The pricing of financial derivatives is a prominent problem that requires extensive computational resources, as the stochastic nature of the underlying assets requires precise modeling. One of the typical financial derivatives is the option, which is a contract that allows the holder to buy or sell assets at a pre-established price (strike) at or before a specified date (maturity date). The payoff of an option relies heavily on the stochastic evolution of asset price. The traditional option pricing model, Black–Scholes–Merton (BSM) [1], usually oversimplifies market dynamics, which limits its practical application to real-life scenarios. As such, numerical methods such as the Monte Carlo method are typically employed for handling more realistic stochastic fluctuations. However, the Monte Carlo method requires extensive computation resources and is slow to predict complicated options. Reducing the computational resources required for models and speeding up option pricing could have significant implications for the financial industry.

Recently, quantum algorithms have shown promise in facilitating computationally hard financial problems such as trading, portfolio optimization, and risk profiling [2,3], and specifically, quantum amplitude amplification can accelerate option pricing with quadratic speedups [4–10]. The unique advantages of quantum algorithms will make up for the shortcomings of classical algorithms to a certain extent, enabling massive high-speed data services in the financial industry. However, current experimental demonstrations using binary approaches and standard quantum circuit models on superconducting devices [11] require dense chip connections and high gate fidelity, making it difficult for practical applications in the near future without a universal quantum computer [12,13]. In addition, superconducting devices require bulky, energy-intensive, and expensive peripherals such as cooling systems, making industrial-scale applications have poor prospects.

For specialized application tasks such as option pricing, there is no need to use universal quantum computers. Photonic circuits can provide fundamental functions that can be combined to implement specific algorithms [14–20], which would be practical and efficient for user-cased application scenarios. Moreover, the reduced energy costs of photonic computing have been a driving force behind works on dedicated photonic chips for machine learning and algebra [16,21–24]. Therefore, we demonstrate a unary (versus binary) approach in a photonic chip for option pricing. Compared to the binary approach, the unary approach [9] has a remarkably simplified structure and reduced depth of quantum circuits and is especially suitable for linear optical circuit realizations in photonic chips. The unary scheme also allows a post-selection strategy for error mitigation. Additionally, we demonstrate generative adversarial learning to upload the probability distribution implicitly given by data samples into the photonic chip. Generative adversarial learning has previously been demonstrated only in superconducting and optoelectronics devices [25–30]. Compared with traditional Monte Carlo methods, our approach shows high accuracy and significant speedup. It provides a promising avenue for interdisciplinary research in quantum machine learning and financial problems, paving the way for the development of practical photonic processors for quantitative financial applications. It can greatly improve the efficiency and quality of financial services, which is of great significance to the rapid and steady development of the financial industry.

In this work, we focus on European option pricing, and the expected payoff of options is given by C(ST,K)=∫K∞(ST−K)dST,(1)where ST is the asset price at time T, and K is the strike price; see European option pricing model in Appendix A. Figure 1 shows the overall scheme of our photonic-chip-based unary approach. The photonic chip [Fig. 1(a)] consists of a generative adversarial network (GAN) and an option pricing part that includes payoff computation and amplitude estimation. In contrast to the classical Monte Carlo approach [Fig. 1(b)] that requires huge computing power to simulate future asset prices to obtain an accurate solution, our approach is expected to show a speedup in the convergence of the standard error of estimated payoff [Fig. 1(c)], which is proved experimentally later in this section results.

The unary approach to option pricing encodes an asset price distribution into the unary basis of a quantum register, as shown in Fig. 2. A binning scheme is applied such that Monte Carlo paths that would belong to the same interval of asset prices end up in the same bin. Each bin is then mapped to an element of the unary basis, whose coefficient is the ratio of the number of Monte Carlo paths in that bin to the total number. The accuracy of unary encoding is bounded by the number of bins that can be stored in a quantum state, i.e., the usable dimension of the high-dimensional unary state. Based on the unary basis, Fig. 3(a) depicts the algorithmic model for unary option pricing, which consists of three modules: a distribution loading module D that loads the asset price distribution into a quantum state, a payoff calculation module P that computes the expected return, and a quantum amplitude estimation module Q to gain quadratic speedup over classical sampling to reach a target accuracy.

