Related papers: Limitations on transversal gates for hypergraph pr…
Braiding defects in topological stabiliser codes has been widely studied as a promising approach to fault-tolerant quantum computing. We present a no-go theorem that places very strong limitations on the potential of such schemes for…
We exhibit nontrivial transversal logical multi-controlled-$Z$ gates on $[\![N,\Theta(N),\tilde\Theta(N)]\!]$ quantum low-density parity-check codes and $[\![N,\Theta(N),\tilde\Theta(N)]\!]$ quantum locally testable codes with soundness…
We give two generalizations of the Clifford theorem to algebraic surfaces. As an application, we obtain some bounds for the number of moduli of surfaces of general type.
The Gottesman-Knill theorem asserts that quantum circuits composed solely of Clifford gates can be efficiently simulated classically. This theorem hinges on the fact that Clifford gates map Pauli strings to other Pauli strings, thereby…
In this article we extend the theory of the binary codes (the strict code $\mathcal{K}$ and the extended code $\mathcal{K}'$), associated to a projective nodal surface, to a coding theory for normal surfaces, with special consideration of…
Clifford gates are a winsome class of quantum operations combining mathematical elegance with physical significance. The Gottesman-Knill theorem asserts that Clifford computations can be classically efficiently simulated but this is true…
We show that single-qudit universality in Clifford-based gate sets follows a trichotomy determined by the prime factorization of the local dimension $d$. For prime $d$, any gate outside the Clifford group is universal. For prime-power…
Fault-tolerant quantum computation is a basic problem in quantum computation, and teleportation is one of the main techniques in this theory. Using teleportation on stabilizer codes, the most well-known quantum codes, Pauli gates and…
The hypergraph product creates a quantum stabilizer code from two input classical linear codes; a paradigmatic example being the surface code as a hypergraph product of two classical repetition codes. Many properties of the hypergraph…
A divisible binary classical code is one in which every code word has weight divisible by a fixed integer. If the divisor is $2^\nu$ for a positive integer $\nu$, then one can construct a Calderbank-Shor-Steane (CSS) code, where…
We study the encoding complexity for quantum error correcting codes with large rate and distance. We prove that random Clifford circuits with $O(n \log^2 n)$ gates can be used to encode $k$ qubits in $n$ qubits with a distance $d$ provided…
The recent proposal (M Planat and M Kibler, Preprint 0807.3650 [quantph]) of representing Clifford quantum gates in terms of unitary reflections is revisited. In this essay, the geometry of a Clifford group G is expressed as a BN-pair, i.e.…
A fundamental problem of fault-tolerant quantum computation with quantum low-density parity-check (qLDPC) codes is the tradeoff between connectivity and universality. It is widely believed that in order to perform native logical…
The color code has been invaluable for the development of the theory of fault-tolerant logic gates using transversal rotations. Three-dimensional examples of the color code have shown us how its structure, specifically the intersection of…
We give a general proof for the existence and realizability of Clifford gates in the Ising topological quantum computer. We show that all quantum gates that can be implemented by braiding of Ising anyons are Clifford gates. We find that the…
Representations of Spin groups and Clifford algebras derived from the structure of qubit trees are introduced in this work. For ternary trees the construction is more general and reduction to binary trees is formally defined by deletion of…
The AND gate is not reversible$\unicode{x2014}$on qubits. However, it is reversible on qutrits, making it a building block for efficient simulation of qubit computation using qutrits. We first observe that there are multiple two-qutrit…
Constructing quantum codes with good parameters and useful transversal gates is a central problem in quantum error correction. In this paper, we continue our work in arXiv:2502.01864 and construct the first family of asymptotically good…
Executing a logical quantum circuit fault-tolerantly incurs a large spacetime overhead. Recent work has proposed and investigated phantom codes, defined by the property that every in-block logical $\mathrm{CNOT}$ circuit can be implemented…
Given some group $G$ of logical gates, for instance the Clifford group, what are the quantum encodings for which these logical gates can be implemented by simple physical operations, described by some physical representation of $G$? We…