Quantum information processing method and system based on quantum walking

Abstract

The invention discloses a quantum information processing method and system based on quantum walk, and the method comprises the steps: firstly introducing phase rotation into a coin operator to obtain standard quantum walk, then connecting the standard quantum walk of two different stages to obtain alternate quantum walk, and then constructing step-by-step alternate quantum walk based on the alternate quantum walk, and finally, quantum walking evolution is carried out by using step-by-step alternating quantum walking to obtain an entanglement relationship between the quantized coin and the particle position, and statistics is carried out on a probability distribution standard deviation-time graph of step-by-step alternating quantum walking, the influence of the phase on the evolution speed in different region ranges and alternating step-by-step quantum walking characteristics of different region boundaries. Quantum entanglement and evolution are completed by researching the evolution speed and the boundary effect, any phase rotation is introduced in coin generalization quantum walking, the efficiency and the application range of quantum information processing are remarkably improved, and higher search speed and higher accuracy can be provided by applying the method to a quantum search algorithm.

Description

Technical Field

[0001] The present application relates to the field of quantum walk technology, and more specifically, to a quantum information processing method and system based on quantum walk. Background Art

[0002] Quantum walk, as a core mechanism in quantum computing and information processing, is rapidly becoming an important field for exploring deep phenomena in quantum physics and developing new quantum algorithms. This concept is based on the wave nature of quantum particles and the superposition principle of quantum states. It can transmit and process information in quantum state space in an unprecedented way, showing processing speed and efficiency that cannot be matched by traditional computing methods. With the breakthrough progress of quantum technology in the fields of quantum optics, cold atom physics, and solid quantum systems, the theoretical models and experimental realizations of quantum walks are constantly deepening and expanding, providing new solutions for quantum search, quantum simulation, and even quantum optimization problems.

[0003] In classical computing, searching for a specific item in an unordered database requires checking every item in the database in the worst case, so the search time is linearly related to the size of the database. The physical realization of quantum walks faces challenges such as quantum decoherence and system noise. These factors cause the information processing capabilities of quantum systems to decay over time, limiting the practical application scope and efficiency of quantum walks. Most current quantum walk experiments use medium-sized noisy quantum systems. Even so, by improving quantum error correction technology and developing new quantum control strategies, scientists are studying how to increase the coherence time of quantum systems, laying the foundation for the realization of more complex quantum walk algorithms. Summary of the invention

[0004] The purpose of this application is to provide a quantum information processing method and system based on quantum walk in order to overcome the existing technical defects. By introducing arbitrary phase rotation in the coin generalized quantum walk, the efficiency and application scope of quantum information processing are significantly improved. Applying it to the quantum search algorithm can provide faster search speed and higher accuracy.

[0005] The purpose of this application is achieved through the following technical solutions:

[0006] In the first aspect, the present application proposes a quantum information processing method based on quantum walk, the method comprising:

[0007] Introducing phase rotation into the coin operator yields the standard quantum walk;

[0008] Connecting two different phases of the standard quantum walk gives an alternating quantum walk;

[0009] Construct step-by-step alternating quantum walk based on alternating quantum walk;

[0010] The quantum walk evolution is carried out using step-by-step alternating quantum walks to obtain the entanglement relationship between the quantized coin and the particle position. The probability distribution standard deviation-time diagram of the step-by-step alternating quantum walk, the influence of the phase on the evolution speed in different regions, and the alternating step-by-step quantum walk characteristics at the boundaries of different regions are statistically analyzed.

[0011] In one possible implementation, a standard quantum walk consists of a walker and two coins on a two-dimensional lattice.

[0012] In one possible implementation, the evolution of the walker is controlled by a unitary operator, U: U = S * G, where the matrix ↑ and ↓ represent the spin-down and spin-up states of the particle, respectively.

[0013] In one possible implementation, a standard quantum walk for:

[0014]

[0015] Where 1 is a matrix of all 1s, is the unit matrix, and θ is the adjustable phase.

