CN111915013B - Nuclear magnetic resonance homonuclear three-bit pseudo-pure state preparation method and system - Google Patents
Nuclear magnetic resonance homonuclear three-bit pseudo-pure state preparation method and system Download PDFInfo
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Abstract
The invention discloses a nuclear magnetic resonance homonuclear multi-bit pseudomorphic pure state preparation method and a nuclear magnetic resonance homonuclear multi-bit pseudomorphic pure state preparation system. Thus, to a good approximation, the gradient field removes all coherent terms of the nuclear magnetic quantum state except for the pseudo-pure state. Moreover, in the same-core three-bit system, only one gradient field is needed to prepare a pseudo-pure state with high theoretical fidelity.
Description
Technical Field
The invention relates to the technical field of quantum computers, in particular to a nuclear magnetic resonance same-core multi-bit pseudo-pure state preparation method and system.
Background
Nuclear magnetic resonance quantum computing requires the ability to prepare the initial state of quantum computing — the pseudo-pure state. The dynamic performance and the measurement performance of the pseudo-pure state are the same as those of the pure state. Pseudo-pure state preparation is a specific step of nuclear magnetic resonance quantum computation, and the position of the pseudo-pure state preparation is the same as that of initialization in other quantum computing systems.
The existing pseudo-pure state preparation method comprises the following steps: time averaging (PRA 57,3348), space averaging (PNAS 94,1634), controlled transfer (PRA 82,032315), relaxation (PRA 94,012312), line-selective (Chemical Physics Letters 340,509), and the like.
The time averaging method needs to run experiments six times in sequence for the three-bit situation, and the experiment results are added together to serve as the final experiment result, so that the time is long, and the spectrometer needs to have good time stability. The spatial averaging method requires multiple use of pulsed gradient fields (three bits case, three times of gradient fields are needed), and the efficiency of preparing pseudo-pure state is low (i.e. the final nmr signal is weak). The control transmission method also needs to use the pulse gradient field for multiple times, such as the case of three qubits, and needs to use the gradient field five times. The relaxation method is a better method, the pseudo-pure state preparation efficiency is higher than that of other methods, the disadvantage is that in a single experiment, the same pulse sequence needs to be circulated for many times (hundreds of times), the time is long, and the pulse circulation times and the waiting time of the circulation gaps need to be optimized on an actual spectrometer. The Line-selective method only needs a gradient field once, but is not suitable for the same-nuclear magnetic system because the gradient field cannot eliminate the zero-order coherent term generated in the method, and further cannot prepare a pseudo pure state in the same-nuclear system.
Disclosure of Invention
The invention mainly aims to provide a nuclear magnetic resonance homonuclear multi-bit pseudo-pure state preparation method and a nuclear magnetic resonance homonuclear multi-bit pseudo-pure state preparation system, which can reduce the generation of zero-order coherent terms as much as possible and improve the fidelity of the pseudo-pure state.
The technical scheme for solving the technical problems is to provide a nuclear magnetic resonance homonuclear multi-bit pseudo-pure state preparation method, which comprises the following steps:
construction of |001 from a thermally balanced state using radio frequency pulses>And 011>010 |>And |110>100, | between>And |101>The first order coherent terms between them and reduce their corresponding diagonal elements to zero, the unitary operation used is
Λ1、Λ2、Λ3Are respectively |001>And 011>、|010>And |110>、|100>And |101>The transition operator of (2): lambda1＝|001><011|+|011><001|，Λ2＝|010><110|+|110><010|，Λ3＝|100><101|+|101><100|；
Shifting thermal equilibrium state by unitary operation from |111 in density matrix>The probability of projection on (-1.5) is partially shifted to |011>，|101>And |110>State, the unitary operation used is U2＝exp(-i*α*(Λ4+Λ5+Λ6) Here) of
Λ4、Λ5、Λ6Are respectively |111>And 011>、|111>And |101>、|111>And |110>Transition operator between states: lambda4＝|111><011|+|011><111|，Λ5＝|111><101|+|101><111|，Λ6＝|111><110|+|110><111|；
Through a unitary operation, p is divided4,4、ρ6,6、ρ7,7Half of the projection probability shifts to ρ2,2、ρ3,3、ρ5,5(ii) a The unitary operation used is U3＝exp(-i*β*(Λ1+Λ2+Λ3) Here) of
Exchanging basis vectors, and exchanging the basis vectors in the spaces |001>, |010>, |100> with the basis vectors in other subspaces to change the zero-order coherent item in the space into a non-zero-order coherent item;
a gradient field is applied.
