Estimating the Jones polynomial for Ising anyons on noisy quantum computers Chris N. Self1 2Sofyan Iblisdir3 4Gavin K. Brennen5and Konstantinos Meichanetzidis6 7y 1QOLS Blackett Laboratory Imperial College London SW7 2AZ United Kingdom

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Estimating the Jones polynomial for Ising anyons on noisy quantum computers

Chris N. Self,1, 2, ∗Sofyan Iblisdir,3, 4 Gavin K. Brennen,5and Konstantinos Meichanetzidis6, 7, †

1QOLS, Blackett Laboratory, Imperial College London SW7 2AZ, United Kingdom

2Quantinuum, Partnership House, Carlisle Place, London SW1P 1BX, United Kingdom

3Dpto. An´alisis Matem´atico y Matem´atica Aplicada,

Facultad de Matem´aticas, Universidad Complutense, 28040 Madrid, Spain

4Dpt. Astrof´ısica i F´ısica Qu`antica & Institut de Ci`encies del Cosmos,

Facultat de F´ısica, Universitat de Barcelona, 08028 Barcelona, Spain

5Center for Engineered Quantum Systems, Department of Physics & Astronomy, Macquarie University, 2109 NSW, Australia

6Department of Computer Science, University of Oxford

7Quantinuum, 17 Beaumont St., OX1 2NA, Oxford, United Kingdom

(Dated: October 21, 2022)

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. We further argue that this setup deﬁnes a standard benchmark for near-term noisy

quantum processors. Furthermore, we study the beneﬁt of performing quantum error mitigation

with the method of zero noise extrapolation.

I. INTRODUCTION

Knot theory is of both theoretical and practical inter-

est to a wide range of areas of research [1,2]. A fun-

damental question of knot theory is distinguishing when

two knot representations, or more generally links which

are multicomponent knots, are topologically equivalent.

This is addressed by the notion of the link invariant, a

mathematical quantity extractable from a link which is

independent of the representation of the link. A famous

link invariant is the Jones polynomial, a polynomial in

one complex variable.

The evaluation of the Jones polynomial of a link at

roots of unity is important for the ﬁeld of quantum com-

putation. Any other problem eﬃciently computable on

a quantum computer can be reduced to it. Speciﬁcally,

the problem reduces to estimating a quantum amplitude,

involving a quantum circuit dictated by the link, up to

additive error [3]. The quantum protocol used for this is

the Hadamard test (H-test).

Currently available quantum devices fall under the

noisy intermediate-scale quantum (NISQ) paradigm;

they are not error-corrected and so the size of circuits

one can run is limited by the amount of decoherence. To

avoid the detrimental eﬀects of decoherence, it is crucial

to adapt abstract circuits to the speciﬁc quantum pro-

cessor and reduce the number of operations required, one

aims to optimise compilation of abstract circuits to cir-

cuits composed of the native gateset of the speciﬁc quan-

∗christopher.self@quantinuum.com

†k.mei@quantinuum.com

tum processor, and also respect the processors qubit-

qubit connectivity. Furthermore, there is a plethora of

error mitigation techniques being developed [4], whose

aim is to amplify and correct the biases of the signal

that one reads-oﬀ a quantum device.

In this work, we are concerned with two theoretical re-

ductions, and we focus on knots, however our pipeline is

readily applicable to links, as well. The ﬁrst reduction

maps the evaluation of the Jones polynomial of a knot,

to the computation of the partition function of an associ-

ated Potts model. The second reduction maps the latter

to the estimation of a quantum amplitude involving a sta-

biliser, or Cliﬀord, circuit. This speciﬁc mapping through

a Potts model results in Cliﬀord circuits which are clas-

sically eﬃciently strongly simulatable, where strong sim-

ulation means that any complex amplitude can be ob-

tained exactly. However, under the assumption that the

quantum device’s low level operations, as well as the co-

herent noise processes, do not diﬀerentiate between Clif-

ford and non-Cliﬀord circuits, the motivation for using

such circuits as benchmarks for NISQ processors still

holds. We also remind the reader that randomised bench-

marking with Cliﬀord circuits is based on the fact that

Cliﬀord circuits are eﬃcient to simulate classically [5,6].

