The Zak transform a framework for quantum computation with the Gottesman-Kitaev-Preskill code Giacomo Pantaleoni1 2Ben Q. Baragiola1 3and Nicolas C. Menicucci1
2025-04-24
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The Zak transform: a framework for quantum computation with the
Gottesman-Kitaev-Preskill code
Giacomo Pantaleoni,1, 2, ∗Ben Q. Baragiola,1, 3 and Nicolas C. Menicucci1
1Centre for Quantum Computation & Communication Technology,
School of Science, RMIT University, Melbourne, VIC 3000, Australia
2Centre for Engineered Quantum Systems, School of Physics,
University of Sydney, Sydney, NSW 2006, Australia
3Center for Gravitational Physics and Quantum Information (CGPQI),
Yukawa Institute for Theoretical Physics, Kyoto University,
Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan
The Gottesman-Kitaev-Preskill (GKP) code encodes a qubit into a bosonic mode using periodic
wavefunctions. This periodicity makes the GKP code a natural setting for the Zak transform, which
is tailor-made to provide a simple description for periodic functions. We review the Zak transform
and its connection to a Zak basis of states in Hilbert space, decompose the shift operators that
underpin the stabilizers and the correctable errors, and we find that Zak transforms of the position
wavefunction appear naturally in GKP error correction. We construct a new bosonic subsystem
decomposition (SSD)—the modular variable SSD—by dividing a mode’s Hilbert space, expressed in
the Zak basis, into that of a virtual qubit and a virtual gauge mode. Tracing over the gauge mode
gives a logical-qubit state, and preceding the trace with a particular logical-gauge interaction gives
a different logical state—that associated to GKP error correction.
I. INTRODUCTION
The vast majority of schemes for quantum compu-
tation require discrete Hilbert spaces constructed from
tensor products of 2-dimensional systems—the discrete-
ness is the basis of quantum error correction and fault
tolerance as well as known quantum algorithms. This
has led to the question of how infinite-dimensional
continuous-variable systems, such as quantized electro-
magnetic fields [1,2], superconducting circuits [3,4], or
mechanical resonators [5,6], can be repurposed into be-
having effectively as 2-level systems (qubits). Clever
methods to do so go beyond simply encoding qubits
into continuous-variable systems; they also endow en-
codings with error-correcting properties, making them
potential building blocks for a fault-tolerant quantum de-
vice. These encodings are collectively known as bosonic
codes.
The Gottesman-Kitaev-Preskill (GKP) code [7] stands
out among bosonic codes due to its noise resilience [8],
all-Gaussian implementation of logical Clifford gates [7]
and magic-state production [9,10], seamless interface
with continuous-variable cluster-state quantum comput-
ing [11,12], and rich set of mathematical properties [13,
14]. The discretization mechanism at the basis of the
GKP code is periodization in phase space: code-word
wavefunctions in the computational and dual bases are
periodic combs of Dirac-delta distributions in position
and momentum, respectively. This introduces a redun-
dancy that protects against small displacements in po-
sition and momentum, as the error correction procedure
gives access to only small-displacement information while
∗g.pantaleoni@pm.me
leaving the logical information untouched.
Pure GKP states, both ideal and approximate, are
often represented as wavefunctions in a quadrature ba-
sis. An appealing alternative description is obtained by
performing a Zak transform [15–17] of the position (or
momentum) wavefunction [18]. A Zak transform, also
known as a Weyl-Brezin transform, takes a function of
a single unbounded real variable to a function of two
bounded real variables. This takes a wavefunction ψ(x)
with x∈Rto a modular wavefunction ψ(u, v)with u, v
lying in a bounded patch of R2. The key property of mod-
ular wavefunctions is that they are particularly suited for
compact representations not only of ideal, infinite-energy
GKP states, but also their approximately periodic, finite-
energy approximations.
