The amplitude encoding of an arbitrary $n$ -qubit state vector requires $Ω(2^n)$ gate operations, owing to the exponential dimension of the Hilbert space. We can, however, form dimensionality-reduced representations of quantum states using matrix product states (MPS). In this article, we illustrate that MPS techniques enable the preparation of quantum states representative of functions with complexity up to low-degree piecewise polynomials via shallow-depth quantum circuits with accuracy exceeding 99.99\%. We extend these results to the approximate amplitude encoding of pixel values. We showcase this approach by efficiently preparing a $128\times 128$ ChestMNIST medical image (https://medmnist.com/) on 14 qubits with fidelity exceeding 99.2\% on a circuit with a total depth of just 425 single-qubit rotation and CNOT gates.The amplitude encoding of an arbitrary $n$-qubit state vector requires $Ω(2^n)$ gate operations, owing to the exponential dimension of the Hilbert space. We can, however, form dimensionality-reduced representations of quantum states using matrix product states (MPS). In this article, we illustrate that MPS techniques enable the preparation of quantum states representative of functions with complexity up to low-degree piecewise polynomials via shallow-depth quantum circuits with accuracy exceeding 99.99\%. We extend these results to the approximate amplitude encoding of pixel values. We showcase this approach by efficiently preparing a $128\times 128$ ChestMNIST medical image (https://medmnist.com/) on 14 qubits with fidelity exceeding 99.2\% on a circuit with a total depth of just 425 single-qubit rotation and CNOT gates.

arXiv:2502.16464Feb 23, 20250 citations

Quantum Image Representations

Fei Yan, Salvador Elías Venegas-Andraca

Springer eBooksJan 1, 20204 citations

Representation Theorems for Matrix Product States

Erdong Guo, David Draper

Abstract. In this work, we investigate the universal representational capacity of matrix product states (MPSs) in the context of Boolean and continuous functions. We show that MPSs can precisely realize arbitrary Boolean functions by presenting a constructive method for designing MPS representations of Boolean gates. Furthermore, we prove that the function space of MPSs, equipped with scale-invariant sigmoidal activation functions, is dense in the space of continuous functions on compact subsets of [Formula: see text]. By examining the connection between MPSs and neural networks (NNs), we establish that an MPS with scale-invariant sigmoidal activation is equivalent to a one-hidden-layer NN equipped with kernel functions. For specific MPS models, we construct equivalent NNs and show that nonlinear kernels, such as polynomial kernels coupling input components, arise naturally in these representations. Finally, we analyze the realization of Gaussian processes via infinitely wide MPSs, leveraging their equivalence to NNs. Our analysis of MPSs highlights the critical role of the kernel function in MPSs, as it encodes the correlations between different components of the input data. Its careful design directly influences the model’s representational capacity.Abstract. In this work, we investigate the universal representational capacity of matrix product states (MPSs) in the context of Boolean and continuous functions. We show that MPSs can precisely realize arbitrary Boolean functions by presenting a constructive method for designing MPS representations of Boolean gates. Furthermore, we prove that the function space of MPSs, equipped with scale-invariant sigmoidal activation functions, is dense in the space of continuous functions on compact subsets of [Formula: see text]. By examining the connection between MPSs and neural networks (NNs), we establish that an MPS with scale-invariant sigmoidal activation is equivalent to a one-hidden-layer NN equipped with kernel functions. For specific MPS models, we construct equivalent NNs and show that nonlinear kernels, such as polynomial kernels coupling input components, arise naturally in these representations. Finally, we analyze the realization of Gaussian processes via infinitely wide MPSs, leveraging their equivalence to NNs. Our analysis of MPSs highlights the critical role of the kernel function in MPSs, as it encodes the correlations between different components of the input data. Its careful design directly influences the model’s representational capacity.

SIAM Journal on Mathematics of Data ScienceOct 13, 20251 citation

A survey of quantum image representations

Fei Yan, Abdullah M. Iliyasu, Salvador Elías Venegas-Andraca

Quantum Information ProcessingDec 11, 2015311 citations

Quantum Image Operations

Fei Yan, Salvador Elías Venegas-Andraca

Springer eBooksJan 1, 20201 citation

Parent Hamiltonian for fully augmented matrix product states

Xiangjian Qian, Mingpu Qin

Density matrix renormalization group (DMRG) or matrix product states (MPS) is the most effective and accurate method for studying one-dimensional quantum many-body systems. However, the application of DMRG to two-dimensional systems is not as successful because of the limited entanglement encoded in the wave-function ansatz. The fully augmented matrix product states (FAMPS), introduced recently in [Chin. Phys. Lett. 40, 057102 (2023)], extends MPS formalism to two dimensions and increases the entanglement in the wave-function ansatz, representing a significant advance in the simulation of two-dimensional quantum many-body physics. In the study of one-dimensional systems, the concept of a parent Hamiltonian for MPS has proven pivotal in the understanding of quantum entanglement. In this work, we extend this framework to two-dimensional systems. We illustrate the procedure to construct a two-dimensional Hamiltonian with given FAMPS as its exact ground state (the parent Hamiltonian for FAMPS). Additionally, through numerical simulations, we demonstrate the effectiveness of the algorithm outlined in [Chin. Phys. Lett. 40, 057102 (2023)] in precisely identifying the FAMPS for the constructed parent Hamiltonian. The introduction of FAMPS and its associated parent Hamiltonian provides a useful framework for the future investigations of two-dimensional quantum many-body systems.Density matrix renormalization group (DMRG) or matrix product states (MPS) is the most effective and accurate method for studying one-dimensional quantum many-body systems. However, the application of DMRG to two-dimensional systems is not as successful because of the limited entanglement encoded in the wave-function ansatz. The fully augmented matrix product states (FAMPS), introduced recently in [Chin. Phys. Lett. 40, 057102 (2023)], extends MPS formalism to two dimensions and increases the entanglement in the wave-function ansatz, representing a significant advance in the simulation of two-dimensional quantum many-body physics. In the study of one-dimensional systems, the concept of a parent Hamiltonian for MPS has proven pivotal in the understanding of quantum entanglement. In this work, we extend this framework to two-dimensional systems. We illustrate the procedure to construct a two-dimensional Hamiltonian with given FAMPS as its exact ground state (the parent Hamiltonian for FAMPS). Additionally, through numerical simulations, we demonstrate the effectiveness of the algorithm outlined in [Chin. Phys. Lett. 40, 057102 (2023)] in precisely identifying the FAMPS for the constructed parent Hamiltonian. The introduction of FAMPS and its associated parent Hamiltonian provides a useful framework for the future investigations of two-dimensional quantum many-body systems.

arXiv:2401.07659Physical review. B./Physical review. BSep 27, 20241 citation

The amplitude encoding of an arbitrary

n

-qubit state vector requires

Ω(2^n)

gate operations, owing to the exponential dimension of the Hilbert space. We can, however, form dimensionality-reduced representations of quantum states using matrix product states (MPS). In this article, we illustrate that MPS techniques enable the preparation of quantum states representative of functions with complexity up to low-degree piecewise polynomials via shallow-depth quantum circuits with accuracy exceeding 99.99\%. We extend these results to the approximate amplitude encoding of pixel values. We showcase this approach by efficiently preparing a

128\times 128

ChestMNIST medical image (https://medmnist.com/) on 14 qubits with fidelity exceeding 99.2\% on a circuit with a total depth of just 425 single-qubit rotation and CNOT gates.