Autonomous quantum error correction and fault-tolerant quantum computation with squeezed cat qubits Qian Xu1Guo Zheng1Yu-Xin Wang1Peter Zoller2 3Aashish A. Clerk1and Liang Jiang1

2025-05-02 0 0 1.23MB 19 页 10玖币

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Autonomous quantum error correction and fault-tolerant quantum computation with

squeezed cat qubits

Qian Xu,1, ∗Guo Zheng,1, ∗Yu-Xin Wang,1Peter Zoller,2, 3 Aashish A. Clerk,1and Liang Jiang1, †

1Pritzker School of Molecular Engineering, The University of Chicago, Chicago 60637, USA

2Institute for Theoretical Physics, University of Innsbruck, Innsbruck A-6020, Austria

3Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, Innsbruck A-6020, Austria

(Dated: October 25, 2022)

We propose an autonomous quantum error correction scheme using squeezed cat (SC) code against

the dominant error source, excitation loss, in continuous-variable systems. Through reservoir engi-

neering, we show that a structured dissipation can stabilize a two-component SC while autonomously

correcting the errors. The implementation of such dissipation only requires low-order nonlinear cou-

plings among three bosonic modes or between a bosonic mode and a qutrit. While our proposed

scheme is device independent, it is readily implementable with current experimental platforms such

as superconducting circuits and trapped-ion systems. Compared to the stabilized cat, the stabilized

SC has a much lower dominant error rate and a signiﬁcantly enhanced noise bias. Furthermore, the

bias-preserving operations for the SC have much lower error rates. In combination, the stabilized SC

leads to substantially better logical performance when concatenating with an outer discrete-variable

code. The surface-SC scheme achieves more than one order of magnitude increase in the threshold

ratio between the loss rate κ1and the engineered dissipation rate κ2. Under a practical noise ratio

κ1/κ2= 10−3, the repetition-SC scheme can reach a 10−15 logical error rate even with a small mean

excitation number of 4, which already suﬃces for practically useful quantum algorithms.

I. INTRODUCTION

Quantum information is fragile to errors introduced

by the environment. Quantum error correction (QEC)

protects quantum systems by correcting the errors and

removing the entropy [1–3]. Based upon QEC, fault-

tolerant quantum computation (FTQC) can be per-

formed, provided that the physical noise strength is below

an accuracy threshold [4–7]. However, realizing FTQC is

yet challenging due to the demanding threshold require-

ment and the signiﬁcant resource overhead [8–11]. Unlike

discrete-variable (DV) systems, continuous-variable (CV)

systems possess an inﬁnite-dimensional Hilbert space.

Encoding the quantum information in CV systems, there-

fore, provides a hardware-eﬃcient approach to QEC [12–

16]. Various bosonic codes have been experimentally

demonstrated to suppress errors in CV systems [17–22].

The standard QEC procedure relies on actively mea-

suring the error syndromes and performing feedback

control [1]. However, such adaptive protocols demand

fast, high-ﬁdelity coherent operations and measurements,

which poses signiﬁcant experimental challenges. At this

stage, the error rates in the encoded level are still

higher than the physical error rates in current devices

due to the errors during the QEC operations [23–26].

To address these challenges, we may implement QEC

non-adaptively via engineered dissipation – an approach

called autonomous QEC (AutoQEC) [27]. Such an ap-

proach avoids the measurement imperfection and over-

head associated with the classical feedback loops. Au-

tonomous QEC in bosonic systems that can magniﬁcently

∗These authors contributed equally.

†liang.jiang@uchicago.edu

suppress the dephasing noise has been demonstrated us-

ing the two-component cat code [20,22,28]. However,

AutoQEC against excitation loss, which is usually the

dominant error source in a bosonic mode, remains chal-

lenging. It requires either large nonlinearities that are

challenging to engineer (e.g., the multiphoton processes

needed for the multi-component cat codes [29]) or cou-

plings to an intrinsically nonlinear DV system [30,31]

that is much noisier than the bosonic mode.

In this work, we propose an AutoQEC scheme against

excitation loss with low-order nonlinearities and acces-

sible experimental resources. Our scheme is, in prin-

ciple, device-independent and readily implementable in

superconducting circuits and trapped-ion systems. The

scheme is based on the squeezed cat (SC) encoding, which

involves the superposition of squeezed coherent state. We

introduce an explicit AutoQEC scheme for the SC against

loss errors by engineering a nontrivial dissipation, which

simultaneously stabilizes the SC states and corrects the

loss errors. We show that the engineered dissipation is

close to the optimal recovery obtained using a semideﬁ-

nite programming [32–34]. Notably, our proposed dissi-

pation can be implemented with the same order of nonlin-

earity as that required by the two-component cat, which

has been experimentally demonstrated in superconduct-

ing circuits [20] and shown to be feasible in trapped-ion

systems [35].

