- PDF ΔΩΡΕΑΝ Λήψη

H = p m + mω x x, p m, ω H = ω ( P + X ) P = p/ mω X = x/ mω p X = x x ZPF, P = p ZPF x ZPF =,p mω mω ZPF = P = 1 i (a a ) X = 1 (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. a) b) p x M C k q L p M k x φ q L C X P i XP T X = e P X i PX T P = e [X, P] = constant e B e A e B = e A+B [A, B] =

D X, P = T 1 X T PT1 X = ( XP PX) ei D X, P = e ( X i P)a ( X +i P)a X P a) b) c) d) α = X + i P D α = e α a αa α = D α 0 α = α e iφ n

α = e α α n n n! n α(t) = e iht α 0 = e iωa a e α 0 α0 n n n! = e α 0 n = α 0 e iωt n (α 0 e iωt ) n n! n α ω n α I Q ( ) ( ) a+a ω I =cosωt a a sin ωt i ( ) ( ) Q = sin ωt cos ωt a a i a+a

a) b) Q ωt ωt I φ = ωt I Q P = e iπa a =( 1) a a. P P ±1 Pa = ap Pa = a P PD = D P ψ cat = N ( α + e iφ α ) 1 N = (1+e α )cosφ

α α = e α N 1 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 L ˆφ + 1 C ˆq φ q

ρ = m,n c mn m n m, n c mn = n n n. 0 n N N a m a n m, n N = 15

= 1 π d α α α 1 π d α α α = 1 π n,m 1 n m n!m! d αe α α n (α ) m d αe α α n (α ) m = πγ( n+m +1)δ nm

a) b) W(α) c) Q(α) 1 π d α α α = n,m Γ( n+m +1) n!m! δ nm n m = n n n = α α = 0 C(λ) C a C s C a (λ) = e λ a e λa C s (λ) = e λa λ a

Q(α) =F{C a (λ)} W (α) =F{C s (λ)} F{C(λ)} = 1 π d λc(λ)e αλ α λ p(α) = α ρ α ρ α Q(α) = 1 π α ρ α ρ ψ = N ( α + α ) α α Q(α) = 1 α ρ α = 1 π π 0 D αρd α 0

a) b) W(α) Q(α) ψ = N ( β β ) β =

a) b) W(α) W(α) β β =4 n n =4 Re(α) = P D α W (α) = π Tr[D αρd α P ]= π D αpd α = π P α P α D α PD α P π α ±1

ρ =π d αw (α)p α Tr[ρO(a, a )] = d αw (α)o(α). O(α) =Tr[D αo(a, a )D α P ] β β W (α) = π e α β F = ψ ρ ψ = 1 π W (α)w (α)d α W t (α) = ψ t P α ψ t F =Tr[ρ t ρ] ρ ρ

I (α ) Q (α ) I Q α = α + iα I (α )= dα W (α) Q (α )= dα W (α). P ρ = ρp C s (λ) =Tr[D λ ρ]=tr[d λ /ρd λ /P ]= π W (λ/). ρ W (α) = 1 π F{W ( λ )} Q(α) = d αe α W (α) Q(α) =e α W (α).

C s (λ) =Tr[ρD λ ] β + β n (n =0 ) 0 1

a σ = ( ) 0 1 0 0 a σ + = ( ) 0 0. 1 0 X P N σ x = ( ) 0 1 1 0 σ y = ( ) 0 i i 0 σ z = ( ) 1 0 0 1 σ + σ = 1 σ z = e e = ( ) 0 0 0 1 e σ x, σ y, σ z N N S = i η i log η i η i

N η i = 1 N S q = N i 1 N log N = N N η i = 1 N+1 N S c = N+1 i 1 N +1 log (N +1)=log (N +1)

a) b) log (N +1) N Γ max Γ max Γ 0 Γ /Γ 0 Γ 0 4Γ 0

I = I c sin πφ Φ 0 Φ 0 I c ϕ =πφ/φ 0

( ) H = ω q a a E J cos ϕ + ϕ E J = I cφ 0 π ϕ = ϕ q (a + a ) ϕ q E J ϕ 6 q ω 70 1 H = ω q a a E J 4 ϕ4 + O(ϕ 6 ) ( ω q a a ) E J 4 ϕ4 q a + a 4 H = ω qa a α a a α = E J 4 ϕ 4 q ω q = ω q α α E = E n+1 E n = ω q α 0, 1 H = ω q e e E C E J H/ = 8E C E J a a E C (a a)

a) b) c) C q 1 0.5 0-0.5-1 π 0 π L J C φ 1 H = ω r a a + ω q e e + g(a + a )σ x

a e H = ω r a a + ω q e e + g(aσ + + a σ ). a, σ σ +,a κ γ g g κ, γ g ω r ω q = H = (ω r χ e e )a a + ω q e e χ = g χ

γ,κ χ nκ, γ n = a a g H quasi = H disp Ka a σ z K = g4 3 σ z = e e Ka a σ z K a a Ka a e e K

ω q a) b) ω c freq ω q 3 ω c e ω q 5 ω q 4 ω q ω q 3 0 ω q 1 ω q ω c e ω c g ω c g ω c g ω c g ω q 1 ω q 0 ω q ω c e ω c e H = i=q,r ( ) ω i a i a i E J cos ϕ + ϕ a q,r ϕ = i=q,r ϕ i(a i + a i ) ϕ q >> ϕ r ϕ H = ( ω i a i a i K i i=q,r a i a i ) χa qa q a ra r K i = E Jϕ 4 i χ = E J ϕ qϕ r K r

