A quantum communication method based on HHL algorithm

Abstract

The application discloses a quantum communication method based on an HHL algorithm, and relates to the technical field of quantum communication.Teams of communication want to transmit the classical information held by each other, and the HHL algorithm is combined to solve quantum linear equations, so that the information of the teams of communication cannot be stolen by a third party, and the HHL quantum circuit constructed by the scheme is combined with a SWAP-TEST gate, a solving result is calculated by calculating fidelity, and instead of directly measuring the result, the solving speed can be improved.The HHL algorithm and the SWAP-TEST circuit are combined, an improved quantum dialogue scheme is realized, the scheme has the characteristics of high safety and high transmission efficiency, the application can resist measurement retransmission, interception retransmission and attacks in extreme cases, and has high transmission efficiency.

Description

Technical Field

[0001] This invention relates to the field of quantum communication technology, and more specifically, to a quantum communication method based on the HHL algorithm. Background Technology

[0002] Quantum communication is a major research focus. Since Bennett and Brassard proposed the quantum teleportation (QT) protocol in 1993, research on quantum communication has attracted widespread attention. Subsequently, many sub-fields of quantum communication research have emerged based on these studies.

[0003] Quantum-Secure Direct Communication (QSDC) is a branch of quantum cryptography that aims to securely transmit secret information from sender to receiver without prior sharing of encryption keys. Early QSDC protocols only allowed for one-way information transmission, not two-way communication. The concept of Quantum Dialogue (QD) was introduced in 2004, solving this problem. Since then, QD has become a branch of quantum cryptography, and various QD protocols have been proposed. Lu et al., in their paper "Quantum Dialogue Protocol Based on Bell Entangled States and Single Photons," proposed a QD protocol with a transmission efficiency of 66.67%. This protocol uses a decoy particle scheme to ensure security. However, this scheme is only secure when the eavesdropper can only eavesdrop on the quantum channel. If the eavesdropper can simultaneously eavesdrop on both the quantum and classical channels, the measurement results of the particles sent by the communicator in the classical channel will be exposed to the eavesdropper, who can then use this information for forgery and theft. Therefore, there is a risk of information leakage.

[0004] The HHL (Harrow-Hassidim-Lloyd) algorithm, proposed by Harrow et al. in 2009, is an algorithm for solving linear equations, given a matrix A and a vector... Find the vector Make The advantage of the HHL algorithm is that after solving it, we do not obtain the solution. It is itself, but with This approximates the expectation value of a certain related operator, representing an exponential improvement over classical algorithms. Current methods for solving linear equations do not use quantum computers, but rather classical computers employ mathematical formulas. For example, to solve the equation Ax = b, where x and b are matrices and do not need to be encoded as quantum states, the formula for solving x is:

[0005] x = A-1 b

[0006] Where A -1 Let A represent the inverse matrix. The key to solving this problem lies in finding A from A. -1 When the dimensions of A and b are relatively low, there is no problem in solving for x. However, as the dimensions increase, the time cost of solving the problem will increase exponentially. Summary of the Invention

[0007] The present invention provides a quantum communication method based on the HHL algorithm, which can alleviate the above-mentioned problems.

[0008] To alleviate the above problems, the technical solution adopted by the present invention is as follows:

[0009] This invention provides a quantum communication method based on the HHL algorithm, comprising the following steps:

[0010] S1, the first communicating party and the second communicating party respectively prepare classical bit information S A S B ;

[0011] S2. The first communication party prepares the Hermitian matrix A and the quantum sequence |b>, |x>, satisfying A|x>=|b>;

[0012] S3. The first communicator sends the Hermitian matrix A to the second communicator through a classical channel, and adds the quantum sequence |b> to the decoy particle before sending it to the second communicator through a quantum channel;

[0013] S4. The second communicating party exchanges information about the decoy particles with the first communicating party through a classical channel and performs eavesdropping detection. If the detection is successful, an HHL quantum circuit is constructed based on the Hermitian matrix A and the quantum sequence |b>. Based on the number of particles in the quantum sequence |b>, possible result data are enumerated. These possible results are determined by |b>. If |b> consists of 2 particles, the enumerated result set should be {|00>, |01>, |10>, |11>}. Similarly, if |b> consists of 3 particles, the enumerated result set should be {|000>, |001>, |010>, |011>, |100>, |101>, |110>, |111>}, and so on.