The Zak transform has been used and re-discovered in
mathematics (differential equations [19,20] and represen-
tation theory [21]), physics [15–17], and signal process-
ing [22–24]. In the early days of the GKP code, modular
wavefunctions were used to describe approximate GKP
states and GKP error correction [7,18], while more re-
cently, they have found application in a wider range of
GKP-related topics [25–30]. Generalizations of the Zak
transform also proved fruitful in the broader context of
quantum information [31–34] and superconducting cir-
cuits [28,35].
In this work, we delve further into connections between
the Zak transform and the GKP code. The results are
organized into two parts: the first part, Section II, is
devoted to the Zak transform, and the second part, com-
prising Sections III and IV, is devoted to the GKP code.
We begin in Section II A by presenting the Zak trans-
form and its relation to the “Zak basis” for representing
quantum states in Hilbert space. We collect a number of
useful facts about modular wavefunctions and elaborate
on their natural 2π-area domain induced by their pe-
arXiv:2210.09494v2 [quant-ph] 4 Apr 2023
2
riodicities. Modular arithmetic plays an important role
throughout this work, especially so in the process of keep-
ing track of the phasing rules that modular wavefunctions
must obey. We provide simple formulas to keep track of
these phases.
In Section II B, we show that the periodicity condi-
tions can be relaxed by defining “stretched Zak bases”
whose domain has arbitrary but finite area. We con-
clude the section by providing the link between modular
variables [34,36]—the operators that are diagonalized
by the Zak-basis eigenstates—and the usual position and
momentum operators of a bosonic mode.
In Section III, we introduce the GKP code in the
modular-variable formalism and show how the Zak frame-
work is inherently present both when considering the
problem of evaluating the logical information of approx-
imate GKP codewords and when performing GKP error
correction. In Section IV, we develop a formalism spe-
cially tailored to deal with the problem of addressing the
logical content of any continuous-variable state with re-
spect to the GKP code. We use the stretched Zak basis
to decompose the CV Hilbert space of a bosonic mode
into that of a two-dimensional qubit and a infinite di-
mensional gauge subsystem. We refer to this change of
basis as a subsystem decomposition (SSD) [37,38]. In
SSDs, logical-qubit information is encoded in a virtual
subsystem rather than in a subspace, which is the typical
framework for quantum error correcting codes. We show
that this decomposition gives a simple interpretation of
the GKP error correction procedure, and also, somewhat
surprisingly, that the logical content is identical to that
the partitioned-position SSD [38].
II. DEFINITIONS AND PROPERTIES
A. The Zak basis
The Zak transform maps a square-integrable function
ψ∈L2(R)into a square-integrable, quasi-periodic func-
tion of two real variables with period a. Denoting with
ψ(x)the function ψevaluated at the point x∈R, its Zak
transform Zψ, evaluated at u, v is [15,17]
(Zψ)(u, v) = ra
2πX
m∈Z
e−iamvψ(u+am).(2.1)
For the moment, we let uand vtake any real value. For
the remainder of this work, we simply indicate ψ(u, v)in-
stead of (Zψ)(u, v), as there is no risk of confusion: ψ(x)
will always refer to a function in L2(R), and ψ(u, v)will
always refer to its Zak transform. Since we are mostly
concerned with quantum states, we use bra-ket nota-
tion wherever possible and interpret square integrable
functions as wavefunctions in the position representation,
ψ(x) = qhx|ψi, where |xiqis a position eigenstate such
that ˆq|xiq=x|xiq,x∈R(we reserve the symbol |xip
to momentum eigenstates). The position and momen-
tum operators are ˆq:=1
√2(ˆa+ ˆa†)and ˆp:=−i
√2(ˆa−ˆa†),
respectively, in terms of creation and annihilation oper-
ators. We refer to ψ(u, v)as a modular wavefunction
when it describes a pure quantum state. We note in
passing that alternative phasings in the definition of the
Zak transform, Eq. (2.1), can be chosen, leading to dif-
ferent periodicity rules for modular wavefunctions—here,
we use Zak’s [15,16].