Furthermore, we show that similar to the stabilized

cat qubits, the stabilized SC qubits also possess a biased

noise channel (with one type of error dominant over oth-

ers), with an even larger bias (deﬁned to be the ratio

between the dominant error rate and the others) ∼e¯n2

(compared to ∼e¯nfor the cat), where ¯ndenotes the mean

excitation number of the codewords. Consequently, we

can concatenate the stabilized SC qubits with a DV code

arXiv:2210.13406v1 [quant-ph] 24 Oct 2022

tailored towards the biased noise to realize low-overhead

fault tolerant QEC and quantum computation [36–41].

We develop a set of operations for the SC that are com-

patible with the engineered dissipation and can preserve

the noise bias needed for the concatenation. Compared

to those for the cat [42], these operations suﬀer less from

the loss errors because of the AutoQEC. Moreover, they

can be implemented faster due to a larger dissipation gap

and a cancellation of the leading-order non-adiabatic er-

rors. In combination, the access to higher-quality oper-

ations leads to much better logical performance in the

concatenated level using the SC qubits. For instance,

we can achieve one-to-two orders of magnitude improve-

ment in the κ1/κ2threshold, where κ1is the excitation

loss rate and κ2is the engineered dissipation rate, for

the surface-SC and repetition-SC scheme (compared to

surface-cat and repetition-cat, respectively). Further-

more, the repetition-SC can easily achieve a logical er-

ror rate as low as 10−15, which already suﬃces for many

useful quantum algorithms [8,43], even using a small SC

with ¯n= 4 under a practical noise ratio κ1/κ2= 10−3.

We note that aspects of the SC encoding were also

recently studied in Ref. [44], with an emphasis on the en-

hanced protection against dephasing provided by squeez-

ing (a point already noted in Refs. [45–47]). Unlike our

work, Ref. [44] neither explored the enhanced noise bias

provided by squeezing, nor exploited the ability to con-

catenate the SC code with outer DV codes using bias-

preserving operations; as we have discussed, these are key

advantages of the SC approach. Our work also goes be-

yond Ref. [44] in providing an explicit, fully autonomous

approach to SC QEC that exploits low-order nonlineari-

ties, and it is compatible with several experimental plat-

forms. In contrast, Ref. [44] studied an approach re-

quiring explicit syndrome measurements and a formal,

numerically-optimized recovery operation. It was unclear

how such an operation could be feasibly implemented in

experiment. We also note that the SC has also been stud-

ied in the context of quantum transduction [48] (a very

diﬀerent setting than that considered here).

II. RESULTS

Squeezed cat encoding

The codewords of the SC are deﬁned by applying a

squeezing along the displacement axis (which is taken

to be real) to the cat codewords:

|SC±

r,α0i:= ˆ

S(r)|C±

α0i(1)

where |C±

α0i:= N±(|α0i+| − α0i) with N±=

√2(1±e−2α02)being normalization factors, and ˆ

S(r) :=

exp1

2r(ˆa2−ˆa†2)is the squeezing operator. The above

codewords with even (|SC+

r,α0i) and odd (|SC−

r,α0i) exci-

tation number parity are deﬁned to be the X-basis eigen-

states. The performance of the code is related to the

mean excitation number ¯nof its codewords. For a code

with ﬁxed ¯n, the amplitude α0of the underlying coherent

states varies with the squeezing parameter ras

α0≈p¯n−sinh2rer,(2)

which holds for the regime of interest where α0>1. Note

that α0is closely related to how separated in phase space

the two computational-basis states are, which determines

their resilience against local error processes. At ﬁxed ¯n,

α02can be written as a concave quadratic function of e2r,

which has a maximum α02

max = ¯n2+ ¯n.

