Floquet codes without parent subsystem codes Margarita Davydova1 2Nathanan Tantivasadakarn3 4and Shankar Balasubramanian5 1Department of Physics Massachusetts Institute of Technology Cambridge MA 02139 USA

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Floquet codes without parent subsystem codes

Margarita Davydova,1, 2 Nathanan Tantivasadakarn,3, 4 and Shankar Balasubramanian5

1Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

2Kavli Institute for Theoretical Physics, University of California, Santa Barbara, California 93106, USA

3Walter Burke Institute for Theoretical Physics and Department of Physics,

California Institute of Technology, Pasadena, CA 91125, USA

4Department of Physics, Harvard University, Cambridge, MA 02138, USA

5Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

We propose a new class of error-correcting dynamic codes in two and three dimensions that has

no explicit connection to any parent subsystem code. The two-dimensional code, which we call

the CSS honeycomb code, is geometrically similar to that of the honeycomb code by Hastings and

Haah, and also dynamically embeds an instantaneous toric code. However, unlike the honeycomb

code it possesses an explicit CSS structure and its gauge checks do not form a subsystem code.

Nevertheless, we show that our dynamic protocol conserves logical information and possesses a

threshold for error correction. We generalize this construction to three dimensions and obtain a

code that fault-tolerantly alternates between realizing two type-I fracton models, the checkerboard

and the X-cube model. Finally, we show the compatibility of our CSS honeycomb code protocol and

the honeycomb code by showing the possibility of randomly switching between the two protocols

without information loss while still measuring error syndromes. We call this more general aperiodic

structure ‘dynamic tree codes’, which we also generalize to three dimensions. We construct a

probabilistic ﬁnite automaton prescription that generates dynamic tree codes correcting any single-

qubit Pauli errors and can be viewed as a step towards the development of practical fault-tolerant

random codes.

CONTENTS

I. Introduction 1

II. 2D CSS honeycomb code 2

III. 3D generalization: Fracton Floquet code 9

IV. Dynamic tree codes 13

V. Discussion 16

Acknowledgements 17

References 17

A. Unitary circuit with measurements framework 19

B. Preparation protocol for Haah’s code 19

I. INTRODUCTION

Any route towards new fault-tolerant schemes for

quantum computing involves ﬁnding qualitatively diﬀer-

ent ways of performing quantum error correction. A

recent approach called operator quantum error correc-

tion [1–3]requires one to recover only a part of the orig-

inal ‘logical’ state, while errors are allowed to aﬀect the

rest of it, which is spanned by ‘gauge qubits’. This can

be accomplished by constructing a subsystem code, which

is speciﬁed by a gauge group Gthat is generically a

non-Abelian subgroup of the Pauli group. The stabi-

lizer group Sof the subsystem code is given by the cen-

tralizer of the gauge group, i.e. the set of the elements

in the gauge group that commute with all elements of

the group, and the logical qubits of the stabilizer code

are split into the logical qubits of the subsystem code

and gauge qubits, which are no longer used for encod-

ing logical information. Subsystem codes thus provide a

generalization of the concept of stabilizer codes [4].

In subsystem codes, syndrome measurement can be

performed using generators of the gauge group only,

which are usually low-weight (non-commuting) opera-

tors. This makes subsystem codes attractive for achiev-

ing fault tolerance and gives rise to several new proposals

for realization of universal quantum computing. A cen-

tral idea in these proposals is a procedure called gauge

ﬁxing, which corresponds to measuring a commuting sub-

set of gauge operators (“checks”), thus ﬁxing the states

of the gauge qubits. The measured gauge operators are

then added to the stabilizer Sof the subsystem code de-

ﬁned by the gauge group G. Diﬀerent ways of perform-

ing gauge ﬁxing allows one to switch between diﬀerent

stabilizer codes S1and S2starting from the same par-

ent gauge group G. This is aptly called ‘code switching’

and a universal transversal set of gates can been real-

ized this way from the gauge color code [5,6], quantum

Reed-Muller code [7], and more [8]. Furthermore, other

methods that allow one to overcome the Eastin-Knill no-

go theorem [9,10], such as lattice surgery and code de-

formation [11,12], can be uniﬁed into the framework of

gauge ﬁxing [13].

