Quantum error correction with dissipatively stabilized squeezed cat qubits Timo Hillmann

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Quantum error correction with dissipatively stabilized squeezed cat qubits

Timo Hillmann ∗

Department of Microtechnology and Nanoscience (MC2),

Chalmers University of Technology, SE-412 96 Gothenburg, Sweden

Fernando Quijandr´ıa

Quantum Machines Unit, Okinawa Institute of Science and

Technology Graduate University, Onna-son, Okinawa 904-0495, Japan

(Dated: April 11, 2023)

Noise-biased qubits are a promising route toward signiﬁcantly reducing the hardware overhead as-

sociated with quantum error correction. The squeezed cat code, a non-local encoding in phase space

based on squeezed coherent states, is an example of a noise-biased (bosonic) qubit with exponential

error bias. Here we propose and analyze the error correction performance of a dissipatively stabilized

squeezed cat qubit. We ﬁnd that for moderate squeezing the bit-ﬂip error rate gets signiﬁcantly

reduced in comparison with the ordinary cat qubit while leaving the phase ﬂip rate unchanged.

Additionally, we ﬁnd that the squeezing enables faster and higher-ﬁdelity gates.

I. INTRODUCTION

The interaction of a quantum system with its environ-

ment leads to the loss of quantum coherence. By tailoring

the coupling of a quantum system to its environment,

typically through an ancilla, well-established reservoir

engineering methods allow overcoming the decoherence

paradigm by creating an eﬀective dissipative dynamics

which evolves in the long time to a target quantum state

or a manifold of quantum states [1–6].

In particular, in the ﬁeld of circuit quantum electrody-

namics (cQED) [7], reservoir engineering has been suc-

cessfully exploited to autonomously protect quantum in-

formation encoded in the inﬁnite Hilbert space of a har-

monic oscillator, i.e., a bosonic code, without the need of

measurement-based feedback. This is achieved through

the engineering of an eﬀective parity-preserving dissipa-

tive evolution which drives the state of a microwave res-

onator to a manifold spanned by even and odd parity

coherent superpositions of coherent states with opposite

displacements also known as Schr¨odinger cat states [8–

11]. In principle, these dissipative dynamics could be

used to prepare the logical states of the cat code [9].

Nevertheless, this is not necessary as universal control

of a microwave resonator ﬁeld using a dispersively cou-

pled qubit is possible using optimal control pulse se-

quences [10] or as it has been recently demonstrated, op-

timized sequences of continuous variables (CV) universal

gate sets [12,13]. Therefore, reservoir engineering is left

for the sole purpose of stabilizing the cat code.

Superpositions of squeezed vacuum states were intro-

duced by Sanders [14]. Later, Hach III and Gerry [15]

and Xin et al. [16] studied the nonclassical properties

of coherent superpositions of squeezed states. The lat-

ter are the states that result from the sequential appli-

cation of displacement and squeezing operations on the

∗timo.hillmann@rwth-aachen.de

photon vacuum with the squeezed vacuum state corre-

sponding to the special case of zero displacement. In

particular, in this work, we will focus on the so-called

squeezed cat states. These are generalizations of the or-

dinary cat states and correspond to coherent superposi-

tions of squeezed states with displacements of opposite

amplitude and equal squeezing. The main interest in

these states was spawned by the fact that they actually

represent superpositions of macroscopic quantum states

as opposed to cat states which correspond to superposi-

tions of nearly classical states. Squeezed cat states were

ﬁrst realized in the optical domain through breeding and

heralding detection operations [17,18]. In Ref. [19] en-

tangled states of two displaced squeezed states of mo-

tion and the spin degrees of freedom of a trapped ion

were realized. This work already highlighted the poten-

tial of these states for metrology. Later, Knott et al. [20]

demonstrated that squeezed cat states provided an ad-

vantage for sensing in the low-photon regime as compared

to more conventional CV states.