K r α = K q ϕ q ϕ r ϕ = i ϕ i(a i + a i ) H 4 = i ( ω i a i a i K i a i a i ) χ ij a i a ia j a j i,j>i K i = E Jϕ 4 i χ ij = E J ϕ i ϕ j K i ϕ 4 i χ ij χ ϕ H 6 = H 4 + i K i 3 6 a i a 3 i + i, j χ ij a i a i a j a j K i = E Jϕ 6 i 6 χ ij = E Jϕ 4 i ϕ i K χ n K i (n) (K + K 3 K 3 n i) χ ij (n i ) (χ ij + χ ij χ ij n i)

ω i /π O(ϕ 4 ) K i /π 1 4!( 4 )( ) EJ ϕ 4 i O(ϕ 4 ) χ ij /π 1 4!( 4 1)( 3 1)( 1)( 1 1) EJ ϕ i ϕ j O(ϕ 6 ) K i/π 1 6!( 6 3)( 3 3) EJ ϕ 6 i O(ϕ 6 ) χ ij/π 1 6!( 6 )( 4 )( 1)( 1 1) EJ ϕ i ϕ 4 j E J ϕ = ϕ i (a i + a i ) cos ϕ O(φ 6 )

a).1 mm phase qubit transmission line resonator 00 µm b) compact resonator Josephson junctions transmon qubit 1 µm 00 µm c) three-dimensional cavity resonator 50 mm 50 µm transmon qubit 6B;m` jxj, _bqm iq`b 7Q` +B`+mBi Z1.X a?qrm Bb b KTH Q7 i? p `Biv Q7 `bqm iq` /bb;mb BMpbiB; i/ 7Q` +B`+mBi Z1.X q?bh i? i` MbKBbbBQM HBM `b@ QM iq` U V M/ +QKT +i `bqm iq` U#V ` 7 #`B+ i/ mbbm; T?QiQ@ Q` H+i`QM # K HBi?Q;` T?v- i? i?`@/bkmbbqm H + pbiv `bqm iq` U+V Bb T`Q/m+/ 7`QK K +?BM/ HmKBMmKX LQiB+ i? HM;i? b+ Hb bbq+b i/ rbi? +? Q7 i?b p `B Mib- ` M;BM; Qp` irq Q`/`b Q7 K ;MBim/X _/m+bm; i? H+i`Q@K ;MiB+ }H/ /MbBiv BM `b@ QM iq` KQ/b?HTb HBKBi i? bm`7 + HQbbb `btqmbb#h 7Q` HBKBi/ `bqm iq` +Q?`@ M+- K FBM; #Qt@KQ/ `bqm M+b M B/ H TH i7q`k 7Q` +` ibm;?b;?hv +Q?`Mi `bqm iq` KQ/bX _T`Q/m+/ 7`QK (>Q7?BMx i HX- kyy3c :`HBM;b i HX- kyrkc S BF i HX- kyrr)x 8k

i` `v [m MimK bi ib + M # +` i/ BM i? `bqm iq` mbbm; i?bb i+?mb[m (G r M/ 1#`Hv- RNNe)X h?bb i+?mb[m? b #M tt`bkmi HHv /KQMbi` i/ M/ b?qr/ i? +` ibqm Q7 +QKTHt [m MimK bi ib mbbm; bmt`+qm/m+ibm; +B`+mBib (>Q7?BMx i HX- kyyn)x a) b) 1 +/π 0.4 0 +1/π 1 W(α ) 0 W(α ) Im(α ) 1 0. 0.0 0. 0 1/π 1 0.4 0 1 0 1 1 Re(α ) 0 1 /π Re(a ) 4 0 Im(a ) 6B;m` jx9, _bqm Mi M/ /BbT`bBp K MBTmH ibqmx lbbm; Bi?` `bqm Mi Q` /BbT`bBp BMi` +ibqmb- +QKTHt [m MimK bi ib + M # +` i/ BM i? + pbivx U V lbbm; bmt`+qm/m+ibm; +B`+mBib rbi? `bqm Mi +QMi`QH- i? T?QiQM MmK#` bi ib M/ i?b` bmt`tqbbibqm + M K MBTmH i/ rbi? [m#bi (>Q7?BMx i HX- kyyn)x qbi? + pbiv Z1. bimt- i? /BbT`bBp BMi` +ibqm HHQrb i? T`QD+iBp K bm`kmib Q7 i? + pbiv iq +` i bmt`tqbbibqm bi ib (.Hû;HBb i HX- kyy3)x qbi? M Qz@`bQM Mi- /BbT`bBp BMi` +ibqm i? [m#bi M/ + pbiv KQ/b rbhh MQi b? ` t+bi ibqmb- #mi #+QK Mi M;H/ /m iq +QM/BiBQM H b?b7ib BM +? KQ/Ƕb i` MbBiBQM 7`[mM+vX qbi? i?bb i+?mb[m- [m MimK MQM@/KQHBiBQM K bm`kmib + M # T`7Q`K/ #v +Q``H ibm; i? T? b Q7 +Q?`Mi }H/ BM i? + pbiv KQ/ rbi? i? ;`QmM/ Q` t+bi/ bi i Q7 i? [m#bix *QmTH/ rbi? M ` [m MimK@HBKBi/ K@ THB}`b ("`; H i HX- kyryc obd v i HX- kyyn) [m MimK DmKTb Q7 [m#bi bi ib? p #M T`7Q`K/ (obd v i HX- kyrrc > i`b/; i HX- kyrj)x "vqm/ [m MimK K @ bm`kmi- i? /BbT`bBp BMi` +ibqm? b b?qrm i? bthbiibm; Q7 M`;v i` MbBiBQMb Q7 i? [m#bi iq # /TM/Mi QM T?QiQM MmK#` (a+?mbi` i HX- kyyd)- r?qb TTHB+ @ ibqmb BM+Hm/ i? Mi M;HBM; Q7 T?QiQM MmK#` bi ib rbi? i? [m#bi KQ/ (CQ?MbQM 8j

