A quantum algorithm for pricing path-dependent options based on quantum amplitude estimation

Abstract

The application discloses a quantum algorithm for pricing path-dependent options based on quantum amplitude estimation, which firstly constructs a quantum circuit for generating a probability distribution condition controlled by a ground state, wherein the quantum circuit comprises a conditional register and a distribution register; the current price is encoded by the conditional register, and the distribution register is controlled to generate a price distribution at a next time point; secondly, a quantum state representing a complete price path probability distribution is prepared by recursively calling a conditional evolution operator; thirdly, a profit function is encoded for different types of path-dependent options; finally, the probability of an auxiliary quantum bit at a specific state is estimated by using a quantum amplitude estimation algorithm, and an expected value of an option profit is obtained through linear mapping. The application solves the technical problems of high-dimensional distribution loading and complex profit function coding of path-dependent options in quantum computing, and provides an efficient quantum solution for financial derivative pricing.

Description

Technical Field

[0001] This invention belongs to the technical fields of artificial intelligence and quantum computing. Specifically, it relates to a quantum algorithm for the efficient pricing of path-dependent financial derivatives, which, under the premise that the underlying asset price process follows a geometric Brownian motion model, realizes path distribution loading and payoff function encoding through quantum circuits, and estimates payoff expectations by combining quantum amplitude estimation. In particular, in the case of no closed-form solution, the payoff expectation calculation is more efficient than traditional methods through quantum amplitude estimation. Background Technology

[0002] Options, as an important financial derivative, have always had a core research focus in the field of financial engineering regarding their pricing. Classical European options can be solved in a closed form using analytical quantum algorithms under certain assumptions. However, in real financial markets, the value realization of many options depends not only on the price of the underlying asset at expiration but also on the entire price evolution path. These options are collectively referred to as path-dependent options, including Asian options, barrier options, and automatic redemption options. Since the payoff function involves integration or conditional judgments over the asset price path, most path-dependent options do not have analytical closed-form solutions. Traditional quantum pricing algorithms are mainly based on Monte Carlo simulations. This algorithm generates a large number of random asset price paths, calculates the payoffs corresponding to each path, and averages them to estimate the expected payoff of the option. Therefore, obtaining a sufficiently accurate option pricing scheme requires a large number of samples, and convergence is a significant challenge. In contrast, quantum amplitude estimation can theoretically provide a secondary acceleration for the calculation of the expected payoff of options. It first uses a distributed loading oracle to prepare a quantum superposition state, allowing all possible price paths to be represented in parallel within the quantum state:

[0003]

[0004] in, Encoding price path The calculation of the ground state, For price path The probability of occurrence. Then, the payoff function corresponding to the path is encoded using an Oracle. Encoding to auxiliary qubits:

[0005] At this point, the auxiliary bit measurement result is The probability corresponds to the expected value of the option payoff. Within the standard quantum amplitude estimation framework, the error can be reduced to:

[0006] in, This reduces the number of Oracle calls, thus achieving a second-order speedup compared to the classical Monte Carlo method. Furthermore, improved variants such as iterative quantum amplitude estimation and maximum likelihood quantum amplitude estimation further reduce the requirements for quantum coherence time and the number of auxiliary qubits while maintaining the advantages of quantum speedup.

[0007] However, for path-dependent options, the asset price path is a high-dimensional stochastic process, and its distribution is difficult to represent directly in quantum terms; at the same time, the payoff function often has a piecewise linear or conditional activation structure, making the encoding implementation quite complex. Summary of the Invention

[0008] To address the aforementioned problems, this invention proposes a learning-based distributed loading oracle. By constructing and training a quantum circuit with a probability distribution condition, and recursively calling its parameterized operator, it achieves the generation of price path distributions that satisfy Markov properties. Simultaneously, this invention proposes a quantum implementation method for a comparator and an adder, and based on this, constructs payoff encoding oracles for Asian options, barrier options, and automatically redeemed options. Combined with quantum amplitude estimation, this invention forms a quantum algorithm suitable for path-dependent option pricing, capable of efficiently calculating expected payoffs. It solves the technical challenges of high-dimensional distributed loading and complex payoff function encoding for path-dependent options in quantum computing, providing an efficient quantum solution for financial derivatives pricing.

