arXiv:2408.03377

Demonstration of magic state power of $\mathbf{D}(\mathbf{S}_{3})$ anyons with two qudits

Lucy Byles, Forbes, Ewan, Jiannis K. Pachos

Published Aug 6, 2024• 0 citations

Abstract

We consider a lattice of $d=6$ qudits that supports $\mathbf{D}(\mathbf{S}_3)$ non-Abelian anyons. We present a method for implementing both braiding and fusion evolutions using only the operators that create and measure anyons, without requiring additional dynamical control. This provides a minimal protocol demonstrating that $\mathbf{D}(\mathbf{S}_3)$ anyons can generate magic states, thereby establishing their universality for quantum computation. Furthermore, we show that the entire scheme can be encoded in just two qudits, offering a compact blueprint that is inherently scalable and readily implementable in current quantum platforms.

DOI: 10.48550/arxiv.2408.03377

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arXiv:2408.03377

Demonstration of magic state power of $\mathbf{D}(\mathbf{S}_{3})$ anyons with two quditsDemonstration of magic state power of $\mathbf{D}(\mathbf{S}_{3})$ anyons with two qudits

Lucy Byles, Forbes, Ewan, Jiannis K. Pachos

Published Aug 6, 2024• 0 citations

arXiv HTML PDF

Abstract

We consider a lattice of

d=6

qudits that supports

\mathbf{D}(\mathbf{S}_3)

non-Abelian anyons. We present a method for implementing both braiding and fusion evolutions using only the operators that create and measure anyons, without requiring additional dynamical control. This provides a minimal protocol demonstrating that

\mathbf{D}(\mathbf{S}_3)

anyons can generate magic states, thereby establishing their universality for quantum computation. Furthermore, we show that the entire scheme can be encoded in just two qudits, offering a compact blueprint that is inherently scalable and readily implementable in current quantum platforms.

DOI: 10.48550/arxiv.2408.03377

Related papers

Demonstration of magic state power of D ( S 3 ) anyons with two qudits

Lucy Byles, E. G. Forbes, Jiannis K. Pachos

Abstract We consider a lattice of d = 6 qudits that supports D ( S 3 ) non-Abelian anyons. We present a method for implementing both braiding and fusion evolutions using only the operators that create and measure anyons, without requiring additional dynamical control. This provides a minimal protocol demonstrating that D ( S 3 ) anyons can generate magic states, thereby establishing their universality for quantum computation. Furthermore, we show that the entire scheme can be encoded in just two qudits, offering a compact blueprint that is inherently scalable and readily implementable in current quantum platforms.Abstract We consider a lattice of d = 6 qudits that supports D ( S 3 ) non-Abelian anyons. We present a method for implementing both braiding and fusion evolutions using only the operators that create and measure anyons, without requiring additional dynamical control. This provides a minimal protocol demonstrating that D ( S 3 ) anyons can generate magic states, thereby establishing their universality for quantum computation. Furthermore, we show that the entire scheme can be encoded in just two qudits, offering a compact blueprint that is inherently scalable and readily implementable in current quantum platforms.

New Journal of PhysicsApr 1, 20260 citations

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Reconstructing non-Abelian braiding and fusion without anyon transport

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Non-Abelian anyons offer a route to fault-tolerant and universal quantum computing, but experimental access to their defining braiding and fusion data remains limited by the resource overhead of implementing extended anyonic processes on quantum hardware. Here we introduce and experimentally realise a measurement-only protocol based on temporally ordered ribbon operations that reconstructs the non-Abelian braiding and fusion primitives of the quantum double model $D(S_3)$ without physical anyon transport. We implement a reduced two-qutrit version of the protocol on Quantinuum's H2 trapped-ion processors, realising ancilla-assisted ribbon operations and anyonic charge projections in a qubit encoding. We reconstruct the squared braiding phases and fusion amplitudes using an adapted Hadamard test and post-selected measurements, respectively. The associated braiding and fusion transformations reproduce their ideal actions with average normalised output-state fidelities of $\overline{\mathcal{F}}_{R}=0.9988$ and $\overline{\mathcal{F}}_{F}=0.9987$ . Combining these primitives produces a non-Clifford braid and a non-stabilizer resource state, supporting measurement-only anyonic encodings as building blocks for larger topologically encoded quantum processors.Non-Abelian anyons offer a route to fault-tolerant and universal quantum computing, but experimental access to their defining braiding and fusion data remains limited by the resource overhead of implementing extended anyonic processes on quantum hardware. Here we introduce and experimentally realise a measurement-only protocol based on temporally ordered ribbon operations that reconstructs the non-Abelian braiding and fusion primitives of the quantum double model $D(S_3)$ without physical anyon transport. We implement a reduced two-qutrit version of the protocol on Quantinuum's H2 trapped-ion processors, realising ancilla-assisted ribbon operations and anyonic charge projections in a qubit encoding. We reconstruct the squared braiding phases and fusion amplitudes using an adapted Hadamard test and post-selected measurements, respectively. The associated braiding and fusion transformations reproduce their ideal actions with average normalised output-state fidelities of $\overline{\mathcal{F}}_{R}=0.9988$ and $\overline{\mathcal{F}}_{F}=0.9987$. Combining these primitives produces a non-Clifford braid and a non-stabilizer resource state, supporting measurement-only anyonic encodings as building blocks for larger topologically encoded quantum processors.

