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arXiv:2210.11127

Estimating the Jones polynomial for Ising anyons on noisy quantum computers

Chris N. Self, S. Iblisdir, Gavin K. Brennen, Konstantinos Meichanetzidis

Published Oct 20, 2022• 0 citations
arXivHTMLPDF

Abstract

The evaluation of the Jones polynomial at roots of unity is a paradigmatic problem for quantum computers. In this work we present experimental results obtained from existing noisy quantum computers for special cases of this problem, where it is classically tractable. Our approach relies on the reduction of the problem of evaluating the Jones polynomial of a knot at lattice roots of unity to the problem of computing quantum amplitudes of qudit stabiliser circuits, which are classically efficiently simulatable. More specifically, we focus on evaluation at the fourth root of unity, which is a lattice root of unity, where the problem reduces to evaluating amplitudes of qubit stabiliser circuits. To estimate the real and imaginary parts of the amplitudes up to additive error we use the Hadamard test, yielding non-Clifford circuits that nevertheless we can always efficiently compute the correct output of. Hence, we further argue that this setup defines a standard benchmark for near-term noisy quantum processors. Additionally, we study the benefit of performing quantum error mitigation with the method of zero noise extrapolation.

DOI
10.48550/arxiv.2210.11127

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The evaluation of the Jones polynomial at roots of unity is a paradigmatic problem for quantum computers. In this work we present experimental results obtained from existing noisy quantum computers for special cases of this problem, where it is classically tractable. Our approach relies on the reduction of the problem of evaluating the Jones polynomial of a knot at lattice roots of unity to the problem of computing quantum amplitudes of qudit stabiliser circuits, which are classically efficiently simulatable. More specifically, we focus on evaluation at the fourth root of unity, which is a lattice root of unity, where the problem reduces to evaluating amplitudes of qubit stabiliser circuits. To estimate the real and imaginary parts of the amplitudes up to additive error we use the Hadamard test, yielding non-Clifford circuits that nevertheless we can always efficiently compute the correct output of. Hence, we further argue that this setup defines a standard benchmark for near-term noisy quantum processors. Additionally, we study the benefit of performing quantum error mitigation with the method of zero noise extrapolation.
The evaluation of the Jones polynomial at roots of unity is a paradigmatic problem for quantum computers. In this work, we present experimental results obtained from existing noisy quantum computers for special cases of this problem, where it is classically tractable. Our approach relies on the reduction of the problem of evaluating the Jones polynomial of a knot at lattice roots of unity to the problem of computing quantum amplitudes of qudit stabilizer circuits, which are classically efficiently simulatable. More specifically, we focus on evaluation at the fourth root of unity, which is a lattice root of unity, where the problem reduces to evaluating amplitudes of qubit stabilizer circuits. To estimate the real and imaginary parts of the amplitudes up to additive error we use the Hadamard test, yielding non-Clifford circuits that nevertheless we can always efficiently compute the correct output of. Hence, we further argue that this setup defines a standard benchmark for near-term noisy quantum processors. Additionally, we study the benefit of performing quantum error mitigation with the method of zero-noise extrapolation.
We present an end-to-end algorithmic pipeline where a noisy digital quantum computer is used to approximate the value of the Jones polynomial at the fifth root of unity for any input link, i.e., a closed braid. This problem is DQC1-complete for Markov-closed braids and BQP-complete for Plat-closed braids, and we accommodate both versions of the problem. Even though it is widely believed that DQC1 is strictly contained in BQP, and so is “less quantum,” the resource requirements of classical algorithms for the DQC1 version are at least as high as for the BQP version, and so we potentially gain “more advantage” by focusing on Markov-closed braids in our exposition. We demonstrate our quantum algorithm on Quantinuum’s H2-2 quantum computer and show the effect of problem-tailored error-mitigation techniques. Further, leveraging that the Jones polynomial is a link invariant, we construct an efficiently verifiable benchmark to characterize the effect of noise present in a given quantum processor. In parallel, we develop and benchmark the state-of-the-art tensor-network-based classical algorithms for computing the Jones polynomial. The reconfigurable tools provided in this work allow for precise resource estimation to identify minimum link sizes for near-term quantum advantage in practice for a meaningful quantum-native problem in knot theory, if a candidate set of links are provided.
