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Andreas Klappenecker

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quant-ph Information Theory math.IT Artificial Intelligence Emerging Technologies Symbolic Computation

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preprint2026arXiv

SCALAR: A Neurosymbolic Framework for Automated Conjecture and Reasoning in Quantum Circuit Analysis

In this paper, we present SCALAR (Symbolic Conjecture and LLM-Assisted Reasoning), a neurosymbolic framework for automated conjecture generation in quantum circuit analysis built on top of the CUDA-Q open source framework. The system integrates quantum simulation, symbolic conjecture generation, and LLM-based interpretation. We evaluate SCALAR on 82 MaxCut instances from the MQLib benchmark dataset and extend the analysis to 2,000 randomly generated graphs across four topologies: regular, Erdos-Renyi, Barabasi-Albert, and Watts-Strogatz. The framework generates conjectured bounds relating optimal QAOA parameters to graph invariants, including known relationships such as periodicity constraints on the phase separation parameter $γ$. SCALAR also recovers previously reported parameter transfer phenomena across structurally similar instances. Additionally, the system identifies correlations between graph structural features and optimization landscape properties, which we characterize through invariant-based descriptors. Using CUDA-Q tensor network simulator, we scale experiments to instances of up to 77 qubits. We discuss the accuracy, generality, and limitations of the generated conjectures, including sensitivity to graph class and quantum circuit depth.

preprint2022arXiv

Encoding Classical Information in Gauge Subsystems of Quantum Codes

We show how to construct hybrid quantum-classical codes from subsystem codes by encoding the classical information into the gauge qudits using gauge fixing. Unlike previous work on hybrid codes, we allow for two separate minimum distances, one for the quantum information and one for the classical information. We give an explicit construction of hybrid codes from two classical linear codes using Bacon-Casaccino subsystem codes, as well as several new examples of good hybrid code.

preprint2020arXiv

Infinite Families of Quantum-Classical Hybrid Codes

Hybrid codes simultaneously encode both quantum and classical information into physical qubits. We give several general results about hybrid codes, most notably that the quantum codes comprising a genuine hybrid code must be impure and that hybrid codes can always detect more errors than comparable quantum codes. We also introduce the weight enumerators for general hybrid codes, which we then use to derive linear programming bounds. Finally, inspired by the construction of some families of nonadditive codes, we construct several infinite families of genuine hybrid codes with minimum distance two and three.

preprint2020arXiv

Nonbinary Error-Detecting Hybrid Codes

Hybrid codes simultaneously encode both quantum and classical information, allowing for the transmission of both across a quantum channel. We construct a family of nonbinary error-detecting hybrid stabilizer codes that can detect one error while also encoding a single classical bit over the residue class rings $\mathbb{Z}_{q}$ inspired by constructions of nonbinary non-additive codes.

preprint2014arXiv

Bird's-eye view on Noise-Based Logic

Noise-based logic is a practically deterministic logic scheme inspired by the randomness of neural spikes and uses a system of uncorrelated stochastic processes and their superposition to represent the logic state. We briefly discuss various questions such as (i) What does practical determinism mean? (ii) Is noise-based logic a Turing machine? (iii) Is there hope to beat (the dreams of) quantum computation by a classical physical noise-based processor, and what are the minimum hardware requirements for that? Finally, (iv) we address the problem of random number generators and show that the common belief that quantum number generators are superior to classical (thermal) noise-based generators is nothing but a myth.

preprint2009arXiv

Degenerate quantum codes and the quantum Hamming bound

The parameters of a nondegenerate quantum code must obey the Hamming bound. An important open problem in quantum coding theory is whether or not the parameters of a degenerate quantum code can violate this bound for nondegenerate quantum codes. In this paper we show that Calderbank-Shor-Steane (CSS) codes with alphabet $q\geq 5$ cannot beat the quantum Hamming bound. We prove a quantum version of the Griesmer bound for the CSS codes which allows us to strengthen the Rains' bound that an $[[n,k,d]]_2$ code cannot correct more than $\floor{(n+1)/6}$ errors to $\floor{(n-k+1)/6}$. Additionally, we also show that the general quantum codes $[[n,k,d]]_q$ with $k+d\leq {(1-2eq^{-2})n}$ cannot beat the quantum Hamming bound.

preprint2009arXiv

Sharing classical secrets with CSS codes

In this paper we investigate the use of quantum information to share classical secrets. While every quantum secret sharing scheme is a quantum error correcting code, the converse is not true. Motivated by this we sought to find quantum codes which can be converted to secret sharing schemes. If we are interested in sharing classical secrets using quantum information, then we show that a class of pure $[[n,1,d]]_q$ CSS codes can be converted to perfect secret sharing schemes. These secret sharing schemes are perfect in the sense the unauthorized parties do not learn anything about the secret. Gottesman had given conditions to test whether a given subset is an authorized or unauthorized set; they enable us to determine the access structure of quantum secret sharing schemes. For the secret sharing schemes proposed in this paper the access structure can be characterized in terms of minimal codewords of the classical code underlying the CSS code. This characterization of the access structure for quantum secret sharing schemes is thought to be new.

preprint2008arXiv

Asymmetric Quantum LDPC Codes

Recently, quantum error-correcting codes were proposed that capitalize on the fact that many physical error models lead to a significant asymmetry between the probabilities for bit flip and phase flip errors. An example for a channel which exhibits such asymmetry is the combined amplitude damping and dephasing channel, where the probabilities of bit flips and phase flips can be related to relaxation and dephasing time, respectively. We give systematic constructions of asymmetric quantum stabilizer codes that exploit this asymmetry. Our approach is based on a CSS construction that combines BCH and finite geometry LDPC codes.

preprint2002arXiv

Quantum Computing and a Unified Approach to Fast Unitary Transforms

A quantum computer directly manipulates information stored in the state of quantum mechanical systems. The available operations have many attractive features but also underly severe restrictions, which complicate the design of quantum algorithms. We present a divide-and-conquer approach to the design of various quantum algorithms. The class of algorithm includes many transforms which are well-known in classical signal processing applications. We show how fast quantum algorithms can be derived for the discrete Fourier transform, the Walsh-Hadamard transform, the Slant transform, and the Hartley transform. All these algorithms use at most O(log^2 N) operations to transform a state vector of a quantum computer of length N.

preprint1999arXiv

Wavelets and Wavelet Packets on Quantum Computers

We show how periodized wavelet packet transforms and periodized wavelet transforms can be implemented on a quantum computer. Surprisingly, we find that the implementation of wavelet packet transforms is less costly than the implementation of wavelet transforms on a quantum computer.