[0009] To achieve the above objectives, the technical solution adopted by the present invention is as follows: This invention provides a quantum algorithm for path-dependent option pricing based on quantum amplitude estimation, comprising the following steps: S1. Construct a quantum circuit that generates the probability distribution condition of the ground state control, and control the evolution to generate the price distribution at the next time point by calculating the current price encoded in the ground state as a condition.

[0010] S2. By recursively calling the operator used for conditional evolution, the quantum state corresponding to the price path probability distribution is prepared.

[0011] S3. Encode the payoff function for different option types: (1) For Asian options, the price at each time point on the price path is accumulated by a quantum adder to obtain the distribution of the price sum; then a comparator is used to determine the relationship between the price sum and the strike price, and the corresponding piecewise linear payoff function is encoded into the amplitude of the auxiliary quantum bit. (2) For barrier options, a comparator is used to record whether the price crosses the barrier at each time point, and the trigger condition of the non-gate is encoded into the flag bit by a multi-bit control to control the activation of the payout encoding operator; (3) For automatic redemption options, the observation date judgment is further introduced on the basis of barrier options, and the comparator is used to mark whether the early redemption conditions are met on each observation date.

[0012] S4. The auxiliary bit measurement for obtaining the encoded profit function value using the quantum amplitude estimation algorithm is... The probability is calculated, and the expected option payoff is obtained through post-processing after linear mapping.

[0013] In one specific implementation, the conditional generation quantum circuit consists of two registers: a condition register, which is used to encode the price at the current time point by calculating the ground state; and a distribution register, which is used to generate the price distribution at the next time point under the control of the condition register, wherein the measurement probability of each calculated ground state in the generated distribution corresponds to the probability of the price occurring.

[0014] In one specific implementation, the evolution operator is a parameterized operator, with the loss between the generating distribution and the target distribution as the objective function, and the parameters are adjusted using a classical optimizer.

[0015] In one specific implementation, the quantum state preparation of the price path probability distribution is as follows: the register with the loaded price distribution is used as a condition register to control the generation of the distribution at the next time point. By introducing a new register and reusing the condition generation operator, the recursive generation of the price path probability distribution is realized.

[0016] In one specific implementation, the adder applies multiple +1 operators based on the bits that are 1 in the binary representation of the addend, thereby performing a modular addition operation on the value represented by the quantum state; the comparator applies multiple bits to control the NOT gate based on the bits that are 0 in the binary representation of a given threshold. When the high-order bit of the 0 bit matches the high-order bit of the register and the current bit of the register is 1, the NOT gate is activated, thereby realizing the conditional judgment.

[0017] In one specific implementation, the expected option payoff is obtained using quantum amplitude estimation. This can be achieved using any quantum amplitude estimation algorithm, including standard quantum amplitude estimation, iterative quantum amplitude estimation, maximum likelihood quantum amplitude estimation, etc. Alternatively, a more complex approach can be to utilize quantum tomography to perform state measurements and estimate the state of the auxiliary qubit. The probability of a state is used to obtain the expected value of the option payout after linear mapping.

[0018] Compared with the prior art, the present invention has at least the following beneficial technical effects: 1. This invention provides a quantum algorithm suitable for path-dependent option pricing, solving the technical challenges of high-dimensional distributed loading and complex payoff function encoding of path-dependent options in quantum computing, and providing an efficient quantum solution for financial derivatives pricing.

[0019] 2. This invention provides a quantum state preparation scheme for loading a multivariate log-normal distribution. Due to its recursive implementation strategy, the method has better versatility and is suitable for efficient estimation of the expected payoff of path-dependent options under different time step settings.

[0020] 3. This invention provides a general implementation scheme for quantum adders and comparators without auxiliary bits, and based on this adder and comparator, constructs an Oracle implementation scheme for payoff function encoding applicable to various option types.

[0021] 4. The quantum computing-based option payoff expectation estimation scheme provided by this invention can reduce the required sampling complexity and improve stability and computational efficiency compared with the classical Monte Carlo method under the same error tolerance conditions.

[0022] Other features and advantages of the invention will be set forth in the following description, and will be apparent in part from the description, or may be learned by practicing the invention. The objects and other advantages of the invention may be realized and obtained by means of the structures particularly pointed out in the written description and the accompanying drawings.