We can construct a Zak ket by applying the Zak trans-
form to the position eigenstates, which gives a superpo-
sition of either position or momentum eigenstates
|u, vi=ra
2πX
m∈Z
eiamv |u+amiq(2.2)
=1
√ae−iuv X
m∈Z
e−i2π
amu
v+2π
amp,(2.3)
where the normalization factor (a/2π)−1/2ensures or-
thonormality in the Dirac-comb sense
hu, v |u0, v0i=X
m
δ(u−u0+am)X
n
δ(v−v0+2π
an).
(2.4)
The reader interested in verifying Eq. (2.3) will find Pois-
son’s formula useful.
In analogy with the position and momentum wave-
functions in their respective bases, we refer to ψ(u, v)
in Eq. (2.1) as the wavefunction in the Zak represen-
tation (or, more simply, the modular wavefunction) by
identifying it with the inner product
ψ(u, v):=hu, v |ψi,(2.5)
One may verify that this expression is consistent by tak-
ing the adjoint of Eq. (2.2) and using the inner product
of L2(R). The dual vector hu, v|is thus the linear func-
tional that gives a wavefunction in Zak representation
evaluated at u, v, in the same sense that qhx|and phx|are
the linear functionals that give the wavefunction in the
position and momentum bases evaluated at the point x.
An example modular wavefunction, that of the vacuum
state, is shown in Fig. 1.
Two important properties of the vectors in Eq. (2.2)
are quasi-periodicity in the first variable and periodicity
in the second,
|u+a, vi=e−iav |u, vi(2.6a)
|u, v + 2π/ai=|u, vi.(2.6b)
which are inherited by modular wavefunctions ψ(u, v),
ψ(u+a, v) = eiavψ(u, v)(2.7)
ψ(u, v + 2π/a) = ψ(u, v).(2.8)
An important consequence of these properties is that the
states |u, viform an overcomplete basis when there are
no restrictions on the domain of uand v. The standard
3
Figure 1. (a)Modular wavefunction ψvac(u, v)and (b)po-
sition wave function ψvac(x)∝exp−x2/2for the vacuum
state of the harmonic oscillator. The modular wavefunction
is given for a Zak transform with period a= 2√π. It is pe-
riodic in the vertical direction, and periodic modulo a phase
(quasi-periodic) in the horizontal direction. Because of the
quasi-periodicity, the values of ψvac(u, v)outside Pare redun-
dant and it is sufficient to restrict ourselves to a fundamental
domain Pwhose center can be freely chosen. The choice of
periodicity and centering here is convenient for representing
states of the square GKP code, whose code words have a 2√π
periodicity and support only on integer multiples of √π.
prescription to construct an orthonormal Zak basis is to
restrict the domain of uand vto a rectangle of area 2π,
which we often refer to as a “Zak patch”, whose sides
are given by the periods aand 2π/a. This is equivalent
to restricting the domain of modular wavefunctions to
a torus. One is free to choose the centering of the Zak
patch; we choose
P=h−a
4,3a
4×h−π
a,π
a(2.9)
so that the two points (u, v) = (0,0) and (u, v)=(a, 0),
which will be important for GKP states, lie within the
patch and not on its boundaries. The fundamental Zak
patch is shown in Fig. 2.
The states |u, viwith u, v ∈ P, span the Hilbert space
of a bosonic mode, and one may write the complete-
ness as (see Ref. [32] for a brief, focused discussion and
Ref. [39] for a rigorous proof)
ZP
du dv |u, vihu, v|=ZR
dx |xiq qhx|=ˆ
I.(2.10)
The Zak transform is then interpreted as an isome-
try from the Hilbert space of complex-valued, square-
integrable functions on the real line to the Hilbert space
of complex-valued, square-integrable functions of two real
variables in the Zak domain P,L2(R)→L2(P)[23,24,
Figure 2. Centered fundamental Zak domain Pwith width a
and height 2π/a.
