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IEEE International Conference on Quantum Computing and Engineering (QCE)

Optimization of non-Gaussian state generation using tensor networks and automatic differentiation

Ryutaro Nagai, Takao Tomono

Published Sep 1, 2022• 1 citation
Publisher

Abstract

Tensor networks have been eagerly developed in recent years as a simulation method for quantum computation on classical computers. The efficiency of tensor networks for simulating quantum computation has been shown by solving a task designed for demonstrating quantum computing supremacy. However, such a task is not useful for real application. Therefore, there is the room for research of finding specific applications suitable for exploiting tensor network simulation. As such an application, we propose optimization of circuits to generate non- Gaussian states. Non-Gaussian state is one of the crucial elements for achieving universal continuous variable quantum computation. We consider this problem as a variational optimization of optical circuits to generate desired non-Gaussian states. Tensor networks efficiently simulate the optical circuits, and their compatibility with automatic differentiation provides a heuristic way to optimize the circuits. We propose and report the result of our approach that combines tensor networks and automatic differentiation to perform variational optimization of circuits such that the desired non-Gaussian state is approximately generated. The complexity of contracting tensor networks is scaled by degree of entanglement, so we consider relatively shallow optical circuit. Such a restriction is beneficial for considering devices without error correction method, which are accessible in near-term.

