Stabilizer subsystem decompositions for single- and multi-mode Gottesman-Kitaev-Preskill codes Mackenzie H. Shaw1Andrew C. Doherty1and Arne L. Grimsmo1 2 3

2025-05-03 0 0 3.68MB 34 页 10玖币

侵权投诉

Stabilizer subsystem decompositions for single- and multi-mode

Gottesman-Kitaev-Preskill codes

Mackenzie H. Shaw,1, ∗Andrew C. Doherty,1and Arne L. Grimsmo1, 2, 3

1ARC Centre of Excellence for Engineered Quantum Systems,

School of Physics, The University of Sydney, Sydney, NSW 2006, Australia.

2AWS Center for Quantum Computing, Pasadena, CA 91125, USA

3California Institute of Technology, Pasadena, CA 91125, USA

(Dated: October 26, 2022)

The Gottesman-Kitaev-Preskill (GKP) error correcting code encodes a ﬁnite dimensional logical

space in one or more bosonic modes, and has recently been demonstrated in trapped ions and su-

perconducting microwave cavities. In this work we introduce a new subsystem decomposition for

GKP codes that we call the stabilizer subsystem decomposition, analogous to the usual approach to

quantum stabilizer codes. The decomposition has the deﬁning property that a partial trace over the

non-logical stabilizer subsystem is equivalent to an ideal decoding of the logical state, distinguishing

it from previous GKP subsystem decompositions. We describe how to decompose arbitrary states

across the subsystem decomposition using a set of transformations that move between the decompo-

sitions of diﬀerent GKP codes. Besides providing a convenient theoretical view on GKP codes, such

a decomposition is also of practical use. We use the stabilizer subsystem decomposition to eﬃciently

simulate noise acting on single-mode GKP codes, and in contrast to more conventional Fock basis

simulations, we are able to consider essentially arbitrarily large photon numbers for realistic noise

channels such as loss and dephasing.

I. INTRODUCTION

Bosonic codes encode digital quantum information in

continuous variable (CV) quantum systems and have

received both theoretical [1–3] and experimental [4–

9] attention. The Gottesman-Kitaev-Preskill (GKP)

codes [10] are one of the most intensively studied encod-

ings of this type, and the single-mode square GKP qubit

code has recently been realized in both trapped ions [11,

12] and superconducting microwave resonators [13–15].

From a theoretical perspective, bosonic codes can be

understood as deﬁning a logical subspace Lof the CV

Hilbert space H=L⊕L∗, with the inﬁnite dimensional

Hilbert space L∗providing the redundancy required for

error correction. However, in the case of GKP codes, the

non-normalizability of the codewords [10] means that the

GKP logical “subspace” is formally not in the CV Hilbert

space.

An alternative formulation, which can be applied to

any error correcting code, is to consider a decomposi-

tion of the Hilbert space such that the logical infor-

mation in the error correcting code forms a subsystem

H=L⊗S [16,17]. In such a decomposition, the partial

trace over the non-logical subsystem corresponds to a de-

coding map H → L. In Ref. [18], Pantaleoni et al. intro-

duced the concept of a bosonic subsystem decomposition,

and deﬁned a subsystem decomposition for single-mode

GKP codes based on a modular quadrature. This sub-

system decomposition has been used in numerical studies

of GKP codes [19–22].

The subsystem decomposition is, however, not unique

and there are good reasons to investigate alternatives.

∗msha2420@uni.sydney.edu.au

Speciﬁcally, the subsystem decomposition of Ref. [18] has

lower symmetry than the GKP code itself: the logical

subsystem diﬀers if one chooses position or momentum

as the “modular quadrature.” More recent work [23] has

also linked the modular position subsystem decomposi-

tion to the Zak basis [24]. In all of these cases the de-

composition does not represent the logical information

one would retrieve by performing noiseless decoding of

the GKP code [23].