100 µs 10 ms

H = ω ra ra r + ω s a sa s + ω q e e χ qr a ra r e e χ qs a sa s e e χ qr (χ qs ) κ r κ s κ r

l h w f mnk = c (m ) ( n ) ( ) k + + l h w c m, n, k f 101

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DQBMBM; + pbiv `bqm iq`bx h?bb +Q tb H KQ/ U HbQ FMQrM b `+i t /m iq Bib `+i M;mH ` /bb;mv Bb +` i/ #v TH +BM; bm#bi` i +QMi BMBM; i? 7 #`B+ i/ /@ pb+ rbi?bm K +?BM/ i`m+? #irm +? + pbiv Ub 6B;X 9XjVX bbm;h CQbT?@ bqm DmM+iBQM Bb HQ+ i/ rbi?bm i?bb i`m+? M/ ; Hp MB+ HHv +QMM+i/ iq irq i` Mb@ KBbbBQM HBMb i? i tim/ BMiQ +? + pbivx h? timbbqm Q7 +? MiMM +` ib + T +BiBp +QmTHBM; iq i? H+i`QK ;MiB+ }H/ BM i? + pbiv HHQrBM; bi`qm; [m#bi@+ pbiv BMi` +ibqm 7Q` #Qi? bt ib HHv bt ` i/ KQ/bX 6B;m` 9Xk, hrq@+ pbiv HM;i? b+ HbX U V a?qrm Bb #Bb+iBQM Q7 irq + pbiv /bb;m K +?BM/ 7`QK H eyerx M BM/BmK i` +F bm``qmm/b #Qi? + pbibb r?b+? T`Q@ pb/b bi`qm; H+i`B+ H +QMM+iBQM i i? /pb+ b KbX U#V h? }`bi BMbi b?qrb i? p`ib+ H i` MbKQM [m#bi #irm #Qi? + pbiv KQ/bX q?bh i? CQbT?bQM DmM+@ ibqm Bb HQ+ i/ rbi?bm i? bk HH i`m+? +QMM+iBM; i? + pbibb- MiMM b 7`QK i? [m#bi tim/ BMiQ #Qi? + pbiv KQ/b T`QpB/BM; bi`qm; [m#bi@+ pbiv +QmTHBM;X U+V 6BM HHv b+qm/ BMbi b?qrb M a1j BK ; Q7 i? i` MbKQM DmM+iBQM rbi? M T@ T`QtBK i DmM+iBQM ` U45 µm V `bmhibm; BM CQbT?bQM BM/m+i M+ Q7 6 nhx *B`+mBi /bb;m Gi mb TT`QtBK i i? /bb;m Q7 i? p`ib+ H i` MbKQM #v +` ibm; +B`+mBi KQ/H iq T`/B+i Bib /TM/M+ QM ;QKi`B+ H 7 im`bx q rbb? iq +` i CQbT?@ bqm DmM+iBQM +B`+mBi r?b+? #? pb HBF i` MbKQM [m#bi #mi rbi?qmi i? mb Q7 HmKT/ HKMi + T +BiQ`b M/ BM/m+iQ`b UiQ KBMBKBx HQbbv bm`7 + z+ibv M/ r?b+? + M? p bb;mb}+ Mi HM;i? BM Q`/` iq +QmTH iq irq bt ib HHv bt ` i/ + p@ ey

Al O 3 ϵ r {9.4, 9.4, 10.} Z line (l) =Z 0 Z L + jz 0 tan(βl) Z 0 + jz L tan(βl) Z 0 Z L β l Z L = 1 jωc Z 0 Z line (l) = jz 0 cot (βl).

l l ( ) ωl Z line (ω, l) = jz 0 cot ν p ν p = c µr ϵ r c µ r,ϵ r ν p (0. 1)c a) Y in (ω) Z 0,ν p b) Y in (ω) L J Z line (ω) E J Z line (ω) l Z 0 ν p Z 0 80Ω ν p 0.4c Y (ω) L J E J = φ 0 L J 1 ω 0 = Leff C eff Y (ω 0 )=0 L Z eff = eff C eff = ω 0 Im[Y (ω 0 )] H/ = ω q a a α a a ϕ

ω q = ω 0 α α = e Z eff L J. Y in (ω) = 1 + j tan jωl J Z 0 Y in (ω 0 )=0 1 = 1 tan ω 0 L J Z 0 ( ω0 l ν p ( ωl ν p ). ). ω 0 Z eff Im[Y (ω)] = 1 + l sec ω L J Z 0 ν p ( ωl ν p ). ( ωl tan ν p ) Y in (ω) 1 + j ωl jωl J Z 0 ν p LC