The base vector exchanging is performed to exchange the base vectors in the |001>, |010>, |100> space with the base vectors in other subspaces, so that the step of changing the zero-order coherent item in the space into the non-zero-order coherent item includes:
using unitary operation U4Will |001>And 011>Exchange, |000>And |010>To change, will |001>And 011>By conversion so that the zero order coherent term ρ2,3,ρ2,5With a first order coherence term ρ4,3,ρ4,5Performing a swap, will |000>And |010>Interchanging, further converting rho2,3、ρ3,5And p of zero2,1、ρ1,5Commute such that the twelve zeroth order coherence terms are zero or approximately zero.
Wherein, after application of the gradient field, diagonal elements of the quantum state shift density matrix are preserved, ρ1,1、ρ2,2、ρ4,4、ρ5,5、ρ6,6、ρ7,7、ρ8,8Are all about-1.5/7, rho3,3Is 1.5; non-diagonal elementIs eliminated, which corresponds to |010>In a pseudo-pure state.
Wherein the method further comprises:
rewriting density matrix into pseudo-pure density matrix
ρpps＝(ρ+I*1.5/7)/(1.5+1.5/7)；
To obtain a pseudo-pure state for the |000> state, a 180 ° rotation pulse of the second kernel is applied.
Wherein the method further comprises:
the theoretical fidelity of the pseudo-pure state is calculated by the formula
Wherein the multi-bit is three bits.
The invention also provides a nuclear magnetic resonance homonuclear multi-bit pseudomorphic preparation system which comprises a memory, a processor and a nuclear magnetic resonance homonuclear multi-bit pseudomorphic preparation program stored on the memory, wherein the nuclear magnetic resonance homonuclear multi-bit pseudomorphic preparation program is executed by the steps of the method when being executed by the processor.
According to the nuclear magnetic resonance homonuclear multi-bit pseudomorphic pure state preparation method and system, the generation of the zero-order coherent term is reduced as much as possible in the scheme, half of the zero-order coherent term can be weakened, and the larger zero-order coherent term is converted into the nonzero-order coherent term before the gradient field. Thus, to a good approximation, the gradient field removes all coherent terms of the nuclear magnetic quantum state except for the pseudo-pure state. Moreover, in the same-core three-bit system, only one gradient field is needed to prepare a pseudo-pure state with high theoretical fidelity.
Drawings
FIG. 1 is a schematic diagram of a nuclear magnetic resonance homonuclear multi-bit pseudomorphic preparation circuit according to the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is described in further detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.
The invention provides a nuclear magnetic resonance homonuclear multi-bit pseudomorphic preparation method, which comprises the following steps:
step S1, constructing |001 from thermal equilibrium state by using RF pulse>And 011>010 |>And |110>100, | between>And |101>The first order coherent terms between them and reduce their corresponding diagonal elements to zero, the unitary operation used is
Λ1、Λ2、Λ3Are respectively |001>And 011>、|010>And |110>、|100>And |101>The transition operator of (2): lambda1＝|001><011|+|011><001|，Λ2＝|010><110|+|110><010|，Λ3＝|100><101|+|101><100|；
Step S2, use unitary operation will |111>The probability of projection on (-1.5) is partially shifted to |011>，|101>And |110>State, the unitary operation used is U2＝exp(-i*α*(Λ4+Λ5+Λ6) Here) of
Λ4、Λ5、Λ6Are respectively |111>And 011>、|111>And |101>、|111>And |110>Transition operator between states: lambda4＝|111><011|+|011><111|，Λ5＝|111><101|+|101><111|，Λ6＝|111><110|+|110><111|；
Step S3, through unitary operation, dividing rho4,4、ρ6,6、ρ7,7Half of the projection probability shifts to ρ2,2、ρ3,3、ρ5,5(ii) a The unitary operation used is U3＝exp(-i*β*(Λ1+Λ2+Λ3) Here) of
Step S4, exchanging basis vectors, exchanging the basis vectors in the space |001>, |010>, |100> with the basis vectors in other subspaces, and changing the zero order coherent item in the space into a non-zero order coherent item;
step S5, a gradient field is applied.