In this work, we introduce a subsystem decomposition

that resolves these issues. In particular, this new decom-

position has the desirable property that tracing over the

non-logical subsystem Scorresponds to a noiseless decod-

ing map for the GKP code. We refer to this decomposi-

tion as the GKP stabilizer subsystem decomposition, as

diﬀerent stabilizer eigenstates correspond to orthogonal

basis states of the subsystem S. The stabilizer subsystem

decomposition for GKP codes is entirely analogous to the

stabilizer/destabilizer formalism of qubit codes [25].

The stabilizer subsystem decomposition can be applied

to all multi-mode qubit or qudit GKP codes (including

the concatenation of GKP and qubit stabilizer codes),

and is closely related to the Zak basis [24]. For any GKP

encoding, we show how to write an arbitrary CV state

in the corresponding stabilizer subsystem decomposition

from the position wavefunction of the state. We use the

subsystem decomposition to provide a description of log-

ical Cliﬀord gates on the subsystem decomposition, and

show that an ideal implementation of a logical Cliﬀord

gate can propagate errors unless a modiﬁed round of de-

coding is performed immediately after the gate.

One practical challenge with GKP codes is the diﬃ-

culty of numerically simulating GKP codes using a trun-

cated Fock basis, since both the mean and variance of

the photon number distribution of physically realizable

arXiv:2210.14919v3 [quant-ph] 9 Jan 2024

GKP codestates increases as the codestates approach the

inﬁnitely squeezed “ideal” codewords. Logical gates can

also increase the photon number of the codestates, pro-

viding a further need to ﬁnd new numerical methods to

eﬃciently store and manipulate GKP states [26].

Using the stabilizer subsystem decomposition we are

able to study realistic noise channels such as loss and

white-noise dephasing for essentially arbitrary photon

numbers. In the case of the single-mode square GKP

qubit code our treatment is analytical. We ﬁnd that

GKP codes are far more resilient against pure loss than

against dephasing: a square single-mode code state with

ten decibels of GKP squeezing achieves an average gate

inﬁdelity below 10−3for a loss rate up to ∼4%, while it

can only tolerate a dephasing rate of ∼0.2% to achieve

the same ﬁdelity. In the case of pure-dephasing, i.e. with

white-noise dephasing as the only noise channel, there is

a threshold value for the GKP squeezing value and de-

phasing rate for the GKP code to “break even”, as the

GKP code only performs better than a qubit deﬁned us-

ing Fock states |0⟩and |1⟩given the GKP squeezing is

above 10 dB and simultaneously the dephasing rate is be-

low 0.1%. We also ﬁnd that for both pure loss and pure

dephasing, there is an optimal ﬁnite photon number that

minimizes the logical error rate, which is much larger for

loss than for dephasing at the same rate, qualitatively

consistent with the results of [13,27]

Our results are organized as follows. Beginning in Sec-

tion II, we present the stabilizer subsystem decomposi-

tion for the single-mode square GKP qubit code. Readers

wishing to quickly learn the key concepts in the paper can

safely begin by reading only Section II, since it provides

a simple explanation of most of the results in the rest of

the paper. Then in Section III, we provide an overview

of the established formalism of multi-mode GKP lattices

and set up the notation we will use in the remainder of

the manuscript. In Section IV, we deﬁne the stabilizer

subsystem decomposition in the general case and show

that the partial trace over the stabilizer subsystem corre-

sponds to noiseless decoding. In Section V, we show how

to transform the states of the stabilizer subsystem decom-

position of one GKP code to any other code, and describe

the method to write the subsystem “wavefunction” of a

state in terms of its position wavefunction. Finally, we

show how to write many practical components of GKP

codes conveniently in the stabilizer subsystem decompo-

sition, namely logical Cliﬀord gates (Section VI), approx-

imate GKP codewords, and noise channels such as pure

loss, Gaussian displacements and white-noise dephasing

(Section VII). Readers focused on applying the stabilizer

subsystem decomposition to model noise can safely skip

Sections III to VI and go straight to Section VII. We

provide concluding remarks in Section VIII.