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We provide a comprehensive systematic method for the numerical computation of elementary braid operations in topological quantum computation (TQC). This {procedure} is systematically applicable to all anyon models, including $SU(2)_k$ . Braiding non-abelian anyons is the essence of TQC, offering a topologically protected implementation of quantum gates. However, obtaining elementary braid matrix representations starting from the fusion and rotation matrices of a specific anyon model is {theoretically guarenteed but no numerical method is given, especially for systems with numerous anyons and complex fusion patterns. Our proposed method addresses this challenge, first in the special case of sparse encoding, allowing for the inclusion of an arbitrary number of anyons per qudit, {and in the general case}. This is accomplished by introducing two methods, one is based on a novel braiding move we call knitting, {the second introduces more general algorithm which is optimal in number of required moves}. The method plays a key role in a broad topological quantum circuit simulator, enabling the examination and study of complex quantum circuits within the TQC framework. Importantly, it proves effective across various anyonic models, accommodating diverse fusion rules. We validate the method by simulating an approximated CNOT gate and present a first-of-a-kind GHZ state simulation on five qubits using three Fibonacci anyons per qubit.We provide a comprehensive systematic method for the numerical computation of elementary braid operations in topological quantum computation (TQC). This {procedure} is systematically applicable to all anyon models, including $SU(2)_k$. Braiding non-abelian anyons is the essence of TQC, offering a topologically protected implementation of quantum gates. However, obtaining elementary braid matrix representations starting from the fusion and rotation matrices of a specific anyon model is {theoretically guarenteed but no numerical method is given, especially for systems with numerous anyons and complex fusion patterns. Our proposed method addresses this challenge, first in the special case of sparse encoding, allowing for the inclusion of an arbitrary number of anyons per qudit, {and in the general case}. This is accomplished by introducing two methods, one is based on a novel braiding move we call knitting, {the second introduces more general algorithm which is optimal in number of required moves}. The method plays a key role in a broad topological quantum circuit simulator, enabling the examination and study of complex quantum circuits within the TQC framework. Importantly, it proves effective across various anyonic models, accommodating diverse fusion rules. We validate the method by simulating an approximated CNOT gate and present a first-of-a-kind GHZ state simulation on five qubits using three Fibonacci anyons per qubit.

arXiv:2307.01892Jul 4, 20230 citations

Demonstration of magic state power of

\mathbf{D}(\mathbf{S}_{3})

anyons with two qudits

We consider a lattice of

d=6

qudits that supports

\mathbf{D}(\mathbf{S}_3)

\mathbf{D}(\mathbf{S}_3)

S_3

, on Quantinuum's H2 quantum processor. This phase of matter exhibits cyclic anyon fusion rules, known to underpin universality, which we evidence by trapping a single non-Abelian anyon on the torus. We encode logical qutrits in the nonlocal fusion space of non-Abelian fluxes and, by combining an entangling braiding operation with anyon charge measurements, realize a universal topological gate set and read-out, which we further demonstrate by topologically preparing a magic state. This work establishes

S_3

TO as simple enough to be prepared efficiently, yet rich enough to enable universal topological quantum computation.

D(S_3)

without physical anyon transport. We implement a reduced two-qutrit version of the protocol on Quantinuum's H2 trapped-ion processors, realising ancilla-assisted ribbon operations and anyonic charge projections in a qubit encoding. We reconstruct the squared braiding phases and fusion amplitudes using an adapted Hadamard test and post-selected measurements, respectively. The associated braiding and fusion transformations reproduce their ideal actions with average normalised output-state fidelities of

\overline{\mathcal{F}}_{R}=0.9988

and

\overline{\mathcal{F}}_{F}=0.9987

. Combining these primitives produces a non-Clifford braid and a non-stabilizer resource state, supporting measurement-only anyonic encodings as building blocks for larger topologically encoded quantum processors.

Universal Quantum Computation with the

S_3

Quantum Double: A Pedagogical Exposition

Non-Abelian topological order (TO) enables topologically protected quantum computation with its anyonic quasiparticles. Recently, TO with

S_3

gauge symmetry was identified as a sweet spot -- simple enough to emerge from finite-depth adaptive circuits yet powerful enough to support a universal topological gate-set. In these notes, we review how anyon braiding and measurement in

S_3

TO are primitives for topological quantum computation and we explicitly demonstrate universality. These topological operations are made concrete in the

S_3

quantum double lattice model, aided by the introduction of a generalized ribbon operator. This provides a roadmap for near-term quantum platforms.

SU(2)_k

. Braiding non-abelian anyons is the essence of TQC, offering a topologically protected implementation of quantum gates. However, obtaining elementary braid matrix representations starting from the fusion and rotation matrices of a specific anyon model is {theoretically guarenteed but no numerical method is given, especially for systems with numerous anyons and complex fusion patterns. Our proposed method addresses this challenge, first in the special case of sparse encoding, allowing for the inclusion of an arbitrary number of anyons per qudit, {and in the general case}. This is accomplished by introducing two methods, one is based on a novel braiding move we call knitting, {the second introduces more general algorithm which is optimal in number of required moves}. The method plays a key role in a broad topological quantum circuit simulator, enabling the examination and study of complex quantum circuits within the TQC framework. Importantly, it proves effective across various anyonic models, accommodating diverse fusion rules. We validate the method by simulating an approximated CNOT gate and present a first-of-a-kind GHZ state simulation on five qubits using three Fibonacci anyons per qubit.