arXiv subjects

Arend-Jan Quist

Publications and source records attributed to Arend-Jan Quist.

6 recordsLinked to original sources

arXiv2026

Faster algorithm for achieving minimal-size quantum decision diagrams

The decision diagram (DD) data structure enables fast linear-algebra calculations by bringing vectors into a normal form and subsequently merging equivalent ones, yielding a minimally-sized DD modulo the equivalence relation. A fruitful application area is quantum-circuit simulation, where the vectors represent quantum states. The Local Invertible Map Decision Diagram (LIMDD) type, merges LIM-equivalent (typically Pauli-gate equivalent) vectors, can efficiently simulate Clifford circuits as well as some high-T-count circuits, and has theoretically been proven exponentially faster for simulation than other well-developed data structures, including other common DD variants. However, these exponential advantages have not fully materialized yet in existing implementations, for which the normal-form procedure, which is a highly complex algorithm, is either absent or only partially implemented. We here present a novel normal-form algorithm for Pauli-LIMDDs, achieving a worst-case speedup from $O(n^3)$ to $O(n^2)$ for an $n$-qubit DD node with a single child node while keeping the $O(n^3)$ run time in case of two distinct children nodes. We implement the algorithm as part of QolDDer, our Pauli-LIMDD simulator for quantum circuits, written from scratch in C/C++. The implementation realizes the theoretically-proven advantages of Pauli-LIMDDs on Clifford circuits, is significantly faster than the existing LIMDD simulators on such circuits, and on a public quantum-circuit data set often outperforms them by an order of magnitude. In the future, we envision that our work will enable further application and development of LIMDD variants, not only for quantum design tasks, but also for analysis of linear-algebra-based systems in general.

quant-ph↗

arXiv2026

From Tensor Networks to Tractable Circuits, and back

Tensor networks and circuits are widely used data structures to represent pseudo-Boolean functions. These two formalisms have been studied primarily in separate communities, and this paper aims to establish equivalences between them. We show that some classes of tensor networks that are appealing in practice correspond to classes of circuits with specific properties that have been studied in knowledge compilation as \emph{tractable circuits}. In particular, we prove that matrix product states (tensor trains) coincide with nondeterministic edge-valued decision diagrams and that tree tensor networks exactly correspond to structured-decomposable circuits. These correspondences enable direct transfer of structural and algorithmic results; for example, canonicity and tractability guarantees known for circuits yield analogous guarantees for the associated tensor networks, and vice versa.

quant-ph↗

arXiv2026

Exact quantum decision diagrams with scaling guarantees for Clifford+$T$ circuits and beyond

A decision diagram (DD) is a graph-like data structure for homomorphic compression of Boolean and pseudo-Boolean functions. Over the past decades, decision diagrams have been successfully applied to verification, linear algebra, stochastic reasoning, and quantum circuit analysis. Floating-point errors have, however, significantly slowed down practical implementations of real- and complex-valued decision diagrams. In the context of quantum computing, attempts to mitigate this numerical instability have thus far lacked theoretical scaling guarantees and have had only limited success in practice. Here, we focus on the analysis of quantum circuits consisting of Clifford gates and $T$ gates (a common universal gate set). We first hand-craft an algebraic representation for complex numbers, which replace the floating point coefficients in a decision diagram. Then, we prove that the sizes of these algebraic representations are linearly bounded in the number of $T$ gates and qubits, and constant in the number of Clifford gates. Furthermore, we prove that both the runtime and the number of nodes of decision diagrams are upper bounded as $2^t \cdot poly(g, n)$, where $t$ ($g$) is the number of $t$ gates (Clifford gates) and $n$ the number of qubits. Our proofs are based on a $T$-count dependent characterization of the density matrix entries of quantum states produced by circuits with Clifford+$T$ gates, and uncover a connection between a quantum state's stabilizer nullity and its decision diagram width. With an open source implementation, we demonstrate that our exact method resolves the inaccuracies occurring in floating-point-based counterparts and can outperform them due to lower node counts. Our contributions are, to the best of our knowledge, the first scaling guarantees on the runtime of (exact) quantum decision diagram simulation for a universal gate set.

quant-ph↗

arXiv2025

Trade-offs between classical and quantum space using spooky pebbling

Pebble games are used to study space/time trade-offs. Recently, spooky pebble games were introduced to study classical space / quantum space / time trade-offs for simulation of classical circuits on quantum computers. In this paper, the spooky pebble game framework is applied for the first time to general circuits. Using this framework we prove an upper bound for quantum space in the spooky pebble game. We also prove that solving the spooky pebble game is PSPACE-complete. Moreover, we present a solver for the spooky pebble game based on satisfiability solvers combined with heuristic optimizers. This spooky pebble game solver was empirically evaluated by calculating optimal classical space / quantum space / time trade-offs. Within limited runtime, the solver could find a strategy reducing quantum space when classical space is taken into account, showing that the spooky pebble model is useful to reduce quantum space.

quant-ph↗

arXiv2025

Numerical Errors in Quantitative System Analysis With Decision Diagrams

Decision diagrams (DDs) are a powerful data structure that is used to tackle the state-space explosion problem, not only for discrete systems, but for probabilistic and quantum systems as well. While many of the DDs used in the probabilistic and quantum domains make use of floating-point numbers, this is not without challenges. Floating-point computations are subject to small rounding errors, which can affect both the correctness of the result and the effectiveness of the DD's compression. In this paper, we investigate the numerical stability, i.e. the robustness of an algorithm to small numerical errors, of matrix-vector multiplication with multi-terminal binary decision diagrams (MTBDDs). Matrix-vector multiplication is of particular interest because it is the function that computes successor states for both probabilistic and quantum systems. We prove that the MTBDD matrix-vector multiplication algorithm can be made numerically stable under certain conditions, although in many practical implementations of MTBDDs these conditions are not met. Additionally, we provide a case study of the numerical errors in the simulation of quantum circuits, which shows that the extent of numerical errors in practice varies greatly between instances.

cs.CE↗

arXiv2024

Advancing Quantum Computing with Formal Methods

This tutorial introduces quantum computing with a focus on the applicability of formal methods in this relatively new domain. We describe quantum circuits and convey an understanding of their inherent combinatorial nature and the exponential blow-up that makes them hard to analyze. Then, we show how weighted model counting (\#SAT) can be used to solve hard analysis tasks for quantum circuits. This tutorial is aimed at everyone in the formal methods community with an interest in quantum computing. Familiarity with quantum computing is not required, but basic linear algebra knowledge (particularly matrix multiplication and basis vectors) is a prerequisite. The goal of the tutorial is to inspire the community to advance the development of quantum computing with formal methods.

quant-ph↗