CN106603086B - A kind of building method of the stud stable structure subcode based on cluster state - Google Patents
A kind of building method of the stud stable structure subcode based on cluster state Download PDFInfo
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- H—ELECTRICITY
- H03—ELECTRONIC CIRCUITRY
- H03M—CODING; DECODING; CODE CONVERSION IN GENERAL
- H03M13/00—Coding, decoding or code conversion, for error detection or error correction; Coding theory basic assumptions; Coding bounds; Error probability evaluation methods; Channel models; Simulation or testing of codes
- H03M13/29—Coding, decoding or code conversion, for error detection or error correction; Coding theory basic assumptions; Coding bounds; Error probability evaluation methods; Channel models; Simulation or testing of codes combining two or more codes or code structures, e.g. product codes, generalised product codes, concatenated codes, inner and outer codes
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Abstract
The building method for the stud stable structure subcode based on cluster state that the invention discloses a kind of, used building method is to select cluster state to be located at two quantum bits on the diagonal line of stud grid to carry out X overturning, compared to the loop chain make of figure state, it is that can construct two Z overturnings on side among two X overturning, therefore just obtain significant promotion to the coding utilization rate of quantum bit that it is markedly improved more.Construction of stable subcode method proposed by the present invention has good scalability, can freely expand in bigger cluster state, select same make that can construct the stabistor with same characteristic features an of race；This building method not only can be adapted for cluster state structure, can also be promoted in other two-dimensional surface polygonal structures, construct Quantum Error Correcting Codes under different symmetry.
Description
Technical field
The invention belongs to safe quantum communication technical fields, and in particular to a kind of quantum based on cluster state stablizes the solution of subcode
Analyse building method.
Background technique
With the development of Quantum Error Correcting Codes, there is the effective ways of a variety of construction of stable subcodes now.Wherein, figure is utilized
The method of state construction of stable subcode is consequently also suggested, and there is construction advantages not available for other building methods.Example
Such as, the conversion characteristic of quantum mistake, code race can be controlled in the identical system of size and can use the geometry of figure
It was found that the new construction etc. of code.Also realized respectively on the basis of X-Y scheme state and higher-dimension figure state the construction of Quantum Error Correcting Codes and
Sufficient verifying is carried out.
Document 1: " Quantum error-correcting codes associated with graphs " is by sudden and violent
Power search method obtains Quantum Error Correcting Codes.In the process based on figure state construction of stable subcode, one is constructed according to figure G and is surpassed
Figure, by code collection searching algorithm, force search goes out qualified code collection in hypergraph, determines a group picture by each code collection
State base, to construct stable subcode.But the disadvantage is that because code collection search problem is a complete np problem, face to face
Face when constructing longer more high-dimensional error correcting code, it is very pessimistic for relying solely on limited search.
Document 2: " Graphical nonbinary quantum error-correcting codes " is to pass through analysis
The structures of Special Graphs obtains Quantum Error Correcting Codes.Based on the low-dimensional code collection and subgraph gone out by computer search, go to analyze
The special graph structure of code collection establishes more high-dimensional Quantum Error Correcting Codes, a series of is suitble to any dimension so as to establish out
Quantum Error Correcting Codes.But the disadvantage is that figure state does not have very high degree of entanglement, safety is relatively weak and tangling for quantum acts on very well
It eliminates.Figure state is not easy to prepare very much in an experiment, and the conclusion much based on figure state is difficult to be verified in an experiment, so it is grasped
Making practicability will not be very high.
Summary of the invention
For above-mentioned problems of the prior art, the object of the present invention is to provide a kind of day words based on cluster state
The building method of shape stable structure subcode goes construction quantum to stablize subcode, has higher entangle with cluster state compared to using figure state
Degree of twining is more easier preparation and the significant advantages such as quantum entanglement effect is difficult to eliminate.And compared to the loop chain structure of figure state
Mode is made, " day " font diagonal line make significant improvement of the present invention is on the side among two X overturnings
Two Z overturnings can be constructed more, therefore apparent promotion is just obtained to the coding utilization rate of quantum bit.