So, sampling knots and estimating the Jones polynomial

on a quantum device and verifying the answer eﬃciently

by classical simulation can be automated, and deﬁnes a

standardised benchmark for NISQ devices over a speciﬁc

distribution of interesting Cliﬀord circuits. We focus on

the quantum engineering aspect of compiling the circuit

for the H-test on IBM Quantum backends and perform-

ing error mitigation with comparison of diﬀerent meth-

ods. Furthermore, we track the performance over days

arXiv:2210.11127v1 [quant-ph] 20 Oct 2022

and check consistency of the devices’ operation.

II. KNOT DIAGRAMS AND

THE JONES POLYNOMIAL

Aknot is a circle embedded in R3. Intuitively, it is a

strand that can be tangled with itself and is closed, i.e.

has no open, or loose, ends. Importantly, strands are not

allowed to intersect nor go through themselves or other

strands. A knot diagram is a projection of a knot in R2

which keeps information about over- and under-crossings.

The following are examples of knot diagrams [7]:

Diﬀerent projections of the same knot, result in dia-

grams that can be smoothly deformed to each other. In

terms of diagrams, these smooth deformations are gener-

ated by the three Reidemeister moves:

The Jones polynomial VK(t) of a knot Kis a Laurent

polynomial in a complex variable t∈Cand it can be

obtained by a diagram of K. It serves as a knot invariant

in the sense that K∼K0⇒VK=VK0. By K∼K0

we denote topological equivalence in the sense that the

diagram of Kcan be rewritten to the diagram of K0by

asequence of Reidemeister moves. In other words, two

diagrams K,K0are topologically equivalent if Kcan

be smoothly deformed to K0by moving, bending and

wiggling the strands, but without cutting or gluing them.

However, note that VKis not a complete invariant, as in

general VK=VK0;K∼K0. The Jones polynomial

is normalised such that Vunknot = 1. Interestingly, it is

conjectured that the unknot is the only knot for which

this is true.

An elegant way of computing the Jones polynomial

from a knot diagram is via the Kauﬀman bracket [2],

which involves breaking every crossing into two types of

avoided crossings, incurring a number of operations ex-

ponential in the number of crossings. In particular, the

Jones polynomial is an invariant of oriented knots, as it is

sensitive to the handedness or the writhe,w. On an ori-

ented diagram, a +1 value is assigned to a right-handed

crossing, a -1 value to a left-handed one, and their sum

is w. For example:

The writhe can be computed eﬃciently, and multiply-

ing the Kauﬀman bracket with a prefactor depending on

wresults in the Jones polynomial. In fact, the max-

imum and minimum degrees of the polynomial can be

inferred from the knot diagram eﬃciently and are upper

bounded by a linear function in the number of crossings.

The exponential cost needed to compute the polynomial

is reﬂected in that its coeﬃcients can get exponentially

large.

III. JONES POLYNOMIAL EVALUATION VIA

POTTS PARTITION FUNCTION

It is in general #P-hard to evaluate the polynomial at a

particular value ton the complex plane. For some values

of t, the problem can be reduced to the computation of

the partition functions of a particular Potts model as

follows [8].

From the diagram of a knot Kwe extract the signed

graph GK, called the Tait graph, by bicolouring the dia-

gram into black and white regions, with the convention

that the background is white and colours on either side of

each strand are diﬀerent. This ”checkerboard” bicolour-

ing is always possible as the crossings are all four-valent.

We assign vertices to the black regions and edges that

connect the black regions through crossings. The edges

are assigned signs, called the Tait signs, according to a

rule that depends on the crossing being over or under

and also the colours around the crossing. The sum of the

Tait signs is the Tait number τ, and it can be obtained

eﬃciently. An example of this procedure is illustrated as

follows:

Note, the crossing signs, which sum to w, and the Tait

signs, which sum to τ, should not be confused, as they

are diﬀerent in general.