[0023] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. Attached Figure Description

[0024] The accompanying drawings are provided to further illustrate the invention and form part of the specification. They are used together with the embodiments of the invention to explain the invention and do not constitute a limitation thereof.

[0025] Figure 1 This invention provides a schematic diagram of a quantum algorithm for path-dependent option pricing based on quantum amplitude estimation.

[0026] Figure 2 The conditionally distributed loading loop structure provided by the present invention; wherein part (a) is The weighted superposition operator implements the loop structure, and part (b) is the one used in this invention. The loop structure consists of a hadamar gate (H), which is controlled. Parametric revolving door , , , It consists of a revolving door.

[0027] Figure 3 After optimization of the parameters provided for this invention, different Comparison of the distribution generation and target distribution when (corresponding to different 3-qubit computational ground states) are used as control conditions.

[0028] Figure 4Provided by the present invention A schematic diagram of the price path loading loop at that time.

[0029] Figure 5 The adder circuit structure provided by the present invention; wherein part (a) is Example of gate implementation, part (b) utilizes controlled Implement the addition of two registers.

[0030] Figure 6 An example of a comparator implementation provided for this invention ( ).

[0031] Figure 7 The encoding provided for this invention Function implementation loop, The value of the variable represented by the control register.

[0032] Figure 8 The payoff function encoding loop structure provided by this invention includes: (a) an Asian option; (b) a European barrier option; and (c) an automatic redemption option.

[0033] Figure 9 In a specific implementation example of the Asian option provided by the present invention, the price distribution result obtained by measuring and summing the register is compared with the distribution result obtained by summing and discretizing the price path through Monte Carlo sampling.

[0034] Figure 10 This diagram illustrates the results of estimating the expected return of an Asian option based on the maximum likelihood quantum amplitude under different Oracle (distributed loading operator and profit encoding operator) call counts provided by this invention.

[0035] Figure 11 This is a schematic diagram of the electronic device structure provided by the present invention. Detailed Implementation

[0036] 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.

[0037] In the description of this invention, it should be noted that some processes described in this application specification and accompanying drawings include multiple operations that appear in a specific order. However, it should be clearly understood that these operations may be performed out of order or in parallel. Therefore, the following detailed description of the embodiments of the invention provided in the accompanying drawings is not intended to limit the scope of the claimed invention, but merely to illustrate selected embodiments of the invention. All other embodiments obtained by those skilled in the art based on the embodiments of this invention without inventive effort are within the scope of protection of this invention.

[0038] See Figure 1 As shown, this invention provides a quantum algorithm for path-dependent option pricing based on quantum amplitude estimation, which mainly includes the following steps: S1. Construct a quantum circuit that generates the probability distribution condition for the ground state control, and control the evolution to generate the price distribution at the next time point by calculating the current price encoded in the ground state as a condition; S2. By recursively calling the conditional evolution operator, the quantum state corresponding to the price path probability distribution is prepared; S3. Encode the payoff function for different option types, where: For Asian options, the price at each time point along the price path is accumulated by a quantum adder to obtain the distribution of the price sum; then a comparator is used to determine the relationship between the price sum and the strike price, and the corresponding piecewise linear payoff function is encoded into the amplitude of the auxiliary qubit. For barrier options, a comparator records whether the price crosses the barrier at each time point, and a multi-bit control is used to encode the non-gate trigger condition into the flag bit to control the activation of the payout encoding operator. For automatic redemption options, an observation date judgment is introduced based on barrier options, and a comparator is used to mark whether the early redemption conditions are met on each observation date. S4. The auxiliary bit measurement for obtaining the encoded profit function value using the quantum amplitude estimation algorithm is... The probability is calculated, and the expected option payoff is obtained after processing with a linear mapping.

[0039] In this embodiment of the invention, the probability distribution conditional generation quantum circuit controlled by the ground state consists of two qubit registers: one is a condition register, used to calculate the ground state encoding condition; the other is a distribution register, which generates different quantum states under the control of the condition register to correspond to different probability distributions. Each calculated ground state in the distribution register corresponds to a preset value point of a discrete spatial variable, and the measurement probability of this ground state represents the probability that the variable takes that preset value.

