Detection Codes, Corresponding Decoding Method, Transmitting, Receiving and Storage Device and Program |
| CN101409564A (en) * | 2008-11-25 | 2009-04-15 | 南京邮电大学 | Construction method for quantum low-density parity check code base on stabilizing subcode |
Cited By (7)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN103873074A (en) * | 2013-11-28 | 2014-06-18 | 西安电子科技大学 | Quantum Viterbi decoding algorithm based on grid chart at decoding end |
| CN103873074B (en) * | 2013-11-28 | 2016-05-25 | 西安电子科技大学 | Quantum Viterbi decoding algorithm based on decoding end grid chart |
| CN109740758A (en) * | 2019-01-09 | 2019-05-10 | 电子科技大学 | A kind of kernel method based on quantum calculation |
| CN109740758B (en) * | 2019-01-09 | 2023-04-07 | 电子科技大学 | Quantum computation-based nuclear method |
| CN110210073A (en) * | 2019-05-10 | 2019-09-06 | 腾讯科技（深圳）有限公司 | Quantum noise process analysis method, device, equipment and storage medium |
| CN114091363A (en) * | 2020-08-04 | 2022-02-25 | 合肥本源量子计算科技有限责任公司 | Computational fluid dynamics simulation method, device and equipment based on quantum algorithm |
| CN114091363B (en) * | 2020-08-04 | 2023-08-08 | 合肥本源量子计算科技有限责任公司 | Quantum algorithm-based computational fluid dynamics simulation method, device and equipment |
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