CN108241778A - The building method of the optimal observation operator set of Quantum Pure reconstruct - Google Patents
The building method of the optimal observation operator set of Quantum Pure reconstruct Download PDFInfo
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Abstract
The present invention discloses a kind of building method of the optimal observation operator set of Quantum Pure reconstruct, belongs to quantum state estimation field.The present invention is observed the diagonal element of quantum state density matrix first with selected observation operator, then it obtains observation and is treated according to observation and estimate that quantum state belongs to eigenstate or superposition state is judged and classified, finally optimal observation operator set is constructed for classified quantum state, and reconstruct the density matrix of quantum state to be estimated according to optimal observation operator set with corresponding observation.The present invention carries out state estimation under the premise of operator is observed selected by guarantee and can uniquely determine the state of being estimated, using part measurement result, can realize the Accurate Reconstruction to Quantum Pure, can effectively reduce pendulous frequency, promotes Quantum Pure reconstruct efficiency.
Description
Technical field
The present invention relates to the building methods of the optimal observation operator set of Quantum Pure reconstruct, belong to quantum state estimation technique neck
Domain.
Background technology
Quantum measurement is an important prerequisite and the basis of quantum state estimation, however, different from the survey in classical system
It measures, the measurement object in quantized system is typically a kind of notional observation operator for including system under test (SUT) partial information.In order to
The accurate state for estimating quantum, people need a different set of observation operator is selected to treat estimated state to take multiple measurements,
To obtain all information of quantum state.Quantum state estimation based on measurement can be divided into the quantum state carried out by offline batch processing
Two methods of estimation and online continuous quantum state estimation, quantum chromatography be in the estimation of batch processing quantum state most common method it
One.The state density matrix ρ of the quantized system of one n quantum bit is d × d (wherein d=in Hilbert space
2n) matrix, have 2n×2n=4nA parameter, traditional quantum chromatography need m=d2- 1=4n- 1 sub-completion measures, and can just obtain
The all information of quantum state.As it can be seen that quantum state chromatographs required observation frequency as the increase of quantum digit n exponentially increases
Long, this to become very difficult to the chromatography of High Dimensional Quantum States.
In order to effectively reduce the observation frequency needed for estimation quantum state, people can be using part observed result come the amount of progress
Sub- state reconstruct, going out a necessary condition of pure state density matrix come Accurate Reconstruction using a small amount of observed result is：Selected observation is calculated
Symbol can uniquely determine estimative pure state (UDP).Existing research shows that：To any amount of d dimensions Hilbert spatially
Sub- pure state, it is only necessary to which being measured in no more than m=4d-5 Pauli observation operator can just uniquely determine.Compression sensing
(CS) theory is pointed out:When quantum state ρ is pure state or approximate pure state, the order r of ρ is far smaller than dimension d, i.e. r ＜ ＜ d, this
When, as long as from d2The measuring configuration number m of stochastical sampling meets m=O (rdlogd) in a overall measurement result, it is ensured that with very big
Probability Accurate Reconstruction go out density matrix ρ compared to quantum chromatography needed for d2- 1 measurement, can pole based on compression sensing theory
The earth reduces the reconstruct required observation frequency of quantum state.
Finkelstein in 2004 is based on a kind of minimum measurement side of measurement proposition that positive operator value measures (POVM) operator
Case measures the reconstruct to realize arbitrarily quantum pure state by 3d-2 times.Since this scheme is using POVM observation operators, it is not
The complete observation operator being made of fixed base vector.The total amount being made of in practical application for one n 2 energy level quantum bits
Subsystem, frequently be Pauli operator, one group of d may be constructed by Pauli operator2A complete observation operator, these observations are calculated
Accord with equal full rank and pairwise orthogonal.When observing operator and the method using step measurement based on Pauli, quantum can be further reduced
Observation frequency needed for state reconstruct.
Invention content
The limitation that the present invention is more for required pendulous frequency in the reconstruct of existing Quantum Pure and brings, proposes a kind of
The building method of the optimal observation operator set of new Quantum Pure reconstruct, this method only need minimal number of measurements can be realized as
The construction of optimal observation operator set.