[0040] In this embodiment of the invention, the method for preparing the quantum state of the discretized valence path distribution is as follows: The discretized price is encoded as the computational ground state. A quantum circuit is generated using probability distribution conditions to prepare the price distribution for the next time step. The generated distribution register is then used as a new condition register to control the quantum state evolution at subsequent time steps. Through this recursive preparation process, the price path probability distribution satisfying Markov properties can be loaded.

[0041] This invention provides an implementation of an adder operator for performing modular addition on discrete values ​​represented by the ground state of a qubit register. The operator can be decomposed into multiple +1 operators, each acting on a qubit in the binary representation of the addend that is 1 and its corresponding higher-order qubit. Each +1 sub-operator can be composed of several multi-control NOT gates with progressively decreasing control bits.

[0042] In this embodiment of the invention, a comparator operator is provided for determining whether the discrete value represented by the ground state of a register exceeds a given threshold. When it does, a flag bit is flipped, thereby implementing conditional logic control in a quantum circuit. This operator can be decomposed into a control NOT gate set based on the bits that are 0 in the binary representation of the threshold. When the high-order bits of the register match the threshold and the current bit is 1, the flip operation is triggered.

[0043] Furthermore, in this embodiment of the invention, a scheme for implementing a payoff encoding operator suitable for Asian options, barrier options, and automatic redemption options is proposed by utilizing the linear function encoding method that approximates a linear function with a sine function and combining adder and comparator operators.

[0044] The specific embodiments of the present invention will be described in detail below: 1. Probability distribution of the price path of the underlying asset in quantum state loading: Considering the discrete-time model, a price path can be represented as a price sequence at finite, equally spaced discrete time points:

[0045] in, Indicates the underlying asset in the Price at a specific point in time. Time interval. , This represents the total duration from the initial moment to the option's expiration time. When using geometric Brownian motion to describe the evolution of asset prices, The distribution follows a log-normal distribution and depends on the prices at earlier times. , Its probability density function can be expressed as:

[0046]

[0047] in, and Representing the risk-free interest rate and volatility, respectively, and the initial price. is a given constant.

[0048] This invention provides a method for preparing the quantum state of the probability distribution of asset price paths. It employs... Each of the qubit registers corresponds to... The price at the 1st point in time, the 2nd In the registers The computational ground state of a qubit , for The binary representation of It is used to represent the discretized target distribution. Price at a specific time , Calculate the ground state The measurement probability needs to be approximated. Falling in the range The probability of the target state can be expressed as:

[0049] in:

[0050]

[0051]

[0052] As can be seen from the above formula, the form of this conditional probability is related to the time point. Irrelevant. Therefore, this invention introduces a condition register to encode conditions using the ground state, proposing a learning-based conditional state preparation scheme. The initial state is... Define parameterized operators as By adjusting parameters Make satisfy:

[0053] To further enhance the expressive power of loops, this invention introduces a multi-parameterized operator weighted superposition strategy: , This indicates the weight. In a specific implementation, each register contains... (in the example below) n =3) Quantum bits are used to load the price distribution at the corresponding time point, with an additional qubit introduced at each time step. One auxiliary bit is used to implement the operator weighted superposition strategy. .

[0054] In one implementation example, the operator The circuit structure is shown in the attached figure. Figure 2 As shown. Settings , , The target distribution is obtained using numerical integration and normalization, with KL divergence as the objective function and the learning rate set to... Adjust using a gradient descent optimizer (the AdamW optimizer is used in this example). common Wheel. Different The distribution of the quantum states prepared below is shown in the attached figure. Figure 3 As shown. Furthermore, a neural network can be introduced and trained to distribute the parameters ( Mapping to loop parameters With weight By leveraging the generalization ability of neural networks, the versatility of the model can be improved.

[0055] Operators completed using training Loadable price path probability distribution. Ignoring auxiliary bits, the evolution process is as follows:

[0056] This implementation case loads... Price path distribution, quantum circuit structure as shown in the appendix Figure 4 As shown.

[0057] 2. Modules required for constructing the payoff function encoding: After preparing the quantum state for the price path distribution, it is necessary to calculate the payoff function in the quantum circuit and embed its value into the amplitude of the auxiliary qubit. The target state can be represented as:

[0058] This quantum state is used for subsequent quantum amplitude estimation, by obtaining the state of the auxiliary qubit. The probability of obtaining the expected return .in, Indicates the price path as The revenue at any given time. The following modules are required during the coding of the revenue function.