For the SC, it is convenient to consider the subsys-

tem decomposition of the oscillator Hilbert space H=

HL⊗ Hg, where HLrepresents a logical sector of di-

mension 2 (which we refer to as a logical qubit) and Hg

represents a gauge sector of inﬁnite dimension (which we

refer to as a gauge mode). Analogous to the modular

subsystem decomposition of the GKP qubit [49], whose

logical sector carries the modular value of the quadra-

tures, the logical sector of the SC carries the parity in-

formation (excitation number modulo 2). We can choose

a basis under the subsystem decomposition spanned by

squeezed displaced Fock states |±iL⊗ |ˆ

˜n=nig≈

N±,n ˆ

S(r)[ ˆ

D(α0)±(−1)nˆ

D(−α0)]|ni(we use ≈since the

right-hand side should be orthonormalized within each

parity branch. See supplement. [50] for details). By

choosing this basis, the SC codewords in Eq. (1) coin-

cide with |±iL⊗|ˆ

˜n= 0ig, i.e., the codespace is the two-

dimensional subspace obtained by projecting the gauge

mode to the ground state. Furthermore, the bosonic an-

nihilation operator ˆacan be expressed as

ˆa=ˆ

ZL⊗(e−rα0+ cosh rˆ

˜a−sinh rˆ

˜a†) + O(e−2α02),(3)

where ˆ

ZLis the Pauli Z operator acting on the logical

qubit, and ˆ

˜a=P∞

n=0 √n+ 1|ˆ

˜n=nighˆ

˜n=n+ 1|is the

annihilation operator acting on the gauge mode.

Typtical bosonic systems suﬀer from excitation loss

(ˆa), heating (ˆa†), and dephasing (ˆa†ˆa) errors, with loss

being the prominent one. To understand why the SC

code can correct excitation loss errors, we consider the

Knill–Laﬂamme conditions [51,52] and evaluate the QEC

matrices for loss errors [16]. Consider a pure loss channel

with a loss probability γ, the leading-order Kraus opera-

tors are {ˆ

I, √γˆa}. The detectability of a single excitation

loss is quantiﬁed by the matrix:

Pcodeˆaˆ

Pcode =e−rα0q+q−1

2ˆ

Zc+ierα0q−q−1

2ˆ

≈p¯n−sinh2rˆ

Zc−ierα0e−2α02ˆ

Yc,

(4)

where ˆ

Pcode is the projection onto the code space, ˆ

(ˆ

Yc) is the Pauli Z(Y) operator in the code space and

q:= q1−e−2α02

1+e−2α02. The approximation in the second

line is made in the regime of interest where α01.

Eq. (4) indicates that a single excitation loss mostly

FIG. 1. The illustration of a SC that suﬀers from a single ex-

citation loss and then approximately corrects it. Each dashed

box represents a state (visualized by the Wigner function) of

the SC, which is decomposed as a product of a logical qubit

and a gauge mode. A single excitation loss corrupts the code-

word |+ic(left) into the state ˆa|+ic/ph+|cˆa†ˆa|+ic(right).

During such a process, a phase ﬂip happens on the logical

qubit, and a fraction 1 −ηof the gauge mode population gets

excited (indicated by the thick orange arrow). The excited

population can be detected and then corrected, as indicated

by the blue arrow.

leads to undetectable logical phase-ﬂip errors, and the

undetectability decreases with the squeezing parameter

r. The increase of the detectability of single excitation

loss events with the squeezing rcan be better under-

stood by considering the action of the decomposed ˆa

operator (Eq. (3)) on the codeword ˆa(|+iL⊗ |00ig) =

|−iL⊗√¯n(√η|00i − √1−η|10i), where

η:= (¯n−sinh2r)/¯n. (5)

As shown in Fig. 1, after a single excitation loss, the

branch of the population (with ratio η) that stays in the

ground state of the gauge mode leads to undetectable

logical phase-ﬂip errors. In contrast, the other branch

(with ratio 1 −η) that goes to the ﬁrst excited gauge

state is in principle detectable. The detectable branch

is also approximately correctable since ˆ

Pcodeˆa†ˆaˆ

Pcode ≈

¯nˆ

Ic+O(e−2α02)ˆ

Xc. Therefore, we expect that we can

suppress the loss-induced phase ﬂip errors by a factor η

that decreases with the squeezing r. Moreover, the ˆ

and ˆ

Ycterms in the QEC matrices for both loss, heating,

and dephasing are exponentially suppressed by α02. As

shown in Eq. (2), α02can be greatly increased by adding

squeezing (with α02

max = ¯n2+¯n). Consequently, we expect

that the SC can also have signiﬁcantly enhanced noise

bias compared to the cat.

