Resource-Efficient Digitized Adiabatic Quantum Factorization

TL;DR

The paper tackles quantum factoring via digitized adiabatic computing and addresses PUBO scalability bottlenecks by introducing a null-subspace QUBO encoding whose zero-energy kernel contains the solution. This reformulation yields a two-body Hamiltonian amenable to gate-efficient QAOA, enabling factorization with substantially reduced circuit depth and improved fidelity for numbers up to $8$-bit. Empirical results for $N=25$ and larger demonstrate a clear gate-count and performance advantage for the QUBO approach, supported by a spectral-density analysis showing lower near-solution state density compared to PUBO. The work suggests a practical path toward resource-efficient quantum factoring on near-term devices and motivates extending the framework to larger composites and deeper investigations of noise and Hamiltonian structure.

Abstract

Digitized adiabatic quantum factorization is a hybrid algorithm that exploits the advantage of digitized quantum computers to implement efficient adiabatic algorithms for factorization through gate decompositions of analog evolutions. In this paper, we harness the flexibility of digitized computers to derive a digitized adiabatic algorithm able to reduce the gate-demanding costs of implementing factorization. To this end, we propose a new approach for adiabatic factorization by encoding the solution of the problem in the kernel subspace of the problem Hamiltonian, instead of using ground-state encoding considered in the standard adiabatic factorization proposed by Peng $et$ $al$. [Phys. Rev. Lett. 101, 220405 (2008)]. Our encoding enables the design of adiabatic factorization algorithms belonging to the class of Quadratic Unconstrained Binary Optimization (QUBO) methods, instead the Polinomial Unconstrained Binary Optimization (PUBO) used by standard adiabatic factorization. We illustrate the performance of our QUBO algorithm by implementing the factorization of integers $N$ up to 8 bits. The results demonstrate a substantial improvement over the PUBO formulation, both in terms of reduced circuit complexity and increased fidelity in identifying the correct solution.

Figures (3)

• Figure 1: Single layer circuit of the (a) PUBO and (b) QUBO algorithms for factoring $N = 25$. While PUBO circuit requires three- and four-qubit gates, which can be decomposed according the circuits in (c), the QUBO algorithm requires two-qubit gates only. We define the circuit gates as $\hat{Z}_{2\gamma_p} = \exp( i \gamma_p \hat{\sigma}^{z}_k )$, $\hat{X}_{2\beta_p} = \exp( i \beta_p \hat{\sigma}^{x}_k )$, $\hat{Z}_{2\gamma_p}^{(2)} = \exp( i \gamma_p \hat{\sigma}^{z}_k \hat{\sigma}^{z}_m )$, $\hat{Z}_{2\gamma_p}^{(3)} = \exp( i \gamma_p \hat{\sigma}^{z}_k \hat{\sigma}^{z}_m \hat{\sigma}^{z}_n )$, $\hat{Z}_{2\gamma_p}^{(4)} = \exp( i \gamma_p \hat{\sigma}^{z}_k \hat{\sigma}^{z}_m \hat{\sigma}^{z}_n \sigma^{z}_l )$. (d) Fidelity of finding the right solution as function o the number of layers for the PUBO and QUBO algorithms, followed by (e) the behavior of the cost function for each model.
• Figure 2: (a) Single-layer circuit scaling of PUBO and QUBO algorithms in terms of the number of CNOTs per layer as a function of $N$. In (b) and (c), respectively, we show the behavior of the smallest (final) value of the cost function $\mathcal{C}_\mathrm{min}$ and the confidence of each algorithm as function of $N$. For the number $N=119$ we present (d) the normalized cost $\mathcal{C}_{l}/\mathcal{C}_{l=1}$ as function $l$, showing the behavior of the cost for each algorithm. Complementarily, we show in (f) and (g) the population distribution in the computational basis ordered from highest to lowest populated states for the PUBO and QUBO algorithms, respectively, where we highlight the corresponding desired solution. Similar data are shown for the number $N=143$ in (e,h,i). "Others" constitutes the sum of populations of all computation states not displayed.
• Figure 3: Normalized energy spectra $|\tilde{E}_{n}| = |E_{n}|/E_{\mathrm{max}}$ of the (a) PUBO Hamiltonian and (b) QUBO Hamiltonian used for factorization of $N=119$, where the $x$-axis shows the decimal encoding of computational basis states. The eigenstate corresponding to the solution ($\widetilde{E}=0$) is marked with red star symbol. Similar plots are shown for the number $N=143$ in (c) and (d). The panels (e) and (f) show the same spectra as (a,b) and (c,d), respectively, but we use a basis ordering based on the energy $|\tilde{E}_{n}|$. This graphs, including the inset plots, allow us to observe how the minimum norm $|\tilde{E}_{n}|$ search is challenged by the high density of states in the PUBO algorithm compared to the QUBO approach.