The present invention provides a kind of building method of the optimal observation operator set of Quantum Pure reconstruct, including to eigenstate and
The construction of the optimal observation operator set of superposition state, wherein：
Step (1.1) select d dimension Hilbert space in arbitrary n-bit Quantum Pure ρ=| ψ><ψ |, dimension d=
2n, wherein state vector | ψ>Meet：
Wherein, ci is the coefficient of eigenstateWherein i=1 ..., d, andFor d
The orthogonal basis of Hilbert space is tieed up, while is also eigenstate；
Step (1.2) select one group by the hermitian operator of full rank that Pauli matrices are formed as observation operator, by d2A sight
The complete observation operator set M that measuring and calculating symbol is formed meets：
Wherein, MjIt is by the observation operator of the direct product composition of I, X, Y and Z；For two-dimentional Pauli matrices, I is
Unit operator, X, Y, Z are Pauli operator, are met：
The complete observation operator set Μ of selected one group of step (1.3)diagTo the diagonal element of the density matrix ρ of quantum state into
Row observation：According to the constraints that the sum of diagonal entries all in density matrix are 1：Unit matrix composition
The observation of operator is observed necessarily for 1, does not need to measure, thus the complete observation operator set of density matrix diagonal element be by
D-1 observation operator composition：
Wherein, M0=II...I is the observation operator of unit matrix composition；
Step (1.4) obtains observation and observation is judged and the classification of eigenstate or superposition state：Due to this
The nonzero element that only one and only one value is 1 on the diagonal in density of states matrix ρ is levied, by n quantum bits sheet to be estimated
Sign state | ei>| i=0 ..., d-1 } observation, observation operator set ΜdiagMiddle whole operator MjObservation half it is another for+1
Half is -1；It is that the location of 1 element is located at this observation calculation if observation is+1, in eigenstate diagonal to be estimated
All diagonal elements are in+1 position in symbol；If observation is -1, the element in eigenstate to be estimated for 1 is located at this sight
All diagonal elements are in -1 position in measuring and calculating symbol；It, can be eigenstate in this way by once observing the observed result of operator
It is the range of 1 element present position in diagonal element, d/2 is narrowed down to from d；N sight of operator is observed by obtaining n times
Measured value can be reduced into d/2 to the range of the element present position for 1 in eigenstaten=1, it is possible thereby to uniquely determine this
It is the position of 1 element in sign state, and estimates eigenstate ρ to be estimated；Only n times measurement in, each time observation be all+
1 or -1, then quantum state to be estimated is eigenstate, as long as there is real number of any observation between+1 and -1, then treat
Estimation quantum state is superposition state；
Step (1.5) when Quantum Pure to be estimated is eigenstate, calculate by the optimal observation of construction quantum eigenstate reconstruct
Symbol collection：From complete diagonal element observation operator set ΜdiagMiddle selection can uniquely determine out n observation of arbitrary eigenstate
Operator forms an optimal observation operator setFor arbitrary eigenstate, optimal observation operator setIt is not unique
, one of which is provided here
When Quantum Pure to be estimated is quantum superposition state, its optimal observation operator set is constructed and according to observation to amount
Sub- superposition state is estimated, includes the following steps：
The selected n-bit quantum superposition state ρ of step (2.1)=| ψ><ψ |, andEigencoefficient ciIn have and
Only l are not 0, are called " l- superposition states ", therefore l × l nonzero element is shared in the density matrix ρ of l- superposition states；
Step (2.2) is to the complete observation operator set Μ of step (1.3)diagIn measured in addition to step (1.4)
N observation operator except other d-n-1 operators be observed, to total observation frequency of diagonal element in l- superposition states
Step (2.3) is 1 according to the sum of the observation of obtained d-1 operator and diagonal element, can be calculated
Whole d diagonal elements in density matrix ρ；
Step (2.4) assumes have the l non-zero diagonal element to be respectively in l- superposition states：ckt=akt+ibkt, t=1 .., l,
akt,bkt∈ R, it is a that nondiagonal element is known as l (l-1), in the case where ignoring global phase, might as well set ck1Imaginary part be zero, so
HaveIt sets up, at this time unknown quantity akt,bktRespectively have that (l-1) is a, unknown number sum is 2 (l-1)；
The suitable observation operator set Μ of step (2.5) selectionnonIn observation operator Mn, n=d+1 ..., d2, to non-right
Angle element is observed, wherein observation operator set ΜnonMeet：
Μnon=Μ-Μdiag-M0
Since unknown number sum is a for 2 (l-1), therefore a observation operators of 2 (l-1) is needed to be observed, i.e. off-diagonal element
Observation frequency
Step (2.6) is by above-mentioned steps (2.1)-(2.5) it is found that reconstructing needed for arbitrary n-bit l- superposition states (2≤l≤d)
Minimum observation frequency be：
Its it is optimal observation operator set be：
Wherein, ΜdiagIt is the complete observation operator set to diagonal element,For the sight to off-diagonal element
Measuring and calculating symbol set.