Recently, a new dynamic error-correcting code, com-

prised of a time-periodic sequence of two-qubit Pauli

measurements, was proposed by Hastings and Haah [14,

15]and dubbed the ‘honeycomb code’. It is considered

the ﬁrst example of a Floquet codebecause of the inher-

arXiv:2210.02468v4 [quant-ph] 25 Oct 2023

ent time periodicity in the measurement protocol. The

honeycomb code is based on a subsystem code with a

gauge group generated by terms in the Hamiltonian of

the Kitaev honeycomb model [16]. Notably, this subsys-

tem code stabilizes no logical qubits [17]. However, the

honeycomb code remedies this and dynamically gener-

ates logical qubits by measuring a commuting subset of

the gauge group at each round, which constitutes one-

third of the full set of two-qubit Pauli checks. This dy-

namic protocol generates a diﬀerent stabilizer group at

each instant in time which diﬀer from that of the original

subsystem code. In particular, the instantaneous stabi-

lizer group of the dynamic code is equivalent to that of a

toric code [18]on a certain superlattice, and the embed-

ded code changes with period 3 while conserving logical

information. Remarkably, the honeycomb code was also

shown to possess a threshold [19,20]. From the quan-

tum matter perspective, the honeycomb code not only

switches between diﬀerent realizations of Z2topological

order but also exhibits a kind of time crystalline behavior

– while the period of the cycling is 3, the period of the

code is 6 because after 3 rounds an e/mautomorphism

occurs. This idea has been more generally explored in

ref. [21].

In this paper, we propose a new class of Floquet codes

in two and three dimensions that are not based on parent

subsystem codes. Our 2D construction is geometrically

similar to that of the honeycomb code, but possesses an

explicit CSS structure; therefore we call our code the CSS

honeycomb code. We show that this code embeds an in-

stantaneous toric code, conserves logical information and

possesses a threshold for error correction. It also turns

out that the CSS honeycomb code performs an automor-

phism every three rounds. Our 3D construction embeds

two distinct type-I fracton models: we show that it cycles

between realizing instances of checkerboard and X-cube

models [22]while preserving logical information and be-

ing error-correcting as well. This is the ﬁrst Floquet code

we are aware of that prepares and cycles between fracton

stabilizer codes.

We argue that our 2D code cannot be reduced to the

honeycomb code. However, we show that it is possible

to fault-tolerantly switch between our CSS protocol and

the honeycomb protocol. Moreover, we consider random

disturbances of the protocol in time, thus generalizing

Floquet codes to a large class of monitored random cir-

cuit codes which we call dynamic tree codes, as the path

of a single instance of such a code is a branch of the

history tree of a probabilistic process. We show that a

special class of these codes, i.e. random-ﬂavor Floquet

codes, is fault-tolerant. Next, we construct a probabilis-

tic ﬁnite automaton (PFA) that allows one to generate

instances of dynamic tree codes that allow detection and

correction of any single-qubit Pauli error. We conjecture

that a large class of PFA-generated dynamic tree codes

is fault-tolerant with an eﬃcient decoder. This construc-

tion advances us one step closer towards fault-tolerant

random codes. Practically, these codes also work well for

FIG. 1. Fragment of a honeycomb lattice with three-colored

plaquettes (Pr,g,b) and edges. The red, blue and green checks

correspond to the edges connecting two plaquettes of the same

color. The red checks (r) which are measured in rounds 3nare

shown by bold lines and the triangular superlattice is shown

by dashed black lines.

error models that are dynamical in time.