40]. With the restriction on the domain to P, the or-
thonormality condition becomes
hu, v |u0, v0i=δ(a)(u−u0)δ(2π/a)(v−v0),(2.11)
where δ(a)(u)and δ(2π/a)(v)are Dirac delta-distributions
in the horizontal and vertical intervals of the Zak patch
respectively. The Zak basis is then
BZ={|u, vi | u, v ∈ P}.(2.12)
Unless otherwise stated, from now on, when we use the
symbols uand v, it is understood that u, v ∈ P.
Zak states that lie outside of Pare phased versions of
the Zak-basis states within P. The periodicity and quasi-
periodicity conditions give the recipe to find this phase.
First, recall that a real number x∈Rcan be written as
a quotient and remainder with respect to a positive real
number δ. Including a centering µ,xdecomposes as
x={x}µ
T+bxeµ
T,(2.13)
where {x}µ
T∈[−µ, T −µ)is the centered fractional part
of xand bxeµ
T=x− {x}µ
Tis the centered closest integer
multiple of Tto x. We will omit the centering superscript
whenever the centering does not matter. Consider now a
Zak state |x, yi, where x, y ∈Rare not limited to P. By
decomposing xand yand using Eqs. (2.6a) and (2.6b),
one finds that
|x, yi=e−ibxea/4
a{y}π/a
2π/a
{x}a/4
a,{y}π/a
2π/aE,(2.14)
with the state on the right-hand side being a bona fide el-
ement of the orthonormal Zak basis within P, Eq. (2.12).
An example that appears often is a Zak basis state |u, vi
with u, v ∈ P that has undergone shifts by unrestricted
values sand t,
|u+s, v +ti=e−ibu+sea{v+t}2π/a
{u+s}a,{v+t}2π/aE.
(2.15)
We discuss the effect of quadrature shifts more in
depth, as well as their Zak space versions. The Weyl-
Heisenberg shift operators
ˆ
Z(t):=eiˆqt ,(2.16)
ˆ
X(t):=e−iˆpt ,(2.17)
4
respectively phase and shift position eigenstates,
ˆ
Z(t)|xiq=eixt |xiqand ˆ
X(t)|xiq=|x+tiq, with com-
plementary actions on momentum eigenstates. In the
Zak basis, their actions are
ˆ
Z(t)|u, vi=eiut |u, v +ti,(2.18)
ˆ
X(t)|u, vi=|u+t, vi.(2.19)
We can see that, once a fundamental comb state is de-
fined, |0,0i:=Pm∈Z|amiq, an equivalent definition of
the Zak transform can be given. We can do so by defining
a Zak vector as a displaced |0,0istate,
|u, vi:=ˆ
X(u)ˆ
Z(v)|0,0i,(2.20)
and ψ(u, v) = hu, v |ψias the Zak transform of |ψi. This
alternative definition highlights the fact that we have
taken the convention where momentum shifts are per-
formed first. Different ordering and phasings of the dis-
placements would give different Zak vectors. There are
two more conventions that appear to be just as natu-
ral or useful as the one above. The second choice is
the one where the shifts are taken in the opposite or-
der |u, viop :=ˆ
Z(u)ˆ
X(v)|0,0i, and in the third one they
are performed symmetrically |u, visym :=ei(vˆq−uˆp)|0,0i.
These Zak states differ from those defined in Eq. (2.20) by
phases, ophu, v |u, vi=e−iuv and symhu, v |u, vi=e−iuv
2.
One could proceed using any of these conventions; we use
Eq. (2.20).