[0014] S5. The second communication party constructs the corresponding SWAP-TEST line based on the result data, and combines the SWAP-TEST line with the HHL quantum line to obtain the fidelity of the result data. Each set of HHL quantum lines corresponds to a fidelity.

[0015] S6. The second communication party compares each HHL quantum circuit and selects the result data with the highest fidelity as the quantum sequence |y>, and |x>=|y>;

[0016] S7, The first communicating party will send the classic bit information S A Encoding is done into a quantum sequence |x>, and after adding a decoy particle, a first encoded sequence is obtained. This first encoded sequence is then transmitted to a second communication party via a quantum channel. The second communication party then transmits the classical bit information S. B The sequence is encoded into a quantum sequence |y>, and after adding a decoy particle, a second encoded sequence is obtained. The second encoded sequence is then transmitted to the first communicator through a quantum channel.

[0017] S8. The first communicating party compares the second encoded sequence with the quantum sequence |x> to obtain the first unitary transformation operation, thus obtaining the classical bit information S of the second communicating party. B The second communicating party compares the first encoded sequence with the quantum sequence |y> to obtain the second unitary transformation operation, thus obtaining the classical bit information S of the first communicating party. A .

[0018] In a preferred embodiment of the present invention, in step S1,

[0019] S A ={(a1,b1),(a2,b2),…,(a i b i ), ..., (a N b N )},

[0020] S B ={(m1,n1),(m2,n2),…,(m i n i ), ..., (m N n N )},

[0021] Among them, a i b i m i n i ∈{0, 1}, i∈{1, 2,…,N}.

[0022] In a preferred embodiment of the present invention, in step S7, it is assumed that |x>=|y> means the following:

[0023] |x>=|y>={|φ>1, |φ>2,…, |φ> N},

[0024] Then the first communicating party will send the classic bit information S A Encoding into a quantum sequence |x>, represented as:

[0025] |x>′={C a1b1 |φ>1,C a2b2 |φ>2,…,C aNbN |φ> N},

[0026] The second communicating party will send the classic bit information S B Encoding into the quantum sequence |y>, represented as:

[0027] |y>′={C m1n1 |φ>1,C m2n2 |φ>2,…,C mNnN |φ> N}

[0028] In a preferred embodiment of the present invention, in step S2, the first communicating party first prepares a vector. Then, for each vector Encode them to obtain quantum sequences |b> and |x> respectively.

[0029] In a preferred embodiment of the present invention, in step S4, if the detection fails, the communication between the first communication party and the second communication party is interrupted.

[0030] In a preferred embodiment of the present invention, in step S8, both the first and second communicating parties need to complete the eavesdropping detection through the classical channel, respectively eliminating the decoy particles of the second coding sequence and the decoy particles of the first coding sequence.

[0031] In a preferred embodiment of the present invention, the first communicating party needs to inform the second communicating party through a classical channel which particles in the first coding sequence have undergone H-gate transformation, and the second communicating party needs to inform the first communicating party through a classical channel which particles in the second coding sequence have undergone H-gate transformation.

[0032] In a preferred embodiment of the present invention

[0033] The first unitary transformation operation includes:

[0034] The first communicator measures the second encoded sequence, performs single-particle measurements on the X (|+>, |->) basis on the particles that have undergone H-gate transformation, and performs single-particle measurements on the Z (|0>, |1>) basis on the remaining particles to obtain the first measurement result, and compares the first measurement result with the quantum sequence |x>.

[0035] The second unitary transformation operation includes:

[0036] The second communicator measures the first encoded sequence, performs single-particle measurements on the X (|+>, |->) basis on the particles that have undergone H-gate transformation, and performs single-particle measurements on the Z (|0>, |1>) basis on the remaining particles to obtain the second measurement result, and compares the second measurement result with the quantum sequence |x>.

[0037] Compared with the prior art, the beneficial effects of the present invention are:

[0038] This invention achieves an improved quantum dialogue scheme by combining the HHL algorithm and the SWAP-TEST circuit, which features high security and high transmission efficiency.