The base vector exchanging is performed to exchange the base vectors in the |001>, |010>, |100> space with the base vectors in other subspaces, so that the step of changing the zero-order coherent item in the space into the non-zero-order coherent item includes:
using unitary operation U4Will |001>And 011>Exchange, |000>And |010>To change, will |001>And 011>By conversion so that the zero order coherent term ρ2,3,ρ2,5With a first order coherence term ρ4,3,ρ4,5Performing a swap, will |000>And |010>Interchanging, further converting rho2,3、ρ3,5And p of zero2,1、ρ1,5Commute such that the twelve zeroth order coherence terms are zero or approximately zero.
Wherein, after application of the gradient field, diagonal elements of the quantum state shift density matrix are preserved, ρ1,1、ρ2,2、ρ4,4、ρ5,5、ρ6,6、ρ7,7、ρ8,8Are all about-1.5/7, rho3,3Is 1.5; the non-zero order coherent term of the non-diagonal elements is eliminated, which corresponds to |010>In a pseudo-pure state.
Wherein the method further comprises:
rewriting density matrix into pseudo-pure density matrix
ρpps＝(ρ+I*1.5/7)/(1.5+1.5/7)；
To obtain a pseudo-pure state for the |000> state, a 180 ° rotation pulse of the second kernel is applied.
Wherein the method further comprises:
the theoretical fidelity of the pseudo-pure state is calculated by the formula
Wherein the multi-bit is three bits.
The invention also provides a nuclear magnetic resonance homonuclear multi-bit pseudomorphic preparation system which comprises a memory, a processor and a nuclear magnetic resonance homonuclear multi-bit pseudomorphic preparation program stored on the memory, wherein the nuclear magnetic resonance homonuclear multi-bit pseudomorphic preparation program is executed by the steps of the method when being executed by the processor.
According to the nuclear magnetic resonance homonuclear multi-bit pseudomorphic pure state preparation method and system, the generation of the zero-order coherent term is reduced as much as possible in the scheme, half of the zero-order coherent term can be weakened, and the larger zero-order coherent term is converted into the nonzero-order coherent term before the gradient field. Thus, to a good approximation, the gradient field removes all coherent terms of the nuclear magnetic quantum state except for the pseudo-pure state. Moreover, in the same-core three-bit system, only one gradient field is needed to prepare a pseudo-pure state with high theoretical fidelity.
The scheme of the invention is explained in detail below:
the present invention contemplates: the three-bit Hilbert space may be defined by |000>，|001>，|010>，|011>，|100>，|101>，|110>，|111>A total of eight basis vectors are spread. The 0 represents a state in which the nuclear spin is parallel to the static magnetic field, and the 1 represents a quantum state in which the nuclear spin is antiparallel to the static magnetic field. The thermal equilibrium density of states matrix can be expressed as,
here, I is an identity matrix of 8 by 8 (only diagonal elements are non-zero and all are 1), Z1、Z2、Z3Is an angular momentum operator in the z direction (static magnetic field direction) of three spins, and has an epsilon of 10-5Small amount of magnitude. In a nuclear magnetic resonance system, it is ε (Z)1+Z2+Z3) This portion produces a nuclear magnetic resonance signal.
ε(Z1+Z2+Z3) The non-diagonal elements are zero and the diagonal elements are epsilon (1.5,0.5,0.5, -0.5,0.5, -0.5, -1.5), corresponding to |000 respectively>，|001>，|010>，|011>，|100>，|101>，|110>，|111>Projection probability of eight basis states. Here, the probability of projection has a negative value because this is not the absolute value of the probability of projection, but rather the probability of projection relative to 1/23The offset value of (2). Pseudo-pure preparation, it is the Z1+Z2+Z3The (called offset density matrix) is prepared in a pseudo-pure form, i.e. the non-diagonal elements remain zero, the diagonal elements become seven equal, one equal to the other seven and larger than the other seven.