Figure 3(b) depicts the optical circuit model, whereby each module of the unary algorithm is mapped to a linear optical operator. We represent the high-dimensional state by path encoding a single photon using n optical waveguides. The superposition of a single photon traveling through different waveguides directly encodes the unary basis. This high-dimensional state can be written as |ψ⟩=∑i=0n−1pi|i⟩, where pi represents the probability of observing a photon in the waveguide mode |i⟩, and these probabilities conform to ∑i=0n−1pi=1. The payoff calculation requires an ancilla qubit to store the expected return for each asset price, expanding the Hilbert space of the algorithm to 2n. To avoid non-local controlled gates in the photonic chip implementation, we instead add an ancillary waveguide to each of the n unary waveguide modes to represent the effect of the ancilla qubit. Each element of the unary basis is now represented by two waveguides. This way, the controlled operations of the original algorithm are converted to linear transformations on the optical circuit. The architecture of the photonic processor with the detailed chip design is shown in Fig. 3(c), which replaces each linear optical operator with the corresponding waveguide structure. The entire chip is reconfigurable via wire bonds and integrated thermo-optic phase shifters; see the experimental setup in Appendix B.

In the distribution loading module D, a single photon is incident into the chip from a waveguide in the middle of the circuit, which encodes the ancilla in its |0⟩ state, e.g., for a three-asset case, the initial input state can be written as the tensor product of the middle unary qudit and the ancilla qubit as [0,1,0]⊗[1,0]=[0,0,1,0,0,0]. The distribution of asset prices is then uploaded to the different waveguides using a linear depth circuit. This distribution loading circuit spreads the superposition to the neighboring basis using swp operators: swp=(Ip1−p1−p−pI)⊗I,(2)where p depends on the target distribution. The procedure is repeated until the edge of the circuit is reached. The distribution loading module can be reconfigured to obtain any target probability distribution in the unary representation. Precisely, given n assets, the depth of the circuit is always ⌊(n+1)/2⌋, and the loading of any known probability distribution onto the unary basis depends on (n−1) splitting parameters p. The generator of a GAN is embedded in this module. The GAN is employed to capture the probability distribution underlying given market data. The details are presented in the next section.

The payoff calculation module P encodes the expected payoff as the probability of measuring the photon in the waveguides encoding the ancilla in state |1⟩, using rotation operations between the two waveguides of each element of the unary basis. The rotations encode the expected return for each asset price in the distribution. This action, labeled P, can be written as P=(M0M1⋱Mn−1),Mi=(cosθi−sinθisinθicosθi)(3)for a 2n-waveguide, n-bin example.

A quantum amplitude estimation module Q is applied to achieve quantum speedups. Various amplitude estimation techniques have been presented that are friendly to NISQ devices [31–33]. Here, we implement an amplitude estimation algorithm without quantum phase estimation in the photonic circuit, following the technique used in Ref. [9]. Increasing steps of amplitude amplification are applied to estimate the relevant amplitudes with up to a square root advantage oversampling from the original distribution. This amplification module Q is performed by applying the following operators. First, Sψ identifies the amplitudes that encode the expected payoff and reverses their signs. Explicitly, for the three-asset example at hand, such an operation is Sψ=diag(1,−1,1,−1,1,−1), and is realized experimentally by applying a phase shift of π on the second waveguide of each element of the unary basis. Then, the original operations are reversed, that is, the inverse of the payoff calculator P† and the distribution loading D† are applied. An operator S0 follows, which reverses the sign of the initial state of the computation. Experimentally, it is applied by introducing a phase shift of π to the waveguide where the photon was introduced. The last step is to repeat the distribution loading D and the payoff calculator P modules. The amplitude amplification operator Q=P·D·S0·D†·P†·Sψ is repeated a different number of times, and the results are processed to estimate the expected payoff. This technique provides up to quadratic speedups over ordinary sampling in the number of calls to the D and P operators to reach the same confidence level; see theoretical derivations in Appendix C.