[0039] Security analysis has demonstrated that this invention can resist attacks such as measurement retransmission, interception of retransmission, and attacks under extreme conditions.

[0040] The transmission efficiency calculation and analysis results from the specific implementation method show that the transmission efficiency of the present invention is very high.

[0041] To make the above-mentioned objects, features and advantages of the present invention more apparent and understandable, embodiments of the present invention are described below in detail with reference to the accompanying drawings. Attached Figure Description

[0042] To more clearly illustrate the technical solutions of the embodiments of the present invention, the accompanying drawings used in the embodiments will be briefly introduced below. It should be understood that the following drawings only show some embodiments of the present invention and should not be regarded as a limitation on the scope. For those skilled in the art, other related drawings can be obtained based on these drawings without creative effort.

[0043] Figure 1 This is a flowchart of the quantum dialogue protocol combining the HHL algorithm provided by the present invention;

[0044] Figure 2 This is a flowchart of the HHL algorithm provided by the present invention;

[0045] Figure 3 This is the quantum circuit diagram provided by the present invention;

[0046] Figure 4 This is the complete quantum circuit diagram constructed based on the |00> state in this invention;

[0047] Figure 5 This invention is based on Figure 4 The resulting simulated numerical plot;

[0048] Figure 6 This is the complete quantum circuit diagram constructed based on the |01> state in this invention;

[0049] Figure 7This invention is based on Figure 6 The resulting simulated numerical plot;

[0050] Figure 8 This is the complete quantum circuit diagram constructed based on the |10> state in this invention;

[0051] Figure 9 This invention is based on Figure 8 The resulting simulated numerical plot;

[0052] Figure 10 This is the complete quantum circuit diagram constructed based on the |11> state in this invention;

[0053] Figure 11 This invention is based on Figure 10 The resulting simulated numerical plot. Detailed Implementation

[0054] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are some embodiments of the present invention, but not all embodiments.

[0055] Please refer to Figure 1 This invention provides a quantum communication method based on the HHL algorithm.

[0056] In this invention, there are two communicating parties: the first communicating party is Alice, and the second communicating party is Bob.

[0057] This invention uses the HHL algorithm to allow Alice and Bob to obtain the same value for |x>. Then, Alice and Bob can use their respective |x> to conduct a quantum dialogue and exchange their classical information.

[0058] The HHL algorithm consists of three parts: quantum phase evaluation, controlled rotation, and inverse quantum phase evaluation. The HHL algorithm flow is as follows: Figure 2 As shown, where |0> n This indicates initializing n |0> particles. H represents the H-gate transformation, a type of fundamental quantum transformation operation. U represents performing a U-gate transformation using particles in quantum register A as control bits and particles in quantum register B as target bits. The construction of the U-gate depends on the A matrix transmitted by Alice. FT + This represents the inverse quantum Fourier transform. R indicates that the auxiliary particle is rotated using the particle in quantum register A as the control bit. The final inverse quantum phase evaluation stage is the inverse transformation of the quantum phase evaluation. Through the inverse transformation, quantum register A is restored to its initial state, and a single-particle measurement is performed on the auxiliary particle. When the measurement result is |1>, the result in quantum register B is the |x> we are looking for.

[0059] In this invention, Alice prepares her own classical bit information S. A Bob prepares his own classical bit information S B Assume S A S B They are respectively:

[0060] S A =1001

[0061] S B =0110

[0062] Alice and Bob negotiate the unitary transformation operation between the two sets of codes.

[0063] The first group of encoding transformations is represented as follows:

[0064] C 00 =I=|0><0|+|1><1|

[0065] C 01 =σ x =|0><1|+|1><0|

[0066] The effect on the particles is as follows:

[0067] C 00 |0>=|0>;C 00 |1>=|1>

[0068] C 01 |0>=|1>;C 01 |1>=|0>

[0069] Before the second set of encoding transformations, an H-gate transformation needs to be performed on the particles, as follows:

[0070]

[0071]

[0072] The second set of encoding transformations is represented as follows:

[0073] C 10 =|+><+|+|-><-|

[0074] C 11 =|+><-|+|-><+|

[0075] The effect on the particles is as follows:

[0076] C 10 |+>=|+>;C 10 |->=|->

[0077] C 11 |+>=|->;C 11 |->=|+>

[0078] Alice prepares the Hermitian matrix A and the vector... satisfy Assume A is:

[0079]

[0080] for:

[0081]

[0082] for:

[0083]

[0084] Will Encoded as |b>, Encode it as |x> and send it to Bob.