Pseudo-pure state preparation means generally comprise radio frequency pulses, gradient fields, free relaxation and the like. The radio frequency pulse can realize unitary operation on the quantum state, and the gradient field and the free relaxation can realize non-unitary operation. Since pseudo-singlet fabrication is a non-unitary process, gradient fields (or the equivalent operation of gradient fields-phase cycling) or free relaxation must be used. The non-unitary operation used by the present invention is a pulsed gradient field. Firstly, using radio frequency pulse to make unitary operation on second to eighth basis vectors of thermal equilibrium state to obtain Z1+Z2+Z3The second to eighth terms of the diagonal become equal and-1.5/7, the first term of the diagonal remains 1.5 constant, and then the non-diagonal generated in the unitary operation is made zero in combination with the basis vector transformation and the gradient field, as shown in fig. 1. This method allows to achieve the highest production efficiency that can be achieved without the aid of relaxation effects.
In the unitary operation design process, the generation of the zeroth order coherent term is considered to be reduced as much as possible. The zeroth order coherence terms are some special terms in the non-diagonal elements in the density matrix and cannot be eliminated by the gradient field. There are twelve zero-order coherence terms in the three-bit density matrix. If the density matrix is denoted as p, then the zeroth order coherence term is p2,3、ρ2,5、ρ3,5、ρ4,6、ρ4,7、ρ6,7And their complex conjugates ρ3,2、ρ5,2、ρ5,3、ρ6,4、ρ7,4、ρ7,6. Eight basic vectors of three bits can be divided into four groups, and the eight basic vectors respectively correspond to four different z-direction angular momentums, so that the eight basic vectors can be regarded as four subspaces. |000>And |111>Each group has 3/2 and-3/2 corresponding angular momentum quanta. |001>，|010>And |100>For one group, the corresponding angular momentum quantum number is 1/2. L 011>，|101>And |110>For one group, the corresponding angular momentum quantum number is-1/2.
ρ2,3,ρ2,5,ρ3,5And their conjugate terms are |001>，|010>，|100>Two-by-two coherent terms of these three terms, i.e. |001 in the density matrix>、|010>、|100>The corresponding cross terms of the row and column.
ρ4,6,ρ4,7,ρ6,7And their conjugate terms are |011>，|101>，|110>Two-by-two coherent terms of the three terms, i.e. |011 in the density matrix>、|101>、|110>The corresponding cross terms of the row and column. The invention does not attenuate all twelve coherence terms, but only half of them, i.e. p4,6,ρ4,7,ρ6,7And its conjugate term (or will be ρ)2,3,ρ2,5,ρ3,5And its conjugate term). The method comprises the following steps:
(1) first, |001 was observed>，|010>，|100>Corresponding diagonal element ρ2,2＝ρ3,3＝ρ5,50.5, and |011>，|101>，|110>Corresponding diagonal element ρ4,4＝ρ6,6＝ρ7,7The opposite is true for-0.5. The |001 can be constructed using radio frequency pulses>And 011>010 |>And |110>100, | between>And |101>The first order coherent terms between them and reducing their corresponding diagonal elements to zero, the effect is achieved: rho2,4＝ρ3,7＝ρ5,6＝0.5i，ρ2,2＝ρ3,3＝ρ5,5＝ρ4,4＝ρ6,6＝ρ7,70. The unitary operation used is
Λ1、Λ2、Λ3Are respectively |001>And 011>、|010>And |110>、|100>And |101>The transition operator of (2): lambda1＝|001><011|+|011><001|，Λ2＝|010><110|+|110><010|，Λ3＝|100><101|+|101><100|。
(2) Will |111 with unitary operation>The probability of projection on (-1.5) is partially shifted to |011>，|101>And |110>And (4) state. The unitary operation used is U2＝exp(-i*α*(Λ4+Λ5+Λ6) Here) of
Λ4、Λ5、Λ6Are respectively |111>And 011>、|111>And |101>、|111>And |110>Transition operator between states: lambda4＝|111><011|+|011><111|，Λ5＝|111><101|+|101><111|，Λ6＝|111><110|+|110><111 |. After this unitary operation, |011>，|101>And |110>Is the projection probability of4,4＝ρ6,6＝ρ7,7＝-3/7，|111>Is the projection probability of8,8-1.5/7. The value of-1.5/7 is all non |000 in the final pseudo-pure state>Projection probability of state, therefore, in U2Then ρ8,8Its target value is reached.