Thus, the dynamic codes we construct in this paper

present a new class of quantum error correcting codes

and suggest a new route towards universal fault-tolerant

schemes for quantum computation, that rely on neither

stabilizer codes, nor subsystem codes, nor Floquet codes

generated from the gauge group of subsystem codes.

The rest of our paper is organized as follows. In sec-

tion II, we introduce the two-dimensional CSS honey-

comb code, discuss it in detail and explain its error-

correction properties. In section III, we elaborate on

an example that generalizes CSS honeycomb codes to

three dimensions and show that the instantaneous code

cycles between diﬀerent realizations of the checkerboard

and X-cube model. In section IV, dynamic tree codes

are introduced and argued to be a more general struc-

ture (that need not be periodic) bridging the honeycomb

code and the CSS honeycomb codes. We propose a PFA

construction of error-correcting protocols and also gener-

alize dynamic tree codes to 3D.

II. 2D CSS HONEYCOMB CODE

We propose a dynamic quantum error correcting code

built solely out of Xand Z-ﬂavored check operators –

for this reason, we refer to this code as the CSS honey-

comb code. Recall that in the honeycomb code of Hast-

ings and Haah [14], one picks a 3-colorable planar graph

and assigns labels of X,Y, and Zto each of the three

orientations of the edges. The edges of the graph are

also three-colorable, and the dynamic measurement pro-

tocol consists of measuring the two-body Pauli operators

(“checks”) of the ﬂavor corresponding to the orientation

of the bond at all the edges of a given color at each round.

The color of the edge is deﬁned by the colors of the two

plaquettes that it connects, see Fig. II.

In the CSS honeycomb code, the protocol is somewhat

simpler and is shown in Table I. It is partially inspired

rISG S(r)Syndrome Logical string Code

Measure m1e m2m1e m2

-3 rXX

-2 gZZ Pb(X)

-1 bXX Pr(Z)Pb(X)

0rZZ Pg(X)Pr(Z)Pb(X)gZZ rXX bZZ TC(r)

1gXX Pb(Z)Pg(X)Pr(Z)Pb(X)bXX gZZ rXX TC(g)

2bZZ Pr(X)Pb(Z)Pg(X)Pr(Z)rZZ bXX gZZ TC(b)

3rXX Pg(Z)Pr(X)Pb(Z)Pg(X)gXX rZZ bXX TC(r)

4gZZ Pb(X)Pg(Z)Pr(X)Pb(Z)bZZ gXX rZZ TC(g)

5bXX Pr(Z)Pb(X)Pg(Z)Pr(X)rXX bZZ gXX TC(b)

6rZZ Pg(X)Pr(Z)Pb(X)Pg(Z)gZZ rXX bZZ TC(r)

TABLE I. Summary of the the CSS honeycomb code. The table features the measurement sequence, the instantaneous stabilizer

group S(r) at each round, the syndrome plaquettes, logical operators, and the instantaneous codes. The checks and plaquette

stabilizers are color-coded for convenience. The ‘syndrome’ column contains the plaquette stabilizers that have been known in

previous rounds but are also contained in the checks of the current round. These measurements are used as syndromes for error

detection (see Sec. II C). The logical operators labeled as electric (e) and magnetic (m1,2) strings correspond to string operators

that violate the superlattice vertex or plaquette stabilizers of the embedded toric code, respectively. The magnetic m1and

m2strings are equivalent up to local operators acting at their ends. The connection between the logical operators of the CSS

honeycomb code and the topological excitations of the embedded codes are explored in Sec. II B. TC(c) with c∈(r, g, b) stands

for a toric code realized on a triangular superlattice with vertices of the superlattice located on plaquettes of color c, while TC

is the same code conjugated by a layer of single-qubit Hadamards, i.e. where stabilizers have ﬂavors exchanged, X↔Z.

by the construction of toric code topological order in

[23,24]. We similarly consider a honeycomb lattice with

periodic boundary conditions and divide the plaquettes

and the edges into three colors, red, green and blue. At

each round of measurements, we apply either red, green,

or blue checks. However, the ﬂavor of the check opera-

tors applied at each round alternates between Xand Z

(i.e. one measures two-qubit operators XX or ZZ on

the edges of the color of the given round). This gives a

measurement schedule whereby we measure the sequence

{rXX, gZZ, bXX, rZZ, gXX, bZZ}periodically in time.