[0085] Bob constructs an HHL quantum circuit, then enumerates possible outcomes, constructs a SWAP-TEST circuit, and combines them into a complete quantum circuit. For example... Figures 3-11 As shown.

[0086] The fidelity is calculated based on the constructed quantum circuit. The fidelity formula is as follows:

[0087] F(|x>,|y>)=p0-p1

[0088] Figure 3 This is the quantum circuit Bob constructed based on A and |b> sent by Alice. Here, q40 and q41 represent the initialization of two particles (quantum register B), and circuit-92 represents the encoding operation on q40 and q41. After encoding, q40 and q41 become |b>. QPE represents the quantum phase evaluation stage, which includes three operations: H-gate transformation, controlled u-gate transformation, and inverse Fourier transform. After the QPE stage, the eigenvalues ​​of matrix A are extracted into q50-q55 (quantum register A). The next step represents a controlled rotation operation on q50-q55, using particles q40 and q41 as control bits to perform a controlled transformation on particles q50-q55. The final step is the quantum inverse phase evaluation stage, which is the inverse transformation of the first stage. Its purpose is to restore q50-q55 to its initial state and obtain the evaluation result, which will be stored in q40 and q41.

[0089] Figures 4-11 All are in Figure 3A complete circuit is formed by adding a swap-test line to the existing circuit. Figure 4 For example, q3590 and q3591 represent quantum register B, q3600-q3605 represent quantum register A, and q361 represents the auxiliary particle. After quantum phase evaluation, controlled rotation, and inverse quantum phase evaluation, the calculated result |x> has been obtained in quantum register B. However, this result is affected by the auxiliary particle; Bob's desired |x> is only obtained when the auxiliary particle's result is |1>. Therefore, a single-particle measurement operation needs to be performed on the auxiliary particle first. Then, the fidelity of the particle state constructed by Bob with |x> is compared using a SWAP-TEST circuit. Figures 4-11 Bob constructed particle states |00>, |01>, |10>, and |11> (these are the enumeration results mentioned earlier) and compared them with |x>. According to the fidelity calculation formula above, |11> has the highest fidelity with |x>. And |11> is precisely the solution to our constructed equation, thus the solution was successful. We denote the solution obtained by Bob as |y>.

[0090] If the encoding is |x>, then |x> is:

[0091] |x>=|11>

[0092] If |y>=|x>, then |y> is:

[0093] |y>=|11>

[0094] Alice and Bob will S A SB encodes the two defined sets of data onto |x> and |y>. This yields |x>′ and |y>′ as follows:

[0095] |x>′=|->|0>

[0096] |y>′=|0>|->

[0097] Alice and Bob add decoy particles to |x>′ and |y>′ and send them to each other through a quantum channel. They then exchange information through a classical channel to complete the eavesdropping detection.

[0098] Alice and Bob communicate the positions of the H-gate transformed particles to each other via a classical channel. Alice tells Bob that the first particle has undergone an H-gate transformation, and Bob tells Alice that the second particle has undergone an H-gate transformation.

[0099] Alice receives |y>′ transmitted by Bob, performs a single-particle measurement on |y>′, performs a Z(|0>,|1>) basis measurement on the first particle, and performs an X(|+>,|->) basis measurement on the second particle.

[0100] Alice obtains |y>′=|0>|->, and the original |x>=|11>, so Alice can deduce that Bob's encoding operation is... Therefore, it can be deduced that Bob's classic information is SB = 0110.

[0101] Bob receives the |x>′ transmitted by Alice, performs a single-particle measurement on |x>′, performs an X(|+>,|->) basis measurement on the first particle, and performs a Z(|0>,|1>) basis measurement on the second particle.

[0102] Bob gets |x>′=|->|0>, and the original |y>=|11>, so Bob can deduce that Alice's encoding operation is... Therefore, it can be inferred that Alice's classic information is S. A =1001.

