The Schmidt decomposition is the go-to tool for measuring bipartite entanglement of pure quantum states. Similarly, it is possible to study the entangling features of a quantum operation using its operator-Schmidt or tensor-product decomposition. While quantum technological implementations of the former are thoroughly studied, entangling properties on the operator level are harder to extract in the quantum computational framework because of the exponential nature of sample complexity. Here, we present an algorithm for unbalanced partitions into a small subsystem and a large one (the environment) to compute the tensor-product decomposition of a unitary the effect of which on the small subsystem is captured in classical memory, while the effect on the environment is accessible as a quantum resource. This quantum algorithm may be used to make predictions about operator nonlocality and effective open quantum dynamics on a subsystem, as well as for finding low-rank approximations and low-depth compilations of quantum circuit unitaries. We demonstrate the method and its applications on a time-evolution unitary of an isotropic Heisenberg model in two dimensions. Published by the American Physical Society 2024
The Schmidt decomposition is the go-to tool for measuring bipartite entanglement of pure quantum states. Similarly, it is possible to study the entangling features of a quantum operation using its operator-Schmidt or tensor-product decomposition. While quantum technological implementations of the former are thoroughly studied, entangling properties on the operator level are harder to extract in the quantum computational framework because of the exponential nature of sample complexity. Here, we present an algorithm for unbalanced partitions into a small subsystem and a large one (the environment) to compute the tensor-product decomposition of a unitary the effect of which on the small subsystem is captured in classical memory, while the effect on the environment is accessible as a quantum resource. This quantum algorithm may be used to make predictions about operator nonlocality and effective open quantum dynamics on a subsystem, as well as for finding low-rank approximations and low-depth compilations of quantum circuit unitaries. We demonstrate the method and its applications on a time-evolution unitary of an isotropic Heisenberg model in two dimensions. Published by the American Physical Society 2024
Quantum computing is fundamentally based on the mathematical structure of complex Hilbert spaces, linear operators, tensor products, and unitary transformations. Among these concepts, tensor products provide the formal mechanism for constructing the state space of composite quantum systems, whereas linear and unitary operators describe quantum transformations, measurements, and computational gates. This article examines the mathematical relationship between linear operators and tensor products and analyzes their role in the formation, manipulation, and characterization of quantum entanglement. Particular attention is given to two-qubit systems, separable and entangled states, Schmidt decomposition, controlled operations, and the role of non-factorizable operators in quantum circuits. The analysis demonstrates that tensor products perform a dual function: they provide the mathematical representation of composite systems and enable the construction of local operations acting on individual subsystems. At the same time, genuinely entangling transformations, such as the controlled-NOT operation, cannot in general be represented as a single tensor product of independent local operators. Recent research further demonstrates that tensor-product and operator-Schmidt decompositions can be used to characterize operator entanglement, reduce circuit complexity, and construct efficient representations of quantum processes. The results establish that linear algebra and tensor-product structures are not merely formal tools but constitute the mathematical foundation connecting quantum entanglement with quantum computation.
Quantum computing is arguably one of the most revolutionary and disruptive technologies of this century. Due to the ever-increasing number of potential applications as well as the continuing rise in complexity, the development, simulation, optimization, and physical realization of quantum circuits is of utmost importance for designing novel algorithms. We show how matrix product states (MPSs) and matrix product operators (MPOs) can be used to express certain quantum states, quantum gates, and entire quantum circuits as low-rank tensors. This enables the analysis and simulation of complex quantum circuits on classical computers and to gain insight into the underlying structure of the system. We present different examples to demonstrate the advantages of MPO formulations and show that they are more efficient than conventional techniques if the bond dimensions of the wave function representation can be kept small throughout the simulation.