II. STABILIZER SUBSYSTEM

DECOMPOSITION FOR THE SQUARE GKP

QUBIT CODE

We begin by constructing the stabilizer subsystem de-

composition in the simplest non-trivial case: the single-

mode square GKP qubit code. To deﬁne the subsystem

decomposition in Eqs. (8), (15) and (17), we will make

use of the Zak states [24], and provide the intuition for

why the stabilizer subsystem decomposition accurately

describes the GKP logical information stored in an arbi-

trary state. Then we will outline the key properties of

the decomposition in Section II C, including examples of

states and operators decomposed in the subsystem de-

composition. In doing so, we foreshadow the numerical

techniques for simulating GKP codes that we develop in

more detail in Section VII.

A. Preliminaries

The square GKP qubit code encodes a qubit into a

single-mode continuous-variable (CV) Hilbert space H,

which is described by position and momentum operators

that satisfy [ˆq, ˆp] = i. We deﬁne the displacement opera-

tors

W(v1, v2) = exp√2πi(v2ˆq−v1ˆp)(1)

for v1, v2∈R, which form an operator basis of L(H), the

space of all linear operators acting on H. The displace-

ment operators obey the commutation relation

qˆ

W(u1, u2),ˆ

W(v1, v2)y=e−2iπ(u1v2−u2v1),(2)

where JA, BK=ABA−1B−1is the group commutator,

and the composition rule

W(u1, u2)ˆ

W(v1, v2) =

e−iπ(u1v2−u2v1)ˆ

W(u1+v1, u2+v2).(3)

W(v1, v2)“displaces” the position and momentum oper-

ators such that

W(v1, v2)†ˆqˆ

W(v1, v2) = ˆq+√2πv1,(4a)

W(v1, v2)†ˆpˆ

W(v1, v2) = ˆp+√2πv2.(4b)

Note that Eqs. (1) and (4) diﬀer by a factor of √πfrom

the more standard deﬁnition ˆ

D(α) = exp αˆa†−α∗ˆa.

The square GKP qubit code is a stabilizer code with

stabilizer group generated by the commuting displace-

ment operators

S1=ˆ

W√2,0=e−2i√πˆp,(5a)

S2=ˆ

W0,√2=e2i√πˆq,(5b)

along with their inverses. The logical Pauli group is gen-

erated by

X=ˆ

W1/√2,0=e−i√πˆp,(6a)

Z=ˆ

W0,1/√2=ei√πˆq,(6b)

which anticommute with each other but commute with

the stabilizer generators. The ideal codespace is the si-

multaneous +1-eigenspace of both stabilizer generators,

and is spanned by the ideal codestates

|¯

0⟩ ∝ X

s∈Z|2s√π⟩q,|¯

1⟩ ∝ X

s∈Z|(2s+ 1)√π⟩q,(7)

where |x⟩qis the x-eigenstate of the position operator ˆq.

A particularly useful set of states for describing GKP

codes is the Zak basis [24], and was ﬁrst applied to GKP

codes in Ref. [28]. The Zak states are parameterized by

two real numbers, k1and k2, and are given in the position

basis by

|k1, k2⟩a=4

√2πa2eiπk1k2X

s∈Z

e2iπak2s√2π(k1+as)q,

(8)

where a > 0is a constant. Note that we have rescaled

some of the constants in our deﬁnition compared to

Ref. [24]. We can interpret the (rescaled) parameter

√2πk1as the quasi-position of the Zak state in the fol-

lowing sense: since a given Zak state has support on

position eigenvalues spaced by √2πa, each Zak state is

an eigenstate of the modular-position operator ˆq(mod

√2πa), with eigenvalue √2πk1. Likewise, it can be shown

that √2πk2represents the quasi-momentum of the Zak

state corresponding to the modular-momentum operator

ˆp(mod √2π/a).