Further, 2 in the step (2.4) in the building method of the optimal observation operator set of above-mentioned Quantum Pure reconstruct
(l-1) selecting step of a non-diagonal observed quantity is as follows：
Step (2.4.1) finds off-diagonal element(i.e.WithElement on corresponding crossover location) corresponding to
By I, the observation operator M of X, Z compositionX12, due to MX12For full rank operator, if MX12Only measureWithValue, then carry out
In next step, if otherwise MX12Also measure other t1Group diagonal element valueThen by converting MX12Middle I, Z are selected again
Go out t1A operator forms { MX12,MXij,MXqp,......}t1+1, according to this t1Observation in+1 operator, calculatesWait t1The real part of+1 off-diagonal element；Then imaginary part is observed, by { MX12,MXij,
MXqp,......}t1+1In an X be transformed to Y, form corresponding t1+ 1 imaginary part observation operator { MY12,MYij,MYqp,......
}t1+1, according to this t1Observation in+1 operator, calculatesWait t1The void of+1 off-diagonal element
Portion, this step select 2 (t altogether1+ 1) a observation operator；
Step (2.4.2) according in step (2.4.1) as a result, rejectWait t1+ 1 withThere is the diagonal item of cross termDeng, it is remaining withThe diagonal item for having cross term is l-t1- 2,
Repeat step in step (2.4.1)；Until find all withThe diagonal item for having cross term corresponds to real part imaginary part observation operator,
Shared observation operator number is 2 (t1+1)+2(t2+ 1)+...=2 (l-1) a ((t1+1)+(t2+ 1)+...=l-1), thus it is non-
Cornerwise total observation frequency is 2 (l-1).
The present invention is measured using part and tied under the premise of operator is observed selected by guarantee and can uniquely determine the state of being estimated
Fruit carries out state estimation, can realize the Accurate Reconstruction to Quantum Pure, can effectively reduce pendulous frequency, promotes Quantum Pure
Reconstruct efficiency.The present invention carries out survey needed for reconstruction of quantum states in contrast to quantum chromatography, pure state UDP and based on compression sensing theory
Based on the measuring method invented, state estimation is carried out using part measurement result for the method for measuring number, and is passed through corresponding excellent
Changing algorithm realizes the minimal number of measurements of Quantum Pure reconstruct and its construction of optimal operator observation collection.Specifically, this item is sent out
Bright selection Pauli observation operator proposes the structural scheme of a kind of minimal number of measurements to Quantum Pure reconstruct and its observation collection,
To d=2nArbitrary n quantum bits pure state ρ in dimension Hilbert space is when being reconstructed, if ρ is eigenstate, then it is required most
Observation frequency is only less：n；For including the superposition state ρ of l (2≤l≤d) a finite eigenvalues, minimum observation frequency needed for reconstruct
For：D+2l-3, measuring configuration number O (rdlogd) needed for the reconstruction of quantum states that this number is provided much smaller than compression sensing theory,
And the pure state provided in existing disclosed document uniquely determines required minimum observation frequency 4d-5.