In order to realize above-mentioned task, the invention adopts the following technical scheme:
A kind of building method of the stud stable structure subcode based on cluster state, comprising the following steps:
Step 1, for the graphic structure of cluster state, one quantum bit of vertex representation of each grid, side indicates quantum
Position is adjacent；It is translation unit with a grid, each quantum bit is starting point i, is to interact across single with two grids
Position constitutes cornerwise quantum bit as terminal m with starting point, links beginning and end, starting point i and point m is made to constitute " day " font
Structure establishes the interaction in cluster state between quantum, the stabistor operator matrix S obtained1Are as follows: { gi|gi=GiGm,i∈n}；Its
In, n indicates to stablize the quantum bit number of subcode, stabistor Gi=XiZN(i)(i ∈ n) indicates the stabistor on the i of vertex, is top
The direct product of X operator on point i and the Z operator on the vertex neighborhood N (i) of i；
Step 2 calculates the mistake whether a certain element for the stabistor operator matrix established in step 1 occurs with possibility
Type constitutes anticommutation, if a certain element and type of error of stabistor operator matrix enable always at anticommutation
Stablize submatrix S=S1, then execute step 4；If all elements of stabistor operator matrix and type of error be not at opposition
Easy relationship, thens follow the steps three；
Step 3 adds out the operation with mistake at anticommutation according to the type of error that may occur in step 2
Operator matrix S2, make S1And S2It is integrally formed the complete stabistor of implementable error correction, this seasonal total stabilization submatrix S=S1∪
S2；
It is steady that (n, a k) quantum can be obtained using following coding formula according to total stabilization submatrix S in step 4
Stator code:
In above formula, C1...CkIndicate the code word of error correcting code, | C1...CkThe code space of > expression error correcting code, I indicate that unit is calculated
Symbol, giIndicate i-th of stabistor,Indicate error correcting code coding codeword C1...CkWhen, take eachI.e. n X operator is multiplied；| 0...0 > expression calculates ground state, and wherein the number i of stabistor is no more than n-
k。
The present invention has following technical characterstic compared with prior art:
1. the make of this stud structure of the invention can apply to the single-row grid of cluster state and single-row grid respectively
In combined and spliced stabilization subcode, complete Quantum Error Correcting Codes can be constructed；As it can be seen that proposed by the present invention based on cluster state
" day " character form structure stablize subcode building method be it is feasible, in a wider context and be easy promote；
2. construction of stable subcode method proposed by the present invention has good scalability, can freely expand to bigger
In cluster state, select same make that can construct the stabistor with same characteristic features an of race；
It, can also be in the polygon knot of other two-dimensional surfaces 3. building method of the invention not only can be adapted for cluster state structure
It is promoted in structure, constructs Quantum Error Correcting Codes under different symmetry；And this cluster state quantum of the method for the present invention construction
Error correcting code has the degree of entanglement of height；If possessing enough degrees of entanglement, they are in the ultra dense coding field of quantum and quantum
Biggish effect can be all played during Teleportation.
Detailed description of the invention
Fig. 1 is the flow chart of the method for the present invention；
Fig. 2 is single-row grid cluster state structure；
Fig. 3 is the single-row grid cluster state structure of 10 quantum bits；
Fig. 4 is the combined and spliced cluster state structure of 10 quantum bit grid；
Specific embodiment
On graphic structure, the form of cluster state is the cubic lattice of n dimension, his one quantum bit of vertex representation, side table
Show the adjacent relationship between different quantum bits.Explain that " day " character form structure based on cluster state stablizes subcode structure in detail below
Make the implementation process of method.
Step 1, for the graphic structure of cluster state, one quantum bit of vertex representation of each grid, side indicates quantum
Position is adjacent, as shown in Figure 2；It is translation unit with a grid, it is mutual with two grids that each quantum bit, which is starting point i,
Unit is crossed in effect, is constituted cornerwise quantum bit as terminal m with starting point, is linked beginning and end, make starting point i and point m structure
At " day " character form structure, and establish the interaction in cluster state between quantum, i.e., in cluster state be located at " day " font grid it is diagonal
Two quantum bits on line carry out X overturning；It can thus be concluded that the stabistor operator matrix S gone out1Are as follows:
{gi|gi=GiGm,i∈n}
In above formula, n indicates to stablize the quantum bit number of subcode, stabistor Gi=XiZN(i)(i ∈ n) is indicated on the i of vertex
Stabistor is the direct product of the Z operator on the vertex neighborhood N (i) of the X operator and i on the i of vertex；GmIndicate the stabilization on terminal m
Son is the direct product of the Z operator on the vertex neighborhood N (m) of the X operator and m on terminal m；
1. for the stabilization subcode of single-row grid
As shown in figure 3, constructed for single-row grid stablizes subcode by 10 quantum bit cluster states, to being located at " day " word in cluster state
Two quantum bits on the diagonal line of shape grid carry out X overturning, obtained stabistor operator matrix are as follows:
G1G6=X Z Z Z Z X I Z I I
G2G5=Z X Z Z X Z Z I I I
G3G8=Z I X Z Z Z Z X I Z
G4G7=I Z Z X Z Z X Z Z I
G5G10=I I Z I X Z Z Z Z X
G6G9=I I I Z Z X Z Z X Z
In above formula, stabistor can be write as: Gi=XiZN(i)(i=1,2,3 ..., 10), that is, indicate the stabilization on the i of vertex
Sub- Gi, it is the direct product of the Z operator on the vertex neighborhood N (i) of the X operator and i on the i of vertex.Wherein selection, which is established in cluster state, is located at
Cornerwise two quantum bits of " day " font (such as the G in Fig. 31With G6, G2With G5, G3With G8Deng) between tangle relationship.?