Y in (ω) 1 jωl J + jωc(l) C(l) = l Z 0 ν p ω 0 = Z0 ν p L J l = 1 LJ C(l) α = e Z eff L J = e Z 0 ν p l = e C(l). ω 0 (l) 1 l α(l) 1 l a) b) Resonance (GHz) Anharmonicity (GHz) Antenna length (mm) Antenna length (mm) ω 0 /π α/π L j = 7 nh,z 0 = 80 Ω, and ν p = 0.4c

tan ( ωl ν p ) ( Y in (ω) 1 + j ( ) ) ωl 1+ ω l. jωl J Z 0 ν p ν p ω 0 = 3 ( νp l ) ( 1+ 8 3 Z 0 L J ) l 1. ν p ω 0 α = e Z 0 ν p l ( 1 Z 0 L J l ν p ).

cavity 1 Y in (ω) cavity Z 1 (ω) Z (ω) 4 mm substrate stripline 0.4 mm Z 1 (ω) Admittance (ms) 4 0 - -4 Y in (ω) cavity 1 7.5 8.0 8.5 9.0 9.5 Frequency (GHz) LC Z 1 (ω), Z (ω) Y in (ω)

O (ϕ 6 ) 150 10

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I + iq ω RF,ω LO V sig V cos (ω IF + δ RF δ LO + δ DUT ) ω IF = ω RF ω LO δ RF, LO, DUT ω RF,ω LO ω IF δ V demod cos ω IF V ref cos (ω IF t + δ RF δ LO )

300K 4K 0mK JPC I I S S 180-H 180-H 10dB Ecco Ecco Ecco Ecco 180-H S S JPC 10dB Ecco Ecco Ecco Ecco 10dB 10dB LP 10GHz LP 10GHz Ecco Ecco LP 10GHz 30dB 30dB LP 10GHz LP 10GHz 30dB LP 1GHz 30dB LP 10GHz 30dB LP 1GHz 30dB 10dB 0dB 10dB 0dB 10dB 10dB 10dB 10dB HEMT 0dB 0dB HEMT 0dB 0dB 0dB 0dB 0dB 0dB 0dB 0dB 0dB 0dB 0dB 0dB A B 1 3 4 5 6 7 8 1 3 4 5 6 7 8

a) b) RF δ RF DUT δ signal RF δ RF DUT δ signal signal reference signal LO δ LO LO δ LO ω RF,ω LO ω IF = ω RF ω LO δ RF,δ LO δ signal

a) I/O setup Qubit and readout input TO FRIDGE 1 S 1 S Switch Switch Qubit ω μw Readout AWG ω μw I DAC Q 1 S LO ω μw ADC Readout output FROM FRIDGE I Storage input Switch ω μw Storage DAC Q DIGITAL b) I/O setup with feedback 1 S Switch AWG ω μw Readout I DAC Q DIGITAL 1 S LO ω μw Readout output FROM FRIDGE Qubit and readout input TO FRIDGE 1 S Switch Qubit FPGA ω μw I DAC Q Feedback SE ADC DIGITAL Storage input Switch FPGA ω μw Storage I DAC Q SE DIGITAL ADC g e

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H/ = ω q e e + ω s a a χa a e e e a a ω q,s χ C Φ = e iφa a e e = g g + e iφa a e e g Φ τ Φ=χτ

C Φ { α ( g + e )} = α, g + αe iφ,e α = e α n=0 α n n! n n α C Φ=π π P a) b) e cavity qubit P g X ψ = e, e iφ β Φ=χτ C Φ χ γ,n κ s γ κ s

n χa a e e K s a a χ a a e e m R mˆn,θ = m m Rˆn,θ + n m n n Rˆn,θ ˆn θ χ

a) P b) P c) P d) P m=0 n max X X X X cavity qubit m n max ωq n = ω q χ n n τ 1/χ m ω m q

H/ = χ(a a m) e e + ϵ(t)σ y ϵ(t) σ y H/ = n = n H n / n n { χ(n m) e e + ϵ(t)σ y } n n. H n / = n ϵ(t)e i n,mt e e σ y e i n,mt e e n,m = ω n q ω m q ψ(t) ψ n (t) = i H n (t) ψ n (t). m Rŷ,θ = e i θ σ y θ = ϵ(t) t ψ n m (t) ψ n (t) { 1 i t } s H n (s) ψ n (0). ψ(0) = n m C n g, n

ψ(t) C n { g, n i n m = C n { g, n n m i t = n m C n { g, n 0 t 0 s H n (s) g, n } sϵ(s)e i n,ms e e σ y e i n,ms e e g, n } t 0 sϵ(s)e i n,ms e, n } n m C n { g, n ˆϵ{ n,m } e, n } ˆϵ{ω = n,m } ϵ(t) n τ ψ(τ) = 1 1+ˆϵ{ n } n m C n { g, n ˆϵ{ n,m } e, n }. ωq m ϵ ψ = n m C n g, n g S = n, g ψ(τ) = n m C n 1+ˆϵ[ n,m ]. m ϵ(t) = Ae σ ωt / σ ω A = 8/πσ ω π ω m q σ ω /π =800 σ t =00

χ/π =3 m (m±1) S =(1+ π 8 e χ /σ ω ) 1 > 99% R m ŷ,π = m m Rŷ,π + n m e iξ n n n m m ξ n σ ω β β R 0 ŷ,π R 0 ŷ,π( β,g + 0,e ) ( β + 0 ) g π 0 n β = n=0 C n n = e β n=0 (β) n n! n σ ω =4 β χ/5 β S = n=1 C n (1 + π 8 e (nχ) /σ ω ) 1 > 99%