Let us start in an arbitrary initial state (alternatively,

one prepare a speciﬁc state in order to encode logical

information in a code) and start measuring checks ac-

cording to the proposed protocol. At each round r, the

state prepared this way is a stabilizer state under an in-

stantaneous stabilizer group (ISG) S(r). The generators

of instantaneous stabilizer groups at each round are listed

in Table I. As a remark, similarly to the honeycomb code,

instead of post-selecting or correcting to the +1 values

of the measured stabilizers, we instead record these signs

and assume a convention where the ground state is eigen-

state of the plaquette stabilizers with eigenvalues deter-

mined by the measured signs.

At initial round r=−3, the red checks shown in

Fig. 1, which we denote rXX, are measured. At the

next round, r=−2, we measure ZZ-checks on green

edges, which anticommute with the measurements at the

previous round. However, at this step, the ISG contains

the stabilizers Pb(X), which corresponds to a product of

Pauli-Xaround blue plaquettes, and belongs to the cen-

ter of the group generated by ⟨rXX, gZZ⟩, i.e. commutes

with checks of both rounds. Measuring bXX in the sub-

sequent round r=−1 produces plaquette stabilizers that

are the center of the group ⟨bXX, gZZ, Pb(X)⟩, which is

Pb(X) and Pr(Z).

After measuring rZZ at round r= 0, the ISG includes

Pg(X), as well as Pb(X) and Pr(Z) from the previous

rounds, as well as current checks rZZ. The prepared

code has a number of stabilizers that matches the num-

ber of qubits on a torus minus two, because the plaquette

operators are not all independent. This instantaneous

code is equivalent to the toric code (TC(r) in the ta-

ble). To see this, consider the superlattice formed by the

dashed black lines in Figure 1. On this triangular super-

lattice, there are two qubits per edge. Constraining to

the subspace where the checks rZZ simply fuse the two

qubits into a single qubit degree of freedom, with eﬀec-

tive qubits located on each red edge which have eﬀective

logical operators ˇ

X=XX and ˇ

Z=ZI =IZ. Then, it

can be seen that Pg(X) and Pb(X) simplify to products

of three ˇ

Xoperators around the triangles of the super-

lattice. Similarly, Pr(Z) corresponds to the product of ˇ

on the star of the edges emanating out of each vertex of

the triangular lattice. For the simplicity of the presenta-

tion, assume that all measured signs of rZZ checks are

+1 (otherwise, the signs would appear as prefactors in

each term in the Hamiltonian without changing the con-

clusions). Thus, the eﬀective stabilizer code corresponds

to the Hamiltonian

Heﬀ

0=X

Av(ˇ

Z) + X

△

B△(ˇ

X) (1)

where Avand B△are the star and plaquette terms on the

triangular lattice, respectively. This Hamiltonian simply

describes the toric code, exhibiting the paradigmatic Z2

topological order.

When we continue implementing the protocol further,

the number of logical qubits does not change, and the

embedded code in each round is a diﬀerent realization

of the toric code; the period of the embedded code is 6.

The logical information is preserved during this cycling,

the details of which we address in the next section. To

see that the embedded code changes each round, con-

sider the subsequent r= 1 step when gXX checks are

measured. The value of the stabilizer Pb(X) from the

previous step is already contained in the values of the

measured green checks, and therefore we do not add it

to the list of generators of the instantaneous stabilizer

group (ISG) (we add it to the table as a syndrome, how-

ever, because the stabilizer value inferred from the green

checks at the current round can be compared with the

one stored earlier). Additionally, measuring gXX turns

the rZZ checks of the previous round to Pb(Z), so the

number of logical qubits in the new code does not change.