The full set of Zak states with k1, k2∈Rspan Hbut

are not linearly independent, obeying the quasi-periodic

boundary conditions

|k1+a, k2⟩a=e−iπak2|k1, k2⟩a,(9a)

|k1, k2+ 1/a⟩a=eiπk1/a |k1, k2⟩a.(9b)

As a result, the Zak states only form a non-overcomplete

basis of Hwhen k1is restricted to an interval of length

aand k2to an interval of length 1/a. Moreover, the Zak

basis is orthonormal, satisfying

a⟨k1, k2|k′

1, k′

2⟩a=δ(k1−k′

1)δ(k2−k′

2)(10)

as long as k1and k2are restricted as above.

An alternative formulation of the Zak states is to de-

ﬁne |0,0⟩aas the unique simultaneous +1-eigenstate of

the displacements ˆ

W(a, 0) and ˆ

W(0,1/a). The remain-

ing Zak states are then given by the property

|k1, k2⟩a=ˆ

W(k1, k2)|0,0⟩a.(11)

Setting a=√2, we observe that the |0,0⟩√2Zak state is

a simultaneous +1-eigenstate of the GKP operators ˆ

FIG. 1. Diagrams representing (a) the stabilizer subsystem

for the single-mode square GKP qubit code, (b) the Zak basis

with a=√2. In subplot (a), each point represents the two-

dimensional stabilizer subspace Vk1,k2; while in (b) each point

represents a single Zak state |k1, k2⟩√2. Applying a random

walk of displacement operators to an ideal GKP codestate

|ψ⟩L⊗ |0,0⟩does not aﬀect the logical subsystem until the

state reaches one of the quasi-periodic boundaries of the cell;

for example causing an ˆ

Xerror as shown in (a). The corre-

sponding path is traced out twice in (b) since each basis state

|ψ⟩⊗|k1, k2⟩of the square GKP code consists of superpo-

sitions of states |k1, k2⟩√2and |k1+ 1/√2, k2⟩√2in the Zak

basis.

and ¯

Z, so we can write

|¯

0⟩=|0,0⟩√2,|¯

1⟩=¯

X|¯

0⟩=|1/√2,0⟩√2.(12)

The remaining a=√2Zak states can be viewed as dis-

placed GKP codestates.

B. Stabilizer Subsystem Decomposition for the

GKP Code

To deﬁne the stabilizer subsystem decomposition we

ﬁrst deﬁne the stabilizer subspaces Vk1,k2, each of which

is a simultaneous eigenspace of the stabilizer generators

S1,ˆ

S2. In particular, we deﬁne Vk1,k2as the set of states

|ϕ⟩∈Hsatisfying

S1|ϕ⟩=e−2i√πˆp|ϕ⟩=e−2√2iπk2|ϕ⟩,(13a)

S2|ϕ⟩=e2i√πˆq|ϕ⟩=e2√2iπk1|ϕ⟩,(13b)

for k1, k2∈−2−3/2,2−3/2. Here, √2πk1and √2πk2

represent the quasi-position ˆq(mod √π) and quasi-

momentum ˆp(mod √π) of |ϕ⟩(respectively). It is

straightforward to show that each subspace Vk1,k2is two-

dimensional and spanned by the a=√2Zak states

|k1, k2⟩√2and |k1+ 1/√2, k2⟩√2. With this connection

to Zak states we can see that the union of subspaces

Vk1,k2for k1, k2∈−2−3/2,2−3/2spans the full Hilbert

space H.

Since each stabilizer subspace Vk1,k2is two-

dimensional, we can deﬁne a qubit within each subspace

labelled by the orthonormal stabilizer states |µ, k1, k2⟩,

where µ= 0,1. The naïve way to do so would be to

deﬁne the |0, k1, k2⟩stabilizer state as |k1, k2⟩√2and

|1, k1, k2⟩as |k1+ 1/√2, k2⟩√2. This is justiﬁed since

|k1, k2⟩√2is “closest” to the ideal codestate |¯

0⟩=|0,0⟩√2,

while |k1+ 1/√2, k2⟩√2is “closest” to |¯

1⟩=|1/√2,0⟩√2,

see Eq. (12) and Fig. 1.

