Sequence and Image Transformations with Monarq: Quantum Implementations for NISQ Devices

TL;DR

Monarq, a unified quantum data processing framework that combines QCrank encoding with the EHands protocol for polynomial transformations, is introduced and its implementation on noisy intermediate-scale quantum (NISQ) hardware is demonstrated.

Abstract

We introduce Monarq, a unified quantum data processing framework that combines QCrank encoding with the EHands protocol for polynomial transformations, and demonstrate its implementation on noisy intermediate-scale quantum (NISQ) hardware. This framework provides fundamental quantum building blocks for signal and image processing tasks, including convolution, discrete-time Fourier transform (DFT), squared gradient computation, and edge detection, serving as a reference for a broad class of data processing applications on near-term quantum devices.

Figures (12)

• Figure 1: EHands primitives. Quantum circuits for multiplication ($\Pi$) and addition ($\Sigma$) of 2 real numbers.
• Figure 2: Example circuit computing the squared gradient on a sequence of 4 real values $I_i \in [-1,1]$.QCrank encodes 4 copies of the input sequence on 6 qubits. EHands negation and weighted sum are computed in 2 copies, which are then multiplied using the EHands product operator. The resulting $\Delta_i^2=(I_{i+1}-I_{i-1})^2 /4$ is measured as an expectation value on the 3rd qubit, conditioned on the bitstring measured on the first two qubits, providing the address $i \in [0,3]$.
• Figure 3: The Monarq framework is an integration of QCrank encoding with EHands polynomial computation. For a degree-$d$ polynomial, $d$ copies of each input values $x_i$ are encoded on data qubits. EHands operations compute $P_d(x_i)$, retrieved via Pauli-$Z$ measurement one one data qubit conditioned on measured address qubits.
• Figure 4: Quantum circuit used for convolution of two lists of length $L = 2^{n_a}$. The multiplication operator $\Pi$ computes the element-wise product.
• Figure 5: Quantum circuit used for DFT. The input signal $h(t_i)$ of length $L=2^{n_a}$ as well as $2k$ sequences of trigonometric function are encoded using QCrank on $n_a+2k+1$ qubits. Next, $2k$EHands multipliers compute $k$ real and $k$ imaginary components. The $n_a$ address qubits are not measured.