Description of the drawings
Fig. 1 is the flow chart of the building method of the optimal observation operator set of Quantum Pure reconstruct that the application provides；
Fig. 2 is three kinds of different reconstruction of quantum states methods：UDP, CS and the m of two step measuring methods with n change curve；
Fig. 3 is three kinds of different reconstruction of quantum states methods：UDP, CS and two step measuring methods and η with n change curve；
Fig. 4 is the pure state in n=4 quantum bit diagonal element non-zero number l=4,8,12,16 of institute's inventive method
Reconstruction result.
Specific embodiment
The building method of the optimal observation operator set of a kind of Quantum Pure reconstruct that the application provides, as shown in Figure 1, packet
It includes：
If the density matrix of Quantum Pure to be estimated is ρ, the diagonal element of quantum state density matrix ρ is seen first
It surveys, obtains observation and treated according to observation and estimate that quantum state belongs to eigenstate or superposition state is judged and classified：
(1.1) quantum state to be estimated for d tie up Hilbert space in arbitrary n-bit Quantum Pure ρ=| ψ><ψ |, dimension
Spend d=2n, wherein state vector | ψ>Meet：
Wherein, ciCoefficient for eigenstateWherein i=1 ..., d, andFor d
The orthogonal basis of Hilbert space is tieed up, while is also eigenstate；
(1.2) select one group by the hermitian operator of full rank that Pauli matrices are formed as observation operator, by d2A observation is calculated
The complete observation operator set M that symbol is formed meets：
Wherein, MjIt is by the observation operator of the direct product composition of I, X, Y and Z；For two-dimentional Pauli matrices, I is
Unit operator, X, Y, Z are Pauli operator, are met：
(1.3) one group of complete observation operator set Μ is selecteddiagThe diagonal element of the density matrix ρ of quantum state is seen
It surveys：According to the constraints that the sum of diagonal entries all in density matrix are 1：The observation of unit matrix composition
The observation of operator is necessarily 1, does not need to measure, therefore the complete observation operator set of density matrix diagonal element is by d-1
A observation operator composition：
Wherein, M0=II...I is the observation operator of unit matrix composition；
(1.4) it obtains observation and observation is judged and the classification of eigenstate or superposition state：Due to eigenstate
The nonzero element that only one and only one value is 1 on the diagonal in density matrix ρ, by n quantum bits eigenstate to be estimated
{|ei>| i=0 ..., d-1 } observation, observation operator set ΜdiagMiddle whole operator MjObservation half for+1 the other half
It is -1；It is that the location of 1 element is located in this observation operator if observation is+1, in eigenstate diagonal to be estimated
All diagonal elements are in+1 position；If observation is -1, the element in eigenstate to be estimated for 1 is located at this observation and calculates
All diagonal elements are in -1 position in symbol；It, can eigenstate is diagonal in this way by once observing the observed result of operator
It is the range of 1 element present position in element, d/2 is narrowed down to from d；N observation of operator is observed by obtaining n times
Value can be reduced into d/2 to the range of the element present position for 1 in eigenstaten=1, it is possible thereby to uniquely determine intrinsic
In state it is the position of 1 element, and estimates eigenstate ρ to be estimated.Only in n times measurement, observation is all+1 each time
Or -1, then quantum state to be estimated is eigenstate, as long as there is real number of any observation between+1 and -1, then wait to estimate
Meter quantum state is superposition state；
When Quantum Pure to be estimated is eigenstate, the optimal observation operator set and basis of the eigenstate reconstruct of construction quantum
Observation estimates quantum eigenstate：
(1.5) from complete diagonal element observation operator set ΜdiagMiddle selection can uniquely determine out arbitrary eigenstate
N observation operator, form an optimal observation operator setFor arbitrary eigenstate, optimal observation operator set
It is not unique, provides one of which here
According toThe observation of middle observation operator carries out quantum eigenstate method of estimation as described in (1.4).