ξ n n = χn σ ω ξ n ξ n = ϵ(t) dt/ n 1/( β ) 1 n ξ n 1 n/(8 β ) n ϵ χa a e e σ z σ y H/ = n { χ(n m) σ z + ϵ(t)σ y} n n. ϵ τ

i H/ τ U(τ) =e = n U n (τ) n n = n e iτ{χ(m n) σz +ϵσ y} n n = n e iφnσ θn n n φ n = ϵτ 1+ [ ] (m n)χ θn = arctan ϵ ( ) (m n)χ σ ϵ θn = cos(θ n )σ y + sin (θ n )σ z U n (τ) =e iφ nσ θ n =cos(φ n ) + i sin(φ n )σ θn =[cos(φ n )+isin(φ n )] sin(θ n ) n, g n, g +[cos(φ n ) i sin(φ n )] sin(θ n ) n, e n, e +sin(φ n )cos(θ n )( n, e n, g n, g n, e ). π/ τ =4n π Rŷ, F = 1 N Tr[R ŷ, U(τ)] 0.96 π n max =0 ω m q = ω q nχ n

ωs g ωs e g e ωs e H / =(ω q ω e s) e e χ g g a a + ϵ(t)a + ϵ(t) a. σ ω χ D α e D e α = e iξ g g + D α e e ξ g Dα e Dα{ 0 e ( g + e )} = e iξ 0,g + α, e Dα e C π Dα e = D α/ C π D α/ D α

ω g s H / =(ω q ω g s) e e χ e e a a + ϵ(t)a + ϵ(t) a. ϵ χ D α=1 6 ns ϵ 170 MHz χ 3 MHz H / =(ω q χ qs a sa s χ qr a qa q ) e e χ qs χ qr τ 1 χ qs, 1 χ qr

a) b) storage cavity - qubit readout cavity- qubit increasing n χ n ω 0n n = ω q n K ω 0n ω 0n n

f 01 = ωq f π 0/ = (ωq α) π K 1.5 1.0 0.5 7.0 7.4 7.8 7.3 Spectroscopy Frequency (GHz) 7.36 f 0 / f 01 K

a) b) 80 storage cavity qubit Tone τ=300μs m=0 Readout Voltage (mv) 60 40 0 storage cavity - readout cavity increasing n 0 8.70 8.7 8.74 8.76 Spectroscopy Frequency (GHz) 8.78 τ 1 χ π Rŷ,π 0 π

K s 0 1 0 n n =, 3 K Readout Signal (mv) 40 50 60 70 80 9.740 9.744 9.748 9.75 Spectroscopy Frequency (GHz) π 0 1 0 n n = 1,, 3 0.5 K/π =163

χ C Φ Φ=χ qs t t ψ(0) = β,g ψ(t) = Rŷ, π C Φ=χ qs trŷ, π β,g = e π 4 ( e g g e ) e iχ qsta a e e e π 4 ( e g g e ) β,g = 1 {( β βe iχqst ) g +( β + βe iχqst ) e } Rŷ, π π/ P e P e = 1 {1+Re( β βeiχ qst )} = 1 {1+e β (cos(χ qst) 1) cos( β sin(χ qs t))}. t e 1 ( β χ qst)

a) cavity qubit b) 0 1.00 0.75 c) 0.50 0 0 00 400 600 Wait time ( ) 800 1000 Displacement ( ).0 1.0 0.0 0 00 400 600 Wait time ( ) 800 1.0 0.8 0.6 0.4 1000 β β =0 β =0.5 β =1.0 β =1.5

t =π/χ qs D β β =0.5 t = µ χ qs a a e e χ qs χ qs a a e e χ qsa a e e n nχ qs χ qs n =5 t = π χ qs β χ qs χ qs χ qs χ qs/χ qs =3.6 10 3 χ qs 3 MHz

a) b) Displacment amplitude ( ) 5 4 3 1 0 300 1.0 0.8 0.6 0.4 Wait time ( ) 440 435 430 45 40 400 500 0 Wait time ( ) 5 10 15 0 5 Mean photon number ( ) a a

C Φ R ṋ n,θ D α p 0 (α) P α P 0 (α) =πq(α) P α = π W (α) p n (α) = n D α 0 = e α α n n!. P = e iπa a P (α) =Tr[PD α 0 0 D α]=e α. P (α) P 1 0.0 δα/α 0.0

a) cavity qubit m or b) Readout signal (mv) c) 4 0 - -4 0 000 4000 Drive amplitude (DAC value) 6000 Photon probability 1.0 0.8 0.6 0.4 0. 0.0 0 1 3 Displacement ( ) 4 R0ˆn,π P n P n α χ sr P 0 P n n = P n

a) cavity qubit or b) Readout signal (mv) c) 4 0 - -4 0 1.0 0.8 0.6 0.4 0. 0.0 0 000 4000 Drive amplitude (DAC value) 1 3 Displacement ( ) 6000 4 C π P α P α α P α P α P α

χ sr R n π,ŷ e g ρ α Q(α) = 1 π α ρ α α

Q(α) = 1 π 0 D αρd α 0 D α α ρ α = D αρd α 0 a) cavity state prep cavity tomography b) Q(α) qubit Im(α) 0 - - 0 Re(α) Q(α) = 1 π 0 D αρd α 0 β

0 0 0 0 σ z g Q Z (α) = 1 π Tr [ ρ qc σ z D α 0 0 D α ] = 1 π 0,g D αρ qc D α 0,g 1 π 0,e D αρ qc D α 0,e = p g Q g g (α) p e Q e e (α) ρ qc p g,p e g ρ qc g, e ρ qc e α σ z Q g g (α) Q e e (α)

a) b) Im(α) 0 0 - Q Z (α) - Q Z (α) - 0 Re(α) - 0 Re(α) Q g g (α) Q e e (α) ψ = N ( g, β + e, e iφ β ) ρ α = D ρd α 0 n Q n (α) = n D αρd α n