[0016] In one possible implementation, the alternating quantum walk is: where θ 1 and θ 2 There are two different adjustable phases. and are respectively 1 and θ 2 The corresponding standard quantum walk.

[0017] In one possible implementation, the step-by-step alternating quantum walk is: in

[0018] In a possible implementation, the step of using step-by-step alternating quantum walk to perform quantum walk evolution to obtain the entanglement relationship between the quantized coin and the particle position includes:

[0019] The quantum state is obtained by quantum walk evolution using step-by-step alternating quantum walk;

[0020] Using quantum states to trace the coin space, we can obtain the essential value of the reduced density matrix

[0021] The essential value of the reduced density matrix is ​​processed to obtain the entanglement relationship between the quantized coin and the particle position.

[0022] In one possible implementation, the entanglement relationship between the quantized coin and the particle position is λj is the essential value of the reduced density matrix, and j is the number of eigenvalues.

[0023] In a possible implementation, the method further includes:

[0024] The relevant G operator is replaced by the matrix G to construct a step-by-step alternating quantum walk and perform quantum walk evolution, where the relevant G operator is: p and q are two interdependent parameters, satisfying 0≤p≤1, and p+q=1.

[0025] In the second aspect, the present application proposes a quantum information processing system based on quantum walk, the system comprising:

[0026] Phase rotation module, used to introduce phase rotation in the coin operator to obtain standard quantum walk;

[0027] A connection module is used to connect two different phases of standard quantum walks to obtain an alternating quantum walk;

[0028] Building blocks for constructing step-by-step alternating quantum walks based on alternating quantum walks;

[0029] The evolution module is used to use step-by-step alternating quantum walks to perform quantum walk evolution to obtain the entanglement relationship between the quantized coin and the particle position, and to statistically analyze the probability distribution standard deviation-time diagram of the step-by-step alternating quantum walk, the influence of the phase on the evolution speed in different regions, and the characteristics of the alternating step-by-step quantum walks at the boundaries of different regions.

[0030] The present application discloses a quantum information processing method and system based on quantum walk. First, phase rotation is introduced into the coin operator to obtain a standard quantum walk. Second, two standard quantum walks of different stages are connected to obtain an alternating quantum walk. Then, a step-by-step alternating quantum walk is constructed based on the alternating quantum walk. Finally, the quantum walk evolution is performed using the step-by-step alternating quantum walk to obtain the entanglement relationship between the quantized coin and the particle position. The probability distribution standard deviation-time diagram of the step-by-step alternating quantum walk, the influence of the phase on the evolution speed in different regions, and the characteristics of the alternating step-by-step quantum walk at the boundaries of different regions are statistically analyzed. The evolution speed and boundary effect are studied to complete quantum entanglement and evolution. Arbitrary phase rotation is introduced into the coin generalized quantum walk, which significantly improves the efficiency and application scope of quantum information processing. Applying it to the quantum search algorithm can provide faster search speed and higher accuracy. BRIEF DESCRIPTION OF THE DRAWINGS

[0031] In order to more clearly illustrate the technical solutions of the embodiments of the present application, the drawings required for use in the embodiments will be briefly introduced below. It should be understood that the following drawings only show certain embodiments of the present application and therefore should not be regarded as limiting the scope. For ordinary technicians in this field, other related drawings can be obtained based on these drawings without paying creative work.

[0032] Figure 1 A flow chart of a quantum information processing method based on quantum walk proposed in an embodiment of the present application is shown.

[0033] Figure 2 It is the standard GQWθ 2 = AGQW and θ of 0.3π 2 Schematic diagram of the evolution rate of the step-by-step AGQW with π = 0.3π.

[0034] Figure 3 This is a diagram of the evolution speed of AGQW.

[0035] Figure 4 This is a schematic diagram of the evolution speed of SSAGQW.

