Measurements of Floquet code plaquette stabilizers James R. Wootton IBM Quantum IBM Research Europe Zurich

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Measurements of Floquet code plaquette stabilizers

James R. Wootton

IBM Quantum, IBM Research Europe – Zurich

(Dated: October 25, 2022)

The recently introduced Floquet codes have already inspired several follow up works in terms of

theory and simulation. Here we report the ﬁrst preliminary results on their experimental imple-

mentation, using IBM Quantum hardware. Speciﬁcally, we implement the stabilizer measurements

of the original Floquet code based on the honeycomb lattice model, as well as the more recently

introduced Floquet Color code. The stabilizers of these are measured on a variety of systems, with

the rate of syndrome changes used as a proxy for the noise of the device.

I. INTRODUCTION

Floquet codes are a concept that has recently emerged

in quantum error correction [1]. The ﬁrst explicit ex-

ample was the honeycomb Floquet code [1–5], based on

Kitaev’s honeycomb lattice model [6]. The concept was

then extended with two additional codes. First was the

e↔mautomorphism code [7], based on a generalization

of honeycomb Floquet code. Next was the Floquet Color

code [8–10], which can be derived from Color codes [11].

Both the original honeycomb Floquet code and the

Floquet Color code share similarities in implementation.

Speciﬁcally, both are based on the measurement of two-

body observables corresponding to edges in a hexagonal

lattice. This makes them both particularly well suited

to the heavy-hexagon architecture of IBM Quantum pro-

cessors [12], in which qubits reside on both the vertices

and links of a hexagonal lattice. These can then be used

as the required code qubits and auxiliary qubits respec-

tively, where the latter are used within the circuits to

measure two-body observables (see Fig. 1).

Most of the available hardware is not suﬃcient to per-

form a sensible analysis of the logical subspace. The 27

qubit Falcon devices, for example allow only two plaque-

ttes to be realized. Nevertheless, the hardware is suf-

ﬁcient to measure the stabilizers, and assess how they

detect noise on the devices. Here we will perform this for

the honeycomb Floquet code and Floquet Color codes.

II. HONEYCOMB FLOQUET CODES

Kitaev’s honeycomb lattice is a well-studied model in

both condensed matter and quantum information [6]. It

is deﬁned on a hexagonal lattice with qubits on the ver-

tices, as shown in Fig. 2. The links are labelled x,yand

zdepending on orientation. Each is assigned a two-body

link operator σα⊗σαon the two adjacent vertex qubits,

where α∈ {x, y, z}is the link type.

Important to any study of this model are the plaquette

operators, since they commute with all the link operators.

Each is deﬁned as the product of link operators around

a corresponding plaquette. They can be expressed as

W=σx

0σy

1σz

2σx

3σy

4σz

5(1)

FIG. 1. Circuits to measure operators for (a) x-links, (b) y-

links and (c) z-links. The two vertex qubits for which the

parity is measured are at the top and bottom of each circuit.

The auxiliary is in the middle. This should be initialized in

the |0istate. If it remains in the output state of a previous

measurement, the end result will be the XOR of the two.

using the numbering of Fig. 2(c).

The most obvious means to convert this model to a

quantum error correcting code is arguably to perform

the two-body parity measurements corresponding to the

link operators, treating them as the gauge operators of

a subsystem code. The plaquettes then form the corre-

sponding stabilizer generator. Unfortunately, such a code

has a trivial logical subspace [13]. However, two separate

means to carve out a non-trivial subspace were intro-

duced in the last year. One of these was the honeycomb

Floquet code, and the other was an approach based on

matching codes [14]. In both, a particular order is used

for the measurement of the gauge operators in order to

eﬀectively create a logical subspace.

For the honeycomb Floquet codes, the measurement

schedule is deﬁned using a tri-colouring of the plaquettes

and links of the hexgaonal lattice. This is done such that

arXiv:2210.13154v1 [quant-ph] 24 Oct 2022

FIG. 2. (a) Portion of the lattice on which the honeycomb

lattice model and derived codes are deﬁned. Vertex qubits

are shown as black dots. Grey dots are auxiliary qubits that

can be used in the derived codes. (b) Deﬁnition of the link

operators. (c) Deﬁnition of the plaquette operators.

any link of a given colour connects two plaquettes of that

colour, and forms part of the boundary of two plaquettes

of diﬀerent colours. The colouring used for the plaquettes

of the devices used are shown in Fig. 3

A. Measuring stabilizer changes

Measurement of the gauge operators is done by mea-

suring the link operators of each colour in succession. We

will consider the order red-green-blue by default. The re-

sults of any two successive rounds can be used to infer

the results for a subset of stabilizers. This is done by

simply adding their results modulo 2. For the initial red

and green rounds, it is the blue plaquettes for which re-

sults can be inferred. This is because the boundary of

blue plaquettes is formed by red and green links. After

the subsequent round of blue link measurements, results

from the green and blue rounds can similarly be used to

infer results for the red plaquettes, and so on.

Assuming that the state begins in a product state, the

initial measurement of each plaquette will typically give

a random result. In the noiseless case, the next measure-

ment would repeat the same result. A mismatch between

subsequent measurements of the same plaquette is there-

FIG. 3. Colouring of plaquettes and links for the

ibm washington device, which is a 127 qubit Eagle processor.

Qubits shown in black are those of the code, those in grey are

auxiliary qubits used during gauge operator measurements,

and those in light grey are unused. The plaquettes of the 27

qubit Falcon devices and the 65 qubit Hummingbird processor

are subsets of those shown here. The Falcon devices support

two plaquettes, corresponding to a red and blue pair. The

Hummingbird processor supports 8 plaquettes, corresponding

to four rows of pairs alternating between red and blue, and

green and red.

fore a detection of an error. To perform such syndrome

comparisons on all plaquettes, at least seven rounds of

link operator measurements must be performed. It is

this minimal seven round case that we will consider here.

We will also consider the case in which resets are not

applied after measurement of the auxiliary qubits. The

results for each link qubit measurement is then actually

the parity of all measurements of that link qubit so far.

The ﬁrst syndrome change for red and blue plaquettes

can then be calculated directly from the results of the

second instance of the pertinent link operators. For the

green plaquettes, only results from the second instance of

the blue link measurements are required, but results from

the ﬁrst and third instance of the red link measurements

must be combined.

B. Results

The seven round process is run for both IBM Quantum

devices and simulated noise. For the former, an example

of a scheduled circuit is shown in Fig. 4 for ibmq cairo.

The circuits typically settle into three distinct layers.

The ﬁrst consists of the two qubit gates required for the

ﬁrst three rounds of link operator measurement, followed

by the measurements on auxilliary qubits for all. This is

then repeated for the next three rounds. The ﬁnal layer

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MeasurementsofFloquetcodeplaquettestabilizersJamesR.WoottonIBMQuantum,IBMResearchEurope{Zurich(Dated:October25,2022)TherecentlyintroducedFloquetcodeshavealreadyinspiredseveralfollowupworksintermsoftheoryandsimulation.Herewereporttherstpreliminaryresultsontheirexperimentalimple-mentation,usingIBMQua...

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Measurements of Floquet code plaquette stabilizers James R. Wootton IBM Quantum IBM Research Europe Zurich

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