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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
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