[0036] Figure 5 This is the probability distribution pattern diagram of walkers (θ2=0.3π, θ1=0.8) in SSAGQW after 80 steps of evolution.

[0037] Figure 6 This is the probability distribution pattern diagram of walkers (θ2=0.3π, θ1=2.8) in SSAGQW after 80 steps of evolution.

[0038] Figure 7 It is the entanglement distribution diagram of SSAGQW for θ1 and θ2.

[0039] Figure 8 It is a schematic diagram of the evolution speed of AGQW after changing the operator.

[0040] Fig. 9 This is a schematic diagram of the evolution speed of SSAGQW after changing the operator. DETAILED DESCRIPTION

[0041] The following describes the embodiments of the present application through specific examples, and those skilled in the art can easily understand other advantages and effects of the present application from the contents disclosed in this specification. The present application can also be implemented or applied through other different specific embodiments, and the details in this specification can also be modified or changed in various ways based on different viewpoints and applications without departing from the spirit of the present application. It should be noted that the following embodiments and features in the embodiments can be combined with each other without conflict.

[0042] Based on the embodiments in this application, all other embodiments obtained by ordinary technicians in this field without making any creative work shall fall within the scope of protection of this application.

[0043] In existing technologies, searching for a specific item in an unordered database requires checking every item in the database in the worst case, so the search time is linearly related to the size of the database. The physical realization of quantum walks faces challenges such as quantum decoherence and system noise. These factors will cause the information processing capabilities of quantum systems to decay over time, limiting the practical application scope and efficiency of quantum walks. Most current quantum walk experiments use medium-sized noisy quantum systems. Even so, by improving quantum error correction technology and developing new quantum control strategies, scientists are studying how to increase the coherence time of quantum systems, laying the foundation for the realization of more complex quantum walk algorithms.

[0044] In classical computing, searching for a specific item in an unordered database requires checking every item in the database in the worst case, so the search time is linearly related to the size of the database. Therefore, in order to solve the above problems, the embodiment of the present application proposes a quantum information processing method and system based on quantum walks, which shortens the search time to approximately the square root of the database size by utilizing the characteristics of quantum computing. If the database contains N items, this method only takes √N steps to find the required items, which greatly improves the search efficiency for large databases.

[0045] Please refer to Figure 1 , Figure 1 A flowchart of a quantum information processing method based on quantum walk proposed in an embodiment of the present application is shown, and the method includes the following steps:

[0046] Step S100: introducing phase rotation into the coin operator to obtain a standard quantum walk.

[0047] The standard quantum walk (GQW, Generic Quantum Walk) is a basic model of quantum walk, especially referring to its implementation in two-dimensional space. The standard quantum walk consists of a walker and two coins on a two-dimensional lattice.

[0048] The evolution of the walker is controlled by the unitary operator U: U = S * G, where the matrix ↑ and ↓ represent the spin-down and spin-up states of the particle, respectively. In addition, the matrix in represents the identity matrix, and 1 represents a matrix of all ones.

[0049] By introducing arbitrary phase rotation in the coin operator, the standard quantum walk is extended to a quantum walk with adjustable parameters, namely the standard quantum walk for:

[0050]

[0051] Where 1 is a matrix of all 1s, is the unit matrix, and θ is the adjustable phase.

[0052] The embodiment of the present application adopts the characteristic of arbitrary phase rotation. By generalizing the coin operator, variable phase rotation is introduced to control the evolution speed and entanglement characteristics of particles. By introducing arbitrary phase rotation in the coin generalized quantum walk, the efficiency and application scope of quantum information processing are significantly improved.

[0053] Step S200, connecting two standard quantum walks in different stages to obtain an alternating quantum walk.