When Quantum Pure to be estimated is quantum superposition state, its optimal observation operator set is constructed and according to observation to amount
Sub- superposition state is estimated, includes the following steps：
(2.1) select n-bit quantum superposition state ρ=| ψ><ψ |, andEigencoefficient ciIn have and only
L are not 0, are called " l- superposition states ", therefore l × l nonzero element is shared in the density matrix ρ of l- superposition states；
(2.2) to the complete observation operator set Μ in (1.3)diagIn the n observations that had measured in addition to step (1.4)
Other d-n-1 operators except operator are observed, to total observation frequency of diagonal element in l- superposition states
(2.3) it is 1 according to the sum of the observation of obtained d-1 operator and diagonal element, density can be calculated
Whole d diagonal elements in matrix ρ；
(2.4) assume there is the l non-zero diagonal element to be respectively in l- superposition states：ckt=akt+ibkt, t=1 .., l, akt,
bkt∈ R, it is a that nondiagonal element is known as l (l-1), in the case where ignoring global phase, might as well set ck1Imaginary part be zero, so havingIt sets up, at this time unknown quantity akt,bktRespectively have that (l-1) is a, unknown number sum is 2 (l-1)；This 2 (l-1) is a
The selecting step of non-diagonal observed quantity is as follows：
(2.4.1) finds off-diagonal element(i.e.WithElement on corresponding crossover location) corresponding to by I,
The observation operator M of X, Z compositionX12, due to MX12For full rank operator, if MX12Only measureWithValue, then carry out next
Step, if otherwise MX12Also measure other t1Group diagonal element valueThen by converting MX12Middle I, Z select t again1It is a
Operator forms { MX12,MXij,MXqp,......}t1+1, according to this t1Observation in+1 operator, calculatesWait t1The real part of+1 off-diagonal element；Then imaginary part is observed, by { MX12,MXij,
MXqp,......}t1+1In an X be transformed to Y, form corresponding t1+ 1 imaginary part observation operator { MY12,MYij,MYqp,......
}t1+1, according to this t1Observation in+1 operator, calculatesWait t1+ 1 off-diagonal element
Imaginary part, this step select 2 (t altogether1+ 1) a observation operator；
As a result, rejecting in (2.4.2) basis (2.4.1)Wait t1+ 1 withThere is intersection
The diagonal item of itemDeng, it is remaining withThe diagonal item for having cross term is l-t1- 2, repeat step
Step in (2.4.1)；Until find all withThe diagonal item for having cross term corresponds to real part imaginary part observation operator, shares observation
Operator number is 2 (t1+1)+2(t2+ 1)+...=2 (l-1) a ((t1+1)+(t2+ 1)+...=l-1), therefore off-diagonal
Total observation frequency is 2 (l-1).
(2.5) the suitable observation operator set Μ of selectionnonIn observation operator Mn, n=d+1 ..., d2, to nondiagonal element
Element is observed, wherein observation operator set ΜnonMeet：
Μnon=Μ-Μdiag-M0
Since unknown number sum is a for 2 (l-1), therefore a observation operators of 2 (l-1) is needed to be observed, i.e. off-diagonal element
Observation frequency
(2.6) as above-mentioned (2.1)-(2.5) it is found that reconstructing the minimum sight needed for arbitrary n-bit l- superposition states (2≤l≤d)
Surveying number is：
Its it is optimal observation operator set be：
Wherein, ΜdiagIt is the complete observation operator set to diagonal element,For the sight to off-diagonal element
Measuring and calculating symbol set.
Carrying out method of estimation to quantum superposition state is：Suitable optimization algorithm is selected, is meeting density matrix positive definite, mark etc.
Under the constraint of 1, hermiticity, by optimal observation operator setIt substitutes into optimization formula and is counted with corresponding whole observations
It calculates, due to the optimal observation operator set constructed by above-mentioned steps (2.1)-(2.5)Include quantum superposition state to be estimated
All information, therefore optimal solution can be uniquely solved, this optimal solution is the reconstruction result of quantum superposition density of states matrix ρ.
Sample rate η is defined as observation frequency m needed for reconstruction of quantum states and total observation operator quantity d2The ratio between η=m/d2.Table 1
For 3 kinds of different conditions reconstructing methods：UDP, CS and the optimal observation frequency m observed needed for operator set and corresponding sample rateη.Its
In optimal observation operator set correspond to observation frequency m and be maximized m=3d-3 and minimum value m=d+1 respectively.Data in contrast table 1
As can be seen that no matter l how value, it is optimal observation operator set needed for observation frequency m always be less than number needed for UDA, this
As a result illustrate：The observation frequency needed for reconstruction of quantum states can be effectively reduced using two step measuring methods progress state estimation.