The usefulness of entanglement as a resource in quantum technologies increases for larger systems, that is, if more particles or higher-dimensional quantum systems are considered. Yet, the interplay between dimensionality and multiparticle entanglement is not well understood. Only for two-particle systems an unambiguous and coherent notion of entanglement dimensionality, based on the Schmidt decomposition, is known. We introduce a concept to characterize the entanglement dimensionality of multiparticle states based on decompositions of pure states into superpositions of states without genuine multiparticle entanglement. We provide constructive methods to characterize the resulting partition rank for pure and mixed states. This allows the identification of novel maximally correlated states as well as a discrete classification of quantum states under stochastic local operations and classical communication. From a mathematical perspective, our approach can be formulated in terms of the slice rank and partition rank of tensors and our results allow to characterize these by connecting them to a generalized injective tensor norm.
Efficient methods to access the entanglement of a quantum many-body state, where the complexity generally scales exponentially with the system size N, have long been a concern. Here we propose the Schmidt tensor network state (Schmidt TNS) that efficiently represents the Schmidt decomposition of finite- and even infinite-size quantum states with nontrivial bipartition boundary. The key idea is to represent the Schmidt coefficients (i.e., entanglement spectrum) and transformations in the decomposition to tensor networks (TNs) with linearly scaled complexity versus N. Specifically, the transformations are written as the TNs formed by local unitary tensors, and the Schmidt coefficients are encoded in a positive-definite matrix product state (MPS). Translational invariance can be imposed on the TNs and MPS for the infinite-size cases. The validity of Schmidt TNS is demonstrated by simulating the ground state of the quasi-one-dimensional spin model with geometrical frustration. Our results show that the MPS encoding the Schmidt coefficients is weakly entangled even when the entanglement entropy of the decomposed state is strong. This justifies the efficiency of using MPS to encode the Schmidt coefficients, and promises an exponential speedup on the full-state sampling tasks.
Entanglement plays a crucial role in quantum physics and is the key resource in quantum information processing. However, entanglement detection and quantification are believed to be hard due to the operational impracticality of existing methods. This work proposes three near-term efficient algorithms that exploit the hybrid quantum-classical technique to address this difficulty. The first algorithm finds the Schmidt decomposition---a powerful tool to analyze the properties and structure of entanglement---for bipartite pure states. While the logarithm negativity can be calculated from the Schmidt decomposition, we propose the second algorithm to estimate the logarithm negativity for bipartite pure states, where the width of the parameterized quantum circuits is further reduced. Finally, we generalize our framework for mixed states, leading to our third algorithm that detects entanglement on specific families of states, and determines distillability in general. All three algorithms share a similar framework where the optimizations are accomplished by maximizing a cost function utilizing local parameterized quantum circuits, with better hardware efficiency and practicality compared to existing methods. The experimental implementation on Quantum Leaf using the Institute of Physics, Chinese Academy of Sciences superconducting quantum processor exhibits the validity and practicality of our methods for analyzing and quantifying entanglement on near-term quantum devices.
Estimating global properties of many-body quantum systems such as entropy or bipartite entanglement is a notoriously difficult task, typically requiring a number of measurements or classical postprocessing resources growing exponentially in the system size. In this work, we address the problem of estimating global entropies and mixed-state entanglement via partial-transposed (PT) moments and show that efficient estimation strategies exist under the assumption that all the spatial correlation lengths are finite. Focusing on one-dimensional systems, we identify a set of approximate factorization conditions (AFCs) on the system density matrix, which allow us to reconstruct entropies and PT moments from information on local subsystems. This identification yields a simple and efficient strategy for entropy and entanglement estimation. Our method could be implemented in different ways, depending on how information on local subsystems is extracted. Focusing on randomized measurements providing a practical and common measurement scheme, we prove that our protocol requires only polynomially many measurements and postprocessing operations, assuming that the state to be measured satisfies the AFCs. We prove that the AFCs hold for finite-depth quantum-circuit states and translation-invariant matrix-product density operators and provide numerical evidence that they are satisfied in more general, physically interesting cases, including thermal states of local Hamiltonians. We argue that our method could be practically useful to detect bipartite mixed-state entanglement for large numbers of qubits available in today’s quantum platforms. Published by the American Physical Society 2024