arXiv subjects

Sami Boulebnane

Publications and source records attributed to Sami Boulebnane.

13 recordsLinked to original sources

arXiv2026

Optimization Using Locally-Quantum Decoders

It was pointed out in [JSW+25] that widely-studied optimization problems such as D-regular max-k-XORSAT can be reduced to decoding of LDPC codes, using quantum algorithms related to Regev's reduction. LDPC codes have very good decoders, such as Belief Propagation (BP), and this therefore makes D-regular max-k-XORSAT an enticing target for this class of quantum algorithms. However, BP was found insufficient to achieve quantum advantage. Here, we develop an intrinsically quantum decoding technique, which decodes classical LDPC codes subject to coherent superpositions of bit flip errors. For average-case instances of D-regular max-k-XORSAT drawn from Gallager's ensemble, this quantum decoder strongly outperforms classical belief propagation at many values of k and D. For some (k,D) the approximate optima achievable using this decoder surpass both Prange's algorithm and simulated annealing. However, we stop short of achieving quantum advantage because we identify an enhancement to Prange's algorithm that recovers a precise tie, much as a precise tie was observed between the standard version of Prange's algorithm and a more limited version of locally-quantum decoding in [CT24].

quant-ph↗

arXiv2026

Quantum Approximate Optimization of Integer Graph Problems and Surpassing Semidefinite Programming for Max-k-Cut

Quantum algorithms for binary optimization problems have been the subject of extensive study. However, the application of quantum algorithms to integer optimization problems remains comparatively unexplored. In this paper, we study the Quantum Approximate Optimization Algorithm (QAOA) applied to integer problems on graphs, with each integer variable encoded in a qudit. We derive a general iterative formula for depth-$p$ QAOA expectation on high-girth $d$-regular graphs of arbitrary size. The cost of evaluating the formula is exponential in the QAOA depth $p$ but does not depend on the graph size. Evaluating this formula for Max-$k$-Cut problem for $p\leq 4$, we identify parameter regimes ($k=3$ with degree $d \leq 10$ and $k=4$ with $d \leq 40$) in which QAOA outperforms the Frieze-Jerrum semi-definite programming (SDP) algorithm, which provides the best worst-case guarantee on the approximation ratio. To strengthen the classical baseline, we introduce a new heuristic algorithm based on the degree-of-saturation that achieves strong results on the \texttt{GSet} benchmark with quasi-linear runtime in the number of edges. It empirically outperforms both the Frieze-Jerrum algorithm and shallow-depth QAOA on regular graphs. Nevertheless, we provide numerical evidence that QAOA may overtake this heuristic at depth $p\leq 20$. Our results show that moving beyond binary to integer optimization problems can open up new avenues for quantum advantage.

quant-ph↗

arXiv2026

Iterative Interpolation Schedules for Quantum Approximate Optimization Algorithm

Quantum Approximate Optimization Algorithm (QAOA) is a promising quantum heuristic with empirical evidence of speedup over classical state-of-the-art for some problems. QAOA uses a parameterized circuit with $p$ layers, where higher $p$ yields better solutions, but requires optimizing $2p$ independent parameters, which is challenging at large $p$. We present an iterative interpolation method that exploits the smoothness of optimal parameter schedules by expressing them in a basis of orthogonal functions, generalizing the work of Zhou et al. By optimizing a small number of basis coefficients and iteratively increasing both circuit depth and coefficient count until convergence, our method constructs high-quality schedules for large $p$. We provide theoretical justification using Jackson's theorem and Lipschitz continuity to bound the required number of basis coefficients for a given accuracy. Our approach achieves better performance with fewer optimization steps than existing methods across three benchmark problems: the Sherrington-Kirkpatrick (SK) model, portfolio optimization, and Low Autocorrelation Binary Sequences (LABS). For the largest LABS instance, we achieve near-optimal merit factors with schedules exceeding 1000 layers, an order of magnitude beyond previous methods. Additionally, we observe that a mild growth in QAOA depth suffices to solve the SK model exactly, a result of independent theoretical interest.

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arXiv2026

Spin-Boson Mapping of the Quantum Approximate Optimization Algorithm

The Quantum Approximate Optimization Algorithm (QAOA) achieves monotonically improving performance with circuit depth $p$, yet the study of the high-depth regime has been obstructed by the exponential in $p$ cost of existing exact evaluation techniques. In this Letter, we prove that, in the infinite-size limit, the depth-$p$ QAOA state for the Sherrington-Kirkpatrick (SK) model converges to the state of a spin coupled to $p$ bosonic modes. We simulate the spin-boson system using matrix product states and provide numerical evidence that QAOA obtains a $(1-ε)$ approximation to the optimal energy of the SK model with circuit depth $O(n/ε^{1.13})$ in the average case. The modest computational cost of our approach allows us to optimize QAOA parameters and observe that QAOA achieves $\varepsilon\lesssim 2.2\%$ at $p=160$ in the infinite-size limit, extending far beyond $p\leq 20$ accessible to prior exact methods. Our mapping provides a many-body route to study and optimize high-depth QAOA in regimes previously inaccessible to exact evaluation.