[0059] 1) Adder: Adders are used to progressively accumulate asset prices in a quantum state, and can be used for applications such as calculating the payoff of Asian options. This invention provides an implementation of an adder operator, allowing for the selection of different adder implementation schemes in practical applications to construct... - Quantum bit unitary gate satisfy:

[0060] make positive integer The binary representation of , then It can be decomposed into multiple Door:

[0061] in, Indicates size is The unit array. hour, Equivalent to X gate; when At that time, there were:

[0062] Right now, Door by -Multi-controllable NOT gates controlled by qubits Composition, inductive reasoning can be obtained It can be decomposed into the number of control bits from Multiple multi-control bit NOT gates, decreasing one by one, as shown in the appendix. Figure 5 As shown in section (a). This circuit can be further decomposed into Tofoli, CX, X gate.

[0063] In addition, as attached Figure 5 As shown in section (b), controlled [technology] is used. (The diagram indicates that it is under control) It can be decomposed into multiple controlled A door can achieve the following: 。

[0064] 2) Comparator: The comparator module is used to determine whether the price exceeds a barrier or the execution price, thereby implementing logical control over the payout conditions. This invention provides a comparator operator. The implementation method satisfies:

[0065] Right now When, flip the bit .make The binary representation of is ,when , ,and hour, Therefore, by controlling the conditions as follows: Multiple control NOT gates can be used to achieve this. Appendix Figure 6 The diagram illustrates the comparator circuit used in a specific case. ).

[0066] 3) Linear function encoding: This embodiment employs a controlled approach. Gate encoding methods are used to implement linear functions on auxiliary qubits. The magnitude mapping. The function of the encoded operator can be expressed as:

[0067] As attached Figure 7 As shown, the encoding process first applies an encoding method to the auxiliary qubit. Subsequently, for all Apply in sequence To control the bits ,in, express The number of qubits. When the hyperparameter When, it can be approximately satisfied that:

[0068] This achieves the linear function. The encoding.

[0069] 3. Encoding of Asian option payoff function: Consider the price path after discretization The payoff functions for Asian call and put options can be expressed as piecewise linear functions:

[0070] in, This indicates the strike price.

[0071] To encode the above profit function, we first introduce a... - A register for qubits Used to store the summation result Then, using a comparator (Corresponding call option) or The value in the register is compared and judged by combining the X-gate (corresponding to a put option) applied to the flag qubit, thus flipping the flag qubit. Finally, the flag qubit is used to selectively activate the encoded linear function 0 and... The operators are used to implement the revenue function encoding. (Appendix) Figure 8 Part (a) illustrates the quantum circuit structure used for encoding the payoff function of Asian call options. The encoded bit measurement results are as follows: The probability is: 。

[0072] 4. European barrier option payoff function encoding: European barrier options, in addition to the basic European options, take into account the barrier price. Its profit function can be expressed as:

[0073] There are four activation conditions: 1) Up-and-In: ; 2) Up-and-Out: ; 3) Down-and-In: ; 4) Down-and-Out: .

[0074] In the implementation process, the first step is to introduce Each flag qubit is used to record the triggering state of the barrier condition. A comparator is applied to each pair of price registers and flag qubits. (For Up-type obstacles) or Combine this with an X-gate applied to the flag bits (for Down-type barriers). Then, apply... - The control NOT gate for the control bit is used to control the flipping of the flag qubit to determine whether the barrier activation condition is met. For Out-type barriers, when the trigger condition is met, the control state is... For In-type barriers, it is Simultaneously, an X gate is added to the controlled qubit. Finally, the linear function 0 and 1 are selectively activated using flag qubits. The operator implements the payoff function encoding. Unlike Asian options, the function variable here comes from... Price register at a specific time point. (Appendix) Figure 8 Section (b) illustrates the quantum circuit structure used for encoding the payoff function of an Up-and-In type European barrier option. The encoded bit measurement results are as follows: The probability is:

[0075] By summing the prices at each time point and using the price in the summation register as a linear function variable, this payoff function encoding method can be extended to Asian barrier options.