Recently, Schlegel et al. introduced the squeezed cat

bosonic code [21]. This is the squeezed counterpart of

the ordinary cat code in which logical states correspond

to squeezed cat states. Contrary to the cat code, in

the squeezed cat code it is possible to approximately

satisfy the Knill-Laﬂamme conditions for both single-

photon loss and dephasing errors simultaneously in the

large squeezing limit as well as the large coherent dis-

placement limit. In other words, the squeezed cat code

merges the most notable quantum error correction fea-

tures of both, cat and Gottesman-Kitaev-Preskill (GKP)

codes, namely, the ability to correct pure dephasing and

single-photon loss errors, respectively [22–24].

In this work, we study the error correction potential

of a squeezed cat qubit under a dissipative stabilization

scheme which conﬁnes the state of the harmonic oscilla-

tor to the squeezed cat qubit manifold. This mechanism

is a generalization of the cat qubit conﬁnement [8,25] and

here we provide a possible implementation using super-

conducting circuits. While the results presented in [21]

arXiv:2210.13359v2 [quant-ph] 10 Apr 2023

indicate an increased performance of the squeezed cat

code for an optimal recovery operation, their chosen met-

ric, the average channel ﬁdelity [26], does not distinguish

between bit- and phase-ﬂip errors. Because the squeezed

cat qubit represents a biased noise qubit, we indepen-

dently evaluate bit- and phase-ﬂip errors in the presence

of single photon losses, photon gain, and pure photon de-

phasing. In addition to these decoherence processes, we

also consider the eﬀect of a residual Kerr interaction.

The subsequent sections of the article are organized

as follows. We begin in Section II by reviewing rele-

vant properties of the squeezing and displacement op-

erations and introduce the necessary notation. Then,

in Section III, we describe a theoretical framework that

allows the dissipative stabilization of coherent superpo-

sitions of Gaussian states from which the stabilization

scheme for squeezed cat states is derived. In Section IV

we utilize the aforementioned stabilization scheme to an-

alyze the error correction capabilities of the squeezed cat

qubit. To this end, we introduce the squeezed cat code in

Section IV A before presenting the main results of this ar-

ticle in Section IV B. Our ﬁndings show an exponential

(in terms of the peak squeezing) reduction of the bit-

ﬂip error rate in comparison with the ordinary cat qubit

without aﬀecting phase-ﬂip error rates. However, at the

same time, our results also highlight the need for small

residual (Kerr) nonlinearities, as is the case for the GKP

code as well. The performance of the single qubit Zgate

is evaluated as well, suggesting exponentially faster and

less noisy gates for the squeezed cat qubit. To pave the

way towards an experimental implementation, we pro-

pose in Section IV C a superconducting circuit based on

Ref. [11] that realizes the dissipative stabilization scheme.

We conclude the article with a discussion of the results

in Section V.

II. DEFINITIONS

We are going to restrict to a single mode bosonic ﬁeld

with annihilation (creation) operator ˆa(ˆa†) obeying the

commutation relation [ˆa, ˆa†] = 1. The unitary displace-

ment operator is deﬁned by

D(α) = exp αˆa†−α∗ˆa,(1)

and the unitary squeezing operator is deﬁned by

S(ξ) = exp 1

2ξ∗ˆa2−ξˆa†2,(2)

with ξ=reiφ. Their action on the annihilation operator

ˆais given by

D†(α) ˆaˆ

D(α) = ˆa+α, (3)

and

S†(ξ) ˆaˆ

S(ξ) = cosh(r) ˆa−e−iφsinh(r) ˆa†(4)

respectively.