(3) At this step, p will be converted into p by unitary operation4,4、ρ6,6、ρ7,7Half of the projection probability shifts to ρ2,2、ρ3,3、ρ5,5. The unitary operation used is U3＝exp(-i*β*(Λ1+Λ2+Λ3) Here) of
After this operation, ρ2,2、ρ3,3、ρ5,5、ρ4,4、ρ6,6、ρ7,7、ρ8,8All are about-1.5/7, and the target values are reached.
(4) In the state prepared in the last step, the zeroth order coherence terms are respectively:
ρ2,3＝ρ3,2＝ρ2,5＝ρ5,2＝ρ3,5＝ρ5,3＝-0.4521，
ρ4,6＝ρ6,4＝ρ4,7＝ρ7,4＝ρ6,7＝ρ7,60.0235. Therefore, of the twelve zeroth order coherence terms, there are six (ρ)4,6,ρ4,7,ρ6,7And its conjugate term) is small, approximately 1/10, which corresponds to the value of the diagonal element, and can be considered approximately zero. Six others (p)2,3,ρ2,5,ρ3,5And its conjugate terms) are large and non-negligible. However, if the gradient field is applied directly on this state, the six zeroth order coherence terms cannot be eliminated. Therefore, another unitary operation is needed before the gradient field to change the six zero-order coherent terms into non-zero-order coherent terms by performing basis vector transposition to convert |001>，|010>，|100>And transposing the basis vector in the space with the basis vectors in other subspaces to change the zero-order coherent term in the space into a non-zero-order coherent term. It should be noted that when transforming the basis vectors, since some non-zero order coherent terms will be shifted to zero order coherent terms in the space, the shift of larger non-zero order coherent terms to the space is avoided. The method adopted by the invention is to use the unitary operation U4Will |001>And 011>Exchange, |000>And |010>And (4) exchanging. Will |001>And 011>By swapping, the zero order coherent term ρ is generated2,3,ρ2,5With a first order coherence term ρ4,3,ρ4,5Carry out the conversion to the original rho4,3,ρ4,5Are all rho4,3＝ρ4,50.0107i, small and negligible, so after swapping, a new ρ2,3And ρ2,5Negligible, but larger, new ρ4,3And rho4,5Is a first order coherence term that will be eliminated by the gradient field that is used next. Will be |000>And |010>Alternatively, one can be furtherStep will rho2,3、ρ3,5And p of zero2,1、ρ1,5And (4) exchanging. So after this step, ρ2,3＝0,ρ2,5＝0.0107i,ρ3,50. Thus, the twelve zeroth order coherence terms are zero or approximately zero.
(5) A gradient field is applied, noted Gz. The diagonal elements of the quantum state shift density matrix are preserved, ρ1,1、ρ2,2、ρ4,4、ρ5,5、ρ6,6、ρ7,7、ρ8,8Are all about-1.5/7, rho3,3Is 1.5. The non-zero order coherent terms of the non-diagonal elements are eliminated. This is a response to |010>The density matrix of the pseudo-pure state can be rewritten into a pseudo-pure state density matrix rhopps(ρ + I1.5/7)/(1.5 + 1.5/7). If desired, |000>The state is pseudo-pure, then a 180 ° rotation pulse of the second kernel may be applied.
The theoretical fidelity of the pseudo-pure state is about 0.9997, and the calculation method is
The closer the fidelity is to 1, the greater the similarity between the two states. Where ρ isthIs a pure density matrix with only one diagonal element being 1 and all other elements being zero, e.g., if this is a |000>Pure state of the state, then ρthIs 1 provided that this is a |010>In the pure state of (1), then ρthThe third diagonal element of (1).