To arrive at a Potts model, we place q-state classi-

cal spins σion the vertices of the obtained graph and

have them interact along the edges. Spins interact only

when they are in the same state and the coupling Jij ∈

{J+, J−}depends on the Tait-sign of the corresponding

edge (i, j). The couplings are related to the Jones vari-

able tby eJ±=−t∓1, which in turn relates to the spin

dimension as q=t+t−1+ 2. Solving for twe obtain

t(q) = 1

2(q+√qpq−4−2),(1)

which tells us at which point t∈Cwe evaluate the Jones

polynomial depending on our choice of the dimension q∈

Nof the spins. The spin-spin interactions are designed

such that the partition function of this Potts model,

ZK(q) = X

{σ}

(i,j)

eδσiσjJij ∈C,(2)

is a knot invariant under the Reidemeister moves, which

now correspond to graph operations:

Then, by this construction, the Jones polynomial is

related to this Potts model’s partition function as

VK(t(q)) = A(t(q), τ, w, n)ZK(q).

The proportionality factor

A= (−t1

2−t−1

2)−(n+1)(−t3/4)wtτ /4

depends on the evaluation point t(q), the number of spins

n, the writhe of the knot w, and the Tait number τof

the Tait graph, so it can be computed eﬃciently. The

computationally expensive quantity here is the partition

function ZK(q) , the exact computation of which, is a

#P-hard problem in general.

IV. POTTS PARTITION FUNCTION AS A

TENSOR NETWORK

For q∈N, the spins can occupy the states σi∈

{0,1, . . . , q −1}. In this case, the scalar quantity ZK(q)

of Eq. 2can be represented as a tensor network TKwith

GKthe underlying graph and qthe bond-dimension [9].

Every vertex is assigned a ‘spider’ tensor and every signed

edge is assigned a sign-dependent matrix. For example:

where the ’spider’ tensors, or ’copy’ tensors, and the ±-

matrices are deﬁned as follows:

The ±-matrices encode the contributions to the par-

tition function due to the spin-spin interactions. The

spiders represent hyperindices, which when summed over

realise the sum over all possible spin conﬁgurations. This

sum is equivalent to the full contraction of TK, i.e. per-

forming all sums over common indices represented by the

edges in the tensor network. Full tensor contraction of

TKthen returns ZK(q). Note that full contraction of

generic tensor networks is a #P-hard problem [10]. Inter-

estingly, tensor contraction algorithms based on graph-

partitioning exhibit subexponential space and time com-

plexity, O(q√n), where nis the number of tensors in the

network, when the underlying graph is planar [11]. This

is indeed the case for tensor networks obtained from Tait

graphs. In Ref. [9], such methods were employed to ex-

actly compute ZK(q) for any q∈Nwhere we refer the

reader for more detail.

However, it is known that at the ‘lattice roots of unity’,

θΛ={eiθ|θ=±π, ±π

2,±π

3,±2π

3}, the problem is in P

[12]. For the speciﬁc values of integer spin dimension we

have that q∈ {2,3,4} ⇔ t(q)∈θΛ, which means that

at these points the computation of ZK(q) is in P. Us-

ing the ZX-calculus [13,14], Ref. [15] presents eﬃcient

simpliﬁcation stategies for these speciﬁc tensor networks,

using rewrite rules of the qubit and qutrit ZX-calculus,

providing an alternative proof of the tractability of eval-

uating the Jones polynomial at lattice roots of unity. We

would like to stress here that the use of formal graphical

languages help bridge existing algorithms for the Jones

polynomial to currently available quantum technology, as

we will see in the next section. We refer the reader to

the work cited above for more detailed expositions.