Following Refs. [27–29] a squeezed state |α, ξi(also

squeezed-coherent or squeezed-displaced state) is the

state that results from the sequential application of the

squeezing operator (2) and the displacement operator (1)

on the photon vacuum state

|α, ξi=ˆ

D(α)ˆ

S(ξ)|0i.(5)

The α= 0 case corresponds to the well-known squeezed

vacuum state. An alternative deﬁnition of a squeezed

state was given by Yuen [30]. This state is called the

two-photon coherent state |αiξand it is deﬁned by ﬁrst

displacing the vacuum state and then squeezing it

|αiξ≡ˆ

S(ξ)ˆ

D(α)|0i.(6)

From the relations (3) and (4) it is straightforward to

show that

D(α)ˆ

S(ξ) = ˆ

S(ξ)ˆ

Dαcosh(r) + α∗e−iφ sinh(r),(7)

which establishes the relation between squeezed and two-

photon coherent states

|α, ξi ≡ αcosh(r) + α∗e−iφ sinh(r)ξ.(8)

In this work, we are going to stick to the squeezed states

as deﬁned by Eq. (5).

The squeezed cat states are deﬁned as the coherent

superposition of two squeezed cat states with opposite

displacement amplitudes and identical squeezing

|C±

α,ξi=1

N±

α,ξ

(|α, ξi ± |−α, ξi),(9)

where N±

α,ξ is a normalization constant. These states can

be thought of as generalizations of the cat states. Sim-

ilarly, these are parity eigenstates with |C+

α,ξi(|C−

α,ξi) a

superposition of even (odd) number states. This property

makes them suitable candidates for designing a bosonic

code as studied in Ref. [21].

III. DISSIPATIVE STABILIZATION OF

COHERENT SUPERPOSITIONS OF GAUSSIAN

STATES

Here we build on the result by Hach III and Gerry [25].

Consider a single-mode bosonic system whose non-

unitary dynamics are described by a Gorini-Kossakowski-

Sudarshan-Lindblad master equation [31,32] of the form

(we set ~= 1 throughout this paper)

dˆρ

dt=−i[Ωˆ

L†+ Ω∗ˆ

L, ρ] + κD[ˆ

L]ˆρ, (10)

where D[ˆ

A]ˆρ=ˆ

Aˆρˆ

A†−1

2ˆ

A†ˆ

Aˆρ−1

2ˆρˆ

A†ˆ

Aand where the

operator ˆ

Lis, in general, a function of the bosonic annihi-

lation (ˆa) and creation (ˆa†) operators. Then, the steady

state ∂tˆρss = 0 of (10) is an eigenstate of the operator ˆ

with eigenvalue z=−2iΩ/κ, i.e.,

Lˆρss =zˆρss.(11)

This allows one to express Eq. (10) in a very concise form

dˆρ

dt=κD(ˆ

L−z)ˆρ. (12)

Following Eq. (11), the most general steady-state of

(10) would be a statistical mixture of eigenstates of ˆ

with a common eigenvalue. This is the basis of the

dissipative stabilization of cat states [8–10]. A similar

approach has been proposed for the stabilization of cat

states in atomic ensembles [33].

Now, starting from ˆ

D(α)ˆ

S(ξ)ˆa|0i= 0, it is straight-

forward to show the relation

b|α;ξi=βα,ξ |α;ξi,(13)

where we have introduced the bosonic operator ˆ

S(ξ) ˆaˆ

S†(ξ) = cosh(r) ˆa+e−iφsinh(r) ˆa†, and the related

complex eigenvalue βα,ξ =αcosh(r) + α∗e−iφ sinh(r).

From Eq. (13) the relation ˆ

bn|α;ξi=βn

α,ξ |α;ξifor an

arbitrary integer nimmediately follows. In turn, from

this relation, it follows that the squeezed cat states (9)

are degenerate eigenstates of the operator ˆ

b2|C±

α,ξi=β2

α,ξ |C±

α,ξi.(14)

Similarly to the case of (Schr¨odinger) cat states, higher-

order superpositions of squeezed states may yield higher-

order powers of the eigenvalue βα,ξ. Nevertheless, in this

work, we are going to restrict to the case n= 2.

Following the above discussion, for ˆ

L=ˆ

b2the steady

state of the dissipative dynamics will be, in general, a

mixture of even and odd parity squeezed cat states. How-

ever, as in this case photons are created and annihilated

in pairs, the parity of an initial state will be preserved

throughout the dynamics. In other words, an initial even

(odd) state will evolve in the long time to the state |C+

α,ξi

(|C−

α,ξi).

