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arXiv:2408.07132 · Physical Review A

Topological quantum compilation of two-qubit gates

Phillip C. Burke, Christos Aravanis, Johannes Aspman, Jakub Mareček, Jiří Vala

Published Nov 25, 2024• 3 citations
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Abstract

We investigate the topological quantum compilation of two-qubit operations within a system of Fibonacci anyons. Our primary goal is to generate gates that are approximately leakage-free and equivalent to the controlled-not (cnot) gate up to single-qubit operations. These gates belong to the local equivalence class [cnot]. Additionally, we explore which local equivalence classes of two-qubit operations can be naturally generated by braiding Fibonacci anyons. We discovered that most of the generated classes are located near the edges of the Weyl chamber representation of two-qubit gates, specifically between the local equivalence classes of the identity $[\mathbb{1}]$ and [cnot], swap and between those of the double-controlled-not [dcnot] and [swap]. Furthermore, we found a numerically exact implementation of a local equivalent of the swap gate using a sequence of only nine elements from the Fibonacci braiding gate set.

DOI
10.1103/physreva.110.052616

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We investigate the topological quantum compilation of two-qubit operations within a system of Fibonacci anyons. Our primary goal is to generate gates that are approximately leakage-free and equivalent to the controlled-not (cnot) gate up to single-qubit operations. These gates belong to the local equivalence class [cnot]. Additionally, we explore which local equivalence classes of two-qubit operations can be naturally generated by braiding Fibonacci anyons. We discovered that most of the generated classes are located near the edges of the Weyl chamber representation of two-qubit gates, specifically between the local equivalence classes of the identity [1][\mathbb{1}][1] and [cnot], swap and between those of the double-controlled-not [dcnot] and [swap]. Furthermore, we found a numerically exact implementation of a local equivalent of the swap gate using a sequence of only nine elements from the Fibonacci braiding gate set.
Abstract Topological quantum computation is an implementation of a quantum computer in a way that radically reduces decoherence. Topological qubits are encoded in the topological evolution of two-dimensional quasi-particles called anyons and universal set of quantum gates can be constructed by braiding these anyons yielding to a topologically protected circuit model. In the present study we remind the basics of this emerging quantum computation scheme and illustrate how a topological qubit built with three Fibonacci anyons might be adopted to achieve leakage free braiding gate by exchanging the anyons composing it. A single-qubit braiding gate that approximates the Hadamard quantum gate to a certain accuracy is numerically implemented using a brute force search method. The algorithms utilized for that purpose are explained and the numerical programs are publicly shared for reproduction and further use.
A method, termed controlled injection, is proposed for compiling three-qubit controlled gates within the non-Abelian Fibonacci anyon model. Building on single-qubit compilation techniques with three Fibonacci anyons, the approach showcases enhanced accuracy and reduced braid length compared to the conventional decomposition method for the controlled three-qubit gates. This method necessitates only four two-qubit gates for decomposition, a notable reduction from the conventional five. In conjunction, the study introduces a class of controlled three-qubit gates and conducts a numerical simulation of the topological i Toffoli gate to validate the approach. In addition, we propose an optimization method for single-qubit gate approximation using algebraic relations and numerical methods, including distributed computing. Published by the American Physical Society 2024
Two applications of gates from the B gate equivalence class can generate all two-qubit gates. This local equivalence class is invariant under the mirror (multiplication with the SWAP gate) operation, inverse (Hermitian conjugate) operation, and the combined inverse and mirror operations. The last two symmetries are associated with the ability of a two-qubit gate to generate the two-qubit local gates and the SWAP gate in two applications. No single local equivalence class of two-qubit gates, except the B gate equivalence class, has these two symmetries. Only the planar regions of the Weyl chamber, describing the mirror operation, contain the local equivalence classes with either one of the two symmetries. We show that there exist one-parameter families of local equivalence classes on these planes, with and without the B gate equivalence class, such that each of them can be used to construct a parameterized universal two-qubit quantum circuit that involves only two nonlocal two-qubit gates. We also discuss the implementation of the gates from a few families of local equivalence classes on superconducting quantum computers for optimal generation of all two-qubit gates. We provide upper bounds on the number of two-qubit gates required to generate an arbitrary nnn-qubit gate for two families, each of which is conjectured to generate all two-qubit gates in two applications. We show that there exists a positive correlation between the area of the convex hull of the squared eigenvalues of the nonlocal part of a parameterized two-qubit gate and the fractional volume of the Weyl chamber covered in two applications of the parameterized two-qubit gate for two families of local equivalence classes.
Fibonacci anyons are attractive for use in topological quantum computation because any unitary transformation of their state space can be approximated arbitrarily accurately by braiding. However, there is no known braid that entangles two qubits without leaving the space spanned by the two qubits. In other words, there is no known ‘leakage-free’ entangling gate made by braiding. In this paper, we provide a remedy to this problem by supplementing braiding with measurement operations in order to produce an exact controlled rotation gate on two qubits.
Fibonacci anyons are attractive for use in topological quantum computation because any unitary transformation of their state space can be approximated arbitrarily accurately by braiding. However there is no known braid that entangles two qubits without leaving the space spanned by the two qubits. In other words, there is no known "leakage-free" entangling gate made by braiding. In this paper, we provide a remedy to this problem by supplementing braiding with measurement operations in order to produce an exact controlled rotation gate on two qubits.
We introduce the Mixed-Integer Quadratically Constrained Quadratic Programming framework for the quantum compilation problem and apply it in the context of topological quantum computing. In this setting, quantum gates are realized by sequences of elementary braids of quasiparticles with exotic fractional statistics in certain two-dimensional topological condensed matter systems, described by effective topological quantum field theories. We specifically focus on a non-semisimple version of topological field theory, which provides a foundation for an extended theory of Ising anyons and which has recently been shown by Iulianelli et al., Nature Communications {\bf 16}, 6408 (2025), to permit universal quantum computation. While the proofs of this pioneering result are existential in nature, the mixed integer programming provides an approach to explicitly construct quantum gates in topological systems. We demonstrate this by focusing specifically on the entangling controlled-NOT operation, and its local equivalence class, using braiding operations in the non-semisimple Ising system. This illustrates the utility of the Mixed-Integer Quadratically Constrained Quadratic Programming for topological quantum compilation.