We can see that on round r= 1 we also obtain an eﬀec-

tive toric code by drawing a triangular lattice centered

on the green plaquettes, and viewing the gXX checks as

a fusion of the two qubits on each green edge, which have

eﬀective logical operators ˇ

X=XI =IX and ˇ

Z=ZZ.

The Hamiltonian corresponding to the embedded code is

Heﬀ

1=X

Av(ˇ

X) + X

△

B△(ˇ

Z),(2)

which is again a triangular lattice toric code.

On the next step, bZZ checks are measured, and the

plaquette Pr(Z) becomes redundant, so we do not list it

in the ISG. A new plaquette Pr(X) is added to the ISG,

and the ISG yields an embeddded toric code centered on

the blue sublattice (TC(b)). The instantaneous stabilizer

groups of the next three rounds are identical to the pre-

vious three apart from X↔Z(and therefore TC code

goes into TC, see Table I); therefore, the period of the

code is 6.

Thus, starting from round r= 0, our CSS honey-

comb code always embeds a toric code in its instanta-

neous stabilizer group. A striking diﬀerence between the

honeycomb code and the CSS honeycomb code is that

while the honeycomb code features ﬁxed plaquette sta-

bilizers after three rounds of measurements, the plaque-

tte stabilizers in the CSS honeycomb code change from

round to round via substitutions where P(Z) is replaced

by another P(X) or vice versa. This suggests a funda-

mental diﬀerence between our code and the honeycomb

code from the perspective of subsystem codes, which we

discuss below. In Appendix A, we also show that this

FIG. 2. The rXX (a) and bZZ (b) rounds of the CSS hon-

eycomb code realized on the three-colorable square-octagon

lattice (periodic boundary conditions are assumed and only

part of the lattice is shown for convenience). Because the al-

gebraic relations between the checks are the same, and the

square-octagon lattice is trivalent with even sided plaque-

ttes, the properties of the square-octagon Floquet code and

its error correction are the same as the honeycomb version.

The left half of each lattice shows the original lattice and the

ISG, and the right half shows the superlattice. At the rXX

step shown in (a), if the two-body checks deﬁne local [[2,1,1]]

codes, the embedded code on the superlattice is the toric code

with qubits on the edges, where Pg,b(Z) become the plaquette

terms and Pr(X) becomes the star term. In (b) (at blue, and

similarly, at green steps), one can view the blue checks to-

gether with Pr(X) plaquettes as stabilizers of a [[4,1,2]] local

code. This results in a Wen plaquette model where the eﬀec-

tive qubits are located on vertices of the square superlattice.

code has a regular representation as the same protocol

where only ZZ-checks are measured at each round and

a layer of single-qubit Hadamard gates is inserted after

each round. This immediately turns it into a period-3

protocol. Formulated this way using only ZZ-checks and

unitary layers, the honeycomb code requires single-qubit

Sand H-gates with a period-3 pattern.

Finally, this protocol does not necessarily require a

honeycomb lattice and will work on any three-colorable

graph, similarly to the honeycomb code [25]. In particu-

lar, if we apply the same protocol to the three-colorable

square-octagon lattice, the embedded code will alternate

between explicitly realizing the Wen plaquette model

[26]and the toric code on a square superlattice, as shown

in Fig. 2.

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FloquetcodeswithoutparentsubsystemcodesMargaritaDavydova,1,2NathananTantivasadakarn,3,4andShankarBalasubramanian51DepartmentofPhysics,MassachusettsInstituteofTechnology,Cambridge,MA02139,USA2KavliInstituteforTheoreticalPhysics,UniversityofCalifornia,SantaBarbara,California93106,USA3WalterBurkeInstit...

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Floquet codes without parent subsystem codes Margarita Davydova1 2Nathanan Tantivasadakarn3 4and Shankar Balasubramanian5 1Department of Physics Massachusetts Institute of Technology Cambridge MA 02139 USA

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