A GAN is implemented in the distribution loading module with on-chip training for real-time noise perception. The goal of the GAN is to obtain an intelligent generator at the chip parameter level that captures the probability distribution behind the given market data without simulating enormous random paths, accumulating data statistics, and then fitting them into the chip architecture. With the GAN, we can efficiently load the classical data, i.e., the probability distribution underlying market data, into quantum states and obtain more precise payoff calculations with the presented unary option pricing methods.

GANs train a generator (G) to synthesize semantically meaningful data from standard signal distributions, as well as a discriminator (D) to distinguish real samples in the training dataset from fake ones produced by the generator [34], as depicted in Fig. 4(a). As its adversary, the generator aims at deceiving the discriminator by producing more realistic samples. Training GANs involve the search for a Nash equilibrium of a two-player game between a generative and a discriminative network, which can be formulated as minGmaxDEx∼preal(log(Dϕ(x)))+Ez∼pz(log(1−Dϕ(Gθ(z)))),(4)where the generative network Gθ takes noisy samples z from a normal or uniform distribution pz as input, and x comes from the real distribution preal. The discriminative network Dϕ tries to distinguish the generated (fake) sample Gθ(z) and real sample x by projecting their output to {0,1}. θ and ϕ are free parameters that construct the generator and discriminator. The training procedure is complete when the generator wins the adversarial game, that is, the discriminator cannot make a better decision than random guesses on the validity of a sample.

We develop a hybrid GAN implementation that consists of a generator network in the photonic chip, a classical discriminator network, and a control system that communicates between the classical computer and photonic chip, all depicted in Figs. 4(b)–4(d). The generator is parameterized by the angles on the phase shifters that are reconfigurable through the thermo-optic effect, induced by applying tiny electrical power to the integrated heaters. Instead of a noise distribution as input, we utilize the uncertainty of photons appearing at different waveguide modes to achieve the equivalent randomness for the generator. The fake samples are the probability distribution of the photons at different waveguide modes. The real samples are drawn from the desired probability distribution, a log-normal or normal distribution for the examples presented in Fig. 5. Fake and real samples sequentially enter the classical discriminator to achieve the classification results. The discriminator is a classical neural network implemented with TensorFlow. We then explicitly discuss the training of the GAN with data samples drawn from the log-normal distribution and the normal distribution.

The training process of the GAN in a photonic chip introduces two challenges, the difficulty of obtaining gradients due to the stochastic nature of measurements, and the phenomenon that the discriminator easily overpowers the generator. To circumvent these problems, we propose a hybrid training strategy, where the generator is optimized under a gradient-free evolutionary algorithm, while the classical discriminator uses a gradient descent optimizer. Additionally, the Wasserstein distance [35,36] is used to train the GAN, which changes the dynamic between the generator and the discriminator. In this new GAN scheme, the discriminator acts as a critic, instead of classifying. It aims to give a high score to real instances over fake ones, effectively alleviating the problem of unstable GAN training.

Our chip can accommodate the entire option pricing process of distribution loading, payoff calculation, and amplitude estimation for three option assets. However, training three bins is too trivial to demonstrate the ability to implement GANs on a photonic chip. Here, to demonstrate the GAN, we employ a chip that supports up to eight bins to demonstrate the generation of probability distribution. Figure 5(a) shows the probability distribution of the generator output compared to the real log-normal distribution. Figure 5(b) shows the convergence of the ℓ2 norm between the fake and real samples, of 100 training iterations. For generator output g and real distribution x, the ℓ2 norm is defined as ℓ2=∑i=1m(xi−gi)2.(5)The results for a target normal distribution are shown in Figs. 5(c) and 5(d). For both examples of log-normal and normal distributions, the final ℓ2 norm between the generator output and the real distribution stabilizes at −18dB.

By training this generative model directly on the photonic chip, we bypass the need to solve the BSM equations while capturing the nuances that the simplified method overlooks. Concurrently, it incorporates environmental elements that are hard to model, such as cross talk and chip imperfections into the GAN training. Another feature of using GANs for the amplitude distribution step is that we can tailor the variational ansatz to construct short-depth circuits for a given degree of accuracy, even in a more general case with multiple photons.