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arXiv2025

Quantum Approximate Optimization Algorithm in Finite Size and Large Depth and Equivalence to Quantum Annealing

The quantum approximate optimization algorithm (QAOA) and quantum annealing are two of the most popular quantum optimization heuristics. While QAOA is known to be able to approximate quantum annealing, the approximation requires QAOA angles to vanish with the problem size $n$, whereas optimized QAOA angles are observed to be size-independent for small $n$ and constant in the infinite-size limit. This fact led to a folklore belief that QAOA has a mechanism that is fundamentally different from quantum annealing. In this work, we provide evidence against this by analytically showing that QAOA energy approximates that of quantum annealing under two conditions, namely that angles vary smoothly from one layer to the next and that the sum is bounded by a constant. These conditions are known to hold for near-optimal QAOA angles empirically. Our proof relies on a series expansion of QAOA energy in sum of angles, which we show converges to quantum annealing limit as QAOA depth grows for constant sum of angles even if angles do not vanish with problem size $n$. A corollary of our results is a quadratic improvement for the bound on depth required to compile Trotterized quantum annealing of the SK model in the average case.

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arXiv2025

Threshold for Fault-tolerant Quantum Advantage with the Quantum Approximate Optimization Algorithm

Optimization is often cited as a promising application of quantum computers. However, the low degree of provable quantum speedups has led prior rigorous end-to-end resource analyses to conclude that a quantum computer is unlikely to surpass classical state-of-the-art on optimization problems under realistic assumptions. In this work, we compile and analyze the Quantum Approximate Optimization Algorithm (QAOA) combined with Amplitude Amplification (AA) applied to random 8-SAT at the satisfiability threshold. Our compilation involves careful optimization of circuits for Hamiltonian simulation, which may be of independent interest. We use the analytical scaling of the time-to-solution for QAOA identified by PRX Quantum 5, 030348 (2024) and find that with QAOA depth $p=623$, QAOA+AA achieves a crossover with state-of-the-art classical heuristics at 179 variables and 14.99 hours of runtime when executed on a surface-code-based fault-tolerant quantum computer with 73.91 million physical qubits, a physical error rate of $10^{-3}$, and a $1~μ$s code cycle time. Notably, we allow the classical solver to be parallelized as long as its total energy consumption is equal to that required for decoding in the surface code. We further show that this restriction on classical solver energy consumption can be relaxed given optimistic but plausible reductions in physical error rates and fault-tolerance overheads, enabling a crossover of 2.94 hours using 8.88 million physical qubits against a classical solver running on a supercomputer with $725,760$ CPU cores. These findings support the hypothesis that large-scale fault-tolerant quantum computers will be useful for optimization.

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arXiv2024

Applying the quantum approximate optimization algorithm to general constraint satisfaction problems

In this work we develop theoretical techniques for analysing the performance of the quantum approximate optimization algorithm (QAOA) when applied to random boolean constraint satisfaction problems (CSPs), and use these techniques to compare the complexity of a variety of CSPs, such as $k$-SAT, 1-in-$k$ SAT, and NAE-SAT. Our techniques allow us to compute the success probability of QAOA with one layer and given parameters, when applied to randomly generated instances of CSPs with $k$ binary variables per constraint, in time polynomial in $n$ and $k$. We apply this algorithm to all boolean CSPs with $k=3$ and a large number of CSPs with $k=4$, $k=5$, and compare the resulting complexity with the complexity of solving the corresponding CSP using the standard solver MapleSAT, determined experimentally. We find that random $k$-SAT seems to be the most promising of these CSPs for demonstrating a quantum-classical separation using QAOA.

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arXiv2022

Solving boolean satisfiability problems with the quantum approximate optimization algorithm

The quantum approximate optimization algorithm (QAOA) is one of the most prominent proposed applications for near-term quantum computing. Here we study the ability of QAOA to solve hard constraint satisfaction problems, as opposed to optimization problems. We focus on the fundamental boolean satisfiability problem, in the form of random $k$-SAT. We develop analytic bounds on the average success probability of QAOA over random boolean formulae at the satisfiability threshold, as the number of variables $n$ goes to infinity. The bounds hold for fixed parameters and when $k$ is a power of 2. We complement these theoretical results with numerical results on the performance of QAOA for small $n$, showing that these match the limiting theoretical bounds closely. We then use these results to compare QAOA with leading classical solvers. In the case of random 8-SAT, we find that for around 14 ansatz layers, QAOA matches the scaling performance of the highest-performance classical solver we tested, WalkSATlm. For larger numbers of layers, QAOA outperforms WalkSATlm, with an ultimate level of advantage that is still to be determined. Our methods provide a framework for analysing the performance of QAOA for hard constraint satisfaction problems and finding further speedups over classical algorithms.