[0076] 5. Automatic redemption options: The payout of an automatic redemption option is divided into two levels. On the one hand, on the observation date... , , If asset prices The contract is terminated and the proceeds are received. On the other hand, if the payout is not triggered on all observation days, it degenerates into a barrier option, where the asset price remains below the barrier price throughout the entire period. And the price at maturity is lower than the strike price. The holder bears the loss. Therefore, the payout of an automatic redemption option can be expressed as:

[0077] Among them, scalar is a given coefficient.

[0078] Because it may incur additional losses, it is first applied to the encoded bits. Used for embedding identity functions Based on the barrier option quantum circuit, further introduction... A number of flag qubits are used, and a comparator is employed. Record whether the benefit is triggered on each observation day. Then, apply controlled... Control bits are The observation date is marked by the qubit, and the control bit state is... The rotating door is activated at the specified time. Finally, a linear function is encoded. The function. (Appendix) Figure 8 Section (c) illustrates the loop of this encoding operation. The encoded bit measurement results are as follows: The probability is: .

[0079] 6. Results Display: This invention uses an Asian option as an example, assuming the initial price is... The ground state is calculated using 3 qubits. Using the attached Figure 4 The circuit loading price path distribution shown is combined with the attached... Figure 8 In section (a), the payoff function is embedded in the amplitude. In this implementation, a 5-qubit register is introduced to record the summation result. Figure 9 A comparison of the probability distributions of the discrete price path sums is presented. It can be seen that although there are some deviations at individual discrete values, both maintain consistency in the overall distribution pattern and the central tendency of probability quality. The positions of the main probability intervals and the decay trend are basically the same, indicating that the path distribution loading method proposed in this invention can effectively characterize the probability distribution trend of the discrete price path sums. Let the strike price... At this time there is That is, the comparator threshold is set to The circuit structure is shown in the attached figure. Figure 6 As shown. When encoding the revenue function, let... Finally, the expected return was estimated using maximum likelihood quantum amplitude estimation, and the results are shown in the appendix. Figure 10 As shown. Figure 10 by The horizontal axis represents the number of Oracle calls, illustrating different... The following is an estimated distribution of the expected value of the Asian option payoff function. The red dashed line represents the distribution obtained using the large-scale classical Monte Carlo method (number of samplings). The reference value was obtained from... Figure 10 It is evident that the mean of the quantum amplitude estimate is close to the classical reference value, and the distribution follows... The increased convergence and significant narrowing indicate that the algorithm of this invention can obtain a stable and controllable accuracy estimate of expected returns with fewer Oracle calls, far fewer than the number of samplings required by the classic Monte Carlo method.

[0080] As described in the above embodiments, those skilled in the art will understand that this invention proposes a quantum algorithm for path-dependent option pricing based on quantum amplitude estimation. First, a probability distribution conditional generation quantum loop controlled by a computational ground state, consisting of a condition register and a distribution register, is constructed. The current price is encoded through the condition register, and the distribution register is used to generate the price distribution for the next time point. Second, a quantum state representing the complete price path probability distribution is prepared by recursively calling the conditional evolution operator. Next, payoff functions are encoded for different types of path-dependent options. Finally, the probability of the auxiliary qubit encoding the payoff being in a specific state is estimated using a quantum amplitude estimation algorithm, and the expected value of the option payoff is obtained through linear mapping. This invention solves the technical challenges of high-dimensional distribution loading and complex payoff function encoding for path-dependent options in quantum computing, providing an efficient quantum solution for pricing financial derivatives.

[0081] Furthermore, refer to Figure 11 As shown, this embodiment of the invention also provides an electronic device that can perform the above-described method. The electronic device may include a processor 10, a memory 11, a communication bus 12, and a communication interface 13, and may also include a computer program stored in the memory 11 and capable of running on the processor 10.

[0082] In some embodiments, the processor 10 may be composed of integrated circuits, such as a single packaged integrated circuit or multiple integrated circuits packaged with the same or different functions, including combinations of one or more central processing units, microprocessors, digital processing chips, graphics processors, and various control chips. The processor 10 is the control core of the electronic device, connecting various components of the entire electronic device through various interfaces and lines. It executes programs or modules stored in the memory 11 and calls data stored in the memory 11 to perform various functions and process data within the electronic device.