[0054] Connecting two quantum walks (GQW) with different phases to define a new quantum walk model, namely the phase-alternating quantum walk (AGQW, Amplitude-Phase Generic Quantum Walk), is a strategy to expand the characteristics of quantum walks. The alternating quantum walk is: where θ 1 and θ 2 There are two different adjustable phases. and are respectively 1 and θ 2 The corresponding standard quantum walk. By cascading two quantum walk operators with different parameters, a new alternating quantum walk operator constructed by the standard quantum walk of the two parameters is realized.

[0055] Step S300: construct a step-by-step alternating quantum walk based on the alternating quantum walk.

[0056] Based on the influence of the extra rotation phase in the AGQW with arbitrary phase, the corresponding evolution speed is considered and the size change of its evolution speed is studied. The universality of AGQW is explored and a Spatially Separated Amplitude-Phase Generic Quantum Walk (SSAGQW) is constructed, which is defined as: in

[0057] Replace the S operator in the alternating quantum walk with S x , S y And add a coin rotation operator Construct a step-by-step alternating quantum walk operator. Try to change the relevant operators to achieve the control of different evolutionary stages and phase transition phenomena. Both generalized models, alternating quantum walk (AGQW) and step-by-step alternating quantum walk (SSAGQW), can show evolutionary characteristics different from standard quantum walks, including sudden changes in evolutionary speed, enhanced entanglement in the evolutionary state, and different evolutionary stage phenomena based on phase rotation.

[0058] Step S400, using step-by-step alternating quantum walk to perform quantum walk evolution to obtain the entanglement relationship between the quantized coin and the particle position, and statistically analyzing the probability distribution standard deviation-time diagram of the step-by-step alternating quantum walk, the influence of the phase on the evolution speed in different regions, and the alternating step-by-step quantum walk characteristics at the boundaries of different regions.

[0059] Driven by the step-by-step alternating quantum walk, the particle continuously evolves and diffuses in the two-dimensional space, and the probability distribution of the particle in the two-dimensional space is obtained. The standard deviation of the particle probability distribution will change with the evolution of the particle. Here, the standard deviation is used to measure the particle evolution speed over time. The distribution of the particle evolution speed in different phase parameter regions is used to study the properties of the step-by-step alternating quantum walk at the boundaries of each parameter region.

[0060] As the walker evolves, the coin space and position space of the walker become entangled. The entanglement between these two degrees of freedom also shows evolutionary phases consistent with the phase distribution found in the evolutionary velocity distribution. After realizing the evolutionary process of quantum walk in an alternating or step-by-step manner, the evolutionary properties are used to perform efficient quantum search and other quantum information processing tasks.

[0061] The speed of particle evolution is measured by statistically analyzing the change in the standard deviation of the probability distribution obtained by the step-by-step alternating quantum walk over time, as well as the influence of the phase of the alternating step-by-step quantum walk on the evolution speed in different regions and the characteristics of the alternating step-by-step quantum walk at the boundaries of different regions.

[0062] In a possible embodiment, the step of using step-by-step alternating quantum walk to perform quantum walk evolution to obtain the entanglement relationship between the quantized coin and the particle position includes:

[0063] The quantum state is obtained by quantum walk evolution using step-by-step alternating quantum walk;

[0064] Using quantum states to trace the coin space, we can obtain the essential value of the reduced density matrix.

[0065] The essential value of the reduced density matrix is ​​processed to obtain the entanglement relationship between the quantized coin and the particle position.

[0066] After completing the evolution process of quantum walk, we further study the eigenvalues ​​of the evolution matrix, study the density matrix of the coin, and quantify the entanglement relationship between the coin and the particle position. The quantified entanglement relationship between the coin and the particle position is λ j is the essential value of the reduced density matrix, and j is the number of eigenvalues.

[0067] The evolution of quantum walks is simulated by applying sequential operation units (such as S and parameterized G operators). Each iteration involves a coin flip (S operation) and a position transfer (G(θ) operation), which gradually advances the evolution of the system from the initial state to the subsequent quantum state. This process produces a quantum state that evolves over time and captures the joint probability distribution of the coin state and the position state.