Table 1.UDP, CS and the optimal observation frequency m observed needed for operator set and corresponding sample rateη
Fig. 2 is that m is with the change curve of n under three kinds of different reconstruction of quantum states methods, and Fig. 3 is is three kinds of different reconstruction of quantum states
Under method η with n change curve.If 3 kinds of reconstructing methods of comparison can be seen that the value that can guarantee diagonal element non-zero number l
Smaller (l=1,2), then optimal observation operator combined method has a clear superiority in 3 kinds of methods, if larger (the l ≈ of the value of l
D), then when n values are smaller (n≤4), CS is slightly better than UDA and optimal observation operator, but as n values increase (n>4), UDP and
Observation frequency m needed for optimal observation operator collection approach reconstruct will be less than CS, when n values are larger (n=12), optimal observation operator
An observation frequency m nearly energy level smaller than CS needed for collection approach reconstruct.
Fig. 4 is based on the method invented in quantum bit n=4, and diagonal element non-zero number l is respectively (Fig. 4 (a)), 8
The reconstruction result of pure state in the case of (Fig. 4 (b)), 12 (Fig. 4 (c)) and four kinds of 16 (Fig. 4 (d)).The state estimation of 4 kinds of different pure state
Fidelity the method more than 98% only with no more than in the case of observation frequency, is obtaining the estimation knot of higher fidelity
Fruit, the quantum state method of estimation of optimal observation operator set that this result illustrates to be invented can realize Quantum Pure well
Estimation.
Claims (2)
1. the building method of the optimal observation operator set of Quantum Pure reconstruct, including the optimal observation calculation to eigenstate and superposition state
Accord with the construction of collection, it is characterised in that：
Step (1.1) select d dimension Hilbert space in arbitrary n-bit Quantum Pure ρ=| ψ><ψ |, dimension d=2n,
Middle state vector | ψ>Meet：
Wherein, ciCoefficient for eigenstateWherein i=1 ..., d, andFor d Vichys
The orthogonal basis in your Bert space, while be also eigenstate；
Step (1.2) select one group by the hermitian operator of full rank that Pauli matrices are formed as observation operator, by d2A observation operator
The complete observation operator set M formed meets：
Wherein, MjIt is by the observation operator of the direct product composition of I, X, Y and Z；For two-dimentional Pauli matrices, I is unit
Operator, X, Y, Z are Pauli operator, are met：
The complete observation operator set Μ of selected one group of step (1.3)diagThe diagonal element of the density matrix ρ of quantum state is seen
It surveys：According to the constraints that the sum of diagonal entries all in density matrix are 1：The observation of unit matrix composition
The observation of operator is necessarily 1, does not need to measure, therefore the complete observation operator set of density matrix diagonal element is by d-1
A observation operator composition：
Wherein, M0=II...I is the observation operator of unit matrix composition；
Step (1.4) obtains observation and observation is judged and the classification of eigenstate or superposition state：Due to eigenstate
The nonzero element that only one and only one value is 1 on the diagonal in density matrix ρ, by n quantum bits eigenstate to be estimated
{|ei>| i=0 ..., d-1 } observation, observation operator set ΜdiagMiddle whole operator MjObservation half for+1 the other half
It is -1；It is that the location of 1 element is located in this observation operator if observation is+1, in eigenstate diagonal to be estimated
All diagonal elements are in+1 position；If observation is -1, the element in eigenstate to be estimated for 1 is located at this observation and calculates
All diagonal elements are in -1 position in symbol；It, can eigenstate is diagonal in this way by once observing the observed result of operator
It is the range of 1 element present position in element, d/2 is narrowed down to from d；N observation of operator is observed by obtaining n times
Value can be reduced into d/2 to the range of the element present position for 1 in eigenstaten=1, it is possible thereby to uniquely determine intrinsic
In state it is the position of 1 element, and estimates eigenstate ρ to be estimated；Only in n times measurement, observation is all+1 each time