Compared with the prior art, the method uses the radio frequency pulse to convert the density matrix of the nuclear magnetic resonance three-bit homonuclear system from the thermal equilibrium state into the density matrix of a diagonal element which has the same characteristics with the diagonal element of the pseudo-pure state and has a negligible zeroth-order coherent term, so that the pseudo-pure state can be prepared by using a gradient field once, and the preparation efficiency of the pseudo-pure state reaches the highest value which can be achieved without the aid of relaxation effect. This is a key difference between the present invention and other technologies.
It should be noted that the steps (1) to (3) in the above description can be generalized to multiple bits (more than three bits). That is to say, a density matrix diagonal element can be prepared by the method, the characteristic of the pseudo pure state density matrix diagonal element is satisfied, and a multi-bit nuclear magnetic resonance homonuclear quantum state with half of zeroth order coherent terms smaller is obtained. It has now been demonstrated that:
assuming N homonuclear bits, the Hilbert space of the system is 2NDimension, which can be divided into N +1 subspaces. We aim to preserve a subspace with an angular momentum of N/2 (|00.. 0)>) The projection probability N/2 of (2) is unchanged, and the basis vectors of the remaining subspace (total 2)NThe projection probabilities at-1) all become-N/2/(2)N-1) such that the diagonal elements of the density matrix are the same as the diagonal elements of the density matrix in the pseudo-pure state. Consider a subspace with an angular momentum of-N/2 (|11.. 1)>) Has a projection probability of-N/2, i.e.. 0>The projection probability summation of all the outer subspace basis vectors can gradually transfer the projection probability of the outer subspace basis vectors to other subspaces. Furthermore, because all the subspaces are in pairs, the angular momentum values of each pair are opposite, and the projection probabilities are also opposite, and because of the symmetry, each pair of subspaces can be regarded as a group, and the projection probabilities of the subspaces of-N/2 are transferred to other subspaces group by group. According to this idea, we consider a subspace (|11.. 1) that involves an angular momentum of-N/2>) Unitary operation with two subspaces of angular momentum-m/2 and m/2. The goal is to get |11.. 1>Transfer a part to the basis vectors in both subspace of-m/2 and m/2, so that the projection probabilities on the basis states in both subspaces are equal and equal to-N/2/(2)N-1); and the zeroth order coherence term of one of the two subspaces-m/2 and m/2 is small.
The first step is to make the projection probability of basis vectors in both-m/2 and m/2 subspaces zero by using unitary operation. And in the thermal equilibrium state, the projection probabilities corresponding to the basis vectors in the two subspaces of-m/2 and m/2 are respectively-m/2 and m/2. Assuming that there are n basis vectors in the-m/2 subspace, there are n basis vectors in the m/2 subspace. May utilize a unitary operation
To achieve this, here, Λ1，Λ2，…，ΛnIs connected in two subspacesn basis vectors. To write out the density matrix, we assume that the basis vectors in the two subspaces-m/2 and m/2 are alternately arranged, | a1>,|b1>,|a2>,|b2>,...,|an>,|bn>，|11...1>Ranking them last. Here, | a1>,|a2>,...,|an>Represents n basis vectors, | b, in m/2 subspace1>,|b2>,...,|bn>Representing n basis vectors in-m/2 subspace. Passing through U1After operation, the density matrix is in the form of
This is a block diagonalized matrix where a is m/2 and B is the probability of projection on |11.. 1 >.