[0068] Once the quantum state of the entire system is known, the next goal is to isolate the dynamics of the coin space. By performing a partial trace operation on the density matrix of the entire system (usually performed on a position basis), a reduced density matrix that describes only the state of the coin can be obtained. This operation effectively ignores the position information and focuses on analyzing the state of the coin (spin).

[0069] The last step is to analyze the obtained reduced density matrix and calculate its essential value. If this value is not zero, it indicates that there is quantum entanglement between the coin state and the ignored position state.

[0070] The method further comprises:

[0071] The relevant G operator is replaced by the matrix G to construct a step-by-step alternating quantum walk and perform quantum walk evolution, where the relevant G operator is: p and q are two interdependent parameters, satisfying 0≤p≤1, and p+q=1.

[0072] Repeat the above steps to study its evolution speed and its boundary effects.

[0073] G acts on the coin space, and S acts on the coin space and the two-dimensional position space of the particle. After the coin is rotated by G, the evolution direction of the particle in the two-dimensional space is determined by the state of the coin and described by S. Therefore, G and S operate continuously to determine the walking evolution of the particle.

[0074] In a possible implementation, in order to verify the feasibility and effect of the quantum information processing method based on quantum walk proposed in the embodiment of the present application, a probability density diagram of a two-dimensional Gaussian distribution is first created in MATLAB, with μ = [0,0] as the mean vector and σ = [100 0; 0 100] as the covariance matrix.

[0075] The code generates grid data through meshgrid, uses mvnpdf to calculate the probability density values ​​at these grid points, and then reshapes these values ​​to match the size of the original grid, and normalizes the probability density values ​​to have unit norm. The processed probability density values ​​are stored in the kxy variables.

[0076] Let c = 1 / 2*kron([1; 1], [1; 1]), and find the tensor product of c and kxy as the core matrix to represent or manipulate the state of the quantum walk. For each position (x, y), apply these two rotation gates and the corresponding phase changes.

[0077] After calculation, the thermal diagram of the evolution speed of AGQW and SSAGQW with the change of θ1 and θ2 was obtained. Figure 3 It is a schematic diagram of the evolution rate of AGQW. The dotted line represents the correlation between the evolution peaks θ1 and θ2. Figure 4 It is a schematic diagram of the evolution rate of SSAGQW. The dotted line represents the correlation between the evolution peaks θ1 and θ2.

[0078] Further study the eigenvalues ​​of the evolution matrix, take the average value of the evolution time t = 50-60, and obtain the heat map of the change of entanglement distribution with θ1 and θ2. Figure 5 This is the probability distribution pattern of walkers in SSAGQW after 80 steps of evolution (θ2=0.3π,θ1=0.8).

[0079] After that, we studied its boundary effect and found that the final result of the quantum gait simulation has a certain boundary, and different θ1 and θ2 present different shapes. Figure 6 This is the probability distribution pattern of walkers in SSAGQW after 80 steps of evolution (θ2=0.3π,θ1=2.8).

[0080] Figure 7 is the entanglement distribution diagram of SSAGQW for θ1 and θ2, and the dotted line represents the correlation between θ1 and θ2 at the entanglement peak. Here the entanglement evolves at time t = 50-60.

[0081] Finally, replace the G operator and repeat the above steps with the changed operator. Figure 8 It is a schematic diagram of the evolution speed of AGQW after changing the operator. Fig. 9 This is a schematic diagram of the evolution speed of SSAGQW after changing the operator.

[0082] There are the following points in the above process:

[0083] First, input the mean vector and covariance matrix, which can be modified as needed.

[0084] Second, the c matrix can also be modified as needed, so the core matrix formed by the tensor product of c and kxy can also be changed.

[0085] Third, the number of squares can be adjusted according to the actual number of evolution steps to achieve better observation effects.