We introduce operators that produce phases and shifts
on Zak eigenstates, analogue to the action of Weyl-
Heisenberg operators on position eigenstates. We define
modular phase operators
ˆ
PU(t)|u, vi:=eiut |u, vi,(2.21a)
ˆ
PV(t)|u, vi:=eivt |u, vi,(2.21b)
for u, v ∈ P and t∈R. The modular phase operators
are generated by the modular variables ˆuand ˆv,eiˆut =
ˆ
PU(t)and eiˆvt =ˆ
PV(t). The modular shift operators are
defined as
ˆ
TU(t)|u, vi:=|u+t, vi,(2.22a)
ˆ
TV(t)|u, vi:=|u, v +ti,(2.22b)
for u, v ∈ P and t∈R. Here, an asymmetry becomes
evident. Scaled integer position aˆmgenerates a mod-
ular momentum translation, eia ˆmt =ˆ
TV(t), but scaled
integer momentum 2π
aˆngenerates a composite action,
e−i2π
aˆnt =ˆ
PV(t)ˆ
TU(t). It is not possible to gener-
ate translations in modular position by exponentiating
only one of the fundamental operators in Eqs. (2.33a)
to (2.33d). This is because, with our choice of conven-
tion for the Zak transform and Zak states (introduced by
Zak [16,17]), modular position and modular momentum
are not on equal footing, at least when interpreting them
in terms of Aharonov’s integer and modular operators.
For this reason, the quadrature shift operators decom-
pose as
ˆ
Z(t) = ˆ
PU(t)ˆ
TV(t),(2.23a)
ˆ
X(t) = ˆ
TU(t)(2.23b)
which are simply Eqs. (2.18) and (2.19) in a basis-
independent form.1We illustrate their action on modular
wavefunctions in Fig. 3.
The modular phase and modular shift operators obey
similar commutation relations to that of the Weyl-
Heisenberg operators, ˆ
Z(s)ˆ
X(t) = eist ˆ
X(t)ˆ
Z(s). That
is, the only non-zero commutators between pairs of mod-
ular operators come from shifts and phases of the same
modular variable: ˆ
PU(s)ˆ
TU(t) = eist ˆ
TU(t)ˆ
PU(s)and
ˆ
PV(s)ˆ
TV(t) = eist ˆ
TV(t)ˆ
PV(s).
The conditions in Eqs. (2.6a) and (2.6b) imply peri-
odicity on the modular shift operators in the following
sense:
ˆ
TU(a) = ˆ
PV(−a),(2.24)
ˆ
TV(2π/a) = I.(2.25)
With these relations, the modular shift operators can be
rewritten using modular arithmetic as
ˆ
TU(t) = ˆ
PV(−btea)ˆ
TU({t}a),(2.26)
ˆ
TV(t) = ˆ
TV({t}2π/a).(2.27)
According to Eq. (2.26), translations that wrap around
the u-domain are accompanied by a phase. Translations
on the v-domain do not exhibit this behavior, as shown
in Eq. (2.27). Note that these relations give us an alter-
native approach to obtaining Eq. (2.14).
1. Relation to modular variables
The modular variables ˆuand ˆvwe encountered are an
important concept in physics. They may be interpreted
as non-local analogues of the position and momentum
of a quantum particle [36]. They are useful quantities
in solid state physics, where useful dynamical variables
are not necessarily local [16,17] (in that context, they
are known as quasi-position and quasi-momentum) and
1It is possible to work with more symmetrical relations using
the Zak kets, |u, visym =eiuv
2|u, vi. We then get a more
symmetrical decomposition of the Weyl-Heisenberg operators,
ˆ
X(t) = ˆ
TU
sym(t)ˆ
PV
sym(−t/2) and ˆ
Z(t) = ˆ
TV
sym(t)ˆ
PU
sym(t/2), with
the modular shift and phase operators acting on symmetric Zak
states as in Eqs. (2.21a) to (2.22b). Exponentiating the scaled
integer operators generates a combination of primed modular
phases and shift, eia ˆmt =ˆ
TV
sym(t)ˆ
PU
sym(−t/2) and ei2π
aˆnt =
ˆ
TU
sym(t)ˆ
PV
sym(t/2). We do not adopt this convention throughout
this work.
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TheZaktransform:aframeworkforquantumcomputationwiththeGottesman-Kitaev-PreskillcodeGiacomoPantaleoni,1,2,BenQ.Baragiola,1,3andNicolasC.Menicucci11CentreforQuantumComputation&CommunicationTechnology,SchoolofScience,RMITUniversity,Melbourne,VIC3000,Australia2CentreforEngineeredQuantumSystems,Schoolof...
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