V. FROM TENSOR NETWORK

TO QUANTUM CIRCUIT

For q∈ {2,3,4}, the tensor network TKprovides

the blueprint for a quantum computation in the form

of an n-qudit quantum circuit CKwith speciﬁc input

states and postselections, representing the quantum am-

plitude ZK(q) = ⊗nh+|CK(q)|+i⊗n∈C[16], where

|+i=Pq−1

i=0 |iiis the unnormalised plus state. To make

this apparent, we use the ‘fusion’ rule obeyed by the spi-

ders [14], according to which any two spiders connected

by at least one ‘wire’, i.e. a common index, they can be

fused to a single spider which inherits the open wires of

the two fused spiders:

The fusion rule immediately follows from the fact that

the spider acts as a copy operation on basis states of the

vector spaces deﬁned on its ‘wires’, and can be veriﬁed

by performing the explicit tensor contraction along the

common indices and the properties of the Kronecker delta

in terms of which the spider is deﬁned. The fusion rule

then allows us to ‘pull out’ the input and output plus-

states and interpret the tensor network as a quantum

circuit. For example:

Reading-oﬀ CKfrom TKis straightforward. Every edge

that goes through a ±-matrix can be interpreted as a

unitary gate K±acting on two q-level qudits:

These gates are diagonal and so they commute, which

is also seen by the fusion rule which allows us to draw

them all in one layer. Thus TKdeﬁnes a special case

of an instantaneous quantum polynomial (IQP) circuit,

simply described by the edge list of GK; the edges can be

read-oﬀ in any order and the edge’s Tait sign indicates

whether to perform a K+or a K−gate.

Evaluating VK(t) at generic roots of unity t=eiθ is

a paradigmatic BQP-complete problem. However, here

we are working in a setting where these amplitudes corre-

spond to the evaluation of the Jones polynomial at lattice

roots of unity. The tractability of the problem for these

cases manifests in CKbeing a Cliﬀord circuit, deﬁned

on qubits for q∈ {2,4}and qutrits for q= 3. Such cir-

cuits can be simulated classically eﬃciently, as stated by

the Gottesman-Knill theorem [17]. The results of Refs.

[15,18] can also be understood as a graphical version of

this result given in terms of diagrammatic rewrite strate-

gies.

Note that the K±-gates are not unitary for integers

q≥5, since then t(q) is not a root of unity and is rather

a real number. In this case the problem of computing

ZK(q) is #P-hard, for which tensor contraction shows

good performance [9].

If one wishes to evaluate the Jones polynomial at non-

lattice roots of unity on a quantum computer, then one

needs to work with unitary representations of the gener-

ators of the braid group [3].

VI. IBM QUANTUM EXPERIMENTS AND

THE H-TEST

The experiments that we implement in this work are

for the case of Ising anyons. This is obtained when we

set q= 2 in Eq. 1and obtain t=i∈θΛ. Now every wire

in the circuit carries one qubit and the gates are

K±= diag(±i, 1,1,±i).(3)

We consider four small knots that are all topologi-

cally equivalent under Reidemeister moves, but which

give rise to diﬀerent quantum circuits on diﬀerent num-

bers of qubits:

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EstimatingtheJonespolynomialforIsinganyonsonnoisyquantumcomputersChrisN.Self,1,2,SofyanIblisdir,3,4GavinK.Brennen,5andKonstantinosMeichanetzidis6,7,y1QOLS,BlackettLaboratory,ImperialCollegeLondonSW72AZ,UnitedKingdom2Quantinuum,PartnershipHouse,CarlislePlace,LondonSW1P1BX,UnitedKingdom3Dpto.Analisi...

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Estimating the Jones polynomial for Ising anyons on noisy quantum computers Chris N. Self1 2Sofyan Iblisdir3 4Gavin K. Brennen5and Konstantinos Meichanetzidis6 7y 1QOLS Blackett Laboratory Imperial College London SW7 2AZ United Kingdom.pdf

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Estimating the Jones polynomial for Ising anyons on noisy quantum computers Chris N. Self1 2Sofyan Iblisdir3 4Gavin K. Brennen5and Konstantinos Meichanetzidis6 7y 1QOLS Blackett Laboratory Imperial College London SW7 2AZ United Kingdom

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