However, we want to ensure that the qubit state rep-

resents the GKP logical information stored in the state.

In particular, we impose the deﬁning property of the sta-

bilizer states that

|ψ, k1, k2⟩=ˆ

W(k1, k2)|¯

ψ⟩(14)

is a displaced ideal codestate for all qubit states |ψ⟩=

α|0⟩+β|1⟩with ideal GKP encoding |¯

ψ⟩=α|¯

0⟩+β|¯

1⟩.

To see the importance of Eq. (14), consider perform-

ing a round of ideal GKP error correction on the state

|ψ, k1, k2⟩as follows. First, we measure the stabilizer

generators, which reveals the values of k1and k2via

Eq. (13). Then, we apply the displacement ˆ

W(k1, k2)†

that returns the state to the ideal codespace. With this

deﬁnition, we ensure that the qubit information |ψ⟩in

the state |ψ, k1, k2⟩is the same as the logical information

one would obtain by performing an ideal round of error-

correction and reading out the resultant ideal codestate.

Equivalently, this enforces that the partial trace over the

stabilizer subsystem correspond to an ideal GKP decod-

ing map, as we show in Section IV D. Strictly enforcing

Eq. (14) is, in fact, the key diﬀerence between our sub-

system decomposition and previous deﬁnitions [18].

Enforcing Eq. (14) gives the stabilizer states in terms

of Zak states

|0, k1, k2⟩=|k1, k2⟩√2,(15a)

|1, k1, k2⟩=eiπk2/√2|k1+ 1/√2, k2⟩√2.(15b)

The additional eiπk2/√2phase on the deﬁnition of

|1, k1, k2⟩arises from the diﬀering geometric phases in

the deﬁnition of the Zak states Eq. (11) and the stabi-

lizer states Eq. (14):

|1, k1, k2⟩=ˆ

W(k1, k2)|¯

1⟩(16a)

=ˆ

W(k1, k2)ˆ

W(1/√2,0) |0,0⟩√2(16b)

=eiπk2/√2|k1+ 1/√2, k2⟩√2.(16c)

This phase has two additional consequences. First, it

ensures that all the logical Pauli operators act as a tensor

product between the logical and stabilizer subsystems,

as we will see in Section II C 3. A similar result is de-

scribed in Ref. [29]. Second, the phase ensures that the

full symmetry of the square GKP code is preserved in the

subsystem decomposition.

It is also interesting to compare Eq. (15) with the Zak-

basis representation of the modular-position subsystem

decomposition [18,23]. Once a rescaling of k1, k2is taken

into account, the only diﬀerence between the two decom-

positions is the k2-dependent phase (see Appendix A). In

this sense the stabilizer subsystem decomposition for the

single-mode square GKP qubit code can be thought of as

a “rephasing” of the modular-position subsystem decom-

position that symmetrizes the treatment of position and

momentum.

The states |µ, k1, k2⟩form a basis for µ= 0,1and

k1, k2∈−2−3/2,2−3/2, so we can deﬁne a subsystem

decomposition

H=L⊗S,|µ⟩⊗|k1, k2⟩=|µ, k1, k2⟩,(17)

where Lis the logical subsystem and Sis the stabilizer

subsystem. Similar to results obtained in [18], Lis a

two-dimensional subsystem while Sis isomorphic to the

full Hilbert space Hby associating the stabilizer sub-

system basis states |k1, k2⟩ ∈ S with a= 1 Zak states

|√2k1,√2k2⟩1∈ H of the full Hilbert space. For this rea-

son we call the basis of the stabilizer subsystem |k1, k2⟩

the Zak basis of S. We note here for clarity that the sta-

bilizer subsystem decomposition Eq. (17) applies to the

square GKP code, which is a stabilizer code and not a

“subsystem code” in the sense of Ref. [30].

