Conference on Lasers and Electro-Optics Europe & European Quantum Electronics Conference (CLEO/Europe-EQEC)
Parameter Shift Rule for Variational Photonic Quantum Circuit
Francesco Hoch, Giovanni Rodari, Taira Giordani, Paul Perret, Nicolò Spagnolo, Gonzalo Carvacho, Ciro Pentangelo, Simone Piacentini, Andrea Crespi, Francesco Ceccarelli, Roberto Osellame, Fabio Sciarrino
In recent years, variational quantum algorithms (VQA) have been receiving a great deal of attention, demonstrating their applicability in a wide range of tasks and implementable even in the noisy quantum processors currently available [1]. A VQA is a hybrid quantum-classical algorithm where a problem is encoded in a cost function that can be efficiently evaluated on a parametric quantum device and the minimum represents the desired solution. A classical optimizer can then be used to tune the parameters of the quantum device in order to minimize the loss function and find the optimal solution (Fig. 1). In the field of quantum computing, one of the main promising platforms is based on the use of single photons and integrated photonic circuits since photons are a natural choice for transferring information and have different degree of freedom that can be to encode the information. Moreover the use of integrated photonic allow the realization of compact and table complex interferometers with a high number of control parameters. Even if there are few applications [2], [3], VQA with photonic platforms remains a relatively unexplored subject. One of the main problems is the application of the classical optimizer to the photonic platform since most of the algorithms are based on the estimation of the gradient of the cost function with the finite difference method but that can not be applied to a noisy apparatus.
In recent years, variational quantum algorithms (VQA) have been receiving a great deal of attention, demonstrating their applicability in a wide range of tasks and implementable even in the noisy quantum processors currently available [1]. A VQA is a hybrid quantum-classical algorithm where a problem is encoded in a cost function that can be efficiently evaluated on a parametric quantum device and the minimum represents the desired solution. A classical optimizer can then be used to tune the parameters of the quantum device in order to minimize the loss function and find the optimal solution (Fig. 1). In the field of quantum computing, one of the main promising platforms is based on the use of single photons and integrated photonic circuits since photons are a natural choice for transferring information and have different degree of freedom that can be to encode the information. Moreover the use of integrated photonic allow the realization of compact and table complex interferometers with a high number of control parameters. Even if there are few applications [2], [3], VQA with photonic platforms remains a relatively unexplored subject. One of the main problems is the application of the classical optimizer to the photonic platform since most of the algorithms are based on the estimation of the gradient of the cost function with the finite difference method but that can not be applied to a noisy apparatus.
In the era of noisy intermediate-scale quantum computers, variational quantum algorithms are promising approaches for solving optimization tasks by training parameterized quantum circuits with the aid of classical routines informed by quantum measurements. In this context, photonic platforms based on reconfigurable integrated optics are an ideal candidate for implementing these algorithms. Among various techniques to train variational circuits, the parameter shift rule enables the exact calculation of cost function derivatives efficiently, facilitating gradient descent-based optimization. In this paper, we derive a formulation of the parameter shift rule for computing derivatives and integrals tailored to reconfigurable optical linear circuits and based on the Boson Sampling paradigm. This allows us to naturally embed common types of experimental noise, such as partial distinguishability and mixedness of the states, thus obtaining a resilient approach. Finally, we employ the developed approach to experimentally test variational algorithms with single-photon states processed in a reconfigurable six-mode universal integrated interferometer. Specifically, we apply the photonic parameter shift rules to the variational implementation, on a photonic platform, of both an eigensolver and a Universal- gate.
Abstract Variational quantum metrology represents a powerful tool to optimize estimation strategies, resulting particularly beneficial for multiparameter estimation problems that often suffer from limitations due to the curse of dimensionality and computational complexity. To overcome these challenges, we develop a variational approach able to efficiently optimize a quantum multiphase sensor. Leveraging the reconfigurability of an integrated photonic device, we implement a hybrid quantum-classical feedback loop able to enhance the estimation performances. The quantum circuit evaluations are used to compute the system partial derivatives by applying the parameter-shift rule, and thus reconstruct experimentally the Fisher information matrix. This in turn is adopted as the cost function of a classical learning algorithm run to optimize the measurement settings. Our experimental results showcase significant improvements in estimation accuracy and noise robustness, highlighting the potential of variational techniques for practical applications in quantum sensing and more generally in quantum information processing using photonic circuits.
