arXiv subjects

Søren Fuglede Jørgensen

Publications and source records attributed to Søren Fuglede Jørgensen.

9 recordsLinked to original sources

arXiv2026

Link-middle-cut lower bounds for Clifford circuit synthesis

Minimizing the number of CNOT gates required to synthesize a Clifford operator is a central problem in quantum circuit optimization. We extend the link--middle--cut (LMC) framework for linear reversible circuits to Clifford operators represented by binary symplectic tableaux. By introducing block-support connectivity graphs and determinantal invariants, we obtain efficiently computable lower bounds on ancilla-free CNOT complexity. We prove that these bounds are tight for qubit permutations: a permutation of $n$ qubits with $k$ cycles requires exactly $3(n-k)$ CNOT gates, even when arbitrary one-qubit Clifford gates are available. We also derive efficiently computable bounds for synthesis up to a permutation of the output qubits, corresponding to free qubit relabeling. Finally, we further demonstrate the utility of the bound by using it as an admissible heuristic for $A^*$-based Clifford synthesis and show that, on benchmark instances, the resulting search can prove optimality beyond what was possible with earlier SAT-based methods, and that it may be used in combination with earlier $A^*$ heuristics to improve the CNOT counts obtained through heuristic search.

quant-ph↗

arXiv2026

Lower bounds for the CNOT-complexity of linear reversible operators

The CNOT-complexity of an invertible matrix over $\mathbb{F}_2$ is the minimum number of CNOT gates needed to synthesize the corresponding linear reversible operator. While the maximum CNOT-complexity over all $n \times n$ matrices is known to be $Θ(n^2 / \log n)$, no explicit family of matrices requiring a superlinear number of CNOT gates is known, and until now the hardest explicitly known family has been the cyclic permutations, with CNOT-complexity $3(n-1)$. We show that lower bounds for the additive complexity of not-necessarily-reversible linear operators can be lifted to the reversible setting with only a small loss. As an application, we use this to describe an explicit family of matrices, constructed from parity-check matrices of error-correcting codes, with CNOT-complexity at least $4n - o(n)$, asymptotically surpassing the cyclic permutations. Moreover, this construction yields an explicit matrix $A \in \mathrm{GL}_{n}(\mathbb{F}_2)$, $n = 17167$, whose CNOT-complexity exceeds that of the cyclic permutation on $n$ symbols.

quant-ph↗

arXiv2026

Variance reduction methods in the estimation of Pauli sums

Accurately estimating expectation values of quantum observables with as few measurements as possible is crucial to many quantum computing applications. We introduce a framework that covers many of existing measurement strategies and introduce heuristics that can be used to enhance randomized schemes, including those based on Pauli grouping with inverse probability weighting and variants of the classical shadow algorithm. We show how to maximize information gain from such schemes, while carefully optimizing the distribution of possible measurements, and show that simple grouping algorithms can get close to, and in some cases exceed, state-of-the-art accuracy for unbiased estimation of expectation values on a standard quantum chemistry benchmark. We show how these randomized methods may be compared to more recent measurement schemes, such as shadow grouping, derandomized shadow, and overlapped grouping measurement, we show how the same strategies can be used to augment these schemes, and we demonstrate that we can reduce measurement costs by up to a factor of two by allowing Clifford measurement circuits for otherwise Clifford-less methods.

