Enhanced Digitized Adiabatic Quantum Factorization Algorithm Using Null-Space Encoding

TL;DR

The work addresses quantum factorization in the NISQ era by proposing an enhanced digitized adiabatic scheme that uses null-space encoding to replace high-order interactions with a two-body Hamiltonian $\hat{H}_{\mathrm{LP}}$. It embeds this linearized Hamiltonian in a QAOA-like framework and demonstrates numerically that the resulting protocols can achieve comparable or higher fidelities with significantly fewer quantum resources, including two-qubit gates, and often converge faster for problem instances up to eight qubits. A key finding is that the linearized protocol yields abrupt fidelity jumps tied to the broader spectral separation in $\hat{H}_{\mathrm{LP}}$, and that choosing the cost function (preferably $\langle |\hat{H}_{\mathrm{LP}}| \rangle$) further enhances performance. The results suggest a practical, resource-efficient pathway toward quantum factoring on near-term devices, with strong implications for hardware feasibility and scalable factorization strategies.

Abstract

Integer factorization is a computational problem of fundamental importance in cybersecurity and secure communications, as its difficulty form the basis of modern public-key cryptography. While Shor's algorithm can solve this problem efficiently on a universal quantum computer, near-term devices require alternative approaches. The Adiabatic Factorization Algorithm and its digitized counterparts offer a promising NISQ-era pathway but suffer from high-order many-body interactions that are difficult to implement. In this work, we propose a modified QAOA-based factorization protocol that simplifies the interacting Hamiltonian to include only two-body terms, significantly reducing its experimental complexity. Numerical simulations show that this method achieves comparable or higher fidelities than the standard protocol, while requiring fewer quantum resources and converging more rapidly for problem instances up to eight qubits. We analyze the characteristic fidelity behavior introduced by the Hamiltonian modification. Additionally, we report on simulations with alternative cost-function definitions that frequently yielded improved performance.

Figures (18)

• Figure 1: Schematic of an adiabatic passage. The orange line represents the Hamiltonian's ground state, while the green a Hamiltonian's eigenstate different from the ground state.
• Figure 2: Diagram of a $p$-layer QAOA circuit. Starting from the initial state $\ket{+}^{\otimes n}$,the circuit alternates between applying the unitaries $e^{-i \gamma_i \hat{H}_C}$ and $e^{-i \beta_i \hat{H}_M}$ for $i = 1$ to $p$. The final state is measured to estimate the expectation value $\langle \hat{H}_C \rangle$, which is passed to a classical optimizer. This process is repeated until convergence.
• Figure 3: One-layer circuit for factorizing the number $N=35$ using the standard QAOA protocol. Notice the presence of three- and four-qubit gates, highlighted in yellow and red, respectively. Rotation angles are omitted for simplicity.
• Figure 4: Decomposition of (a) two-, (b) three-, and (c) four-qubit Z-rotation gates in CNOTs and single-qubit Z-rotations.
• Figure 5: One-layer circuit for factorizing the number $N=35$ using our protocol, evolving the state with the linear Hamiltonian $H_{\mathrm{LP}}$. Notice the simplification with respect to Fig. \ref{['fig:standard_circuit']} due to the absence of three- and four-qubit gates.