Q n (α) N W (α) = ρd αpd α = n ( 1) n n D αρd α n = n ( 1) n Q n (α) Im(α) a) b) m = 0 Q 0 (α) Re(α) m = 1 Q 1 (α) Re(α) c) d) m = Q (α) Re(α) m = 3 Q 3 (α) Re(α) Q(α) = 1 π 0 D αρd α 0 Q m (α) = 1 π m D αρd α m 0 Q 0 (α), Q 1 (α), Q (α), Q 3 (α) 0, 1,, 3

W (α) = π Tr[D αρd α P ] D αρd α α P P = e iπa a U = π Rŷ, C π Φ=πRŷ, = R ŷ, π a e e e iπa π Rŷ, U U U = n U n n n = n = n = n even = n even Rŷ, π e iπn e e Rŷ, π Rŷ, π n n { (1+( 1) n ) (1 ( 1) + σ n ) z Rŷ, π R ŷ, π n n + Rŷ,π n n + n odd n odd n n } Rŷ, π n n Rŷ, π σ zrŷ, π n n

a) state cavity b) prep tomography cavity W(α) qubit Im(α) 0 - - 0 Re(α) (τ π χ ) P α = D α PD α β β = 3 W i = π σ ip α P α σ i {I, X, Y, Z}

a) cavity state prep qubit tomography cavity tomography qubit b) W I (α) W X (α) W Z (α) W Y (α) Im(α) Re(α) ψ = N ( g e ) β β = 3 W Z (α) W Y (α) W X (α) {X, Y, Z} P α = D α PD α

ρ nm ij ρ = 1 N i,j=0 n,m=0 ρ nm ij i j n m i, j n, m AB =Tr[ABρ] A B σ i = {I,σ x,σ y,σ z } P α = D α PD α N max = 1 α max,min = ±3.4 α =0.085 W i (α) = σ π ip α A A = i A iσ i A i =Tr[Aσ i ] B = 1 B(α)Pα d α π B(α) =Tr[BP α ] ρ = π i W i (α)σ i P α d α ρ = ρ q ρ c

ρ = 1 i Tr[ρ q σ i ]σ i π π Tr[ρ cp α ]P α d α ρ AB =Tr[ABρ] [ =Tr i,j A i B(α)W j (α )σ i σ j P α P α d αd α ] Tr[σ i σ j ]=δ ij Tr[P α P α ]=δ (α α ) AB = i A i B(α)W i (α)d α D α P W (α) = π Tr[D αρd α P ] ρ

W (α) = π Tr[D αpd αρ] =Tr[M(α)ρ] = i,j M ji (α)ρ ij M(α) =D α PD α M ji (α) ρ ij M(α) ρ ρ ij W (α) ρ Tr[ρ] =1 n max

φ Kerr = KIτ I τ K ω s H = ω s a a K a a

K κ s β RHR R = e i(ω s K )a at H kerr / = K (a a) U(t) =e ih kerr t ψ(t) = U(t) β = e ikt (a a) β = e β n β e ikn t n. n n! n β(t) βe iφ Kerr(t) φ Kerr = K β t n t π T col = π nk T rev = π K U(T rev) =e iπ(a a) =( 1) (a a) =( 1) a a ψ(t rev ) = β

a) P b) P c) P X φ Kerr nkt X X P n=0,1,,3 P φ=n Kt P X X X β = n c n n n c n = c n e iφn φ n =0 c n φ n = n Kt t = π K t = Trev q ψ( T rev ) = 1 q q q 1 p=0 q 1 k=0 e ik(k p) π q βe ipπ q. T rev q q q =

cavity state prep evolution t cavity tomography qubit β t U(t) =e ih kerr t t ψ = 1 ( β + i β ). β P β 0 1 κ s /π = ω q /π = K q /π =(ω 01 q ω 1 q )/π = 50 MHz K

Re(α) - 0 Experiment time Im(α) 0-15 ns 65 ns 440 ns 815 ns Theory - Re(α) 0 Experiment time Im(α) 0-1065 ns 1565 ns 565 ns 3065 ns Theory β β =4 A κ s T 1 =10µ T =8µ ω s /π =9.7 GHz

H = ω q e e +(ω s χ) a a e e K a a. χ/π = ω s K/π = K χ /4K q K > 30κ s β = n =4 Q 0

H Kerr β 15 ns β = βe iφ Kerr =.0e i0.13 n T col =385ns T rev β t = T rev /q q>1 q = q =3, 4 T rev =3065 β = 1.78 β = κ/π =10 ω s µ

- Re(α) 0-0 - 0 Im(α) 1 0-1 0-0 - Q n (α) n = 0 8 t π q =, 3, 4 qk q A = e n q>0 Q n (α)

Q n χ - Re(α) 0 a) Im(α) 0 - Theory Reconstruction b) c) t = π q = 3 4 qk

t = π/k, π/3k, π/4k F = Ψ id ρ m Ψ id ρ m Ψ id β = e κt/ F = 0.71,F 3 = 0.70,F 4 = 0.71 K >> κ

0, 1 β, β ψ = 1 N { cos( θ ) β +sin(θ )eiφ β } θ, φ N = 1+sin(θ)cos(φ)e β N 1 β β +Z c Z c X c Y c ±Z

Z β, β +X c, +Y c, +Z c ±Z c = ±β ±X c = 1 N ( β ± β ) ±Y c = 1 N ( β ±j β ) N (β) β β β = e β. S = j η j log η j.