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arXiv2022

Peptide conformational sampling using the Quantum Approximate Optimization Algorithm

Protein folding -- the problem of predicting the spatial structure of a protein given its sequence of amino-acids -- has attracted considerable research effort in biochemistry in recent decades. In this work, we explore the potential of quantum computing to solve a simplified version of protein folding. More precisely, we numerically investigate the performance of a variational quantum algorithm, the Quantum Approximate Optimization Algorithm (QAOA), in sampling low-energy conformations of short peptides. We start by benchmarking the algorithm on an even simpler problem: sampling self-avoiding walks, which is a necessary condition for a valid protein conformation. Motivated by promising results achieved by QAOA on this problem, we then apply the algorithm to a more complete version of protein folding, including a simplified physical potential. In this case, based on numerical simulations on 20 qubits, we find less promising results: deep quantum circuits are required to achieve accurate results, and the performance of QAOA can be matched by random sampling up to a small overhead. Overall, these results cast serious doubt on the ability of QAOA to address the protein folding problem in the near term, even in an extremely simplified setting. We believe that the approach and conclusions presented in this work could offer valuable methodological insights on how to systematically evaluate variational quantum optimization algorithms on real-world problems beyond protein folding.

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arXiv2021

Predicting parameters for the Quantum Approximate Optimization Algorithm for MAX-CUT from the infinite-size limit

Combinatorial optimization is regarded as a potentially promising application of near and long-term quantum computers. The best-known heuristic quantum algorithm for combinatorial optimization on gate-based devices, the Quantum Approximate Optimization Algorithm (QAOA), has been the subject of many theoretical and empirical studies. Unfortunately, its application to specific combinatorial optimization problems poses several difficulties: among these, few performance guarantees are known, and the variational nature of the algorithm makes it necessary to classically optimize a number of parameters. In this work, we partially address these issues for a specific combinatorial optimization problem: diluted spin models, with MAX-CUT as a notable special case. Specifically, generalizing the analysis of the Sherrington-Kirkpatrick model by Farhi et al., we establish an explicit algorithm to evaluate the performance of QAOA on MAX-CUT applied to random Erdos-Renyi graphs of expected degree $d$ for an arbitrary constant number of layers $p$ and as the problem size tends to infinity. This analysis yields an explicit mapping between QAOA parameters for MAX-CUT on Erdos-Renyi graphs of expected degree $d$, in the limit $d \to \infty$, and the Sherrington-Kirkpatrick model, and gives good QAOA variational parameters for MAX-CUT applied to Erdos-Renyi graphs. We then partially generalize the latter analysis to graphs with a degree distribution rather than a single degree $d$, and finally to diluted spin-models with $D$-body interactions ($D \geq 3$). We validate our results with numerical experiments suggesting they may have a larger reach than rigorously established; among other things, our algorithms provided good initial, if not nearly optimal, variational parameters for very small problem instances where the infinite-size limit assumption is clearly violated.

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arXiv2020

Improving the Quantum Approximate Optimization Algorithm with postselection

Combinatorial optimization is among the main applications envisioned for near-term and fault-tolerant quantum computers. In this work, we consider a well-studied quantum algorithm for combinatorial optimization: the Quantum Approximate Optimization Algorithm (QAOA) applied to the MaxCut problem on 3-regular graphs. We explore the idea of improving the solutions returned by the simplest version of the algorithm (depth-1 QAOA) using a form of postselection that can be efficiently simulated by state preparation. We derive theoretical upper and lower bounds showing that a constant (though small) increase of the fraction of satisfied edges is indeed achievable. Numerical experiments on large problem instances (beyond classical simulatability) complement and support our bounds. We also consider a distinct technique: local updates, which can be applied not only to QAOA but any optimization algorithm. In the case of QAOA, the resulting improvement can be sharply quantified theoretically for large problem instances and in absence of postselection. Combining postselection and local updates, the theory is no longer tractable but numerical evidence suggests that improvements from both methods can be combined.

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arXiv2019

Approximate quantum non-demolition measurements

With the advent of gravitational wave detectors employing squeezed light, quantum waveform estimation---estimating a time-dependent signal by means of a quantum-mechanical probe---is of increasing importance. As is well known, backaction of quantum measurement limits the precision with which the waveform can be estimated, though these limits can in principle be overcome by "quantum nondemolition" (QND) measurement setups found in the literature. Strictly speaking, however, their implementation would require infinite energy, as their mathematical description involves Hamiltonians unbounded from below. This raises the question of how well one may approximate nondemolition setups with finite energy or finite-dimensional realizations. Here we consider a finite-dimensional waveform estimation setup based on the "quasi-ideal clock" and show that the estimation errors due to approximating the QND condition decrease slowly, as a power law, with increasing dimension. As a result, we find that good QND approximations require large energy or dimensionality. We argue that this result can be expected to also hold for setups based on truncated oscillators or spin systems.

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arXiv2018

Cold light dark matter in extended seesaw models

We present a thorough discussion of light dark matter produced via freeze-in in two-body decays A -> B DM. If A and B are quasi-degenerate, the dark matter particle has a cold spectrum even for keV masses. We show this explicitly by calculating the transfer function that encodes the impact on structure formation. As examples for this setup we study extended seesaw mechanisms with a spontaneously broken global U(1) symmetry, such as the inverse seesaw. The keV-scale pseudo-Goldstone dark matter particle is then naturally produced cold by the decays of the quasi-degenerate right-handed neutrinos.

hep-ph↗