quant-ph↗

arXiv2025

On Exact Sizes of Minimal CNOT Circuits

Computing a minimum-size circuit that implements a certain function is a standard optimization task. We consider circuits of CNOT gates, which are fundamental binary gates in reversible and quantum computing. Algebraically, CNOT circuits on $n$ qubits correspond to $GL(n,2)$, the general linear group over the field of two elements, and circuit minimization reduces to computing distances in the Cayley graph $G_n$ of $GL(n,2)$ generated by transvections. However, the super-exponential size of $GL(n,2)$ has made its exploration computationally challenging. In this paper, we develop a new approach for computing distances in $G_n$, allowing us to synthesize minimum circuits that were previously beyond reach (e.g., we can synthesize optimally all circuits over $n=7$ qubits). Towards this, we establish two theoretical results that may be of independent interest. First, we give a complete characterization of all isometries in $G_n$ in terms of (i) permuting qubits and (ii) swapping the arguments of all CNOT gates. Second, for any fixed $d$, we establish polynomials in $n$ of degree $2d$ that characterize the size of spheres in $G_n$ at distance $d$, as long as $n\geq 2d$. With these tools, we revisit an open question of [Bataille, 2022] regarding the smallest number $n_0$ for which the diameter of $G_{n_0}$ exceeds $3(n_0-1)$. It was previously shown that $6\leq n_0 \leq 30$, a gap that we tighten considerably to $8\leq n_0 \leq 20$. We also confirm a conjecture that long cycle permutations lie at distance $3(n-1)$, for all $n\leq 8$, extending the previous bound of $n\leq 5$.

quant-ph↗

arXiv2025

On the clique covering numbers of Johnson graphs

We initiate a study of the vertex clique covering numbers of Johnson graphs $J(N, k)$, the smallest numbers of cliques necessary to cover the vertices of those graphs. We prove identities for the values of these numbers when $k \leq 3$, and $k \geq N - 3$, and using computational methods, we provide explicit values for a range of small graphs. By drawing on connections to coding theory and combinatorial design theory, we prove various bounds on the clique covering numbers for general Johnson graphs, and we show how constant-weight lexicodes can be utilized to create optimal covers of $J(2k, k)$ when $k$ is a small power of two.

math.CO↗

arXiv2023

Sublinear Time Shortest Path in Expander Graphs

Computing a shortest path between two nodes in an undirected unweighted graph is among the most basic algorithmic tasks. Breadth first search solves this problem in linear time, which is clearly also a lower bound in the worst case. However, several works have shown how to solve this problem in sublinear time in expectation when the input graph is drawn from one of several classes of random graphs. In this work, we extend these results by giving sublinear time shortest path (and short path) algorithms for expander graphs. We thus identify a natural deterministic property of a graph (that is satisfied by typical random regular graphs) which suffices for sublinear time shortest paths. The algorithms are very simple, involving only bidirectional breadth first search and short random walks. We also complement our new algorithms by near-matching lower bounds.

cs.DS↗

arXiv2016

The Witten-Reshetikhin-Turaev invariant for links in finite order mapping tori I

We state Asymptotic Expansion and Growth Rate conjectures for the Witten-Reshetikhin-Turaev invariants of arbitrary framed links in 3-manifolds, and we prove these conjectures for the natural links in mapping tori of finite-order automorphisms of marked surfaces. Our approach is based upon geometric quantisation of the moduli space of parabolic bundles on the surface, which we show coincides with the construction of the Witten-Reshetikhin-Turaev invariants using conformal field theory, as was recently completed by Andersen and Ueno.

math.GT↗

arXiv2014

On the Witten--Reshetikhin--Turaev invariants of torus bundles

By methods similar to those used by Lisa Jeffrey, we compute the quantum $\mathrm{SU}(N)$-invariants for mapping tori of trace $2$ homeomorphisms of a genus $1$ surface when $N = 2,3$ and discuss their asymptotics. In particular, we obtain directly a proof of a version of Witten's asymptotic expansion conjecture for these 3-manifolds. We further prove the growth rate conjecture for these 3-manifolds in the $\mathrm{SU}(2)$ case, where we also allow the 3-manifolds to contain certain knots. In this case we also discuss trace $-2$ homeomorphisms, obtaining -- in combination with Jeffrey's results -- a proof of the asymptotic expansion conjecture for all torus bundles.

math.QA↗

arXiv2014

The homological content of the Jones representations at $q = -1$

We generalize a discovery of Kasahara and show that the Jones representations of braid groups, when evaluated at $q = -1$, are related to the action on homology of a branched double cover of the underlying punctured disk. As an application, we prove for a large family of pseudo-Anosov mapping classes a conjecture put forward by Andersen, Masbaum, and Ueno by extending their original argument for the sphere with four marked points to our more general case.

math.GT↗