Related papers: With a Few Square Roots, Quantum Computing is as E…
Non-semisimple extensions of the Ising anyon model developed in our previous work enable universal topological quantum computation via braiding alone, overcoming the Clifford-only limitation of semisimple theories. The non-semisimple theory…
There have been significant recent advances in constructing theoretical and practical quantum error correcting codes that function well as quantum memories; however, performing fault-tolerant logical gates on these codes is less studied,…
We use our Clifford algebra technique, that is nilpotents and projectors which are binomials of the Clifford algebra objects $\gamma^a$ with the property $\{\gamma^a,\gamma^b\}_+ = 2 \eta^{ab}$, for representing quantum gates and quantum…
In this work, we develop a graphical calculus for multi-qudit computations with generalized Clifford algebras, building off the algebraic framework developed in our prior work. We build our graphical calculus out of a fixed set of graphical…
A popular universal gate set for quantum computing with qubits is Clifford+T, as this can be readily implemented on many fault-tolerant architectures. For qutrits, there is an equivalent T gate, that, like its qubit analogue, makes…
This is an exposition of some basic mathematical aspects of quantum logic gates. At first we established some general formulas for the case of arbitrary quantum gate A with unique restriction A^2=I. The explicit form of the generators and…
Free categorical constructions characterise quantum computing as the combination of two copies of a reversible classical model, glued by the complementarity equations of classical structures. This recipe effectively constructs a…
Let ${\cal H}$ be the state-space of a quantum computer coupled with the environment by a set of error operators spanning a Lie algebra ${\cal L}.$ Suppose ${\cal L}$ admits a noiseless quantum code i.e., a subspace ${\cal C}\subset{\cal…
The Clifford+T gate set is a topological generating set for PU(2), which has been well-studied from the perspective of quantum computation on a single qubit. The discovery that it generates a full S-arithmetic subgroup of PU(2) has led to a…
Quantum circuits consisting of Clifford and matchgates are two classes of circuits that are known to be efficiently simulatable on a classical computer. We introduce a unified framework that shows in a transparent way the special structure…
Qubits encoded in a decoherence-free subsystem and realized in exchange-coupled silicon quantum dots are promising candidates for fault-tolerant quantum computing. Benefits of this approach include excellent coherence, low control…
There are various gate sets used for describing quantum computation. A particularly popular one consists of Clifford gates and arbitrary single-qubit phase gates. Computations in this gate set can be elegantly described by the ZX-calculus,…
Quantum computing relies on quantum error correction for high-fidelity logical operations, but scaling to achieve near-term quantum utility is highly resource-intensive. High-rate quantum LDPC codes can reduce error correction overhead, yet…
One learned from Gottesman-Knill theorem that the Clifford model of quantum computing \cite{Clark07} may be generated from a few quantum gates, the Hadamard, Phase and Controlled-Z gates, and efficiently simulated on a classical computer.…
We provide a universal construction of the category of finite-dimensional C*-algebras and completely positive trace-nonincreasing maps from the rig category of finite-dimensional Hilbert spaces and unitaries. This construction, which can be…
We present an entirely 2D transversal realization of phase gates at any level of the Clifford hierarchy, and beyond, using non-Abelian surface codes. Our construction encodes a logical qubit in the quantum double $D(G)$ of a non-Abelian…
We present two deterministic algorithms to approximate single-qutrit gates. These algorithms utilize the Clifford + $\mathbf{R}$ group to find the best approximation of diagonal rotations. The first algorithm exhaustively searches over the…
This paper explores the representation of quantum computing in terms of unitary reflections (unitary transformations that leave invariant a hyperplane of a vector space). The symmetries of qubit systems are found to be supported by…
We introduce Graphical Algebraic Geometry (GAG), a family of diagrammatic languages extending the Graphical Linear Algebra programme. We construct several languages within this family and prove that they are universal and complete for the…
Kliuchnikov, Maslov, and Mosca proved in 2012 that a $2\times 2$ unitary matrix $V$ can be exactly represented by a single-qubit Clifford+$T$ circuit if and only if the entries of $V$ belong to the ring $\mathbb{Z}[1/\sqrt{2},i]$. Later…