Autonomous quantum error correction

While we have shown that the SC encoding can, in prin-

ciple, detect and correct the loss errors, it remains a

non-trivial task to ﬁnd an explicit and practical recov-

ery channel. In this section, we provide such a recovery

channel, showing surprisingly that it requires only ex-

perimental resources that have been previously demon-

strated. As shown by the blue arrow in Fig. 1, we can,

in principle, perform photon counting measurement on

a probe ﬁeld that is weakly coupled to the gauge mode,

and apply a feedback parity ﬂip ˆ

ZLon the logical qubit

upon detecting an excitation in the probe ﬁeld [53]. Such

measurement and feedback process can be equivalently

implemented by applying the dissipator:

F= ( ˆ

ZL⊗ˆ

I)ˆ

S(r)(ˆa2−α02)ˆ

S†(r).(6)

When α01, ˆ

F∝ˆ

ZL⊗ˆ

˜arepresents a logical phase ﬂip

conditioned on the gauge mode losing an excitation. In

the Fock basis, such an operator can be approximately

given by

F≈er

α0(c1ˆa+c2ˆa†)ˆ

S(r)(ˆa2−α02)ˆ

S†(r),(7)

with c1+c2= 1. In Methods, we propose two reservoir-

engineering approaches to implement such a nontriv-

ial dissipator using currently accessible experimental re-

sources. We sketch the main ideas here. The ﬁrst

approach utilizes three bosonic modes that are nonlin-

early coupled. As shown in Fig. 5(a), a high-quality

mode band a lossy mode c, together, serve as a non-

reciprocal bath that provides a directional interaction

e−iθ ˆ

ZL⊗ˆ

˜afrom the gauge mode to the logical qubit in

the storage mode a. Such a coupled system can be phys-

ically realized in, e.g., superconducting circuits [20,54].

The second approach couples a bosonic mode nonlin-

early to a qutrit {|gi,|ei,|fi}. As shown in Fig. 6,

the bosonic mode is coupled to the gf transition via

S(r)(ˆa2−α02)ˆ

S†(r)|fihg|+h.c. and to the ef transition

via ˆ

ZL|eihf|+h.c.. By enhacing the decay from |eito

|gi, we can obtain the eﬀective dissipator ˆ

Fby adiabati-

cally eliminating both |eiand |fi. Such a system can be

physically realized in, e.g., trapped-ion system [35].

With the engineered dissipator in Eq. (6), the SC can

be autonomously protected from excitation loss, heat-

ing and dephasing. The dynamics of the system are de-

scribed by the Lindblad master equation:

dˆρ

dt =κ2D[ˆ

F]ˆρ+κ1(1 + nth)D[ˆa]ˆρ

+κ1nthD[ˆa†]ˆρ+κφD[ˆa†ˆa]ˆρ,

(8)

where D[ˆ

A]ˆρ:= ˆ

Aˆρˆ

A†−1

2{ˆ

A†ˆ

A, ˆρ}. The logical phase-

ﬂip and bit-ﬂip error rates of the SC under the dynamics

described by Eq. (8) can be analytically obtained (see

Methods for the derivations):

γZ= [κ1(1 + 2nth) + κφe−2r](¯n−sinh2r),(9)

γX,Y =κφ(¯n−sinh2r)e2r(sinh22r/4+cosh 4r)

2 sinh(2(¯n−sinh2r)e2r),(10)

where γX,Y denotes the sum of the logical Xand Yerror

rates, which we refer to as the bitf-ﬂip rate for simplic-

ity [55]. We only consider the dephasing for γX,Y since

the loss-induced bit-ﬂip rate has a more favorable scal-

ing ∼e−4α02with α0[54,56]. The loss and the heating

contribute to γZin the same way (both suppressed by a

factor η) since their undetectable portion (η) is the same

(see Eq. (3) and its hermitian conjugate). The dephasing

also contributes to γZ, but with an extra e−2rsuppres-

sion, when combined with the parity-ﬂipping dissipator

F. See Methods for details. Setting r= 0 and remov-

ing the κφterm in γZ, we restore the error rates of the

dissipative cat [42].

In the regime where e−r1 and γZis mainly con-

tributed by excitation loss, we can simplify Eqs. (9) and

(10) as

γZ≈η¯nκ1, γX,Y ≈9

16κφα02e−2α02e4r,(11)

where

α0≈p4η(1 −η)¯n. (12)