As a proof of principle, the fabricated photonic chip supports an option pricing problem with three asset values, whose schematic diagram is shown in Fig. 6(a). The chip has six waveguide inputs: each pair represents one element of the unary basis and the ancilla qubit state. The chip is divided into distribution loading, payoff calculation, and then m runs of amplitude amplification. To stay within the depth constraints of this proof of concept, the circuit’s unitary matrix is multiplied and uploaded into the photonic chip at a constant depth. The single-photon measurement is performed at the waveguide modes that represent ancilla state |1⟩ for asset prices larger than the strike value. The comparison between the theoretical payoff and estimation achieved experimentally is shown in Fig. 6(b), with increasing iterations of amplitude estimation. The performance of the amplitude estimation is shown in Figs. 6(c) and 6(d). In Fig. 6(c), the dotted line represents the theoretical payoff expectation, the solid line with data points represents the experimental results, and the shaded area represents the standard deviation (STD) of 50 measurements performed in each step of amplitude estimation. The progression of m from 0 to 50 (m=0 being a classical sampling of payoff calculation) demonstrates the convergence of the STD. Similarly, in Fig. 6(d), we visualize the convergence of the payoff error with more amplitude estimation runs. The amplitude estimation improves the accuracy of the expected payoff for a certain number of circuit runs.

The structure of the unary algorithm allows a simple but efficient design of photonic chips, especially when loading probability distributions into quantum registers since only local interactions between neighboring waveguides are required. This, however, is inaccessible for the binary alternative, where high connectivity is required to offset the exponential Hilbert space available; see the unary and binary comparison in Appendix D. The optical circuit that implements the unary approach requires a linear number of waveguides scaling with the required precision, which coincides with the remarkable scalability of photonic chips. Instead, avoiding the use of controlled operations via ancilla waveguides bypasses one of the main bottlenecks of photonic chips in quantum computing: the obstacle of realizing photon interactions.

Speedup is achieved in our work much akin to proposals for quantum search without entanglement [37,38], whereby polynomial speedup of unstructured search is achieved with a single photon at the cost of exponential resources. In particular, isomorphisms exist between a system of n qubits and a qudit residing in a 2n Hilbert space (systems presented in this thought experiment and our implementation) [39]; thereby the unary implementation on a photonic chip displays entanglement in the path encoding of the single photon. Coherent light can achieve a similar effect with a high sampling rate, which is advantageous in near-term use case scenarios, with some trade-offs for the random behavior of single photons in the generator part of the GAN.

The presented avenue to achieve speedup in option pricing is scalable in the photonic chip. It transforms the unary algorithm’s need for increasing qubits into a need for waveguide paths, which are highly scalable in photonic chips. For further scalability, in the presented experiment, the photon detectors placed in ancilla waveguides could be combined into a single one, as only the counts of photons in any ancilla qubit are needed, hence significantly reducing the resources needed to scale this approach to meaningful problems. The energy efficiency of photonic chips also promises a relevant advantage beyond a complexity separation between quantum and classical algorithms. Given an energy budget instead of a shot budget, the photonic implementation of the unary approach to option pricing can yield a significant advantage in the number of operations performed.

This work is the first demonstration of photonic chips for financial applications. As a proof of concept, we implement the unary option pricing algorithm in a photonic chip for European options, which includes the generation of amplitude distribution of the asset value, evaluation of expected return, and amplitude estimation. We prove the high accuracy in calculating the payoff function, as well as the effectiveness of amplitude estimation in reducing the number of evaluations to reach the same degree of accuracy when compared to classical sampling. The unary representation remarkably simplifies the structure and reduces the depth of quantum circuits in the linear optical circuit implementation. Such photonic devices could eventually be an eco-friendly alternative to electronic circuits. Furthermore, we demonstrate an on-chip training of a GAN that successfully captures important market dynamics in real-life scenarios, bypassing the simplified assumptions in the BSM model that limit its accuracy, as well as the computational burden in solving differential equations. The photonic chip could be potentially employed for other options pricing, paving the way for developing dedicated processors in finance applications.