A competitive NISQ and qubit-efficient solver for the LABS problem

TL;DR

These findings point at PCE-based solvers as a promising quantum-inspired classical heuristic for hard problems as well as a tool to reduce the resource requirements for actual quantum algorithms.

Abstract

Pauli Correlation Encoding (PCE) is as a qubit-efficient variational approach to combinatorial optimization problems. The method offers a polynomial reduction in qubit count and a super-polynomial suppression of barren plateaus. Here, we extend the PCE-based framework to solve the Low Autocorrelation Binary Sequences (LABS) problem, a notoriously hard problem often used as a benchmark for classical and quantum solvers. To illustrate this,we simulate two variants of the PCE quantum solver for LABS instances of up to $N=45$ binary variables: one with commuting and one with maximally non-commuting sets of Pauli operators. The simulations use $4$ qubits and a circuit Ansatz with a total of $30$ two-qubit gates. We benchmark our method against the state-of-the-art classical solver and other quantum schemes. We observe improved scaling in the total time to reach the exact solution, outperforming the best-performing classical heuristic while using only a fraction of the quantum resources required by other quantum approaches. In addition, we perform proof-of-principle demonstrations on IonQ's Forte quantum processor, showing that the final solution is resilient to noise. Our findings point at PCE-based solvers as a promising quantum-inspired classical heuristic for hard problems as well as a tool to reduce the resource requirements for actual quantum algorithms.

Figures (8)

• Figure 1: Time-to-solution benchmark. Scaling comparison of the TTS for the quantum LABS solver based on Pauli Correlation Encoding using $\Pi^{(\text{NC})}$ (PCE, red) vs. Tabu Search (Tabu, blue) for even (left) and odd (right) values of $N$. Here, TTS is defined as the total number of cost function evaluations required to reach the exact solution BOSKOVIC2017262. Dashed-dotted lines denote linear regression over the list of median values for each $N$, while dotted curves correspond to linear regression over $50$ points per instance size (except for $N\in[41,45]$, with $10$ runs each). The distribution of points regarding different initializations is depicted by violin plots. The fits show a clear exponential scaling in all the cases, with both median and ensemble statistics consistent with a power-law advantage of PCE over the classical baseline. The precise scaling exponents and confidence intervals are shown in Table \ref{['tab:fits_full_PCE']} in App. \ref{['app:approx']}.
• Figure 2: TTS scaling for different solvers. Fitted basis coefficient $b$ of the exponential scaling in Eq. \ref{['eq:scaling']} for our PCE quantum solver using $\Pi^{(\text{NC})}$, QAOA Shaydulin_2024, and Tabu Search (classical) LABS solvers. The corresponding fit data appears in Table \ref{['tab:fits_full_PCE']} in App. \ref{['app:approx']}. We report the results for TTS, $\text{TTS}_{1\text{st}}$, and $\text{TTS}_{2\text{nd}}$ results, corresponding respectively to the exact solution (Exact) and to approximate solutions given by the first (1st) and second (2nd) excited energy levels.
• Figure 3: LABS state-of-the-art. Best merit factor vs. problem size $N$ for classical solvers. Red dots indicate the optimal merit factor $F_N\coloneq \max_{x\in\{-1,+1\}^{\otimes N}}F(x)$ obtained via exact solvers, while blue dots indicate the best known approximation to $F_N$ obtained via heuristic solvers. The dashed curve is Golay's conjectured asymptotic upper bound $\approx12.3248/(8\pi N)^{\frac{3}{2N}}$Golay1982.
• Figure 4: Circuit depth scaling.(Top) Linear scaling of the optimal circuit depth with the number of qubits $n$ for the qubit-efficient solver with quadratic compression using $\Pi^{(\text{C})}$. This corresponds to a $\mathcal{O}(\sqrt{N})$ scaling with the problem size $N$. (Bottom) For each $n$ (here illustrated for $n=4$), the optimal circuit depth was determined by increasing the number of circuit layers until no significant improvement in the average approximation ratio was observed over $1000$ runs.
• Figure 5: Hyperparameters tuning. (Left) Tuning of the rescaling parameter $\alpha$; (Right) Tuning of the penalization constant $\beta$. In each case, we fixed the number of qubits to $n=4$, the circuit depth to $10$, and the optimal value was determined by increasing the corresponding parameter until no significant improvement in the average approximation ratio was observed over $1000$ runs.