As plotted in Fig. 2(a), ﬁxing ¯n,γZdecreases monotoni-

cally with the squeezing r(unless rapproaches the max-

imum squeezing rmax ≈sinh−1(√¯n). See Methods for

details) as the undetectable portion ηof the loss-induced

errors decreases (see Eq. (5)). The change of γX,Y with r

(or equivalently, η) is roughly captured by the change in

the displacement amplitude α0(see Eq. (12)), and γX,Y

takes the minima roughly when α0reaches the maxima

α0

max =√¯n2+ ¯n. Note that the minimal bit-ﬂip rate

of the SC enjoys a more favorable scaling γX,Y ∝e−2¯n2

with ¯n, compared to γX,Y ∝e−2¯nfor the cat, so that

the SC can have a much larger noise bias under the same

excitation number constraint. In principle, one needs to

consider the tradeoﬀ between γZand α0and choose the

optimal ηdepending on the tasks of interest. Smaller η

leads to better protection from excitation losses, which

is preferred by, e.g., the idling operations. Larger α0,

on the other hand, leads to a larger noise bias and a

widened dissipation gap [57] (∝α02), which can support

faster operations, e.g., the bias-preserving CX gate intro-

duce in the next section. In the following, we ﬁx ¯n= 4

and η= 1/4 if not speciﬁed otherwise, which corresponds

to a squeezing of r= 1.32 (11.5 dB). Such a parameter

choice leads to γZ≈κ1, which removes the enhancement

factor ¯npresent for the cat (for ¯n= 4). Meanwhile,

α02≈3

4¯n2provides a suﬃciently large noise bias and

dissipation gap.

In Fig. 2(b), we benchmark the performance of our

Auto-QEC scheme against loss errors by comparing it

to the optimal recovery channel given by a semideﬁnite

programming (SDP) method [32–34]. We consider the

joint channel N=D · Nγ· E, where Edenotes the en-

coding map from a qubit to the SC, Nγdenotes a Gaus-

sian pure loss channel (corresponding to Eq. (8) with

κ2=κφ=nth = 0) with loss probability γ:= κ1t,

and Ddenotes the recovery channel either using the au-

tonomous QEC with the dissipator Eq. (6) or the op-

timal recovery channel. We evaluate the entanglement

inﬁdelity (EI) 1−Fe, where Fedenotes the entanglement

ﬁdelity, of the joint channel N, as a function of the loss

(b)

(a)

FIG. 2. (a) The phase γZ(orange) and bit γX,Y (cyan) er-

ror rate of the dissipatively stabilized SC as a function of

squeezing runder the parameters ¯n= 4, κ1= 100κφ=

κ2/100, nth = 0.01. The solid lines represent the analytical

expressions Eqs. (9) and (10) while the diamonds represent

the numerically extracted values. All the error rates are nor-

malized by those of the dissipative cat γZ,c,(γX,Y )c, which are

given by Eqs. (9) and (10) with r= 0. (b) The entanglement

inﬁdelity of a joint loss and recovery channel varying with

the loss probability γfor the SC encoding with ¯n= 4. The

recovery channel is either the engineered dissipation (the cir-

cles) or the optimal recovery channel determined by an SDP

program [32–34] (the stars).

probability γ. Note that the EI is the objective function

for the SDP. As shown in Fig. 2(b), the EI obtained us-

ing the Auto-QEC is close to the optimal EI, especially in

the low-γregime, demonstrating that our proposed au-

tonomous QEC scheme is close to optimal for correcting

excitation loss errors. We note that it is crucial to have

the phase-ﬂip ˆ

ZLcorrection in the dissipator ˆ

Fin order

to correct the loss-induced phase-ﬂip errors. Otherwise, a

simple dissipator ˆ

S(r)(ˆa2−α02)ˆ

S†(r) directly generalized

from the dissipative cat would still give an unsuppressed

phase-ﬂip rate γZ=κ1¯n.

We note that the SC encoding also emerges as the

optimal or close-to-optimal single-mode bosonic code

through a bi-convex optimization (alternating SDP) pro-

cedure for a loss and dephasing channel with dephasing

being dominant, as shown in Ref. [58].

Bias-preserving operations

To apply the autonomously protected SC for compu-

tational tasks, we need to develop a set of gate op-

erations that are compatible with the engineered dis-

sipation. Furthermore, the operations should preserve

the biased noise channel of the SC, which can be uti-

lized for resource-eﬃcient concatenated QEC and fault-

tolerant quantum computing [40,41,54,59,60]. Fol-

lowing the literature for the cat and the pair-cat [42,

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Autonomousquantumerrorcorrectionandfault-tolerantquantumcomputationwithsqueezedcatqubitsQianXu,1,GuoZheng,1,Yu-XinWang,1PeterZoller,2,3AashishA.Clerk,1andLiangJiang1,y1PritzkerSchoolofMolecularEngineering,TheUniversityofChicago,Chicago60637,USA2InstituteforTheoreticalPhysics,UniversityofInnsbruck,...

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Autonomous quantum error correction and fault-tolerant quantum computation with squeezed cat qubits Qian Xu1Guo Zheng1Yu-Xin Wang1Peter Zoller2 3Aashish A. Clerk1and Liang Jiang1

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