arXiv subjects

Quentin Buzet

Publications and source records attributed to Quentin Buzet.

3 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

IQP circuits for 2-Forrelation

The 2-Forrelation problem provides an optimal separation between classical and quantum query complexity and is also the problem used for separating $\mathsf{BQP}$ and $\mathsf{PH}$ relative to an oracle. A natural question is therefore to ask what are the minimal quantum resources needed to solve this problem. We show that 2-Forrelation can be solved using Instantaneous Quantum Polynomial-time ($\mathsf{IQP}$) circuits, a restricted model of quantum computation in which all gates commute. Concretely, two $\mathsf{IQP}$ circuits with two quantum queries and efficient classical processing suffice. For the signed variant of 2-Forrelation, even a single $\mathsf{IQP}$ circuit and query suffices. This answers a recent open question of Girish (arXiv:2510.06385) on the power of commuting quantum computations. We use this to show that $(\mathsf{BPP}^{\mathsf{IQP}})^O \not\subseteq \mathsf{PH}^O$ relative to an oracle $O$, strengthening the result of Raz and Tal (STOC 2019). Our results show that $\mathsf{IQP}$ circuits can be used for classically hard decision problems, thus providing a new route for showing quantum advantage with $\mathsf{IQP}$ circuits, avoiding the verification difficulties associated with sampling tasks. We also prove Fourier growth bounds for $\mathsf{IQP}$ circuits in terms of the size of their accepting set. The key ingredient is an algebraic identity of the quadratic function $Q(x) = \sum_{i < j} x_ix_j$ that allows extracting inner-product phases within an $\mathsf{IQP}$ circuit.

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arXiv2025

Fine-Grained Unambiguous Measurements

Unambiguous measurements play an important role in quantum information, with applications ranging from quantum key distribution to quantum state reconstruction. Recently, such measurements have also been used in quantum algorithms based on Regev's reduction. The key problem for these algorithms is the S-$|LWE>$ problem in the lattice setting and the Quantum Decoding Problem in the code setting. A key idea for addressing this problem is to use unambiguous measurements to recover $k$ coordinates of a code (or lattice) element $x$ from a quantum state $|\psi_x\rangle$, which corresponds to a noisy word $x$ with errors in quantum superposition. However, a general theoretical framework to analyze this approach has been lacking. In this work, we introduce the notion of fine-grained unambiguous measurements. Given a family of states $\{\,|\psi_x\rangle\,\}_{x\in\{0,1\}^n}$, we ask whether there exist measurements that can return, with certainty, $k$ bits of information about $x$. We study this question in the setting of symmetric states, which naturally arises in the Quantum Decoding Problem. We show that determining the maximal number of parities that a measurement can output can be formulated as a linear program, and we use its dual formulation to derive several upper bounds. In particular, we establish necessary and sufficient conditions for the existence of fine-grained unambiguous measurements and prove impossibility results showing, in particular, that such measurements cannot improve upon the approach of arXiv:2310.20651. Finally, we discuss the implications of these findings for the Quantum Decoding Problem.

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