[0083] Those skilled in the art will understand that embodiments of the present invention can be provided as methods, systems, electronic devices, or computer program products, etc. Therefore, the present invention can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, the present invention can take the form of a computer program product embodied on one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0084] It should be noted that the word "comprising" does not exclude the presence of components or steps not listed in the claims. The words "a" or "an" preceding a component do not exclude the presence of a plurality of such components. This invention can be implemented by means of hardware comprising several different components and by means of a suitably programmed computer.

[0085] The various embodiments in this specification are described in a progressive manner, with each embodiment focusing on the differences from other embodiments. The same or similar parts between the various embodiments can be referred to each other.

[0086] The above description of the disclosed embodiments enables those skilled in the art to implement or use the present invention. The above description is merely a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the technical scope disclosed in the present invention should be included within the scope of protection of the present invention.

Claims

1. A quantum algorithm for path-dependent option pricing based on quantum amplitude estimation, characterized in that, Includes the following steps: S1. Construct a quantum circuit that generates the probability distribution condition for the ground state control, and control the evolution to generate the price distribution at the next time point by calculating the current price encoded in the ground state as a condition; S2. By recursively calling the conditional evolution operator, the quantum state corresponding to the price path probability distribution is prepared; S3. Encode the payoff function for different option types, where: For Asian options, the price at each time point along the price path is accumulated by a quantum adder to obtain the distribution of the price sum; then a comparator is used to determine the relationship between the price sum and the strike price, and the corresponding piecewise linear payoff function is encoded into the amplitude of the auxiliary qubit. For barrier options, a comparator records whether the price crosses the barrier at each time point, and a multi-bit control is used to encode the non-gate trigger condition into the flag bit to control the activation of the payout encoding operator. For automatic redemption options, an observation date judgment is introduced based on barrier options, and a comparator is used to mark whether the early redemption conditions are met on each observation date. S4. Use the quantum amplitude estimation algorithm to obtain the probability that the auxiliary bit of the encoded payoff function value is in a specific state, and then obtain the expected option payoff after processing by linear mapping.

2. The quantum algorithm for path-dependent option pricing based on quantum amplitude estimation according to claim 1, characterized in that, In S1, the conditional generation quantum circuit includes a condition register and a distribution register, wherein: the condition register is used to encode the price at the current time point by calculating the ground state; the distribution register is used to generate the price distribution at the next time point under the control of the condition register, and the measurement probability of each calculated ground state in the generated distribution corresponds to the probability of the price occurring.

3. The quantum algorithm for path-dependent option pricing based on quantum amplitude estimation according to claim 2, characterized in that, In S2, the evolution operator is a parameterized operator, with the loss between the generated distribution and the target distribution as the objective function, and an optimizer is used to adjust the parameters.

4. The quantum algorithm for path-dependent option pricing based on quantum amplitude estimation according to claim 3, characterized in that, In step S2, the quantum state corresponding to the price path probability distribution is prepared by: using the register with the loaded price distribution as a condition register to control the generation of the distribution at the next time point; and by introducing a new register and reusing the condition generation operator to realize the recursive generation of the price path probability distribution.

5. The quantum algorithm for path-dependent option pricing based on quantum amplitude estimation according to claim 1, characterized in that, In step S3, the adder applies multiple +1 operators based on the bits that are 1 in the binary representation of the addend, thereby performing a modular addition operation on the value represented by the quantum state; the comparator applies multiple bit control gates based on the bits that are 0 in the binary representation of a given threshold, and activates the NOT gates when the high-order bits of the 0 bit match the high-order bits of the register and the current bit of the register is 1, thereby performing a conditional judgment.

6. The quantum algorithm for path-dependent option pricing based on quantum amplitude estimation according to claim 1, characterized in that, In S4, the quantum amplitude estimation algorithm includes any one of the following: standard quantum amplitude estimation, iterative quantum amplitude estimation, and maximum likelihood quantum amplitude estimation.

7. The quantum algorithm for path-dependent option pricing based on quantum amplitude estimation according to claim 1, characterized in that, In step S4, a state measurement may be performed using quantum tomography to estimate the state of the auxiliary bit. The probability of a state.

8. An electronic device, characterized in that, It includes a processor and a memory, the memory storing machine-executable instructions that can be executed by the processor, the processor executing the machine-executable instructions to implement a quantum algorithm capable of executing any one of claims 1 to 7.

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