In recent years, as datasets have grown larger, quantum computing has shown potential in addressing the rapidly increasing costs of data processing due to its superposition and entanglement capabilities. Limited by quantum hardware and the number of controllable qubits, variational quantum algorithms (VQAs) are algorithms suitable for NISQ devices, capable of computation on medium-scale noisy devices. VQAs are hybrid classical-quantum algorithms that use parameterized quantum circuits to prepare parameterized quantum states and employ classical optimizers to minimize objective functions, enabling efficient solutions for complex problems. Classical optimizers reduce the stringent requirements on quantum hardware and can achieve quantum advantage on current devices. This paper systematically elaborates on the core foundational theories of quantum computing and neural networks, with a focus on reviewing the research and application status of VQAs in three main scenarios: machine learning (supervised, unsupervised, and reinforcement learning), solving systems of equations (linear and nonlinear), and network attack detection. It analyzes the current technical bottlenecks in this field and provides prospects for future development, offering a reference for further research on variational quantum algorithms.
We present a method for gradient computation in quantum algorithms implemented on linear optical quantum computing platforms. While parameter-shift rules have become a staple in qubit gate-based quantum computing for calculating gradients, their direct application to photonic platforms has been hindered by the infinite dimensionality of phase-shift operators in Fock space. We introduce a photonic parameter-shift rule that overcomes this limitation, providing an exact formula for gradient computation in linear optical quantum processors. Our method scales linearly with the number of input photons and utilizes the same parametrized photonic circuit with shifted parameters for each evaluation. This advancement bridges a crucial gap in photonic quantum computing, enabling efficient gradient-based optimization for variational quantum algorithms on near-term photonic quantum processors. We demonstrate the efficacy of our approach through numerical simulations in quantum chemistry and generative modeling tasks, showing superior optimization performance as well as robustness to noise from finite sampling and photon distinguishability compared to other gradient-based and gradient-free methods.
Variational quantum algorithms (VQAs) offer some promising characteristics for carrying out optimization tasks in noisy intermediate-scale quantum devices. These algorithms aim to minimize a cost function by optimizing the parameters of a quantum parametric circuit. Thus, the overall performance of these algorithms, heavily depends on the classical optimizer which sets the parameters. In the last years, some gradient-based and gradient-free approaches have been applied to optimize the parameters of the quantum circuit. In this work, we follow the second approach and propose the use of estimation of distribution algorithms for the parameter optimization in a specific case of VQAs, the quantum approximate optimization algorithm. Our results show an statistically significant improvement of the cost function minimization compared to traditional optimizers.
Parameterized quantum circuits (PQCs) play an essential role in the application of variational quantum algorithms (VQAs) in noisy intermediate-scale quantum (NISQ) devices. The PQCs are a leading candidate to achieve a quantum advantage in NISQ devices and have already been applied in various domains such as quantum chemistry, quantum machine learning, combinatorial optimization, and many others. There is no single definitive way to optimize PQCs. The most commonly used methods are based on computing the gradient via the parameter-shift rule to use classical gradient descent (GD) optimizers like Adam, stochastic GD, and others. In addition, sequential single-qubit optimizers have been proposed, such as Rotosolve, Free-Axis Selection (Fraxis), Free-Quaternion Selection (FQS), and hybrid algorithms from the aforementioned optimizers. We further develop hybrid algorithms than those represented in the previous work by drawing inspiration from the early stopping method used in classical machine learning. The switch between the optimizers depends on the previous cost function values compared to the previous ones. We introduce two new hybrid algorithms that are more robust and scalable, and they outperform previous hybrid methods in terms of convergence towards the global minima across various cost functions. In addition, we find that they are feasible for NISQ devices with different noise profiles.