For convenience, in this work we will restrict to the case

of a superposition of two squeezed states with squeez-

ing ξalong the xquadrature, i.e., φ= 0, and displaced

along the xaxis, i.e., αreal. In this case, βα,ξ re-

duces to βα,ξ =αexp(r). By setting the drive amplitude

Ω = i¯

Ω, with ¯

Ω real, we can ﬁx the steady-state eigen-

value z= 2¯

Ω/κ to be real as well. Therefore, in order

to stabilize a coherent superposition of xsqueezed states

displaced along the xaxis, we choose the drive amplitude

to be ¯

Ω = κα2exp(2r)/2.

IV. APPLICATION: SQUEEZED CAT QUBIT

A. Introduction

We have seen above that the dynamics of an oscillator

described by the Lindblad equation

dˆρ

dt=κ2D[ˆ

b2−β2

α,r],(15)

with ˆ

b=ˆ

S(r)ˆaˆ

S†(r), ˆathe annihilation operator of the

oscillator mode and ˆ

S(r) the squeezing operator, are

restricted to the two-dimensional subspace spanned by

the orthogonal squeezed cat states {|C+

α,ri,|C−

α,ri} with

βα,r =αer. This motivates the deﬁnition of the squeezed

cat-qubit (SCQ) logical basis as

|C0

α,ri=1

√2|C+

α,ri+|C−

α,ri≈ |α, ri(16)

|C1

α,ri=1

√2|C+

α,ri − |C−

α,ri≈ |−α, ri,(17)

where the approximation sign occurs because in contrast

to |C±

α,ri, the squeezed coherent states |±α, riare only

quasi-orthogonal, that is, their ﬁnite overlap is given by

h−α, r|α, ri= exp−2α2e2r.(18)

A Bloch sphere representation of the SCQ is shown in

Fig. 1(a). The squeezed cat code is related to the ordi-

nary cat code in the sense that one recovers the ordinary

cat code from the squeezed version in the limit of zero

squeezing (r→0). In the opposite limit, r→+∞,

the code becomes translation invariant with respect to

phase space translations of amplitude s= 2πk/|α|, k ∈Z

along the pquadrature. The translation invariance in

this limit relates the code to the GKP code. In fact, one

can interpret the squeezed cat code as a (bad) approx-

imation of the GKP code that has only two (inﬁnitely)

squeezed peaks. This view explains intuitively the ﬁnite

error correction capabilities of the SCQ against phase

ﬂips the authors of Ref. [21] found through an analysis

of the Knill-Laﬂamme conditions. From the perspective

of the ordinary cat code, the increased error correction

capabilities arise as the squeezed coherent state is not

an eigenstate of the annihilation operator anymore such

that for r > 0 the state ˆa|C±

α,rihas a ﬁnite component

that is orthogonal to the code space, i.e., it lies outside

of the code space in the error space.

In Ref. [21] the authors also demonstrate numeri-

cally the increased error correction performance of the

squeezed cat qubit over the ordinary cat qubit by com-

puting and applying the optimal recovery operation ob-

tained from a semideﬁnite program. While one can argue

that the optimal recovery operation allows one to com-

pute the maximally achievable performance of a given

quantum code, physically implementing the required re-

covery is in many cases non-trivial. Thus, dissipative

stabilization schemes such as the one described here are

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QuantumerrorcorrectionwithdissipativelystabilizedsqueezedcatqubitsTimoHillmannDepartmentofMicrotechnologyandNanoscience(MC2),ChalmersUniversityofTechnology,SE-41296Gothenburg,SwedenFernandoQuijandraQuantumMachinesUnit,OkinawaInstituteofScienceandTechnologyGraduateUniversity,Onna-son,Okinawa904-04...

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Quantum error correction with dissipatively stabilized squeezed cat qubits Timo Hillmann

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