[0103] At this point, this specific implementation method realizes the mutual exchange of classical information between the two communicating parties, thus completing the quantum dialogue task.

[0104] The following is a safety analysis of the method of the present invention:

[0105] If an eavesdropper named Eve exists, she can eavesdrop on the quantum channel and intercept, measure, and then transmit the quantum information within it. In this invention, all quantum information passing through the quantum channel is infused with decoy particles. These decoy particles require correct measurement basis measurements to yield the correct results. If the eavesdropper performs arbitrary measurements, it will trigger quantum state collapse, causing errors when the receiver uses the correct measurement basis. Finally, when the communicating parties verify the measurement results, inconsistencies are found, leading to the discovery of the eavesdropper, interruption of communication, and ensuring security.

[0106] If an eavesdropper named Eve exists, he can intercept the quantum channel and obtain the quantum information within it. However, he won't perform a measurement because decoy particles exist, and a direct measurement would inevitably be detected. Instead, he will reconstruct a false quantum message and send it out. However, the communicating parties will later verify the measurement results of the decoy quantum. The results from the false quantum message will always be inconsistent, which will also lead to the eavesdropper being discovered.

[0107] In extreme cases, if an eavesdropper named Eve exists, she can simultaneously eavesdrop on both the quantum and classical channels. She can intercept both quantum and classical information, breaking through the decoy particle defenses. This is because she can intercept the decoy particle's information in the classical channel, thereby eliminating the decoy particle and stealing the quantum information. She can also tamper with the classical information exchanged between the two parties. For example, Alice might inform Bob via the classical channel that the decoy particle verification failed due to an eavesdropper and request a communication interruption. Eve could then steal this information, modify it to indicate that the verification passed, allowing Bob to continue communication and deceiving him into encoding the classical information onto an insecure quantum sequence. In this extreme case, the security measures involving decoy particles are no longer secure.

[0108] In this invention, only |b> and the encoded |x>′, |y>′ are transmitted through the quantum channel. In extreme cases, an eavesdropper could intercept |b>, but |b> is used to calculate |x> and is useless on its own. Even if the eavesdropper intercepts |x>′, |y>′, without knowing |x>, they cannot deduce what encoding transformations |x>′, |y>′ underwent. The eavesdropper cannot obtain Alice and Bob's classical information.

[0109] The following is a protocol comparison analysis of the method of the present invention:

[0110] The formula for calculating transmission efficiency is as follows:

[0111]

[0112] Among them, c t q represents the classical number of bits transmitted. t b represents the number of qubits used in the scheme. t This indicates the number of extra particles.

[0113] According to the transmission efficiency formula, the transmission efficiency of this scheme is expressed as:

[0114]

[0115] Where 4N represents the number of classical bits transmitted between the two sides, 2log22N represents the number of qubits required to construct |b> and |x>, and x represents the number of qubits required in the phase evaluation phase of the HHL algorithm, which depends on the complexity of matrix A. y represents the number of qubits required to construct the SWAP-TEST circuit. Since the number of particles used in the phase evaluation phase is restored to its initial state after the HHL algorithm process is completed, these particles can be reused when constructing the SWAP-TEST circuit. The value of y depends on whether x is sufficient to construct the SWAP-TEST circuit. If it is insufficient, y needs to be introduced to compensate for the lack of particles in constructing the SWAP-TEST circuit. 1 indicates that an additional particle is needed for HHL circuit construction.

[0116] Based on the efficiency formula, taking the classic information in

[0043] and the matrix in

[0048] as examples, we can derive the transmission efficiency as follows:

[0117]

[0118] The transmission efficiency reaches 72.73%, which is very high compared to other quantum dialogue protocols (such as the existing dialogue protocols mentioned in the background art).

