Quantum Encoding of Structured Data with Matrix Product States

TL;DR

The paper addresses the bottleneck that amplitude encoding of an arbitrary $n$-qubit state requires $Ω(2^n)$ gate operations, and shows that Matrix Product States (MPS) provide dimensionality-reduced representations by exploiting entanglement structure. It introduces the Matrix Product Disentangler (MPD) and Tensor Network Optimisation (TNO), composing them into MPD+TNO to produce shallow-depth quantum circuits that approximate target states corresponding to low-degree piecewise polynomials and discretised functions with fidelity exceeding 99.99%. The authors demonstrate approximate amplitude encoding of a $128×128$ ChestMNIST image on $n=14$ qubits with fidelity $F>0.992$ using a total circuit depth of 425 gates, and show that MPD+TNO yields shallower circuits than exact MPS decompositions for comparable fidelity. The results indicate MPD-based MPS encoding is a viable near-term approach for encoding structured data into quantum amplitudes, while more expressive tensor networks may be required for highly unstructured data, with potential applications to quantum image encoding and other structured-data tasks.

Abstract

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.

Figures (11)

• Figure 1: The transition from a highly structured smooth function target vector with low entanglement entropy to fitting a smooth function to an arbitrary unstructured state vector.
• Figure 2: The Grover-Rudolph Algorithm: (a) The resolution of the target state doubles with each Grover-Rudolph iteration and (b) The circuit for the $m^{\text{th}}$ iteration of the algorithm.
• Figure 3: A visualisation of the structure of an open boundary condition MPS for an $n=7$ qubit state. The MPS cores $A^{[j]}$ have physical indices $i_j$ with dimension $d=2$ and are connected by virtual indices $\alpha_j$ with dimension at most $\chi$.
• Figure 4: The sequential circuit for the MPD and MPD+TNO algorithms. Each entangling layer $\hat{U}_j^\dagger$ for $j=1,2,...,L$ corresponds to a $\chi=2$ MPO as computed via Algorithm 2. The first circuit layer $\hat{U}_1^\dagger$ exactly prepares the $\chi=2$ approximation of the target state, with subsequent layers designed to progressively increase entanglement and improve upon the $\chi=2$ approximation. The maximum number of layers $L$ is a hyperparameter.
• Figure 5: Visual improvement in the fidelity of the n = 12-site MPS representation of a discretised Gaussian function $f(x)\sim N(0,0.3^2)$ with increasing bond dimension $\chi$.