It is worth brieﬂy reiterating why the subsystem de-

composition Eq. (17) is non-trivial. The key feature of

the stabilizer subsystem decomposition is that the state

in the logical subsystem is the information one would

obtain if one performed a round of ideal quantum error

correction and logical read-out. This feature is enforced

by Eq. (14) and appears as a state-dependent eiπk2/√2

phase when deﬁning the subsystem basis states in terms

of Zak states, see Eq. (15). We can use the connection

to error correction to justify the use of the stabilizer sub-

system decomposition in the analysis of GKP codes, as

we will now do in the rest of the manuscript.

C. Properties

Now that we have deﬁned the stabilizer subsystem de-

composition for the square GKP code, we outline its key

properties. We begin by presenting the quasi-periodic

boundary conditions of Sin Section II C 1, which pro-

vide an intuitive picture of how uncorrectable errors on

the oscillator cause logical errors in the logical subsystem

L. In Section II C 2, we provide examples of states de-

composed into the subsystem decomposition. In partic-

ular, the decomposition of approximate GKP codestates

follows a simpliﬁed version of the general method devel-

oped in Section VII for numerical simulations of GKP

codestates. Finally in Section II C 3, we decompose ex-

amples of operators, including logical Cliﬀord operators,

into the subsystem decomposition, and discuss how op-

erators that do not decompose into tensor products can

spread errors in the GKP code.

1. Boundary Conditions

We begin by noting that the stabilizer states |µ, k1, k2⟩

obey quasi-periodic boundary conditions given by

|µ, k1+ 1/√2, k2⟩=e−iπk2/√2|µ⊕1, k1, k2⟩,(18a)

|µ, k1, k2+ 1/√2⟩=eiπk1/√2(−1)µ|µ, k1, k2⟩,(18b)

where here ⊕denotes addition mod 2. These are anal-

ogous to the Zak state boundary conditions Eq. (9), ex-

cept that the boundary conditions also aﬀect the logical

information. In particular, Eq. (18a) applies a Pauli ˆ

operator to the logical information while Eq. (18b) ap-

plies a Pauli ˆ

For illustrative purposes, consider a toy error model

consisting of a random walk of displacement errors ap-

plied to an ideal square GKP codestate |¯

ψ⟩, as depicted

in Fig. 1. The logical information in the state remains

unchanged as long as the random walk does not cross a

boundary, i.e. while the error remains correctable. Once

it crosses a boundary, applying Equation (18a) or (18b)

causes a logical Pauli operator to be applied to the logical

subsystem, corresponding to a logical error on the state,

reﬂecting the fact that the correctable error has now be-

come uncorrectable. The applied logical Pauli operator

is identical to the logical error that would be applied if

an ideal decoder acted on the displaced codestate.

2. States

Arbitrary single-mode CV states can be decomposed

into the square subsystem decomposition using Equa-

tions (8) and (12). For example, ideal GKP codestates

|¯µ⟩sq =|µ⟩⊗|0,0⟩are tensor product states by deﬁnition.

Position eigenstates |x⟩qand momentum eigenstates |x⟩p

are also tensor product states given by

|x⟩q=1

√π|µx⟩ ⊗ ˆ1

2√2

−1

2√2

dk2e−iπ(kx+√2nx)k2|kx, k2⟩,

(19a)

|x⟩p=1

√π|±x⟩ ⊗ ˆ1

2√2

−1

2√2

dk1eiπ(kx+√2nx)k1|k1, kx⟩,

(19b)

where we decompose x=√2πkx+√πnxsuch that kx∈

−2−3/2,2−3/2and nx∈Z,µx=nx(mod 2), and we

write |±x⟩=|+⟩if µx= 0 and |±x⟩=|−⟩ if µx=

1. Intuitively, the position eigenstate |x⟩qcorresponds

to a product state with logical subsystem state |0⟩(|1⟩,

respectively) if xrounds to an even (odd) multiple of √π.