η j ρ = j η j j j ρ = 1 ( β β + β β ) ρ E, O ρ = 1 (1 + e β ) E E + 1 (1 e β ) O O S = 1+e β log [ 1+e β ] 1 e β [ ] 1 e β log. S β 0 =0 S β =1 β =0 β β =1 S =0.99 a) P d X b) Entropy (bits) Displacement β β, β β β = 0 d =(β β) β S S 1 β β =1 e d

β, β d =β d d β β = e d ρ ρ(t) = 1 [ ] β(t) β(t) + β(t) β(t) + e β(t) (1 e κt) ( β(t) β(t) + β(t) β(t) ) β(t) =βe 1 κt κ e 1 d κt

ψ 0 = β ( g + e ) β π ψ 1 = C π ψ 0 = β,g + β,e ψ = D β ψ 1 = β,g + 0,e π 0 ψ 3 Rŷ,π 0 ψ =( β + 0 ) g ψ 4 = D β ψ 3 =( β + β ) g 0 { cos( θ ) g +sin(θ )eiφ e } { cos( θ ) β +sin(θ )eiφ βe iφ } g θ φ β β iφ 1

ω s π κ s =7. = 1 π π.1 µ ωr π κ r = 330 = 1 π π 480 = 8.18 = 9.36 ω q π γ =7.46 =36 = 1 π π 4.4 µ χ qs π =.4 K s χ qs n n =min[χ qs /χ qs =560,χ qs /K s =650,χ qs /κ s =330] β C Φ Q = α ρ α α = β

βe iφ α = β Q(β) = β βe iφ = e β (1 cos Φ) Φ Φ χ qs β 1 n Q(α) = 1 α ρ α ρ π ψ = 1 { 0 ( g + e )} ψ = N{( β + β ) g } β = 7 N 1 0,g P e Dβ e ρ = 0 0 {P g g g + P e e e } D e β ρde β = P g 0,g 0,g + P e β,e β,e

a) mapping cavity qubit m=0 tomography b) 8 4 (1) () (3) (4) Im( ) 0-4 -8 8 4 (5) (6) (7) (8) Im( ) 0-4 -8-8 -4 0 4 8 Re( ) -8-4 0 4 8-8 -4 0 4 8-8 -4 0 4 8 Re( ) Re( ) Re( ) ϵ/π =990.5µ β,e β 17 α =6

[ρ] =1 W (α) α =1 P e =0.1 P e 0.01 W (α) = [ D α PD αρ ] ρ F = W W α W

a) Im( ) Re( ) -4-0 4 4 b) Z Re( ) Im( ) 1.0-4 0 4 1.0-0 Z Y 0.5 Y 0.5 X X 0.0 0.0 0.6 0.0 0 Z -0.5-1.0 Y X 1.0 0.5 0.0-0.5 Z -0.5-1.0 Y X 1.0 0.5 0.0-0.5-0.6 - Z 1.0 Y 0.5 X 0.0-1.0 Z 1.0 Y 0.5 X 0.0-1.0-4 0.6 0.4 0. 0.0 0.6 0.0-0.6 Z -0.5-1.0 Y X 1.0 0.5 0.0-0.5-1.0 Z -0.5-1.0 Y X 1.0 0.5 0.0-0.5-1.0 ψ = N ( β + β ) β, β ψ = { cos( θ ) g +sin(θ ) e } ψ = 1 { g + e iφ e }

β ± β β P n ( β ) = n β = e β β n n! P n ( β ± β ) (1 ± e iπn ) e β β n n! β,g { β + β } g { β β } g β =.3 d = β 1 β β 1 β d W (Re(α) =0, Im(α)) W (0, Im(α)) Ae Im(α) cos(d Im(α) +δ) A δ 111 +0 d

φ Z Y X Ramsey angle (φ) θ Z Y X Rabi angle (θ) Im(α) Re(α) Φ C π/3 C π/ F A = 0.60 F B = 0.58 F C = 0.5

α cal = α act (1 + δα) α cal α act δα =( (n th +1) n th W (α) e α n th +1 n th 0.01 d d (1 n th ) <d act d d act 109 <d 111 β δφ 1 n n = β

1 n g g + e β 0 + β n = β Φ δφ = 1/ P e Φ P e Φ δφ D = e/ n Φ δφ C =1/ n δφ C nκτ 1 κ τ δφ C = e nκτ / n n = 15.5.5

σ z F F recov F F recov

a) b) Spectroscopy frequency (GHz) 7.435 7.445 7.455 0.8 0. 0.0 Normalized spectroscopy signal 0.1 0.0 0.4 0. 0.0 0.4-0.8 0.7 0.0-0.7 0.5 0.0-0.5 0.3 0. 0.0 0.0 10 9 8 7 6 5 4 3 Photon number 1 0-0.3 - -1 0 1 Im( ) β β + β β β β =.3 111 +0 d Ae Im(α) cos(d Im(α) +δ) S A δ

