Quantum pixel representations and compression for N-dimensional images

Abstract

We introduce a novel and uniform framework for quantum pixel representations that overarches many of the most popular representations proposed in the recent literature, such as (I)FRQI, (I)NEQR, MCRQI, and (I)NCQI. The proposed QPIXL framework results in more efficient circuit implementations and significantly reduces the gate complexity for all considered quantum pixel representations. Our method scales linearly in the number of pixels and does not use ancilla qubits. Furthermore, the circuits only consist of $$R_y$$ R y gates and $$\text {CNOT}$$ CNOT gates making them practical in the NISQ era. Additionally, we propose a circuit and image compression algorithm that is shown to be highly effective, being able to reduce the necessary gates to prepare an FRQI state for example scientific images by up to 90% without sacrificing image quality. Our algorithms are made publicly available as part of , a Quantum Image Pixel Library.

Quantum-parallel vectorized data encodings and computations on trapped-ions and transmons QPUs

“…The (improved) flexible representation of quantum images ([I]FRQI) [15,16,14], the (improved) novel enhanced quantum representation ([I]NEQR) [38,12], the multi-channel representation of quantum images (MCRQI/MCQI) [31,30], and the (improved) novel quantum representation of color digital images ([I]NCQI) [25,29] are among the most powerful existing QIR methods. In our previous work, we proposed an overarching encoding framework called QPIXL [2] that unifies all QIRs mentioned above. In the QPIXL framework, every QIR can be written as…”

“…where |c i is an encoding of the pixel colors in the qubit state and |i an encoding of the pixel positions in the qubit state [2]. All QIRs that are commonly considered in the literature use a straightforward basis encoding for the pixel position information.…”

“…( 3) consists of 2 na fullycontrolled R y gates, where, for i ∈ [2 na ], the rotation angle is given by α i and the n a address qubits are controlled on the state |i . An optimized circuit implementation existing of a depth-2 na sequence of two-qubit CX gates and uncontrolled single-qubit R y rotations [21] reduces the quantum resources [2] at the cost of an increased classical overhead to solve the linear system…”

“…The linear system ( 6) is a Walsh-Hadamard transformation with Gray ordering, explained in more details in Appendix A. It can be solved efficiently classically in O(N log N ) operations through a Fast Walsh-Hadamard Transform (FWHT) [2].…”

“…Figure 1: Different types of uniformly controlled rotations (UCR) with optimal connectivity graphs for qubits shown on the right. (a) Standard compact circuit implementation for a UCR y gate that was used in the QPIXL framework [2] for 5 address and 1 data qubits. Square boxes denote single qubit R y rotations.…”