We analyze relationships between quantum computation and a family of generalizations of the Jones polynomial. Extending recent work by Aharonov et al., we give efficient quantum circuits for implementing the unitary Jones-Wenzl representations of the braid group. We use these to provide new quantum algorithms for approximately evaluating a family of specializations of the HOMFLYPT two-variable polynomial of trace closures of braids. We also give algorithms for approximating the Jones polynomial of a general class of closures of braids at roots of unity. Next we provide a self-contained proof of a result of Freedman et al. that any quantum computation can be replaced by an additive approximation of the Jones polynomial, evaluated at almost any primitive root of unity. Our proof encodes two-qubit unitaries into the rectangular representation of the eight-strand braid group. We then give QCMA-complete and PSPACE-cornplete problems which are based on braids. We conclude with direct proofs that evaluating the Jones polynomial of the plat closure at most primitive roots of unity is a #P-hard problem, while learning its most significant bit is PP-hard, circumventing the usual route through the Tutte polynomial and graph coloring.
Abstract Topological quantum computers promise a fault tolerant means to perform quantum computation. Topological quantum computers use particles with exotic exchange statistics called non-Abelian anyons, and the simplest anyon model which allows for universal quantum computation by particle exchange or braiding alone is the Fibonacci anyon model. One classically hard problem that can be solved efficiently using quantum computation is finding the value of the Jones polynomial of knots at roots of unity. We aim to provide a pedagogical, self-contained, review of topological quantum computation with Fibonacci anyons, from the braiding statistics and matrices to the layout of such a computer and the compiling of braids to perform specific operations. Then we use a simulation of a topological quantum computer to explicitly demonstrate a quantum computation using Fibonacci anyons, evaluating the Jones polynomial of a selection of simple knots. In addition to simulating a modular circuit-style quantum algorithm, we also show how the magnitude of the Jones polynomial at specific points could be obtained exactly using Fibonacci or Ising anyons. Such an exact algorithm seems ideally suited for a proof of concept demonstration of a topological quantum computer.
Topological quantum computers promise a fault tolerant means to perform quantum computation. Topological quantum computers use particles with exotic exchange statistics called non-Abelian anyons, and the simplest anyon model which allows for universal quantum computation by particle exchange or braiding alone is the Fibonacci anyon model. One classically hard problem that can be solved efficiently using quantum computation is finding the value of the Jones polynomial of knots at roots of unity. We aim to provide a pedagogical, self-contained, review of topological quantum computation with Fibonacci anyons, from the braiding statistics and matrices to the layout of such a computer and the compiling of braids to perform specific operations. Then we use a simulation of a topological quantum computer to explicitly demonstrate a quantum computation using Fibonacci anyons, evaluating the Jones polynomial of a selection of simple knots.
We analyze relationships between quantum computation and a family of generalizations of the Jones polynomial. Extending recent work by Aharonov et al., we give efficient quantum circuits for implementing the unitary Jones-Wenzl representations of the braid group. We use these to provide new quantum algorithms for approximately evaluating a family of specializations of the HOMFLYPT two-variable polynomial of trace closures of braids. We also give algorithms for approximating the Jones polynomial of a general class of closures of braids at roots of unity. Next we provide a self-contained proof of a result of Freedman et al. that any quantum computation can be replaced by an additive approximation of the Jones polynomial, evaluated at almost any primitive root of unity. Our proof encodes two-qubit unitaries into the rectangular representation of the eight-strand braid group. We then give QCMA-complete and PSPACE-complete problems which are based on braids. We conclude with direct proofs that evaluating the Jones polynomial of the plat closure at most primitive roots of unity is a #P-hard problem, while learning its most significant bit is PP-hard, circumventing the usual route through the Tutte polynomial and graph coloring.