Variational Quantum Algorithm for Schmidt Decomposition

Abstract

Entanglement plays a crucial role in quantum physics and is the key resource in quantum information processing. In entanglement theory, Schmidt decomposition is a powerful tool to analyze the fundamental properties and structure of quantum entanglement. This work introduces a hybrid quantum-classical algorithm for Schmidt decomposition of bipartite pure states on near-term quantum devices. First, we show that the Schmidt decomposition task could be accomplished by maximizing a cost function utilizing bi-local quantum neural networks. Based on this, we propose a variational quantum algorithm for Schmidt decomposition (named VQASD) of which the cost function evaluation notably requires only one estimate of expectation with no extra copies of the input state. In this sense, VQASD outperforms existent approaches in resource cost and hardware efficiency. Second, by further exploring VQASD, we introduce a variational quantum algorithm to estimate the logarithm negativity, which can be applied to efficiently quantify entanglement of bipartite pure states. Third, we experimentally implement our algorithm on Quantum Leaf using the IoP CAS superconducting quantum processor. Both experimental implementations and numerical simulations exhibit the validity and practicality of our methods for analyzing and quantifying entanglement on near-term quantum devices.

Modularized and Scalable Compilation for Double Quantum Dot Quatum Computing

“…Compared to schemes such as [32][33][34] where ansatz's parameters are updated in a closed-loop style, the application of random states reduces the susceptibility of the optimization to local optima akin to the usage of random samples in the training landscape of classical neural networks [26]. In addition, this utilization of quantum states is more resource efficient than commonly used approaches based on matrices in loss calculation, such as the Hilbert-Schmidt distance or other customized metrics between the associated unitaries [32][33][34]. It is worth noting that this training approach is scalable and applicable generally to the construction or decomposition of unitary transformations with any number of qubits.…”

“…In this paper, we enlist another emerging and powerful technique to compile a desired unitary into a native gate sequence without changed length -the variational quantum algorithm (VQA) [31], which has recently been used for decomposing complex unitary transformations into ordered universal gates [32,33] or Schmidt decomposition [34], etc.…”

“…Additionally, we suggest training the Ansatz, a circuit of parametric native gates, with random quantum states. Compared to the widely used matrix-based approaches [32][33][34], this training scheme offers reduced overhead in loss calculation and smaller possibility of being stucked in local optimums during optimization. Considering singlequbit rotations and entangling two-qubit gates are crucial ingredients for universal quantum computation, a series of them are compiled and high fidelities are reached, underling the foundations for accurately implementing quantum programs in DQDs.…”

Modularized and scalable compilation for double quantum dot quantum computing

“…In this paper, we enlist another emerging and powerful technique to compile a desired unitary into a native gate sequence without changed length-the variational quantum algorithm (VQA) [31], which has recently been used for decomposing complex unitary transformations into ordered universal gates [32,33] or Schmidt decomposition [34], etc. Utilizing the adjoint differentiation [31], the VQA permits gradients collection of all parameters with only one forward pass and one recursive backward pass [31]-a significant calculation saving compared to traditional gradient-based methods.…”

Fundamental limitations on optimization in variational quantum algorithms

“…The hybrid quantum-classical computation framework, including variational quantum algorithms (VQAs) [10][11][12][13], is widely believed to be promising in making use of NISQ devices to deliver meaningful quantum applications. Specifically, VQAs use classical optimizers to train a parameterized quantum circuit (PQC) in order to solve problems in various topics such as ground state preparation [14], quantum linear algebra [15][16][17][18], quantum metrology [19][20][21], quantum entanglement [22][23][24][25], and machine learning [26][27][28].…”