[0119] The above description is merely a preferred embodiment of the present invention and is not intended to limit the invention. Various modifications and variations can be made to the present invention by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A quantum communication method based on the HHL algorithm, characterized in that, Includes the following steps: S1, the first communicating party and the second communicating party respectively prepare classical bit information S A S B ; S2. The first communication party prepares the Hermitian matrix A and the quantum sequence |b>, |x>, satisfying A|x>=|b>; S3. The first communicator sends the Hermitian matrix A to the second communicator through a classical channel, and adds the quantum sequence |b> to the decoy particle before sending it to the second communicator through a quantum channel; S4. The second communication party exchanges information about the decoy particles with the first communication party through a classical channel and performs eavesdropping detection. If the detection is successful, the HHL quantum circuit is constructed based on the Hermitian matrix A and the quantum sequence |b>, and the possible result data is enumerated based on the number of particles in the quantum sequence |b>. S5. The second communication party constructs the corresponding SWAP-TEST line based on the result data, and combines the SWAP-TEST line with the HHL quantum line to obtain the fidelity of the result data. Each set of HHL quantum lines corresponds to a fidelity. S6. The second communication party compares each HHL quantum circuit and selects the result data with the highest fidelity as the quantum sequence |y>, and |x>=|y>; S7, The first communicating party will send the classic bit information S A Encoding is done into a quantum sequence |x>, and after adding a decoy particle, a first encoded sequence is obtained. This first encoded sequence is then transmitted to a second communication party via a quantum channel. The second communication party then transmits the classical bit information S. B The sequence is encoded into a quantum sequence |y>, and after adding a decoy particle, a second encoded sequence is obtained. The second encoded sequence is then transmitted to the first communicator through a quantum channel. S8. The first communicating party compares the second encoded sequence with the quantum sequence |x> to obtain the first unitary transformation operation, thus obtaining the classical bit information S of the second communicating party. B The second communicating party compares the first encoded sequence with the quantum sequence |y> to obtain the second unitary transformation operation, thus obtaining the classical bit information S of the first communicating party. A .

2. The quantum communication method based on the HHL algorithm according to claim 1, characterized in that, In step S1, S A ={(a1,b1),(a2,b2),...,(a i ,b i ),...,(a N ,b N )}, S B ={(m1,n1),(m2,n2),...,(m i ,n i ),...,(m N ,n N )}, Among them, a i b i m i n i ∈{0, 1}, i∈{1, 2,…,N}.

3. The quantum communication method based on the HHL algorithm according to claim 2, characterized in that, In step S7, assume that |x>=|y> means the following: |x>=|y>={|φ>1,|φ>2,…,|φ> N }, Then the first communicating party will send the classic bit information S A Encoding into a quantum sequence |x>, represented as: |x>′={C a1b1 |φ>1,C a2b2 |φ>2,…,C aNbN |φ> N }, The second communicating party will send the classic bit information S B Encoding into the quantum sequence |y>, represented as: |y>′={C mln1 |φ>1,C m2n2 |φ>2,…,C mNnN |φ> N }。 4. The quantum communication method based on the HHL algorithm according to claim 1, characterized in that, In step S2, the first communicating party first prepares a vector. Then, for each vector Encode them to obtain quantum sequences |b> and |x> respectively.

5. The quantum communication method based on the HHL algorithm according to claim 1, characterized in that, In step S4, if the detection fails, the communication between the first and second communicating parties is interrupted.

6. The quantum communication method based on the HHL algorithm according to claim 1, characterized in that, In step S8, both the first and second communicating parties need to complete the eavesdropping detection through the classical channel, respectively eliminating the decoy particles of the second coding sequence and the decoy particles of the first coding sequence.

7. The quantum communication method based on the HHL algorithm according to claim 6, characterized in that, The first communicating party needs to inform the second communicating party via a classical channel which particles in the first encoded sequence have undergone H-gate transformation, and the second communicating party needs to inform the first communicating party via a classical channel which particles in the second encoded sequence have undergone H-gate transformation.

8. The quantum communication method based on the HHL algorithm according to claim 7, characterized in that, The first unitary transformation operation includes: The first communicator measures the second encoded sequence, performs single-particle measurements on the X (|+>, |->) basis on the particles that have undergone H-gate transformation, and performs single-particle measurements on the Z (|0>, |1>) basis on the remaining particles to obtain the first measurement result, and compares the first measurement result with the quantum sequence |x>. The second unitary transformation operation includes: The second communicator measures the first encoded sequence, performs single-particle measurements on the X (|+>, |->) basis on the particles that have undergone H-gate transformation, and performs single-particle measurements on the Z (|0>, |1>) basis on the remaining particles to obtain the second measurement result, and compares the second measurement result with the quantum sequence |x>.

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