Optimization by Decoded Quantum Interferometry (2408.08292v4)

Abstract: Whether quantum computers can achieve exponential speedups in optimization has been a major open question in quantum algorithms since the field began. Here we introduce a quantum algorithm called Decoded Quantum Interferometry (DQI), which uses the quantum Fourier transform to reduce optimization problems to decoding problems. For approximating optimal polynomial fits to data over finite fields, DQI efficiently achieves a better approximation ratio than any polynomial time classical algorithm known to us, thus suggesting exponential quantum speedup. Sparse unstructured optimization problems such as max-k-XORSAT are reduced to decoding of LDPC codes. We prove a theorem which allows the performance of DQI to be calculated instance-by-instance based on the empirical performance of classical decoders. We use this to construct an instance of max-XORSAT for which DQI finds an approximate optimum that cannot be found by simulated annealing or any of the other general-purpose classical heuristics that we tried, unless given five orders of magnitude more compute time than the decoding problem requires. Although we subsequently design a tailored classical solver that beats DQI within reasonable runtime, our results nevertheless demonstrate that the combination of quantum Fourier transforms with powerful decoding primitives provides a promising new approach to finding quantum speedups for hard optimization problems.

• The paper generalizes core DQI formulas to accommodate scenarios with high degree polynomials where standard assumptions fail.
• It derives novel normalized states and satisfied constraint counts in finite-size Max-LINSAT problems, highlighting the impact of dual code weight distributions.
• The study expands the scope of quantum optimization frameworks, laying a theoretical foundation for more robust quantum algorithm design.

Overview of "DQI semicircle law for high degree polynomials"

The paper, "DQI semicircle law for high degree polynomials," authored by Adam Zalcman, investigates the mathematical underpinnings of the semicircle law within the framework of Discrete Quantum Information (DQI), specifically focusing on scenarios where traditional assumptions may not hold. The semicircle law in this context relates to the asymptotic optimal satisfied fraction for certain computational problems. The paper extends the applicability of DQI semicircle laws to problems that fail to satisfy the condition $2\ell + 1 < d_\text{min}$, a crucial criterion in previous literature.

Key Contributions

1. Generalization of DQI Formulas: The paper begins by generalizing three central formulas that typically support the semicircle law when $2\ell + 1 < d_\text{min}$. These formulas are:
   • The norm of the DQI state, expressed in terms of its coefficient vector.
   • The count of satisfied constraints for finite-size Max-LINSAT problems, derived from the spectrum of a specific matrix.
   • The limit of these constraints as the problem size approaches infinity under a fixed ratio condition.

The author revises these formulas under a different scenario where the condition $2\ell + 1 \geq d_\text{min}$ applies, thus broadening the scope of the semicircle law in DQI.

1. Novel Derivations for Inapplicable Assumptions: When the standard assumption of $2\ell + 1 < d_\text{min}$ does not apply, the problem requires handling higher degree polynomials and potentially variable-sensitive cases. The paper includes derivations under the condition $p=2$ and assumptions relevant to solutions within max-LINSAT instances. The results demonstrate that the dependency on problem constraints becomes more intricate, involving the weight distribution of the dual code $C^\perp$.
2. Expanded Formula Interpretations: The research incorporates adjustments in the normalization process for the DQI state, ensuring the mathematical consistency of newly derived formulas with established theories when traditional assumptions are validated ($2\ell < d_\text{min}$). The treatments related to elementary symmetric polynomials contribute to understanding the expected number of constraints satisfied by strings sampled from DQI states.

Implications and Future Work

The paper pushes the boundary of applicability for DQI models, ensuring these frameworks remain robust and relevant when typical assumptions are breached. This work holds theoretical significance as it reveals deeper insights into how quantum information principles can be applied across broader problems. The dependence of derived formulas on weight distributions of dual codes implies an enriched understanding of how quantum states interact with combinatorial structures.

Practically, these findings could inform algorithm design within quantum computing fields, particularly in optimizing problem instances that align better with physical quantum systems. Future work could explore extending these analyses to different prime $p$ values or generalizing the formulation to encompass even broader classes of computational problems. There remains potential in exploring how these formulations influence heuristics or performance improvements in quantum algorithms, especially in constraint satisfaction problems and related applications.

By providing broader conditions under which the semicircle law and related quantum state evaluations hold, this paper enhances the flexibility and applicability of quantum informatic theories, paving the way for developing more adaptable quantum computational frameworks.