In above formula, I is unit operator, and X is quantum bit overturning operator, and Z is quantum phase overturning operator, I, X, Z in matrix below
Meaning be identical with this.
2. the stabilization subcode combined and spliced for grid
As shown in figure 4, equally by taking 10 quantum bit cluster states construct the combined and spliced stable subcode of grid as an example.Two quantum ratios
Spy is located at the side of single-row grid, labels it as a and b in order to facilitate classification.Still it selects to be located at " day " reticle font in cluster state
Two quantum bits on the diagonal line of son carry out X overturning, obtained stabistor operator matrix are as follows:
G1G6=X Z Z Z Z X I Z I I
G2G5=Z X Z Z X Z Z I Z I
G3G8=Z I X Z Z Z Z X I I
G4G7=I Z Z X Z Z X Z I Z
G1Gb=X Z Z Z I I I I Z X
G3Ga=Z Z X Z Z I I I X Z
G3Ga=Z Z X Z Z I I I X Z
In above formula, stabistor can be write as: Gi=XiZN(i)(i=1,2,3 ..., 8, a, b), that is, indicate steady on the i of vertex
Stator Gi, it is the direct product of the Z operator on the vertex neighborhood N (i) of the X operator and i on the i of vertex.Wherein position in cluster state is established in selection
In cornerwise two quantum bits of " day " font (such as the G in Fig. 41With G6, G2With G5, G1With Gb, G3With GaDeng) between tangle
Relationship.
Selection cluster state of the present invention is located at the progress of two quantum bits on the diagonal line of " day " font grid X and turns over
Turn, compared to the loop chain make of figure state, significant improvement is can to construct two on side among two X overturnings more
A Z overturning, therefore significant promotion is just obtained to the coding utilization rate of quantum bit.
Step 2 calculates the mistake whether a certain element for the stabistor operator matrix established in step 1 occurs with possibility
Type constitutes anticommutation, that is, judges stabistor matrix main body part S1Whether satisfaction stablizes error correction item required by subcode
Part, wherein the type of error being likely to occur is XX, YY, ZZ, XZ mono- shares 4 kinds, may be constructed anticommutation:
XX and XZ；
There is I on YY and any bit；
ZZ and XZ；
XZ and ZZ.
If a certain element and type of error of stabistor operator matrix enable total stabilization submatrix S at anticommutation
=S1, then execute step 4；If all elements of stabistor operator matrix and type of error are held not at anticommutation
Row step 3；Specific example is as follows:
1. for the stabilization subcode of single-row grid
Opposed that easy constituent relation detects that the not detectable mistake of the stabistor operator matrix that step 1 obtains has Z3Z8,
Z4Z7, Y5Y6Three kinds, but stabistor operator matrix only has six rows, and the n-1 quantity principle by stablizing sub-operation operator limits, i.e., most
Three rows operation operator can also be mostly added again, and error correction is carried out to it；
2. the stabilization subcode combined and spliced for grid
Opposed that easy constituent relation detects that the not detectable mistake of the stabistor operator matrix that step 1 obtains has Z2Z5,
Z4Z7, Y3Y4Three kinds, but stabistor operator matrix only has six rows, and the n-1 quantity principle by stablizing sub-operation operator limits, i.e., most
Three rows operation operator can also be mostly added again, and error correction is carried out to it.