The second step is to combine |11.. 1>Is transmitted to m/2 subspace, so that the projection probability of all basis vectors in the m/2 subspace is-N/(2)N-1). The unitary operation used is U2＝exp(-i*α*(Λn+1+Λn+2+...+Λ2n) Here Λ /)n+1、Λn+2、…、Λ2nIs |11.. 1>Transition operators of states to all n basis vectors in the m/2 subspace. U shape2In the form of a matrix
U2Jth row and kth column matrix element: if j and k belong to the-m/2 and m/2 subspaces, the element is
If one of j and k corresponds to |11.. 1>State, the other is m/2 subspace, then the matrix element is
If one of j and k corresponds to |11.. 1>State of anotherOne belongs to the subspace of-m/2, and the matrix element is 0; if j is k and belongs to m/2 subspace, the matrix element is
If j is k and belongs to-m/2 subspace, then the element is 1; if j ═ k and corresponds to |11.. 1>State of which the matrix element is
Passing through U2In operation, the diagonal elements of the density matrix are (-BE)2,0,-BE2,0,…,BF2). By selecting an appropriate angle alpha, the-BE can BE made2＝-N/(2N-1)。
Thirdly, the projection probability is transferred from m/2 subspace to half to m/2 subspace, and the used unitary operation is U3＝exp(-i*β*(Λ1+Λ2+...+Λn)). In the form of a matrix of
The diagonal elements of the density matrix after this operation are
(-BE2G2-2ACHGi,BE2H2+2ACGHi,-BE2G2-2ACHGi,BE2H2+2ACGHi,…,BF2)，
The zero order coherent terms of the m/2 subspaces are all-BE2G2-2ADHGi, the zeroth order coherence terms of the-m/2 subspaces are all BE2H2+2 ADGHi. The purpose of the method is to enable the zeroth order coherent item of one subspace to be small, and the zeroth order coherent item of a subspace of m/2 is not taken as an investigation object. We can next demonstrate that the absolute value of the ratio of these zeroth order coherence terms to the diagonal is less than an upper bound ξ. First, we find this ratio
In the above formula, the denominator is
This is because, the projection probability of the basis vectors in the last m/2 subspace and the-m/2 subspace needs to be the same,
the method is easy to verify the operation of the device,
then
Here, will
Note for Δ. We want | R | < 1, then
Since A > 0 and Δ < 0, sin2 β < 0 is required. sin2 beta < 0 is achievable because the value of beta is completely dependent on
This requirement, from which can be derived
C is not less than zero under the condition that n is more than or equal to 2, so that tan2 beta is less than 0, and 2 beta can be taken as a quadrant angle to meet the requirement that sin2 beta is less than 0. From
Can find out
And then to
To find the range of R, we first look at the range of C.
Its value is determined by α. Due to the fact that in U3In the diagonal elements of the density matrix after the operation,
should be true, it is easy to derive
Namely, it is
Therefore, if
Is positive, then
And C is in the range
If it is not
Is negative, then
And C is in the range
Let us next see
Since A is the subspace angular momentum amplitude and Delta is the diagonal element of the density matrix in the last pseudo-pure state, the range of (A) is easy to obtain
Range of binding C and
in the range of
Time of flight
In the above formula, it is assumed that the minimum values of N and N are both 3, and N is 3, which means that the minimum homonuclear system we discuss is three homonuclei, and N is 3, which means that the subspace considered has at least 3 basis vectors. Therefore, R is less than 0.2616 and is more than or equal to 0. When in use
When the temperature of the water is higher than the set temperature,
so that in this case it is possible to,
when the comparison of n is small, the method can be used,
is greater than 0.2616. In conclusion, we prove that the value of alpha is reasonably selected, so that
The absolute value of the ratio of the zeroth order coherent term to the pseudo-pure diagonal of the-m/2 subspace may be made smaller than an upper bound ξ -0.2616.
According to the above steps, the angular momentum is a subspace of-N/2 (|11.. 1)>) The projection probability of (a) can be gradually shifted to |00.. 0>All but the subspace basis vectors, and finally except |00.. 0>Besides the basis vectors, the projection probability of all the basis vectors is N/2/(2)N-1)，Meets the requirement of pseudo pure state on diagonal elements of the density matrix, and half of the absolute values of zeroth order coherent terms are less than N/2/(2)N-1)*0.2616。
FIG. 1 is a circuit diagram of the present invention. RhoeqRefers to the thermal equilibrium state of the same-core three-bit system. RhoppsIndicating the pseudo-pure state prepared. The three horizontal lines represent three bits. U shape1、U2、U3、U4Refers to four unitary operations implemented in sequence using radio frequency pulses. Gz refers to the gradient field applied after four unitary operations.