[0086] Fourth, the operator G can also be changed, and different operators will exhibit different evolution speeds and entanglement effects. This operator can be adjusted to meet relevant needs.

[0087] Compared with the prior art, the embodiments of the present application have the following beneficial effects:

[0088] First, it improves the efficiency and accuracy of the quantum search algorithm and improves its controllability.

[0089] Second, it enhances the entanglement properties of the evolving state, providing new tools for quantum information processing.

[0090] Third, the control of the evolution stage is achieved, providing more flexibility and application possibilities for quantum information processing.

[0091] The above description is only a preferred embodiment of the present application and is not intended to limit the present application. Any modifications, equivalent substitutions and improvements made within the spirit and principles of the present application should be included in the protection scope of the present application.

Claims

1. A quantum information processing method based on quantum walk, characterized in that: The method comprises: Introducing phase rotation into the coin operator yields the standard quantum walk; Connecting two different phases of the standard quantum walk gives an alternating quantum walk; Construct step-by-step alternating quantum walk based on alternating quantum walk; The quantum walk evolution is carried out using step-by-step alternating quantum walks to obtain the entanglement relationship between the quantized coin and the particle position. The probability distribution standard deviation-time diagram of the step-by-step alternating quantum walk, the influence of the phase on the evolution speed in different regions, and the alternating step-by-step quantum walk characteristics at the boundaries of different regions are statistically analyzed.

2. The quantum information processing method according to claim 1, characterized in that: A standard quantum walk consists of a walker and two coins on a two-dimensional lattice.

3. The quantum information processing method according to claim 2, characterized in that: The evolution of the walker is controlled by the unitary operator U: U = S * G, where the matrix ↑ and ↓ represent the spin-down and spin-up states of the particle, respectively.

4. The quantum information processing method according to claim 3, characterized in that: Standard quantum walk for: Where 1 is a matrix of all 1s, is the unit matrix, and θ is the adjustable phase.

5. The quantum information processing method according to claim 4, characterized in that: The alternating quantum walk is: Where θ1 and θ2 are two different adjustable phases, and are the standard quantum walks corresponding to θ1 and θ2 respectively.

6. The quantum information processing method according to claim 5, characterized in that: The step-by-step alternating quantum walk is: in 7. The quantum information processing method according to claim 1, characterized in that: The steps of using step-by-step alternating quantum walk to perform quantum walk evolution to obtain the entanglement relationship between the quantized coin and the particle position include: The quantum state is obtained by quantum walk evolution using step-by-step alternating quantum walk; Using quantum states to trace the coin space, we can obtain the essential value of the reduced density matrix. The essential value of the reduced density matrix is ​​processed to obtain the entanglement relationship between the quantized coin and the particle position.

8. The quantum information processing method according to claim 7, characterized in that: The entanglement relationship between the quantized coin and the particle position is λ j is the essential value of the reduced density matrix, and j is the number of eigenvalues.

9. The quantum information processing method according to claim 1, characterized in that: The method further comprises: The relevant G operator is replaced by the matrix G to construct a step-by-step alternating quantum walk and perform quantum walk evolution, where the relevant G operator is: p and q are two interdependent parameters, satisfying 0≤p≤1, and p+q=1.

10. A quantum information processing system based on quantum walk, characterized in that: The system comprises: Phase rotation module, used to introduce phase rotation in the coin operator to obtain standard quantum walk; A connection module is used to connect two different phases of standard quantum walks to obtain an alternating quantum walk; Building blocks for constructing step-by-step alternating quantum walks based on alternating quantum walks; The evolution module is used to use step-by-step alternating quantum walks to perform quantum walk evolution to obtain the entanglement relationship between the quantized coin and the particle position, and to statistically analyze the probability distribution standard deviation-time diagram of the step-by-step alternating quantum walk, the influence of the phase on the evolution speed in different regions, and the characteristics of the alternating step-by-step quantum walks at the boundaries of different regions.

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