Step 3 adds out the operation with mistake at anticommutation according to the type of error that may occur in step 2
Operator matrix S2, make S1And S2It is integrally formed the complete stabistor of implementable error correction, this seasonal total stabilization submatrix S=S1∪
S2；
1. for the stabilization subcode of single-row grid
According to not detectable wrong Z3Z8, Z4Z7, Y5Y6Three kinds, then add and operate the operation operator that operator is constituted by two rows
Matrix S2Error correction is carried out to it:
G3G7=Z I X Z I I X Z Z I
G4G5=I Z I X X I Z I I I
2. the stabilization subcode combined and spliced for grid
According to not detectable wrong Z2Z5, Z4Z7, Y3Y4Three kinds, then add and the operation operator that operator is constituted is operated by a line
Matrix S2Error correction is carried out to it:
G4G5=I Z I X X I Z I I Z
Step 4 can be obtained one using following coding formula and can detect and take the post as according to total stabilization submatrix S
Wrong (n, k) quantum on two quantum bits of anticipating stablizes subcode:
In above formula, C1...CkIndicate the code word of error correcting code, | C1...CkThe code space of > expression error correcting code, I indicate that unit is calculated
Symbol, giIndicate i-th of stabistor,Indicate error correcting code coding codeword C1...CkWhen, take eachI.e. n X operator is multiplied；| 0...0 > expression calculates ground state, and wherein the number i of stabistor is no more than n-
k；Wherein n is physical number of bits, and k is logical bits number.
This new " day " character form structure stabistor code constructing method based on cluster state, compared with general figure state code structure before
Method is made, many advantages are possessed.This method not only reduces general by the higher multiple of force search method structural map state code
Miscellaneous degree also overcomes the deficiency that operation in other excitation state physical models does not constitute a complete base.And this method is in physics
Level has obtained stability, rather than by accurate error correction step, simplify error correction procedure.Therefore, this method has very
Good scalability, can freely expand in bigger cluster state, select same make that can construct a race
Stabistor with same characteristic features.Perhaps, this building method that we search out not only can be adapted for cluster state structure, perhaps
It can be promoted in other two-dimensional surface polygonal structures, construct Quantum Error Correcting Codes under different symmetry.And I
This cluster state Quantum Error Correcting Codes that design there is the degree of entanglement of height.If possessing enough degrees of entanglement, they are being measured
Certain effect may be all played during the ultra dense coding field of son and quantum teleportation.
Claims (1)
1. a kind of building method of the stud stable structure subcode based on cluster state, which comprises the following steps:
Step 1, for the graphic structure of cluster state, one quantum bit of vertex representation of each grid, side indicates quantum phase
It is adjacent；It is translation unit with a grid, it is that unit is crossed in interaction with two grids that each quantum bit, which is starting point i, with
Starting point constitutes cornerwise quantum bit as terminal m, links beginning and end, starting point i and point m is made to constitute " day " character form structure,
The interaction in cluster state between quantum is established, the stabistor operator matrix S obtained1Are as follows: { gi|gi=GiGm,i∈n}；Wherein, n table
Show the quantum bit number of stable subcode, stabistor Gi=XiZN(i)(i ∈ n) indicates the stabistor on starting point i, is on starting point i
The direct product of Z operator on the vertex neighborhood N (i) of X operator and i；Wherein, X is quantum bit overturning operator, and Z is that quantum phase is turned over
Turn operator；
Step 2 calculates the type of error whether a certain element for the stabistor operator matrix established in step 1 occurs with possibility
Anticommutation is constituted, if a certain element and type of error of stabistor operator matrix enable total stabilization at anticommutation
Submatrix S=S1, then execute step 4；If all elements of stabistor operator matrix and type of error do not close easily at opposition
System, thens follow the steps three；The type of error being wherein likely to occur is XX, YY, ZZ, and XZ mono- shares 4 kinds, and Y is quantum bit
Phase overturns operator simultaneously；
Step 3 adds out the operation with type of error at anticommutation according to the type of error that may occur in step 2
Operator matrix S2, make S1And S2It is integrally formed the complete stabistor of implementable error correction, this seasonal total stabilization submatrix S=S1∪
S2；
Step 4 obtains (n, a k) quantum using following coding formula and stablizes subcode according to total stabilization submatrix S,
In, n indicates physics number of bits, and k indicates logical bits number；
In above formula, C1...CkIndicate the code word of error correcting code, | C1...CkThe code space of > expression error correcting code, I indicate unit operator, gi
Indicate i-th of stabistor,Indicate error correcting code coding codeword C1...CkWhen, take eachI.e.
N X operator is multiplied；| 0...0 > expression calculates ground state, and wherein the number of stabistor is no more than n-k.
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| CN103067093A (en) * | 2012-12-26 | 2013-04-24 | 中国电力科学研究院 | Quantum error correction coding method applicable to high-voltage overhead power lines |
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