Similarly, the momentum eigenstate |x⟩pcorresponds to

a product state with logical subsystem state |+⟩(|−⟩,

respectively) if xrounds to an even (odd) multiple of

√π.

In contrast, approximate codestates are “entangled”

across the two subsystems. We deﬁne approximate

codestates by |¯

ψ∆⟩ ∝ e−∆2ˆa†ˆa|¯

ψ⟩with constant of pro-

portionality such that |¯

ψ∆⟩is normalized, and where

e−∆2ˆa†ˆais the non-unitary envelope operator. To ﬁnd

the analytical form of |¯

ψ∆⟩in the subsystem decompo-

sition, we ﬁrst utilize the characteristic function of the

envelope operator (see Appendix F)

e−∆2ˆa†ˆa∝ˆR

dv1dv2e−π

2coth∆2

2(v2

1+v2

2)ˆ

W(v1, v2).(20)

With the envelope operator written in this form it is

straight-forward to apply it to an ideal codestate |ψ⟩ ⊗

|0,0⟩using Eq. (14). However, since the integral in

Eq. (20) is over v1, v2∈R, we must apply the boundary

conditions Eq. (18) to obtain a valid subsystem decom-

position, giving

|¯

ψ∆⟩ ∝ X

s∈Z2

P(s)|ψ⟩ ⊗ ˆd2ve−π

2coth∆2

2|v+s/

√2|2

×eiπ(v1s2−v2s1)/

√2|v1, v2⟩,(21)

where ˆ

P(s) = eiπs1s2/2ˆ

Xs1ˆ

Zs2, the region of integration

is v1, v2∈−2−3/2,2−3/2, and we have written v=

(v1, v2). Note that the boundary conditions introduce

logical Pauli operators acting on the logical subsystem,

reﬂecting the fact that the envelope operator introduces

errors on the ideal codestate.

To quantify the logical information stored in a state,

we can apply the partial trace over S, which gives an

expression of the form

trS|¯

ψ∆⟩⟨¯

ψ∆|∝X

s,t∈Z2

I∆

s,tˆ

P(s)|ψ⟩⟨ψ|ˆ

P(t).(22)

We derive Eq. (22) and provide the analytical form of

I∆

s,tin Appendix Cdue to the length of the equations.

To numerically evaluate Eq. (22) we can truncate the

inﬁnite sums over s,t∈Z2, which is justiﬁed as long as

|I∆

s,t| → 0suﬃciently fast as |s|,|t|→∞. Importantly,

numerically evaluating Eq. (22) also becomes easier as

∆→0since |I∆

s,t|converges to zero faster as ∆becomes

small, requiring fewer terms in the sum to be included.

Intuitively, this is because the characteristic function of

the envelope operator Eq. (20) decays exponentially away

from the origin, and the rate of decay increases as ∆→0.

In Fig. 2, we plot the logical state given by Eq. (22),

where we have quoted ∆in decibels using the formula

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

Stabilizersubsystemdecompositionsforsingle-andmulti-modeGottesman-Kitaev-PreskillcodesMackenzieH.Shaw,1,∗AndrewC.Doherty,1andArneL.Grimsmo1,2,31ARCCentreofExcellenceforEngineeredQuantumSystems,SchoolofPhysics,TheUniversityofSydney,Sydney,NSW2006,Australia.2AWSCenterforQuantumComputing,Pasadena,CA911...

展开>> 收起<<

Stabilizer subsystem decompositions for single- and multi-mode Gottesman-Kitaev-Preskill codes Mackenzie H. Shaw1Andrew C. Doherty1and Arne L. Grimsmo1 2 3.pdf

共34页,预览5页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Stabilizer subsystem decompositions for single- and multi-mode Gottesman-Kitaev-Preskill codes Mackenzie H. Shaw1Andrew C. Doherty1and Arne L. Grimsmo1 2 3

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: