*QMi`QHHBM; +Q?2`2Mi bi i2 bmT2`TQbBiBQMb rBi? bmT2`+QM/m ...

1

µ

2

H =p2

2m+

mω2

2x2

x, p m,ω

H = !ω(P2 + X 2

)

P = p/√2mω! X = x/

√mω2!

X = x2xZPF

,P = p2pZPF

xZPF =√

!2mω , pZPF =

√!mω2

P = 12i(a− a†) X = 1

2(a+ a†)[a, a†

]=

H = !ωa†a

H |n⟩ = !ωn |n⟩

|n⟩

a |n⟩ =√n |n− 1⟩ a† |n⟩ =

√n+ 1 |n+ 1⟩ a†a |n⟩ = n |n⟩ .

M

xp

k

C Lq

a) b)

p2

2Mk2x2

φ2

2Lq2

2C

X

P

TX = e−2iXP

P X

TP = e2iPX

[X ,P] = constant eB2 eAe

B2 = eA+B

[A,B] =

DX ,P = T 12 X

TPT 12 X

= e2i(XP−PX)

DX ,P = e(X−iP)a−(X+iP)a†

X P

b)a) c) d)

α = X + iP

Dα = eα∗a−αa†

|α⟩ = Dα |0⟩

α = |α|eiφ

|n⟩

|α⟩ = e−|α|22

∑

n

αn

√n!

|n⟩

|α(t)⟩ = e−iHt

! |α0⟩

= e−iωa†ae−|α0|

2

2

∑

n

αn0√n!

|n⟩

= e−|α0|

2

2

∑

n

(α0e−iωt)n√n!

|n⟩

= |α0e−iωt⟩

|α|

ω |n⟩

|α⟩

I Q

ω I = cosωt(

a+a†

2

)− sinωt

(a−a†

2i

)

Q = − sinωt(

a−a†

2i

)− cosωt

(a+a†

2

)

a)

I

Q

ωt

b)

ωt

φ = −ωt

I Q

P = eiπa†a = (−1)a

†a.

P P

±1

Pa = −aP Pa† = −a†P PD = D†P

|ψ⟩cat = N(|α⟩+ eiφ |−α⟩

)

N = 1√2(1+e−2|α|2 ) cosφ

⟨α|−α⟩ = e−2|α|2 N = 1√2

φ = 0, π

|ψ⟩even = N+ (|α⟩+ |−α⟩) |ψ⟩even = N− (|α⟩ − |−α⟩)

P |ψ⟩even =

+ |ψ⟩even P |ψ⟩odd = − |ψ⟩odd

H =1

2Lφ2 +

1

2Cq2

φ q

ρ =∑

m,n

cmn |m⟩ ⟨n|

|m⟩ , |n⟩ cmn

=∑

n

|n⟩ ⟨n| .

0 ≤ n ≤ N

N

⟨a†man⟩ m,n

N = 15

= 1π

∫d2α |α⟩ ⟨α|

1π

∫d2α |α⟩ ⟨α| = 1

π

∑

n,m

1√n!m!

|n⟩ ⟨m|∫

d2αe−|α|2αn(α∗)m

∫d2αe−|α|2αn(α∗)m = πΓ(n+m

2 + 1)δnm

W(α) Q(α)a) b) c)

1π

∫d2α |α⟩ ⟨α| =

∑

n,m

Γ(n+m2 + 1)

√n!m!

δnm |n⟩ ⟨m|

=∑

n

|n⟩ ⟨n| =

⟨α|α′⟩ = 0

C(λ)

Ca Cs

Ca(λ) = ⟨e−λ∗aeλa†⟩ Cs(λ) = ⟨eλa†−λ∗a⟩

Q(α) = FCa(λ) W (α) = FCs(λ)

FC(λ) = 1π2

∫d2λC(λ)eαλ

∗−α∗λ

p(α) = ⟨α|ρ|α⟩

ρ |α⟩

Q(α) = 1π ⟨α|ρ|α⟩

ρ

|ψ⟩ = N (|α⟩ + |−α⟩)

|α⟩ |−α⟩

Q(α) = 1π ⟨α|ρ|α⟩ =

1π ⟨0|D

†αρDα|0⟩

W(α) Q(α)

a ) b )

|ψ⟩ = N (|β⟩ − |β⟩) β = 2

a) b)

W(α)W(α)

|β⟩ |β|2 = 4 |n⟩ n = 4Re(α) = 2

P D†α

W (α) = 2πTr[D

†αρDαP ] = 2

π ⟨DαPD†α⟩ = 2

π ⟨Pα⟩

Pα

DαPD†α

2π ⟨Pα⟩

±1

ρ = 2π

∫d2αW (α)Pα

Tr[ρO(a, a†)] =

∫d2αW (α)O(α).

O(α) = Tr[D†αO(a, a†)DαP ]

|β⟩ ⟨β|

W (α) = 2πe

−2|α−β|2

F = ⟨ψ |ρ|ψ ⟩ = 1

π

∫W (α)W (α)d2α

Wt(α) = ⟨ψt|Pα|ψt⟩

F = Tr [ρtρ] ρ ρ

I(α′) Q(α′′) I Q

α = α′ + iα′′

I(α′) =

∫dα′′W (α) Q(α

′′) =

∫dα′W (α).

P

ρ = ρP

Cs(λ) = Tr[Dλρ] = Tr[Dλ/2ρD†λ/2P ] = π

2W (λ/2).

ρ

W (α) = 12πFW (λ2 )

Q(α) =

∫d2αe−2α2

W (α)

Q(α) = e−2α2 ∗W (α).

Cs(λ) = Tr[ρDλ]

|β⟩ + |−β⟩ |n⟩ (n = 0 2)

|0⟩ |1⟩

a → σ− =

(0 10 0

)a† → σ+ =

(0 01 0

).

X P

N

σx =

(0 11 0

)σy =

(0 −ii 0

)σz =

(1 00 −1

)

σ+σ− =1− σz

2= |e⟩ ⟨e| =

(0 00 1

)

|e⟩

⟨σx⟩ , ⟨σy⟩ , ⟨σz⟩

N

N

S = −∑

i

ηi log2 ηi

ηi

2N

ηi =12N

Sq =2N∑

i

1

2Nlog2 2

N = N

N ηi = 1N+1

N

Sc =N+1∑

i

1

N + 1log2 (N + 1) = log2 (N + 1)

a) b)

log2(N + 1) NΓmax

Γmax

Γ0

Γ /Γ0

Γ0 4Γ0

3

I = Ic sin2πφ

Φ0

Φ0 Ic

ϕ = 2πφ/Φ0

H = !ωqa†a− EJ

(cosϕ+

ϕ2

2

)

EJ = IcΦ02π ϕ =

∑ϕq(a + a†)

ϕq

EJ!ω

ϕ6q

720 ≪ 1

H = !ωqa†a− EJ

24ϕ4 +O(ϕ6)

≈ !ωqa†a− EJ

24ϕ4q

(a+ a†

)4

H = !ω′qa

†a− !α2a†

2a2

α = EJ4 ϕ

4q ω′

q = ωq − α α

∆E = En+1 −En = !ωq − !α

|0⟩ , |1⟩

H = !ω′q |e⟩ ⟨e|

EC

EJ H/! =√8ECEJa†a− EC

2 (a†a)2

b)

-1

-0.5

0.5

1

0

0 π−π

c)

Cq

a)

LJ

Cφ ≪ 1

H = !ωra†a+ !ωq |e⟩ ⟨e|+ !g(a+ a†)σx

a |e⟩

H = !ωra†a+ !ωq |e⟩ ⟨e|+ !g(aσ+ + a†σ−).

a, σ− σ+, a†

κ γ

g

g ≫ κ, γ

g ≫ |ωr −

ωq| = ∆

H = !(ωr − χ |e⟩ ⟨e|)a†a+ !ωq |e⟩ ⟨e|

χ = g2

2∆ χ

γ,κ χ ≫ nκ, γ n =

⟨a†a⟩

g∆

Hquasi = Hdisp −Ka†2a2σz

K = g4

∆3

σz = − 2 |e⟩ ⟨e|

Ka†2a2σz →

K

2a†

2a2 −Ka†

2a2 |e⟩ ⟨e|

K

ωcgωc

e

ωq ωc

ωq5ωq4 ωq

2

ωq1ωq0

ωq3

ωce

ωce

ωce

ωcg

ωcg

ωcg

ωq0

ωq1

ωq2

ωq3freq

a) b)

H = !∑

i=q,r

ωia†iai − EJ

(cosϕ+

ϕ2

2

)

aq,r

ϕ =∑

i=q,r ϕi(ai + a†i )

ϕq >> ϕr

ϕ

H =∑

i=q,r

(!ωia†iai −

Ki

2a†i

2a2i )− χa†qaqa

†rar

Ki = EJϕ4i

2 χ = EJϕ2qϕ

2r

Kr

Kr α = Kq

ϕq ∼ ϕr

ϕ =∑

i ϕi(ai + a†i )

H4 =∑

i

(!ωia†iai −

Ki

2a†i

2a2i )−

∑

i,j>i

χija†iaia

†jaj

Ki =EJϕ4

i2 χij = EJϕ2

iϕ2j

Ki ∝ ϕ4i χij

χ ϕ

H6 = H4 +∑

i

K ′i

6a†i

3a3i +

∑i, j

χ′ij

2a†i

2a2i a

†jaj

K ′i =

EJϕ6i

6 χ′ij =

EJϕ4iϕ

2i

2 K ′ χ′

n

Ki(n) → (K +K ′

3− K ′

3ni)

χij(ni) → (χij +χ′ij

2−χ′ij

2ni)

ωi/2π

O(ϕ4) Ki/2π14!

(42

)(22

)EJϕ4

i

O(ϕ4) χij/2π14!

(41

)(31

)(21

)(11

)EJϕ2

iϕ2j

O(ϕ6) K ′i/2π

16!

(63

)(33

)EJϕ6

i

O(ϕ6) χ′ij/2π

16!

(62

)(42

)(21

)(11

)EJϕ2

iϕ4j

EJ ϕ =∑ϕi(ai+ a†i )

cosϕ

O(φ6)

1 µm

Josephsonjunctions

compact resonator

200 µm

transmon qubit

phase qubit2.1 mm

transmissionline resonator

50 mm250 µm

transmon qubit

three-dimensional cavity resonator

a)

b)

c)

200 µm

–2 –1 0 1 2 –2 –1 0 1 2

0

+1/π

+2/π

–1/π

–2/π

W( ) α

Re ( )α

Im

() α

–2–1

01

2–2

–10

12

0.4

–2

–20

202

4

0.2

0.0

–0.2

–0.4

Im(a )Re(a )W( ) α

a) b)

4

100 µs

10 ms

H

! = ωra†rar + ωsa

†sas + ωq |e⟩ ⟨e|

− χqra†rar |e⟩ ⟨e|− χqsa

†sas |e⟩ ⟨e|

χqr (χqs)

κr

κs ≫ κr

l h w

fmnk =c

2

√(ml

)2+(nh

)2+

(k

w

)2

c m, n, k

f101

∼ 1 GHz

7− 10 GHz

45 µm2 ∼ 6 nH

Al2O3

ϵr ≈ 9.4, 9.4, 10.2

Zline(l) = Z0ZL + jZ0 tan(βl)

Z0 + jZL tan(βl)

Z0 ZL

β l

|ZL| = | 1jωC | ≫

Z0

Zline(l) = −jZ0 cot (βl).

Zline(ω, l) = −jZ0 cot

(ωl

νp

)

νp = c√µrϵr

c µr, ϵr

νp ≈ (0.2− 1)c

b)a)

Z0,νpEJ

⎫ ⎬ ⎭l ⎫ ⎬ ⎭lYin(ω)

Yin(ω) LJ

Zline(ω)

Zline(ω)

l Z0

νp

Z0 ∼ 80Ω νp ∼ 0.4c

Y (ω)

LJ

EJ = φ20

LJ

ω0 = 1√LeffCeff

Y (ω0) = 0

Zeff =√

LeffCeff

= 2ω0Im[Y ′(ω0)]

H/! = ωqa†a− α

2a†

2a

ϕ

ωq = ω0 − α

α =e2Z2

eff

2!LJ.

Yin(ω) =1

jωLJ+

j

2Z0tan

(ωl

νp

).

Yin(ω0) = 0

1

ω0LJ=

1

2Z0tan

(ω0l

νp

).

ω0 Zeff

Im[Y ′(ω)] =1

ω2LJ+

l

2Z0νpsec2

(ωl

νp

).

tan(

ωlνp

)

Yin(ω) ≈1

jωLJ+

j

2Z0

ωl

νp

LC

Yin(ω) ≈1

jωLJ+ jωC(l)

C(l) = l2Z0νp

ω0 =

√2Z0νpLJ l

=1√

LJC(l)

α =e2Z2

eff

2!LJ=

e2Z0νp!l =

e2

2!C(l).

ω0(l) ∝ 1√l

α(l) ∝ 1l

Reso

nanc

e (GHz)

Anha

rmon

icity

(GHz)

Antenna length (mm)

a)

Antenna length (mm)

b)

ω0/2πα/2π

Lj = 7 nH, Z0 = 80 Ω, and νp = 0.4c

tan(

ωlνp

)

Yin(ω) ≈1

jωLJ+

j

2Z0

ωl

νp

(1 +

ω2

2

(l

νp

)2).

ω20 =

3

2

(νpl

)2(√

1 +8

3

Z0

LJ

l

νp− 1

).

ω0

α =e2Z0νp!l

(1− 2Z0

LJ

l

νp

).

LJ

Yin(ω)

Z1(ω)

cavity 1 cavity 2Z2(ω)

4 m

m

0.4 mm

7.5 8.0 8.5 9.0 9.5

0

-2

-4

2

4

Frequency (GHz)

Adm

ittan

ce (mS)

Yin(ω)

cavi

ty 1

stripline

substrate

Z1(ω)

LC

Z1(ω), Z2(ω)

Yin(ω)

O (ϕ6)

15010

Ω

∼

I + iQ

ωRF, ωLO Vsig

V ∝ cos (ωIF + δRF − δLO + δDUT)

ωIF = ωRF − ωLO

δRF, LO, DUT

ωRF, ωLO

ωIF

δ

Vdemod ∝ cosωIF

Vref ∝ cos (ωIFt+ δRF − δLO)

300K4K

20mK

1 2

3 4 5

6 7

81

2 3

4 5 6

7 8

20dB

20dB

20dB

20dB

20dB

20dB

20dB

20dB

20dB

20dB

20dB

20dB

20dB

20dB

HEM

T30dB

30dB

30dB

30dB

30dB

30dB

10dB

10dB

10dB

LP 12GHz

LP 10GHzEcco

AB

LP 10GHz

LP 10GHz

LP 10GHz

20dB

10dB

20dB

10dB

10dB

LP 12GHz

LP 10GHz

EccoEcco

10dB10dB

Ecco

HEM

TLP 10GHz

JPC

180-H

SS

10dB

Ecco

Ecco

JPC

180-H

180-H

Ecco 10dB

SS

II

EccoEccoEcco

DUT

reference signal

LO

RF

δRF

δLO

δsignal

DUT

signal

LO

RF

δRF

δLO

δsignal

a) b)

ωRF, ωLO

ωIF = ωRF − ωLO

δRF, δLOδsignal

Feedback

ωμw

Storage

Storage input

Qubit and

readout input

Readout

output

I/O setup with feedback

I

Q

12

S

ωμw

Qubit

ADCSE

DA

C

I

Q

ADCSE

DIGITAL

DA

C

12

S ωμw

Readout

I

Q

DA

C

DIGITAL

FPGA

FPGA

Sw

itch

TO FRIDGE

FROM

FRIDGE

ωμw

LOS

witc

h

DIGITAL

Sw

itch

AWG1 2

S

ωμw

StorageStorage input

Qubit and

readout input

Readout

output

I/O setup

12

S

ωμw

Qubit

12

S ωμw

Readout

Sw

itchTO FRIDGE

FROM

FRIDGE

ωμw

LO

Sw

itch

1 2S

I

Q

DA

C

DIGITAL

I

Q

DA

C

AWG

Sw

itch

ADC

a)

b)

|g⟩ |e⟩

5

H/! = ωq |e⟩ ⟨e|+ ωsa†a− χa†a |e⟩ ⟨e|

|e⟩ a† a

ωq,s

χ

CΦ = eiΦa†a|e⟩⟨e| = ⊗ |g⟩ ⟨g|+ eiΦa†a ⊗ |e⟩ ⟨e|

|g⟩ Φ

τ Φ = χτ

CΦ |α⟩ ⊗ (|g⟩+ |e⟩) = |α, g⟩ + |αeiΦ, e⟩

|α⟩ = e−|α|22∑∞

n=0αn√n!|n⟩ |n⟩

α

CΦ=π π

⟨P ⟩

cavity

qubit

b)

a) P

X

e

g

|ψ⟩ = |e, eiΦβ⟩Φ = χτ

CΦ

χ ≫

γ, n κs γ κs

n

χa†a |e⟩ ⟨e|

Ksa†2a2 χ′a†2a2 |e⟩ ⟨e|

m

Rmn,θ = |m⟩ ⟨m|⊗ Rn,θ +

∑

n =m

|n⟩ ⟨n|⊗

Rn,θ n θ

χ

P

X

P

X

b)a) c) d)P

X

cavity

qubit

P

X

m=0nmax

m

nmax

ωnq =

ωq − χn n

τ ≫ 1/χ

m

ωmq

H/! = −χ(a†a−m) |e⟩ ⟨e|+ ϵ(t)σy

ϵ(t)

σy

H/! =∑

n

Hn/! |n⟩ ⟨n|

=∑

n

−χ(n−m) |e⟩ ⟨e|+ ϵ(t)σy |n⟩ ⟨n| .

Hn/! =∑

n

ϵ(t)ei∆n,mt|e⟩⟨e|σye−i∆n,mt|e⟩⟨e|

∆n,m = ωnq − ωm

q |ψ(t)⟩

|ψn(t)⟩ = − i

!Hn(t) |ψn(t)⟩ .

m

Ry,θ = eiθ2σy θ = 2

∫ϵ(t) t

|ψn =m(t)⟩

|ψn(t)⟩ ≈1− i

!

∫ t

sHn(s)

|ψn(0)⟩ .

|ψ(0)⟩ =∑

n =m Cn |g, n⟩

|ψ(t)⟩ ≈∑

n =m

Cn|g, n⟩ −i

!

∫ t

0

sHn(s) |g, n⟩

=∑

n =m

Cn|g, n⟩

− i

∫ t

0

sϵ(s)ei∆n,ms|e⟩⟨e|σye−i∆n,ms|e⟩⟨e| |g, n⟩

=∑

n =m

Cn|g, n⟩ −∫ t

0

sϵ(s)ei∆n,ms |e, n⟩

≈∑

n =m

Cn|g, n⟩ − ϵ∆n,m |e, n⟩

ϵω = ∆n,m ϵ(t)

∆n τ

|ψ(τ)⟩ = 1√1 + ϵ∆n2

∑

n =m

Cn|g, n⟩ − ϵ∆n,m |e, n⟩.

ωmq

ϵ

|ψ⟩ =∑

n=m Cn |g, n⟩

|g⟩

S = |⟨n, g|ψ(τ)⟩|2 =∑

n=m

|Cn|2

1 + ϵ[∆n,m]2.

m

ϵ(t) = Ae−σ2ωt

2/2

σω A =√

8/πσω

π ωmq

σω/2π = 800 σt = 200

χ/2π = 3 m (m±1)

S = (1 + π8 e

−χ2/σ2ω)−1 > 99%

Rmy,π = |m⟩ ⟨m|⊗ Ry,π +

∑

n =m

eiξn |n⟩ ⟨n|⊗

|m⟩ m ξn

σω

|β⟩ |−β⟩

R0y,π

R0y,π(|2β, g⟩+ |0, e⟩) → (|2β⟩+ |0⟩)⊗ |g⟩

π |0⟩

|n⟩ |2β⟩ =∑∞

n=0 Cn |n⟩ =

e−|2β|2

2∑∞

n=0(2β)n√

n!|n⟩

σω = 4|β|2χ/5

|2β⟩ S =∑∞

n=1 |Cn|2(1 + π8 e

−(nχ)2/σ2ω)−1 > 99%

ξn

∆n = χn ≫ σω

ξn

ξn =∫ϵ(t)2dt/∆n 1/(2|β|) ≪ 1 |n⟩

ξn ∝ 1 − n/(8|β|2)

n

ϵ

χa†a |e⟩ ⟨e|

σz σy

H/! =∑

n

−χ(n−m)σz2

+ ϵ(t)σy |n⟩ ⟨n| .

ϵ τ

U(τ) = e−iH/!τ

=∑

n

Un(τ) |n⟩ ⟨n|

=∑

n

e−iτχ(m−n)σz2 +ϵσy |n⟩ ⟨n|

=∑

n

e−iφnσθn |n⟩ ⟨n|

φn = ϵτ

√1 +

[(m−n)χ

2ϵ

]2θn = arctan

((m−n)χ

2ϵ

)σθn = cos (θn)σy +

sin (θn)σz

Un(τ) = e−iφnσθn

= cos(φn) + i sin(φn)σθn= [cos(φn) + i sin(φn)] sin(θn) |n, g⟩ ⟨n, g|

+ [cos(φn)− i sin(φn)] sin(θn) |n, e⟩ ⟨n, e|+ sin(φn) cos(θn)(|n, e⟩ ⟨n, g|− |n, g⟩ ⟨n, e|).

π/2

τ = 4 n

Ry,π2F = | 1N Tr[R†

y,π2U(τ)]|2 0.96

nmax = 20

ωmq = ωq − nχ

n

ωgs ωe

s

|g⟩ |e⟩

ωes

H /! = (ωq−ωes) |e⟩ ⟨e|− χ |g⟩ ⟨g| a†a+ ϵ(t)a† + ϵ(t)∗a.

σω ≪ χ Dα

|e⟩

Deα = ⊗ eiξ |g⟩ ⟨g|+Dα ⊗ |e⟩ ⟨e|

ξ |g⟩

Deα

Deα|0⟩ ⊗ (|g⟩ + |e⟩) = eiξ |0, g⟩ + |α, e⟩

Deα

Cπ Deα = D−α/2CπDα/2

Dα

ωgs

H /! = (ωq−ωgs) |e⟩ ⟨e|− χ |e⟩ ⟨e| a†a+ ϵ(t)a† + ϵ(t)∗a.

ϵ ≫ χ

Dα=1 6 ns ϵ ≈ 170 MHz ≫ χ ≈ 3 MHz

H /! = (ωq − χqsa†sas − χqra

†qaq) |e⟩ ⟨e|

χqs χqr

τ ≫ 1χqs

, 1χqr

storage cavity - qubit readout cavity- qubit

incr

easin

g n

a) b)

χ

n−

ω0n

n= ωq − n

K

2

ω0nω0nn

f01 = ωq

2π f02/2 = (ωq−α)2π

K

7.367.327.287.247.20

Spectroscopy Frequency (GHz)

1.5

1.0

0.5

f02/2f01

K

storage cavity - readout cavity

incr

easin

g n

80

60

40

20

0

Readout V

oltage (

mV

)

8.2788.2768.2748.2728.270

Spectroscopy Frequency (GHz)

b)

storagecavity

qubit

a)

m=0

Toneτ=300μs

τ ≫ 1χ

π R0y,π π

Ks

|0⟩ → |1⟩

|0⟩ |n⟩ n = 2, 3

K

80

70

60

50

40

Readout S

ignal (m

V)

9.27529.27489.27449.2740

Spectroscopy Frequency (GHz)

π|0⟩ → |1⟩

|0⟩ |n⟩ n = 1, 2, 30.5K/2π = 163

χ

CΦ Φ = χqst

t |ψ(0)⟩ = |β, g⟩

|ψ(t)⟩ = Ry,π2CΦ=χqstRy,π2

|β, g⟩

= eπ4 (|e⟩⟨g|−|g⟩⟨e|)e−iχqsta†a|e⟩⟨e|e

π4 (|e⟩⟨g|−|g⟩⟨e|) |β, g⟩

=1

2(|β⟩ − |βe−iχqst⟩)⊗ |g⟩+ (|β⟩+ |βe−iχqst⟩)⊗ |e⟩

Ry,π2π/2

Pe

Pe =1

21 + Re(⟨β|βeiχqst⟩)

=1

21 + e|β|

2(cos(χqst)−1) cos(|β|2 sin(χqst)).

t

e−12 (|β|χqst)2

b)

c)

2.0

1.0

0.08006004002000

Dis

plac

emen

t (

)

Wait time ( )

a)

1000

8006004002000Wait time ( )

1000

1.00

0.75

0.50

1.00.80.60.4

0

0

cavity

qubit

β

β = 0 β = 0.5 β = 1.0β = 1.5

t = 2π/χqs

Dβ

|β| = 0 2.5 t = µ

χ′qs

2 a†2a2 |e⟩ ⟨e|

χqs

χqsa†a |e⟩ ⟨e|

χ′qsa

†2a2 |e⟩ ⟨e|

n nχ′qs

χqs

n = 25

t = 2πχqs−|β|2χ′

qs

χqs χ′qs

χ′qs/χqs = 3.6× 10−3

χqs ≈ 3 MHz

a) b)

440

435

430

425

4202520151050

5

4

3

2

1

0500400300

Wait time ( )

Disp

lacm

ent a

mpl

itude

( )

Wai

t tim

e (

)

Mean photon number ( )

1.00.80.60.4

a†2a

6

CΦ Rnn,θ Dα

p0(α)

⟨Pα⟩

P0(α) = πQ(α) ⟨Pα⟩ = π2W (α)

pn(α) = | ⟨n|Dα |0⟩ |2 = e−|α|2 |α|2n

n!.

P = eiπa†a

⟨P (α)⟩ = Tr[PDα |0⟩ ⟨0|D†α] = e−2|α|2 .

⟨P (α)⟩

P1 ≈ 0.02

δα/α ≈ 0.02

a)

b)

-4-2024

6000400020000

Rea

dout

sig

nal (

mV)

Drive amplitude (DAC value)

43210

1.00.80.60.40.20.0Ph

oton

pro

babi

lity

Displacement ( )

c)

cavity

qubit

m

or

R0n,π

Pn

Pn αχsr

P0

Pn n =Pn

4

2

0

-2

-46000400020000

Rea

dout

sig

nal (

mV)

Drive amplitude (DAC value)

1.00.80.60.40.20.0

43210Displacement ( )

b)

c)

a)

cavity

qubitor

Cπ

⟨Pα⟩⟨Pα⟩ α

⟨Pα⟩

⟨Pα⟩ ⟨Pα⟩

χsr

Rnπ,y

|e⟩ |g⟩

ρ |α⟩

Q(α) =1

π⟨α|ρ|α⟩

α

Q(α) =1

π⟨0|D†

αρDα|0⟩

Dα α

ρα = D†αρDα |0⟩

a)

cavity

qubit

2

0

-2

20-2

b)cavitytomography

Re(α)

Im(α)

Q(α)stateprep

Q(α) = 1π ⟨0|D

†αρDα|0⟩

|β⟩

|0⟩ ⟨0|

|0⟩ ⟨0| ⊗ σz

|g⟩

QZ(α) =1

πTr[ρqcσzDα |0⟩ ⟨0|D†

α

]

=1

π⟨0, g|D†

αρqcDα |0, g⟩ −1

π⟨0, e|D†

αρqcDα |0, e⟩

= pgQ|g⟩⟨g|(α)− peQ|e⟩⟨e|(α)

ρqc pg, pe

⟨g|ρqc|g⟩ , ⟨e|ρqc|e⟩

|α⟩

σz

Q|g⟩⟨g|(α)

Q|e⟩⟨e|(α)

a) b)

2

0

-2

20-2

2

0

-2

20-2

Re(α) Re(α)

Im(α)

QZ(α) QZ(α)

Q|g⟩⟨g|(α)Q|e⟩⟨e|(α)

|ψ⟩ = N(|g, β⟩+ |e, eiΦβ⟩

)

ρα = D†ρDα |0⟩

|n⟩

Qn(α) = ⟨n|D†αρDα|n⟩

Qn(α) N

W (α) = ⟨ρD†αPDα⟩

=∑

n

(−1)n ⟨n|D†αρDα|n⟩

=∑

n

(−1)nQn(α)

Im(α)

a)m = 0

Re(α)

Q0(α)

b)m = 1

Re(α)

Q1(α)

c)m = 2

Re(α)

Q2(α)

d)m = 3

Re(α)

Q3(α)

Q(α) = 1π ⟨0|D

†αρDα|0⟩

Qm(α) = 1π ⟨m|D†

αρDα|m⟩

|0⟩

Q0(α), Q1(α), Q2(α), Q3(α)

0, 1, 2, 3

W (α) =2

πTr[D†

αρDαP ]

D†αρDα α P

P = eiπa†a

U = Ry,π2CΦ=πRy,π2

= Ry,π2e−iπa†a|e⟩⟨e|Ry,π2

U

U

U =∑

n

Un |n⟩ ⟨n|

=∑

n

Ry,π2e−iπn|e⟩⟨e|Ry,π2

|n⟩ ⟨n|

=∑

n

Ry,π2

(1+(−1)n)

2 + σz(1−(−1)n)

2

Ry,π2

|n⟩ ⟨n|

=∑

n even

Ry,π2Ry,π2

|n⟩ ⟨n|+∑

n odd

Ry,π2σzRy,π2

|n⟩ ⟨n|

=∑

n even

Ry,π |n⟩ ⟨n|+∑

n odd

|n⟩ ⟨n|

a) b)

2

0

-2

20-2

Re(α)

Im(α)

W(α)cavity

qubit

cavitytomography

stateprep

(τ ≈ πχ)

Pα = DαPD†α

|β⟩ β =√3

Wi =2

π⟨σiPα⟩

Pα σi

I, X, Y, Z

a)

b)

Re(α)

Im(α)

WI(α)

WZ(α)

WX(α)

WY(α)

stateprep

qubittomography

cavitytomography

cavity

qubit

|ψ⟩ = N (|g⟩ − |e⟩) ⊗ |β⟩ β =√3

WZ(α) WY (α)WX(α)

X, Y, Z

Pα = DαPD†α

ρ =1∑

i,j=0

N∑

n,m=0

ρnmij |i⟩ ⟨j|⊗ |n⟩ ⟨m|

ρnmij |i, j⟩

|n,m⟩

⟨AB⟩ = Tr [ABρ]

A B

σi = I, σx, σy, σz

Pα = DαPD†α

Nmax = 12

αmax,min = ±3.4 ∆α = 0.085

Wi(α) =2π ⟨σiPα⟩

A A =∑

i Aiσi

Ai = Tr[Aσi]

B = 1π

∫B(α)Pαd2α

B(α) = Tr[BPα]

ρ = π∑

i

∫Wi(α)σiPαd

2α

ρ = ρq ⊗ ρc

ρ =1

2

∑

i

Tr[ρqσi]σi ⊗ 2π

∫2

πTr[ρcPα]Pαd

2α

ρ

⟨AB⟩ = Tr [ABρ]

= Tr

[∑

i,j

∫AiB(α)Wj(α

′)σiσjPαPα′d2αd2α′

]

Tr[σiσj] = δij Tr[PαPα′ ] = δ2(α − α′)

⟨AB⟩ =∑

i

∫AiB(α)Wi(α)d

2α

Dα

P

W (α) =2

πTr[D†

αρDαP ]

ρ

W (α) =2

πTr[DαPD†

αρ]

= Tr[M(α)ρ]

=∑

i,j

Mji(α)ρij

M(α) = DαPD†α Mji(α) ρij M(α)

ρ

ρij W (α)

ρ Tr[ρ] = 1

nmax

7

φKerr = KIτ I τ K

ωs

H = !ωsa†a− !K

2a†

2a2

K

κs

|β⟩

RHR† R = e−i(ωs−K2 )a†at

Hkerr/! = −K2 (a

†a)2

U(t) = e−iHkerrt

!

|ψ(t)⟩ = U(t) |β⟩

= eiKt2 (a†a)2 |β⟩

= e−|β|2∑

n

βn

√n!e

iKn2t2 |n⟩ .

|n⟩

|β(t)⟩ ≈ |βeiφKerr(t)⟩

φKerr = K|β|2t n2

t

π Tcol =π

2√nK

Trev = 2πK U(Trev) = eiπ(a

†a)2 = (−1)(a†a)2 = (−1)a

†a

|ψ(Trev)⟩ = |−β⟩

P

X

b)a) c)P

X

P

X

P

X

P

X

P

X

n=0,1,2,3φ=n2Kt

φKerr≈nKt

|β⟩ =∑

n cn |n⟩|n⟩

cn = |cn|eiφn φn = 0

cnφn = n2Kt

t = πK

t = Trevq

|ψ(Trevq )⟩ = 1

2q

2q−1∑

p=0

2q−1∑

k=0

eik(k−p)πq |βeipπq ⟩ .

Trevq q

q = 2

cavity

qubit

cavitytomography

stateprep

evolutiont

|β⟩t U(t) = e

iHkerrt!

t

|ψ⟩ = 1√2(|β⟩+ i |−β⟩) .

⟨β|P |β⟩ ≃ 0

∼ 1

κs/2π =

ωq/2π =

Kq/2π = (ω01q − ω12

q )/2π =

∼ 250 MHz

K

Im(α

)

20-2

2

0

-2

Experiment

Theory

15 ns 65 ns 440 ns 815 ns

1065 ns 1565 ns 2565 ns 3065 ns

Re(α) time

20-2

2

0

-2

Im(α

)

Re(α)

Experiment

Theory

time

|β⟩ |β|2 = 4

A

κs

T1 = 10 µ T ∗2 = 8 µ

ωs/2π = 9.27 GHz

H

! = ωq |e⟩ ⟨e|+ (ωs − χ) a†a |e⟩ ⟨e|− K

2a†

2a2.

χ/2π =

ωs K/2π =

K ≈ χ2/4Kq

K > 30κs

|β|2 = n = 4 Q0

HKerr |β⟩

15 ns

β = 2 βeiφKerr = 2.0ei0.13

n2

Tcol = 385 ns

Trev

|−β⟩ t = Trev/q

q > 1

q = 2

q = 3, 4

Trev = 3065

|β| = 1.78

|β| = 2

κ/2π = 10

ωs

µ

Im(α)

Re(α)20-2

1

0

-1

2

0

-2

2

0

-2

20-2 20-2

Qn(α) n = 0 → 8

t ≈ 2πqK q = 2, 3, 4

qA = e−n

q > 0

Qn(α)

Qn

χ

ReconstructionTheory

a)

b)

c)

20-2

2

0

-2

Im(α)

Re(α)

t = 2πqK q = 2 3 4

t = 2π/2K, 2π/3K, 2π/4K

F = ⟨Ψid| ρm |Ψid⟩ ρm

|Ψid⟩

|β| = 2e−κt/2 F2 = 0.71, F3 = 0.70, F4 = 0.71

K >> κ

8

|0⟩ , |1⟩

|β⟩ , |−β⟩

|ψ⟩ = 1N

cos( θ2) |β⟩+ sin( θ2)e

iφ |−β⟩

θ,φ

N =√

1 + sin(θ) cos(φ)e−2|β|2

N → 1

|β⟩ |−β⟩ +Zc

−Zc

Xc Yc

±Z

Z

β, −β

+Xc,+Yc,+Zc

|±Zc⟩ = |±β⟩ |±Xc⟩ = 1N

√2(|β⟩± |−β⟩) |±Yc⟩ = 1

N√2(|β⟩± j |−β⟩)

N (β)

|β⟩

⟨β|− β⟩ = e−2|β|2 .

S = −∑

j

ηj log2 ηj.

ηj ρ =∑

j ηj |j⟩ ⟨j|

ρ = 12(|β⟩ ⟨β| + |−β⟩ ⟨−β|) ρ

|E⟩ , |O⟩

ρ = 12(1 + e−2|β|2) |E⟩ ⟨E|+ 1

2(1− e−2|β|2) |O⟩ ⟨O|

S = −1 + e−2|β|2

2log2

[1 + e−2|β|2

2

]− 1− e−2|β|2

2log2

[1− e−2|β|2

2

].

Sβ→0 = 0 Sβ→∞ = 1

β = 0 β → ∞

β = 1 S = 0.99

P

X

a)

⎫ ⎬ ⎭d

b)

Displacement β

Entr

opy

(bits

)

|β⟩ , |−β⟩⟨β|−β⟩ == 0

d2 = (β − −β)2

βS

S ≈ 1− | ⟨β|−β⟩ |2 = 1− e−d2

|β⟩ , |−β⟩

d = 2β

d2 d2

| ⟨β|−β⟩ |2 = e−d2

ρ

ρ(t) = 12

[|β(t)⟩ ⟨β(t)|+ |−β(t)⟩ ⟨−β(t)|+ e−2|β(t)|2(1−e−κt) (|−β(t)⟩ ⟨β(t)|+ |β(t)⟩ ⟨−β(t)|)

]

β(t) = βe−12κt κ

e−12d

2κt

|ψ0⟩ = |β⟩ ⊗ (|g⟩ + |e⟩)

|β⟩

π

|ψ1⟩ = Cπ |ψ0⟩ = |β, g⟩ + |−β, e⟩

|ψ2⟩ = Dβ |ψ1⟩ = |2β, g⟩ + |0, e⟩ π

|0⟩

|ψ3⟩ ≈ R0y,π |ψ2⟩ = (|2β⟩ + |0⟩) ⊗ |g⟩

|ψ4⟩ = D−β |ψ3⟩ = (|β⟩ + |−β⟩) ⊗ |g⟩

|0⟩ ⊗cos( θ2) |g⟩+ sin( θ2)e

iφ |e⟩→cos( θ2) |β⟩+ sin( θ2)e

iφ |βeiΦ⟩⊗ |g⟩

θ φ

|⟨β|βiΦ⟩|2 ≪ 1

ωs2π = 8.18

κs2π = 7.2 = 1

2π×22.1µ

ωr2π = 9.36

κr2π = 330 = 1

2π×480

ωq

2π = 7.46 γ2π = 36 = 1

2π×4.4µ

χqs

2π = 2.4

Ks χ′qs

n ≪ n = min[χqs/χ′qs = 560,χqs/Ks = 650,χqs/κs = 330]

|−β⟩

CΦ

Q = ⟨α|ρ|α⟩

α = |β|

|βeiΦ⟩ α = β

Q(β) = |⟨β|βeiΦ⟩|2

= e−2|β|2(1−cosΦ)

Φ

Φ χqs

|−β⟩

1 n

Q(α) = 1π ⟨α|ρ|α⟩ ρ

|ψ⟩ = 1√2|0⟩ ⊗ (|g⟩ + |e⟩)

|ψ⟩ = N(|β⟩+ |−β⟩)⊗ |g⟩ |β| =√7 N ≈ 1√

2

|0, g⟩

Pe

Deβ ρ = |0⟩ ⟨0| ⊗ Pg |g⟩ ⟨g| + Pe |e⟩ ⟨e|

DeβρD

e†β = Pg |0, g⟩ ⟨0, g| + Pe |β, e⟩ ⟨β, e|

(5)

Re( )

Im()

-8

-4

0

4

8

-4 0 4 8-4 0 4 8-8 -4 0 4 8 -8 -8Re( ) Re( ) Re( )

Im()

(1) (2) (3) (4)

b)

(6) (7)

-8

-4

0

4

8

-8 -4 0 4 8

(8)

a)

cavity

qubit

m=0

mapping

tomog

raph

y

ϵ/2π = 990 2.5µ

|β, e⟩ |β| ≈ 17

|α| = 6

[ρ] = 1∫W (α) 2α = 1

Pe = 0.12

Pe ≤ 0.01

W (α) =[DαPD†

αρ]

ρ

F =∫W W α W

-4

-2

0

2

4-4 -2 0 2 4

0.60.40.20.0

0.6 0.0 -0.6

Re( )

Im(

)

a)

Z

XY

Z

XY

Z

XY

Z

XY

Z

XY

Z

XY

Z

XY

Z

XY

-1.0

-0.5

0.0

0.5

1.0-4 0 4

-1.0

-0.5

0.0

0.5

1.0

-1.0

-0.5

0.0

0.5

1.0-2 0 2

-1.0

-0.5

0.0

0.5

1.0

-1.0

-0.5

0.0

0.5

1.0

-1.0

-0.5

0.0

0.5

1.0

-1.0

-0.5

0.0

0.5

1.0

-1.0

-0.5

0.0

0.5

1.0

Re( ) Im( )

0.0

0.6

-0.6

b)

|ψ⟩ = N (|β⟩+ |−β⟩)

|β⟩ , |−β⟩

|ψ⟩ =cos( θ2) |g⟩+ sin( θ2) |e⟩

|ψ⟩ = 1√

2

|g⟩+ eiφ |e⟩

|β⟩ ± |−β⟩

|β⟩

Pn(|β⟩) = |⟨n|β⟩|2 = e−|β|2 |β|2nn!

Pn(|β⟩ ± |−β⟩) ∝ (1 ± eiπn) e−|β|2 |β|2n

n!

|β, g⟩ |β⟩+ |−β⟩⊗ |g⟩

|β⟩ − |−β⟩ ⊗ |g⟩ |β| = 2.3

d2 = |β1 − β2|2

|β1⟩ |β2⟩

d2

W (Re(α) = 0, Im(α))

W (0, Im(α)) Ae−2|Im(α)|2 cos(2d Im(α) + δ) A δ

111+0−2

d2

Ram

sey

angl

e (φ)

Rabi

ang

le (θ)

Z

XY

φ

Z

XY

θ

Im(α) Re(α)

Φ

C2π/3 Cπ/2

FA = 0.60 FB = 0.58 FC = 0.52

αcal = αact(1 + δα) αcal

αact δα = (√(2nth + 1)

nth

W (α) ∝ e− 2|α|2

2nth+1

nth ≤ 0.01

d2

d2(1 − 2nth) < d2act ≤ d2 dact

109 < d2 ≤ 111

|β⟩

δΦ 1√n

n = |β|2

1n

|g⟩ |g⟩ + |e⟩

|β⟩ |0⟩ + |√2β⟩

n = |β|2 Φ

δΦ = 1/ PeΦ

Pe

Φ δΦD =√

e/n

Φ

δΦC = 1/n δΦC

nκτ ≪ 1 κ τ

δΦC = enκτ/n

n = 15.5

22.5

σz

F

Frecov F ≈√Frecov

a) b)

0.0

0.5

-0.5

0.0

0.3

-0.3

-2 0 2

Im( )

0.0

0.7

-0.7

0.0

0.8

-0.8

1-1

0.2

0.1

0.0

7.4557.4457.435

0.4

0.2

0.0

Norm

alized s

pectr

osco

py s

igna

l

Spectroscopy frequency (GHz)

Photon number

012345678910

0.4

0.2

0.0

|β⟩ |β⟩ + |−β⟩ |β⟩ − |−β⟩|β| = 2.3

111+0−2 d2

Ae−2|Im(α)|2 cos(2d Im(α) + δ)S A δ

4

-4

-2

0

2Im()

Re( )4-4 -2 0 2 4-4 -2 0 2

4

-4

-2

0

2

4-4 -2 0 2

4

-4

-2

0

2

2-2 -1 0 1

2

-2

-1

0

1

b)a)

d)c)

0.0

-0.4

0.4

C2π/3 Cπ/2

|β⟩ + eiλ1 |βei2π/3⟩ +eiλ2 |βei4π/3⟩ |β| =

√7 λ1 = 0.6π λ2 = −0.3π |0⟩ + eiµ1 |−iβ⟩ +

eiµ2 |βeiπ/3⟩+ eiµ3 |βei2π/3⟩ |β| =√7 µ1 = 0.5π µ2 = −0.4π µ3 = −0.2π

|β⟩ + eiν1 |iβ⟩ + |−β⟩ + eiν2 |−iβ⟩ |β| =√7 ν1 = π ν2 = −0.2π

3456

0.1

2

3456

1

4 5 6 7 8 910

2 3 4

0.50

0.25

0.00

0.50

0.25

0.00

0.50

0.25

0.00

0

0

c)b)

Im(

)

Re( )0

0

Im(

)

Re( )

a)

cavity 1

qubitReadoutor or

Phase (radians)Phase (radians)

d)

f )

0.8

0.8

0.6

0.4

0.2

0.0

0.8

0.6

0.4

0.2

0.0

0.6

0.4

0.2

0.0

e)

(radi

ans)

Energy (photons)

δΦn

Φ

Pe

1/√n 1/n

δΦC ∝ enκτ/nκ

τ

nκτ > 1

d)

0

0.5

1

IX

Y ZI X Y Z

0

0.5

1

IX

Y ZI X Y Z

0

0.5

1

IX

Y ZI X Y Z

0

0.5

1

IX

Y ZI X Y Z

a)

cavity 1

qubitReadout

b)

cavity 1

qubit

QPT

0

0.5

1

c)

0

0.5

1

0

0.5

1

0

0.5

1

0

16 photons

28 photons

40 photons

100 photons

16 photons 28 photons

40 photons 100 photons

Rotation Angle ( )

Re(

)

Re(

)

Re(

)

Re(

)

< |0.06|

90%

9

|β⟩ |−β⟩

|β|

|ψ⟩ = 1√2(|g⟩ + |e⟩) ⊗ |β⟩ |g⟩ , |e⟩

|β⟩

t = πχ

|ψB⟩ = 1√2(|g, β⟩+ |e,−β⟩)

|ψ⟩ = 1√2(|gg⟩ + |ee⟩)

|ψB⟩ ⟨ψB| = IIc +XXc − Y Yc + ZZc

I, X, Y, Z Ic, Xc, Yc, Zc

statepreparation

qubittomography

cavitytomography

cavity

qubit

|ψ⟩ = 1√2(|g⟩ + |e⟩) ⊗ |β⟩ Dβ

β Ryπ2

π2 y

|ψB⟩ = 1√2(|g, β⟩ + |e,−β⟩)

Ri X Y ZPα

F C

|ψtarget⟩ =

1√2(|gg⟩+ |ee⟩)

F = ⟨ψtarget|ρ|ψtarget⟩ = 14 (⟨II⟩+ ⟨XX⟩ − ⟨Y Y ⟩+ ⟨ZZ⟩) .

II, XX, Y Y, ZZ

W = 14 (⟨II⟩ − ⟨XX⟩+ ⟨Y Y ⟩ − ⟨ZZ⟩)

W

F > 12

±1

−2 ≤ O = ⟨AAc⟩+ ⟨ABc⟩ − ⟨BAc⟩+ ⟨BBc⟩ ≤ 2

A,B Ac, Bc

τs = 55 µs

τr = 30 ns

T1, T2 ∼ 10 µs

5 8 GHz

H/! = ωsa†a+ (ωq − χa†a) |e⟩ ⟨e|

a |e⟩ ⟨e|

ωs, ωq χ

2π ∗ 1.4 MHz

X, Y, Z

|g⟩

Pα

Pα = DαPD†α Dα

P

W (α) = 2π ⟨Pα⟩

α

Wi(α) =2π ⟨σiPα⟩

σi I, X, Y, Z

Wi(α)

F = ⟨ψB| ρ |ψB⟩ = π2

∑i

∫WB

i (α)Wi(α)d2α WBi (α)

|ψB⟩ Wi(α)

F = (87 ± 2)%

β =√3

|⟨β|−β⟩|2 = 6 × 10−5 ≪ 1

V = 2π

∫⟨IPα⟩ d2α = (85 ± 1)%

V

F V

(a)

Mea

n Va

lue

(b)

1.0

-1.0

0.5

-0.5

0.0

Re( )

Im(

)

-2 0 2

-2

0

2

0 2 4 6 8 10 0 2 4 6 8 10

eg0

2

4

6

8

10

0

2

4

6

8

10

g

e

Fock state basis

Re( )

ge g

e

0.1

-0.1

0.0

Re( )0.5

0.0

(c)

Encoded basis

Wi(α) =2π ⟨σiPα⟩

σi = I ,X , Y , Z Pα

|ψB⟩ β =√3

⟨XPα⟩ ⟨Y Pα⟩ρ

|β⟩ ⟨β|+ |−β⟩ ⟨−β|

W (α)

| ⟨β|−β⟩ |2 ≪ 1

Xc = P0 Ic = Pβ + P−β

Yc = P jπ8β

Zc = Pβ − P−β

Ic, Xc, Yc, Zc

|ψB⟩

|β| =√3 FDFE = 1

4(⟨IIc⟩+ ⟨XXc⟩ − ⟨Y Yc⟩+ ⟨ZZc⟩) =

(72 ± 2)%

FDFE ≈ V × F

(a)

(b) Re( )

0.0

0.5

1.0

1.5

2.00.0

0.5

1.0

1.52.0

-3 -2 -1 0 1 2 3 -2 -1 0 1 2

-3 -2 -1 0 1 2 3

1.0

0.5

0.0

-0.5

-1.01.0

0.5

0.0

-0.5

-1.0

Re( )

Im( )

Im( )

Mean Value

1.0

-1.0

0.5

-0.5

0.0

-2 -1 0 1 2

Mea

n Va

lue

|ψB⟩ β = 0⟨IPα⟩ ⟨ZPα⟩ Im(α) = 0

⟨XPα⟩ ⟨Y Pα⟩ Re(α) = 0

β =√3

IIc, XXc, Y Yc, ZZc

X(θ), Z(θ), Xc, Zc θ

β

O1 = 2.30 ± 0.04 θ = −π4 β = 1

X, Y,Xc(α), Yc(α)

α

O2 = 2.14 ± 0.03 β = 1

|±⟩ ±M q1 |±⟩ ∓M q

1

0

0

1

2

3

2

3

Cat amplitude ( )Rotation ( )0.50.0 1.0 1.5 2.0

Cat amplitude ( )0.50.0 1.0 1.5 2.00.0 1.0-1.0

Displacement ( )

(a)

(b)

idealphoton loss

visibility

idealphoton loss

visibility

X(θ) Z(θ) Zc Xc

O = ⟨AAc⟩ + ⟨ABc⟩ − ⟨BAc⟩ + ⟨BBc⟩θ

X Y Xc(α) Yc(α)α

β

O = 2√2

|±⟩ ±M c2 |±⟩ ∓M c

2

|g⟩

AB A,B

AB = (A+ − A−)B A+ + A− = I

⟨A+B⟩ (1− 2pc)⟨A+B⟩ pc

⟨AB⟩ →(1− 2pc) ⟨A+B⟩ − ⟨A−B⟩=(1− pc) ⟨A+B − A−B⟩ − pc ⟨A+B + A−B⟩=(1− pc) ⟨AB⟩ − pc ⟨B⟩

B = Xc, Yc, Zc |ψc⟩ ⟨B⟩ = 0

⟨AB⟩ (1 − pc)

pc = 1 − e−τwaitT1 ≈

0.06 V

Vpred = (1− pc)V = 82%

V 85%

⟨σiPα⟩

V ∈ [0, 1]

Wmeasi (α) = VW ideal

i (α) V

∫W ideal

I (α)d2α

V =

∫Wmeas

I (α)d2α

I V = 85%

I,X, Y, Z

Ic, Xc, Yc, Zc

A,B

Ac, Bc

O = ⟨AAc⟩+ ⟨ABc⟩ − ⟨BAc⟩+ ⟨BBc⟩

|ψB⟩

|ψB⟩

Zc, Xc

Z(θ), X(θ)

Z(θ) = Z cos θ2 −X sin θ

2 X(θ) = X cos θ2 + Z sin θ

2

θ

O

θ = −π4

A = X+Z√2; B = X−Z√

2

Ac = Zc; Bc = Xc

⟨AZc⟩ ⟨BZc⟩

Oideal =√2(2− e−8|β|2)

V

Ovis =√2V(2− e−8|β|2)

⟨AXc⟩ ⟨BXc⟩

Oloss =√2(1− e−8|β|2 − e−2|β|2γ)

γ = teffτs

τs teff

Opred =√2V(1− e−8|β|2 − e−2γ|β|2)

V = 0.85 teff = 1.24 µs

X, Y

Xc(α), Yc(α)

Xc(α) = DjαP0D†jα ≈ Xc cos

α4β + Yc sin

α4β

Yc(α) = DjαP jπ8βD†

jα ≈ Yc cosα4β −Xc sin

α4β

α

O

α = 0.15 β = 1

A = X; B = YAc =

Xc+Yc√2

Bc =Xc−Yc√

2

Oideal = 2(cos 4α0β + sin 4α0β)e−2|α0|2

α0

Opred = 2Ve−2γ|β|2(cos 4α0β + sin 4α0β)e−2|α0|2

V = 0.85 teff = 1.24 µs

|β| ≫ 1

P±jα0 ∼ 1√2(Xc ± Yc)

β − α0

β + α0= tan 4α0β

α0 Djα0

Pjα0 β

β

1√2(Xc + Yc) Pα= jπ

16β

W =

IIc−XXc+Y Yc−ZZc |ψ⟩ = 1√2(|gg⟩+ |ee⟩)

β

⟨W⟩ < 0

W F

β

β = 0

1√2(|g⟩+ |e⟩)⊗ |0⟩

W = II −ZZ −XX + Y Y

F = II + XX − Y Y + ZZF > 0.5

a)

b)

Mm

|ψm⟩ =Mm |ψ⟩√

⟨ψ|M †mMm|ψ⟩

X, Y, Z

X : 12

(1 11 1

)⊗ c,

12

(1 −1−1 1

)⊗ c

Y : 12

(1 −jj 1

)⊗ c,

12

(1 j−j 1

)⊗ c

Z :

(1 00 0

)⊗ c,

(0 00 1

)⊗ c

|ψm⟩ = |ψq⟩ ⊗ |ψc⟩

|ψcav⟩ → X : N (|β⟩+ |β⟩) N (|β⟩ − |β⟩)Y : N (|β⟩ − j |β⟩) N (|β⟩+ j |β⟩)Z : |β⟩ |−β⟩

|ψB⟩

|ψB⟩ = 1√2(|g, β⟩ + |e,−β⟩)

X Y

|e⟩ mth

|m⟩ |β⟩ m = 3 β =√3

|ψ⟩ = Cm |e,m⟩+∑

n =m

Cn |g, n⟩

Cm = ⟨m|β⟩ Z

+1

|ψcav⟩ = N (|β⟩ − Cm |m⟩) −1

|ψcav⟩ = |m⟩

|β⟩ β =√3

mth m = 3|ψ⟩ = Cm |e,m⟩ +

∑n =m Cn |g, n⟩ Cn

nth Cn = ⟨n|β⟩−Z

+Z

|ψc⟩ = N∑

n=3 Cn |n⟩

10

a

a |α⟩ = α |α⟩

aP = −Pa

|0L⟩ = N (|β⟩+ |−β⟩) |1L⟩ = N (|jβ⟩+ |−jβ⟩)

N 1√2

β |0L⟩ ⟨0L| , |1L⟩ ⟨1L|

P

Sn(θ) = eiθ|n⟩⟨n|

S(θ)

S(θ) =∞∏

n=0

Sn(θn)

θ = θn∞n=0

|ψ⟩ = N (|β, β⟩+ |−β,−β⟩) .

a) P

X

|0L〉

|1L〉

c)

20-2

2

0

-2Im

(α)

Re(α)

-40

0

40

2001000Time (μs)

Read

out (m

V)

-1

0

1

Parity

b)

|0L⟩ |1L⟩

N (|0L⟩ + |1L⟩)P

xy

xy

xy

-40-30-20-100Qubit drive detuning (MHz)

0

4

8

12

Sign

al (mV)

d)b) c)

a)

initial finalmanipulation

|ψc⟩ =∑

n cn |n⟩ cn|n⟩

|g⟩

cn

A

σx2 = σy

2 = σz2 = 1

σxσy = iσz σyσz = iσx σzσx = iσyeiθσn = 1 cos θ + iσn sin θ

eiπ2 σnei

π2 σm = −σnσm

H /! = 12(ξ + ξ∗)σx +

12i(ξ − ξ∗)σy +

12∆σz

ξ

∆

ξ σx σy δt

U = ei!! δt0 H(t) dt

ξ(t) ξ(t) = 0 t < 0 t > δt

Ax = eiA2 σx By = ei

B2 σy

2ξ

Ω(t)σx

Ω(t)2 σy

π/2

Uxπ/2 =

(Xπ/2Xπ/2

)NXπ/2 =

(ei

π4 σxei

π4 σx

)Nei

π4 σx

N

π2 → π

2 (1 + ϵ)

U′xπ/2 =

[ei

π4 (1+ϵ)σxei

π4 (1+ϵ)σx

]Nei

π4 (1+ϵ)σx

= ei

"Nπ2 (1+ϵ)+

π4 (1+ϵ)

#σx

eiθ/2σx ⟨Z⟩ = cos θ

Uxπ/2 |0⟩ → ⟨Z⟩ = cos

[Nπ(1 + ϵ) + π

2 (1 + ϵ)]

= (−1)N+1 sin[π2 ϵ+Nπϵ

]

|ϵ| ≪ 1 ⟨Z⟩

ϵ

⟨Z⟩ ≈ (−1)N+1 [Nπϵ+ π2 ϵ]

π

π/m

Uxπ/m =

(Xπ/m

)mNXπ/2 =

(ei

π2mσx

)mN

eiπ4 σx

σx σy

σy σ′y = cosφσy − sinφσx

X

Y

U = Yπ/2 (XπY−πXπYπ)N Xπ/2 = ei

π4 σy

(ei

π2 σxe−i

π2 σyei

π2 σxei

π2 σy

)Nei

π4 σx

Y σ′y

X Y π

eiπ2 σxe−i

π2 σ

′yei

π2 σxei

π2 σ

′y = −σxσ′

yσxσ′y

= −σx [cosφσy − sinφσx] σx [cosφσy − sinφσx]

= − [cosφσxσy + sinφ] [cosφσxσy + sinφ]

= −1− i sin(2φ)σz= 1 cos(π + sin(2φ)) + iσz sin(π + sin(2φ))

= eiσz(π+sin(2φ))

Z π/2

X/Y

⟨Z⟩

⟨Z⟩ = (−1)N+1 sin(N sin(2φ))

φ/(2π) ≪ 1

⟨Z⟩ ≈ (−1)N+12Nφ

U = (XπY−πXπYπ)N Xπ/2 =

(ei

π2 σxe−i

π2 σyei

π2 σxei

π2 σy

)Nei

π4 σx

σx σy σx + δσz σy + δσz

δ

eiπ2 σ

′xe−i

π2 σ

′yei

π2 σ

′xei

π2 σ

′y = −σ′

xσ′yσ

′xσ

′y

= − [σx + δσz] [σy − δσz] [σx + δσz] [σy + δσz]

= −[σxσy + δ(σzσy − σxσz) + δ2

] [σxσy + δ(σzσy + σxσz) + δ2

]

≈ 1− 2δiσx= 1 cos(−2δ) + iσx sin(−2δ)

= e−2δiσx

X N

⟨Z⟩ ≈ −4Nδ

Vout = (1 + ϵ) [cos(ωIFt− φ) + γ] cos(ωLOt) + (1− ϵ) [sin(ωIFt+ φ) + γ] sin(ωLOt)

ϵ φ γ

Vout = cos(ωIFt) cos(ωLOt) + sin(ωIFt) sin(ωLOt)

= cos([ωLO − ωIF]t)

Vout = (1 + ϵ) cos(ωIFt) cos(ωLOt) + (1− ϵ) sin(ωIFt) sin(ωLOt)

= cos([ωLO − ωIF]t) + ϵ cos([ωLO + ωIF]t)

ϵ

ϵ = 10PdBc/20 PdBc

Vout = cos(ωIFt+ φ) cos(ωLOt) + sin(ωIFt+ φ) sin(ωLOt)

= cos(ωLOt) [cos(ωIFt) cos(φ)− sin(ωIFt) sin(φ)]

+ sin(ωLOt) [cos(ωIFt) cos(φ)− sin(ωIFt) sin(φ)]

= cos(φ) cos([ωLO − ωIF]t)− sin(φ) sin([ωLO + ωIF]t)

tan(φ)

tan(φ) = 10PdBc/20 PdBc

Vout = [cos(ωIF) + γ] cos(ωLOt) + [sin(ωIFt) + γ] sin(ωLOt)

= cos([ωLO − ωIF]t) + γ [cos(ωLOt) + sin(ωLOt)]

= cos([ωLO − ωIF]t) + γ sin(ωLOt+ π/4)

F = [χ χ ]

α/2π = 250MHz

−αq

∆ 8.453

ξ2π

φ 2π ∗ 5.25× 10−3

ϵ 0.02

σ

τ

X Y π/2 /π

a) b)

(1 − 3.5e−3) (1 − 1.0e−3)

a)

b)

N = 15

⟨Z⟩ ≈ (−1)N+1 [Nπϵ+ π2 ϵ]

X

Uxπ/2 =

(Xπ/2Xπ/2

)NXπ/2

Uxπ = (Xπ)

N Xπ/2

Y

Uyπ/2 =

(Yπ/2Yπ/2

)NYπ/2

Uyπ = (Yπ)

N Yπ/2

⟨Z⟩ ≈

(−1)N+12Nφ

U = Yπ/2 (XπY−πXπYπ)N Xπ/2

⟨Z⟩ ≈ 4Nδ

a)

b)

ππ/2

a)

b)

a)

b)

U = (XπY−πXπYπ)N Xπ/2

104

2.63e−05 ±3.4e−05

−7.9e−06 ±3.7e−05

−2.0e−06 ±6.3e−05

−2.6e−05 ±3.4e−05

4.9e−05 ±9.7e−05

1 − 1e−04

2.1e−06%

250

> 60

σz

2.8e−5 8.9e−5

X π/2 π

X Y

(1 − 8.2e−4)

(1− 1.4e−3)

1− 8.2e−04 1− 1.4e−3

(0IF)

1.1e−05 ±2.7e−05

1.4e−06 ±8.5e−05

−8.1e−05 ±1.3e−04

B

ρ

Q(α) = F Ca(λ)

Ca(λ) = Tr[ρe−λ∗aeλa

†]

F = 1π2

∫d2λeαλ

∗−α∗λ

Q(α) =1

π2

∫d2λeαλ

∗−α∗λTr[ρe−λ∗aeλa

†]

1π

∫d2 |β⟩ ⟨β| =

Q(α) =1

π3Tr

[ρ

∫d2λd2βeλ

∗(α−β)−λ(α∗−β∗) |β⟩ ⟨β|]

∫λ2eλ

∗µ−λµ∗= π2δ(µ)

Q(α) =1

πTr

[ρ

∫d2βδ(α− β) |β⟩ ⟨β|

]

=1

πTr [ρ |α⟩ ⟨α|]

=1

π⟨α|ρ|α⟩ .

|α⟩

ρ

W (α) = F Cs(λ)

Cs(λ) = Tr [ρD(λ)] F = 1π2

∫d2λeαλ

∗−α∗λ

W (α) =1

π2

∫d2λeαλ

∗−α∗λTr [ρD(λ)] .

α, λ α′+ iα′′, λ′+ iλ′′

eαλ∗−α∗λ = e2i(α

′′λ′−α′λ′′)

D(λ) = eλa†−λ∗a

= e2iλ′′

$a†+a2

%−2iλ′

$a†−a2i

%

= e−iλ′λ′′TP=λ′′TX=λ′

|x⟩

Cs(λ) = Tr [ρD(λ)]

=

∫dx ⟨x|ρD(λ)|x⟩ .

W (α) =1

π2

∫d2λdxe2i(α

′′λ′−α′λ′′) ⟨x|ρD(λ)|x⟩ .

D(λ) |x⟩ = e−iλ′λ′′TP=λ′′TX=λ′ |x⟩= e−iλ′λ′′TP=λ′′ |x+ λ′⟩= e−iλ′λ′′

e2iλ′′(x+λ′) |x+ λ′⟩ .

W (α) =1

π2

∫d2λdxe2i(α

′′λ′−α′λ′′)e−iλ′λ′′e2iλ

′′(x+λ′) ⟨x|ρ|x+ λ′⟩

=1

π2

∫d2λdxeiλ

′′(λ′+2x−2α′)e2iα′′λ′ ⟨x|ρ|x+ λ′⟩ .

∫dµeiµν = 2πδ(ν)

W (α) =2

π

∫dλ′dxδ(λ′ + 2x− 2α′)e2iα

′′λ′ ⟨x|ρ|x+ λ′⟩

=2

π

∫dxe2iα

′′(2α′−2x) ⟨x|ρ|x+ 2α′ − 2x⟩

=2

π

∫dxe4iα

′′(α′−x) ⟨x|ρ|2α′ − 2x⟩

u = 2(x− α′)

D(α) |−u2 ⟩ = eiα

′α′′e−iα′′u |α′ − u

2 ⟩|u2 ⟩D

†(α) = ⟨α′ + u2 | e

−iα′α′′e−iα′′u

W (α) =1

π

∫due−2iα′′ueiα

′α′′e−iα′α′′

eiα′′u ⟨u2 |D

†(α)ρD(α)|− u2 ⟩

=1

π

∫du ⟨u2 |D

†(α)ρD(α)|− u2 ⟩ .

P

P |−x⟩ = |x⟩

W (α) =1

π

∫du ⟨u2 |D

†(α)ρD(α)P |u2 ⟩

=2

π

∫dv ⟨v|D†(α)ρD(α)P |v⟩

=2

πTr[D†(α)ρD(α)P

]

=2

πTr[D(α)PD†(α)ρ

]

Pα = D(α)PD†(α)

W (α) = Tr[DαPD†αρ] Qn(α) = Tr[Dα |n⟩ ⟨n|D†

αρ]

W (α) =∑

i,j

W(α)i,jρi,j Qn(α) =∑

i,j

Q(α)i,jρi,j

W(α) = DαPD†α, Q(α) = Dα |n⟩ ⟨n|D†

α

W(α) Wi,j(α) = ⟨j|DαPD†α|i⟩

Dαa = (a− α)Dα Pa = −aP

D†αa = (a+ α)D†

α Pa† = −a†P.

aDαPD†α = 2αDαPD†

α −DαPD†αa

DαPD†αa

† = 2α∗DαPD†α − a†DαPD†

α

W(α)

W0,0(α) = ⟨0|DαPD†α|0⟩ = ⟨0|2α⟩ = e−2|α|2

Wk,0(α) = ⟨0|DαPD†α|k⟩

=1√k⟨0|DαPD†

αa†|k − 1⟩

=2α∗√kWk−1,0(α).

W(α) WT (α) = W∗(α)

W0,k(α) =2α∗√kW0,k−1(α) = W∗

k,0(α).

Wk,l(α) = ⟨l|DαPD†α|k⟩

=1√k⟨l|DαPD†

αa†|k − 1⟩

=1√k

(2α∗Wk−1,l(α)−

√lWk−1,l−1(α)

).

Wl,k(α) = ⟨k|DαPD†α|l⟩ = W∗

k,l(α).

nmax(nmax − 1)

α nmax

ρ

α

import numpy as np

def designW(basis = 10, alpha = np.zeros([10,10]) ):”””Returns the design matrix to build a Wigner function from a givendensity matrix.

Parameters----------basis : integer

The truncation number of the density matrix which will be used to determine theWigner function.

alpha : complex matrixAn array of complex values which represent the displacement amplitude fora set of measurements

Returns-------

Wmat : complex 4-dim arrayValues representing the design matrix to create a Wigner functiongiven an arbitrary cavity state density matrix.

”””

rho_shape = [basis, basis]Wmat = np.zeros(np.append(rho_shape, alpha.shape), dtype = complex)

#initial ’seed’ calculation for |0><0|Wmat[0][0] = np.exp(-2.0 * np.abs(alpha) ** 2)

for n in range(1,basis):# calculate |0><n| and |n><0|Wmat[0][n] = (2.0 * alpha * Wmat[0][n-1]) / np.sqrt(n)Wmat[n][0] = np.conj(Wmat[0][n])

for m in range(1,basis):for n in range(m , basis):

# calculate |m><n| and |n><m|Wmat[m][n] = (2.0 * alpha * Wmat[m][n - 1]

- np.sqrt(m) * Wmat[m - 1][n - 1]) / np.sqrt(n)Wmat[n][m] = np.conj(Wmat[m][n])

return Wmat

Qn(α)

Qn(α) = Tr [Qn(α)ρ] Qni,j(α) = ⟨j|Dα |n⟩ ⟨n|D†

α|i⟩

aDα |0⟩ ⟨0|D†α = αDα |0⟩ ⟨0|D†

α

Dα |n⟩ ⟨n|D†α =

1

nDa† |n− 1⟩ ⟨n− 1| aD†

α

=1

n(a† − α∗)D |n− 1⟩ ⟨n− 1|D†

α(a− α)

=1

n(a†D |n− 1⟩ ⟨n− 1|D†

αa− α∗D |n− 1⟩ ⟨n− 1|D†αa

− αa†D |n− 1⟩ ⟨n− 1|D†α + |α|2D |n− 1⟩ ⟨n− 1|D†

α).

Qni,j(α)

Q00,0(α) = ⟨0|Dα |0⟩ ⟨0|D†

α|0⟩ = e−|α|2

Q0k,l(α) = ⟨l|Dα |0⟩ ⟨0|D†

α|k⟩

=1√l⟨l − 1|aDα |0⟩ ⟨0|D†

α|k⟩

=α√l⟨l − 1|Dα |0⟩ ⟨0|D†

α|k⟩

=α√lQ0

k,l−1(α)

QT (α) = Q∗(α)

Qnl,k(α) = Qn

k,l∗(α).

Qnk,l =

1

n(√lkQn−1

k−1,l−1(α)− α∗√kQn−1

k−1,l(α)− α√lQn−1

k,l−1(α) + |α|2Qn−1k,l )

nth Qn(α)

(0, 1, ..., n − 1)

import numpy as np

def designQ(basis = 10, alpha = np.zeros([10,10]), photon_proj = 0):”””Returns the design matrix to build a generalized Q function from a givendensity matrix.

Parameters----------basis : integer

The truncation number of the density matrix which will be used to determine thegeneralized Q function.

alpha : complex matrixAn array of complex values which represent the displacement amplitude fora set of measurements

Returns-------

Qmat : complex 5-dim arrayValues representing the design matrix to create a generalized Q-functiongiven an arbitrary cavity state density matrix.

”””

rho_shape = [basis, basis]photon_array = np.arange(photon_proj + 1)Q_size = np.append(rho_shape, photon_array.shape)Q_size = np.append(Q_size, alpha.shape)

Qmat = np.zeros(Q_size,dtype = complex)

#initial ’seed’ calculation for |0><0|, 0 photonQmat[0][0][0] = np.exp( -np.abs(alpha) ** 2)

for k in np.arange(1,basis):# calculate |k><0| for 0 photonQmat[0][k][0] = (alpha * Qmat[0][k-1][0]) / np.sqrt(k)Qmat[k][0][0] = np.conj(Qmat[0][k][0])

for k in np.arange(1,basis):for l in np.arange(k, basis):

# calculate |k><l| for n photonQmat[k][l][0] = (alpha * Qmat[k][l-1][0]) / np.sqrt(l)Qmat[l][k][0] = np.conj(Qmat[k][l][0])

for n in np.arange(1, photon_proj+1):# calculate |0><0| for n photonQmat[0][0][n] = np.abs(alpha)**2 * Qmat[0][0][n-1] / n

for k in np.arange(1, basis):# calculate |k><0| for n photonQmat[0][k][n] = ( (1./n) * (np.abs(alpha)**2 * Qmat[0][k][n-1] -

alpha * Qmat[0][k-1][n-1] * np.sqrt(k) ) )Qmat[k][0][n] = np.conj(Qmat[0][k][n])

for k in np.arange(1, basis):for l in np.arange(k, basis):

# calculate |k><l| for n photonQmat[l][k][n] = ( (1./(n)) * ( 1.*np.sqrt(l*k) * Qmat[l-1][k-1][n-1]

- (alpha) * Qmat[l][k-1][n-1] * np.sqrt(k)- np.conj(alpha) * Qmat[l-1][k][n-1] * np.sqrt(l)

+ np.abs(alpha)**2 * Qmat[l][k][n-1] ) )Qmat[k][l][n] = np.conj(Qmat[l][k][n])

return Qmat

|β⟩

|ψ(t)⟩ = U(t) |β⟩ = e−iKt2 (a†a)2 |β⟩

=∑

n

e−iKtn2

2 e−|β|2

2β2

√n!

|n⟩

tq =2πqK q

|ψ(tq)⟩ =∑

n

Fne−|β|2

2β2

√n!

|n⟩

Fn = e−iπn2

q Fq 2q Fn+2q =

e−iπq (n+2q)2 = e−

iπn2

q e−4πnie−4πqi = e−iπn2

q = Fn Fn

Fn =2q−1∑

p

fpeiπpn

q

fp =1

2q

2q−1∑

k

Fke−iπkp

q =1

2q

2q−1∑

k

eiπk2

q e−iπkp

q =1

2q

2q−1∑

k

eiπq k(k−p)

|ψ(tq)⟩ =2q−1∑

p

fp

(∑

n

e−|β|2

2βne

iπknq

√n!

|n⟩)

=2q−1∑

p

fp |βeipπq ⟩

=1

2q

2q−1∑

p=0

2q−1∑

k=0

eiπq k(k−p) |βe

ipπq ⟩

q = 2

|ψ(t2)⟩ = 1√2

(e

iπ4 |β⟩+ e

−iπ4 |−β⟩

)

C

d

X =

⎛

⎜⎜⎜⎜⎝

0 1 0 · · · 00 0 1 · · · 00 0 · · · 1 0· · · · · · · · · · · · · · ·1 0 0 · · · 0

⎞

⎟⎟⎟⎟⎠Z =

⎛

⎜⎜⎜⎜⎝

1 0 0 · · · 00 ω 0 · · · 00 0 ω2 · · · 0· · · · · · · · · · · · · · ·0 0 0 · · · ω(d−1)

⎞

⎟⎟⎟⎟⎠

ω = e2πid X, Z d

d = 2 d

X Z

ZX = ωXZ Zd = Xd = I.

|j⟩

X |j⟩ = |(j + 1) mod d⟩ Z |j⟩ = ωj |j⟩

Y

Y = ωXZ

d

d = 2

G2 ≡ ±I,±X,±Y,±Z,

g1, ..., gk G

G G

g1, ..., gk G = ⟨g1, ..., gk⟩

G2 = ⟨X,Z,−I⟩ .

d

Gd = ⟨X,Z,ωI⟩

d

X2, Z3,

Gd S VS

S = ⟨g1, ..., gl⟩ VS

S S VS

VS S

VS P

|ψ⟩

S

P = N∏

l

(I + gl).

N 12

S P

VS

Gd

S C(S) Ej

Ejgl = −glEj C(S)

d

gl ∈ S glgk =

gkgl

P VS

gl

Ei

S

C(S)

Ei VS

d = 4, S = ⟨Z2⟩

C(S)

d = 4

G4 = ⟨X,Z,ωI⟩

X =

⎛

⎜⎜⎝

0 1 0 00 0 1 00 0 0 11 0 0 0

⎞

⎟⎟⎠ Z =

⎛

⎜⎜⎝

1 0 0 00 ω 0 00 0 ω2 00 0 0 ω3

⎞

⎟⎟⎠

ω = eiπ2 S = ⟨Z2⟩

Z2

Z2 =

⎛

⎜⎜⎝

1 0 0 00 −1 0 00 0 1 00 0 0 −1

⎞

⎟⎟⎠

S P

VS

P =1

2(I + Z2) =

⎛

⎜⎜⎝

1 0 0 00 0 0 00 0 1 00 0 0 0

⎞

⎟⎟⎠ .

P = |0⟩ ⟨0| + |2⟩ ⟨2|

|0L⟩ = |0⟩ |1L⟩ = |2⟩ .

VS Z2

G4

XZ2 = ω−2Z2X = −Z2X

C(S) |0⟩ , |2⟩

Z2 X

d = 4, S = ⟨X2⟩

S = ⟨X2⟩

X2 =

⎛

⎜⎜⎝

0 0 1 00 0 0 11 0 0 00 1 0 0

⎞

⎟⎟⎠

X2

P =1

2(I +X2) =

1

2

⎛

⎜⎜⎝

1 0 1 00 1 0 11 0 1 00 1 0 1

⎞

⎟⎟⎠ .

P = 1√2(|0⟩+ |2⟩)⊗ c.c.+ 1√

2(|1⟩+ |3⟩)⊗ c.c.

|0L⟩ = 1√2(|0⟩+ |2⟩) |1L⟩ = 1√

2(|1⟩+ |3⟩).

ZX2 = ω2X2Z = −X2Z

C(S) 1√2(|0⟩+ |2⟩) 1√

2(|1⟩+

|3⟩) X2

Z

d = 4, S = ⟨X2, Z2⟩

Z2 X2

S

Z2X2 = ω4X2Z2 = X2Z2

VS

P =1

2(I + Z2)(I +X2) =

1

2

⎛

⎜⎜⎝

1 0 1 00 0 0 01 0 1 00 0 0 0

⎞

⎟⎟⎠ .

P = 1√2(|0⟩ + |2⟩) ⊗ c.c.

|ψ⟩ = 1√2(|0⟩ + |2⟩)

S = ⟨X2, Z2⟩

d = 8, S = ⟨X4, Z4⟩

Z4 X4

(X4)† =

X4, (Z4)† = Z4

Z4X4 = ω16X4Z4 = X4Z4

ω = eiπ4

VS

P =1

2(I + Z4)(I +X4)

P = 1√2(|0⟩ + |4⟩) ⊗ c.c. + 1√

2(|2⟩ + |6⟩) ⊗ c.c.

|0L⟩ = 1√2(|0⟩+ |4⟩) |1L⟩ = 1√

2(|2⟩+ |6⟩).

C(S)

1√2(|0⟩ + |4⟩), 1√

2(|2⟩ + |6⟩) S = ⟨X4, Z4⟩

X, Z

d = 18 9

|j⟩ = |βωj⟩

ω = e2πid d

|βωj⟩ β

⟨j|k⟩ ≈ δj,k

X |j⟩ → |(j + 1) mod d⟩

X = e2πid a†a

a†, a X

|j⟩

|j + 1⟩ Z |j⟩ →

ωj |j⟩

d = 4, S = ⟨Z2⟩

d = 4

Z2

Z2 = (|β⟩ ⟨β|+ |−β⟩ ⟨−β|)− (|iβ⟩ ⟨iβ|+ |−iβ⟩ ⟨−iβ|)

|0L⟩ = |β⟩ |1L⟩ = |−β⟩ .

X

X = eπi2 a†a

d = 4, S = ⟨X2⟩

d = 4 S = ⟨X2⟩

X2 = (|β⟩ ⟨−β|+ |−β⟩ ⟨β|) + (|iβ⟩ ⟨−iβ|+ |−iβ⟩ ⟨iβ|)

=(

1√2(|β⟩+ |β⟩)⊗ c.c.+ 1√

2(|iβ⟩+ |−iβ⟩)⊗ c.c

)

−(

1√2(|β⟩ − |β⟩)⊗ c.c.+ 1√

2(|iβ⟩ − |−iβ⟩)⊗ c.c

)

X2

P = eiπa†a

|0L⟩ = 1√2(|β⟩+ |−β⟩) |1L⟩ = 1√

2(|iβ⟩+ |−iβ⟩).

Z

Z

a

aX2 = aeiπa†a = aP = −Pa = −X2a.

d = 4

S = ⟨X2⟩

C(S)

d = 8, S = ⟨X4, Z4⟩

d = 8

X Z

X4 = eiπa†a = P Z4

Z4

|0L⟩ = 1√2(|β⟩+ |−β⟩) |1L⟩ = 1√

2(|iβ⟩+ |−iβ⟩)

X Z

X4

a

d = 18