The above description is only for the preferred embodiment of the present invention, but the scope of the present invention is not limited thereto, and any changes or substitutions that can be easily conceived by those skilled in the art within the technical scope of the present invention are included in the scope of the present invention. Therefore, the protection scope of the present invention shall be subject to the protection scope of the claims.
Claims (7)
1. A nuclear magnetic resonance homonuclear multi-bit pseudo-pure state preparation method is characterized by comprising the following steps:
construction of |001 from a thermally balanced state using radio frequency pulses>And 011>010 |>And |110>100, | between>And |101>The first order coherent terms between them and reduce their corresponding diagonal elements to zero, the unitary operation used is
Λ1、Λ2、Λ3Are respectively |001>And 011>、|010>And |110>、|100>And |101>The transition operator of (2):
Λ1＝|001><011|+|011><001|，Λ2＝|010><110|+|110><010|，
Λ3＝|100><101|+|101><100|；
shifting thermal equilibrium state by unitary operation from |111 in density matrix>The probability of projection on (-1.5) is partially shifted to |011>，|101>And |110>State, unitary operation usedIs U2＝exp(-i*α*(Λ4+Λ5+Λ6) Here) of
Λ4、Λ5、Λ6Are respectively |111>And 011 and 111>And |101>、|111>And |110>Transition operator between states:
Λ4＝|111><011|+|011><111|，Λ5＝|111><101|+|101><111|，
Λ6＝|111><110|+|110><111|；
through a unitary operation, p is divided4,4、ρ6,6、ρ7,7Half of the projection probability shifts to ρ2,2、ρ3,3、ρ5,5(ii) a The unitary operation used is U3＝exp(-i*β*(Λ1+Λ2+Λ3) Here) of
Exchanging basis vectors, and exchanging the basis vectors in the spaces |001>, |010>, |100> with the basis vectors in other subspaces to change the zero-order coherent item in the space into a non-zero-order coherent item;
and applying a gradient field to prepare a pseudo-pure state in a homonuclear system.
2. The method for preparing the homonuclear multi-bit pseudomorphic homonuclear magnetic resonance according to claim 1, wherein the step of performing basis vector exchange to exchange basis vectors in |001>, |010>, |100> spaces with basis vectors in other subspaces so that the zero-order coherent term in the space becomes a nonzero-order coherent term comprises:
using unitary operation U4Will |001>And 011>Exchange, |000>And |010>And (3) interchanging |001> and |011 > to make the zero-order coherent term rho2,3,ρ2,5With a first order coherence term ρ4,3,ρ4,5Performing a swap, swap |000> and |010>Further dividing p2,3、ρ3,5And p of zero2,1、ρ1,5Commute such that the twelve zeroth order coherence terms are zero or approximately zero.
3. The method of claim 1, wherein diagonal elements of the quantum state shift density matrix are preserved, p, after application of the gradient field1,1、ρ2,2、ρ4,4、ρ5,5、ρ6,6、ρ7,7、ρ8,8Are all about-1.5/7, rho3,3Is 1.5; the non-zero order coherent terms of the non-diagonal elements are eliminated, which is a pseudo-pure state corresponding to |010 >.
4. The method of claim 3, further comprising:
rewriting density matrix into pseudo-pure density matrix
ρpps＝(ρ+I*1.5/7)/(1.5+1.5/7)；
To obtain a pseudo-pure state for the state |000>, a 180 ° rotation pulse of the second kernel is applied.
5. The method of claim 4, further comprising:
the theoretical fidelity of the pseudo-pure state is calculated by the formula
Where ρ isthThe matrix is a pure density matrix, only one diagonal element is 1, and all other matrix elements are zero.
6. The method of claim 4, wherein the plurality of bits is three bits.
7. A nmr homonuclear multi-bit pseudomorphic preparation system, comprising a memory, a processor, and a nmr homonuclear multi-bit pseudomorphic preparation program stored on the memory, wherein the nmr homonuclear multi-bit pseudomorphic preparation program, when executed by the processor, performs the steps of the method of any of claims 1-6.
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