DOI
10.1109/qce53715.2022.00130

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Tensor networks have been eagerly developed in recent years as a simulation method for quantum computation on classical computers. The efficiency of tensor networks for simulating quantum computation has been shown by solving a task designed for demonstrating quantum computing supremacy. However, such a task is not useful for real application. Therefore, there is the room for research of finding specific applications suitable for exploiting tensor network simulation. As such an application, we propose optimization of circuits to generate non- Gaussian states. Non-Gaussian state is one of the crucial elements for achieving universal continuous variable quantum computation. We consider this problem as a variational optimization of optical circuits to generate desired non-Gaussian states. Tensor networks efficiently simulate the optical circuits, and their compatibility with automatic differentiation provides a heuristic way to optimize the circuits. We propose and report the result of our approach that combines tensor networks and automatic differentiation to perform variational optimization of circuits such that the desired non-Gaussian state is approximately generated. The complexity of contracting tensor networks is scaled by degree of entanglement, so we consider relatively shallow optical circuit. Such a restriction is beneficial for considering devices without error correction method, which are accessible in near-term.
The study of quantum circuit simulation using classical computers is a key research topic that helps define the boundary of verifiable quantum advantage, solve quantum many-body problems, and inform development of quantum hardware and software. Tensor networks have become forefront mathematical tools for these tasks. Here we introduce a method to approximately simulate quantum circuits using sparsely-populated tensors. We describe a sparse tensor data structure that can represent quantum states with no underlying symmetry, and outline algorithms to efficiently contract and truncate these tensors. We show that the data structure and contraction algorithm are efficient, leading to expected runtime scalings versus qubit number and circuit depth. Our results motivate future research in optimization of sparse tensor networks for quantum simulation.
Converting classical information into a quantum circuit is an inevitable step in utilizing quantum computing. Current quantum computers are referred to as NISQ devices, which have a high level of noise without error correction, imposing strict limitations on the number of qubits and quantum gates. Therefore, when constructing quantum circuits, it becomes crucial to generate shallower circuits. Specifically, when amplitude encoding is employed for classical-quantum information conversion, analytical methods often necessitate an exponential number of quantum gates relative to the number of qubits, rendering them unsuitable for current NISQ devices. Here, we explore a method for constructing an approximately improved circuit for amplitude encoding, considering classical information from an array composed of floating-point numbers. In our method, we optimize the matrix elements of a two-qubit unitary gate using a technique similar to that used in the tensor network method. We then analytically decompose the obtained unitary matrix into a product of elementary unitary gates. Furthermore, using this as an element of the basic optimization technique, we explore a method for selectively adapting better quantum gates. We apply this method to the problem of constructing a circuit that approximately generates the ground state of the quantum spin model. Consequently, we find that better circuits can be automatically constructed without prior knowledge of classical information.
Parametrized quantum optical circuits are a class of quantum circuits in which the carriers of quantum information are photons and the gates are optical transformations. Classically optimizing these circuits is challenging due to the infinite dimensionality of the photon number vector space that is associated to each optical mode. Truncating the space dimension is unavoidable, and it can lead to incorrect results if the gates populate photon number states beyond the cutoff. To tackle this issue, we present an algorithm that is orders of magnitude faster than the current state of the art, to recursively compute the exact matrix elements of Gaussian operators and their gradient with respect to a parametrization. These operators, when augmented with a non-Gaussian transformation such as the Kerr gate, achieve universal quantum computation. Our approach brings two advantages: first, by computing the matrix elements of Gaussian operators directly, we don't need to construct them by combining several other operators; second, we can use any variant of the gradient descent algorithm by plugging our gradients into an automatic differentiation framework such as TensorFlow or PyTorch. Our results will find applications in quantum optical hardware research, quantum machine learning, optical data processing, device discovery and device design.
Efficient quantum circuit optimization schemes are central to quantum simulation of strongly interacting quantum many body systems. Here, we present an optimization algorithm which combines machine learning techniques and tensor network methods. The said algorithm runs on classical hardware and finds shallow, accurate quantum circuits by minimizing scalar cost functions. The gradients relevant for the optimization process are computed using the reverse mode automatic differentiation technique implemented on top of the time-evolved block decimation algorithm for matrix product states. A variation of the ADAM optimizer is utilized to perform a gradient descent on the manifolds of charge conserving unitary operators to find the optimal quantum circuit. The efficacy of this approach is demonstrated by finding the ground states of spin chain Hamiltonians for the Ising, three-state Potts and the massive Schwinger models for system sizes up to L=100. The first ten excited states of these models are also obtained for system sizes L=24. All circuits achieve high state fidelities within reasonable CPU time and modest memory requirements.
Tensor networks are highly promising methods for efficiently simulating quantum systems using classical computers. In recent years, the efficiency of tensor networks has been eagerly harnessed to simulate NISQ devices in the era of quantum supremacy. However, as of now, quantum advantage with NISQ devices has not been clearly demonstrated in practical applications. This provide impetus for us to intensify our effort in fault tolerant quantum computation (FTQC). Various physical systems and schemes for FTQC have been proposed, with GKP encoding in bosonic systems being one such example. The preparation of GKP states serves as an important building block for realizing FTQC with the GKP code. We introduce tensor networks for simulating and optimizing the GKP state preparation circuit. It is known that the generation of approximated GKP states is achievable through photon counting and post-selection applied to properly prepared multimode Gaussian states. However, it requires optimization of numerous circuit parameters, which is typically computationally challenging on classical computers due to the involvement of the photon counting measurement process. We attempt to utilize tensor networks for more efficient parameter optimization. Additionally, we explore further efficient approach in terms of tensor network structure. We propose a multi-cutoff dimension approach combined with a tree tensor network structure.
The continuous variable quantum computing platform constitutes a promising candidate for realizing quantum advantage, as exemplified in Gaussian Boson Sampling. While noise in the experiments makes the computation attainable for classical simulations, it has been suggested that the addition of non-linear elements to the experiment will help retain the quantum advantage. We propose a classical simulation tool for general continuous variable sampling problems, including Gaussian Boson Sampling and beyond. We reformulate the sampling problem as that of finding the ground state of a simple few-body Hamiltonian. This allows us to employ powerful variational methods based on tensor networks and to read off the simulation error directly from the expectation value of the Hamiltonian. We validate our method by simulating Gaussian Boson Sampling, where we achieve results comparable to the state of the art. We also consider a non-Gaussian sampling problem, for which we develop novel local basis optimization techniques based on a non-linear parameterization of the implicit basis, resulting in high effective cutoffs with diminished computational overhead.