Or -1, then quantum state to be estimated is eigenstate, as long as there is real number of any observation between+1 and -1, then wait to estimate
Meter quantum state is superposition state；
Step (1.5) is when Quantum Pure to be estimated is eigenstate, the optimal observation operator set of construction quantum eigenstate reconstruct：
From complete diagonal element observation operator set ΜdiagMiddle selection can uniquely determine out n observation operator of arbitrary eigenstate,
Form an optimal observation operator setFor arbitrary eigenstate, optimal observation operator setBe not it is unique, this
In provide one of which
When Quantum Pure to be estimated is quantum superposition state, constructs its optimal observation operator set and quantum is folded according to observation
State is added to be estimated, is included the following steps：
The selected n-bit quantum superposition state ρ of step (2.1)=| ψ><ψ |, andEigencoefficient ciIn have and only
L are not 0, are called " l- superposition states ", therefore l × l nonzero element is shared in the density matrix ρ of l- superposition states；
Step (2.2) is to the complete observation operator set Μ of step (1.3)diagIn the n that had measured in addition to step (1.4) see
Other d-n-1 operators except measuring and calculating symbol are observed, to total observation frequency of diagonal element in l- superposition states
Step (2.3) is 1 according to the sum of the observation of obtained d-1 operator and diagonal element, can calculate density
Whole d diagonal elements in matrix ρ；
Step (2.4) assumes have the l non-zero diagonal element to be respectively in l- superposition states：ckt=akt+ibkt, t=1 .., l, akt,
bkt∈ R, it is a that nondiagonal element is known as l (l-1), in the case where ignoring global phase, might as well set ck1Imaginary part be zero, so havingIt sets up, at this time unknown quantity akt,bktRespectively have that (l-1) is a, unknown number sum is 2 (l-1)；
The suitable observation operator set Μ of step (2.5) selectionnonIn observation operator Mn, n=d+1 ..., d2, to nondiagonal element
Element is observed, wherein observation operator set ΜnonMeet：
Μnon=Μ-Μdiag-M0
Since unknown number sum is a for 2 (l-1), therefore a observation operators of 2 (l-1) is needed to be observed, i.e. the sight of off-diagonal element
Survey number
Step (2.6) is as above-mentioned steps (2.1)-(2.5) it is found that reconstructing needed for arbitrary n-bit l- superposition states (2≤l≤d) most
Observation frequency is less：
Its it is optimal observation operator set be：
Wherein, ΜdiagIt is the complete observation operator set to diagonal element,For the observation operator to off-diagonal element
Set.
2. the building method of the optimal observation operator set of Quantum Pure reconstruct according to claim 1, it is characterised in that：Institute
The selecting step of a non-diagonal observed quantities of 2 (l-1) is as follows in the step of stating (2.4)：
Step (2.4.1) finds off-diagonal element(i.e.WithElement on corresponding crossover location) corresponding to by I,
The observation operator M of X, Z compositionX12, due to MX12For full rank operator, if MX12Only measureWithValue, then carry out next
Step, if otherwise MX12Also measure other t1Group diagonal element valueThen by converting MX12Middle I, Z select t again1
A operator forms { MX12,MXij,MXqp,......}t1+1, according to this t1Observation in+1 operator, calculatesWait t1The real part of+1 off-diagonal element；Then imaginary part is observed, by { MX12,MXij,
MXqp,......}t1+1In an X be transformed to Y, form corresponding t1+ 1 imaginary part observation operator { MY12,MYij,MYqp,......
}t1+1, according to this t1Observation in+1 operator, calculatesWait t1The void of+1 off-diagonal element
Portion, this step select 2 (t altogether1+ 1) a observation operator；
Step (2.4.2) according in step (2.4.1) as a result, rejectWait t1+ 1 withHave
The diagonal item of cross termDeng, it is remaining withThe diagonal item for having cross term is l-t1It -2, repeats
Step in step (2.4.1)；Until find all withThe diagonal item for having cross term corresponds to real part imaginary part observation operator, shares
It is 2 (t to observe operator number1+1)+2(t2+ 1)+...=2 (l-1) a ((t1+1)+(t2+ 1)+...=l-1), therefore non-diagonal
Total observation frequency of line is 2 (l-1).
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