Im( ) Re( ) -4-0 4 4-4 4-0 4 a) b) 0 0-0.4 - - 0.0-4 4 c) -4 d) 0.4 1 0 0 - -1-4 -4-0 4 - - -1 0 1 C π/3 C π/ β + e iλ 1 βe iπ/3 + e iλ βe i4π/3 β = 7 λ 1 =0.6π λ = 0.3π 0 + e iµ 1 iβ + e iµ βe iπ/3 + e iµ 3 βe iπ/3 β = 7 µ 1 =0.5π µ = 0.4π µ 3 = 0.π β + e iν 1 iβ + β + e iν iβ β = 7 ν 1 = π ν = 0.π

a) cavity 1 qubit or or Readout b) Re( ) 0 c) Re( ) 0 Im( ) Im( ) (radians) d) 0.50 0.5 0.00 0.50 0.5 0.00 f) 0 1 0.1 6 5 4 3 6 5 4 Phase (radians) 0.50 0.5 0.00 3 4 5 6 7 8 9 3 4 10 Energy e) 0 0.8 0.6 0.4 0. 0.0 0.8 0.8 0.6 0.4 0. 0.0 (photons) Phase (radians) 0.6 0.4 0. 0.0 δφ n Φ P e 1/ n 1/ n δφ C e nκτ / n κ τ nκτ > 1

a) cavity 1 b) cavity 1 qubit Readout qubit QPT c) 16 photons 1 0.5 0 8 photons 1 0.5 0 40 photons 1 0.5 0 100 photons 1 0.5 0 0 Rotation Angle ( ) d) Re( ) Re( ) 16 photons 8 photons 1 1 0.5 0.5 0 0 I I X X Y Y Z I X Y Z Z 40 photons 100 photons 1 0.5 0 I X Y Z I X Y Z Re( ) Re( ) 1 0.5 0 I X Y Z I X Y Z I X Y Z < 0.06 90%

β ψ = 1 ( g + e ) β g, e β t = π χ ψ B = 1 ( g, β + e, β ) ψ = 1 ( gg + ee ) ψ B ψ B = II c + XX c YY c + ZZ c {I, X, Y, Z} {I c,x c,y c,z c }

cavity state preparation qubit tomography cavity tomography qubit ψ = 1 ( g + e ) β D β β Rŷπ π ŷ ψ B = 1 ( g, β + e, β ) R i X Y Z P α

F C ψ target = 1 ( gg + ee ) F = ψ target ρ ψ target = 1 ( II + XX YY + ZZ ). 4 II, XX, Y Y, ZZ W = 1 ( II XX + YY ZZ ) 4 W F > 1 ±1 O= AA c + AB c BA c + BB c A, B A c,b c

τ s =55µs τ r =30ns T 1,T 10 µs 5 8 GHz H/ = ω s a a +(ω q χa a) e e a e e ω s,ω q χ π 1.4 MHz {X, Y, Z} g P α P α = D α PD α D α

P W (α) = π P α α W i (α) = π σ ip α σ i {I, X, Y, Z} W i (α) W B i (α)w i (α)d α Wi B (α) F = ψ B ρ ψ B = π i ψ B W i (α) F =(87± )% β = 3 β β = 6 10 5 1 V = IPα d α =(85± 1)% π V F V

(a) Im( ) 0 - Re( ) - 0 (b) g e g e 0 4 6 8 10 0 4 6 8 10 0 4 6 8 10 0 4 6 8 10 Re( ) 0.1 0.0-0.1 Mean Value 1.0 0.5 0.0-0.5 (c) Fock state basis Re( ) 0.5-1.0 g e g Encoded basis e 0.0 W i (α) = σ π ip α σ i = {I,X,Y,Z} P α ψ B β = 3 XP α YP α ρ β β + β β

W (α) β β 1 X c = P 0 I c = P β + P β Y c = P jπ 8β Z c = P β P β {I c,x c,y c,z c } ψ B β = 3 F DFE = 1( II 4 c + XX c YY c + ZZ c )= (7 ± )% F DFE V F

(a) 0.0 0.5 1.0 1.5.0 0.0 0.5 1.0 1.5.0-3 - -1 0 1 3 - -1 0 1 Re( ) Im( ) (b) 1.0 0.5 0.0 Mean Value -0.5-1.0 1.0 0.5 0.0-0.5-1.0-3 - -1 0 1 3 - -1 0 1 Re( ) Im( ) 1.0 0.5 0.0-0.5-1.0 Mean Value ψ B β =0 IP α ZP α Im(α) =0 XP α YP α Re(α) =0 β = 3 {II c,xx c,yy c,zz c }

X(θ),Z(θ),X c,z c θ β O 1 =.30 ± 0.04 θ = π 4 β =1 X, Y, X c (α),y c (α) α O =.14 ± 0.03 β =1 ± ±M q 1 ± M q 1

(a) 3 ideal photon loss visibility 0 Rotation ( ) 1 0.0 0.5 1.0 1.5.0 Cat amplitude ( ) (b) 3 ideal photon loss visibility 0-1.0 0.0 1.0 Displacement ( ) 0.0 0.5 1.0 1.5.0 Cat amplitude ( ) X(θ) Z(θ) Z c X c O = AA c + AB c BA c + BB c θ X Y X c (α) Y c (α) α β O =

± ±M c ± M c g AB A, B AB =(A + A )B A + + A = I A + B (1 p c ) A + B p c AB (1 p c ) A + B A B =(1 p c ) A + B A B p c A + B + A B =(1 p c ) AB p c B B = X c,y c,z c ψ c B =0 AB (1 p c ) p c =1 e τ wait T 1 0.06 V V pred =(1 p c )V =8% V 85%

σ i P α V [0, 1] W meas i (α) =VW ideal (α) V W ideal I (α)d α i V = W meas I (α)d α I V =85% {I,X,Y,Z} {I c,x c,y c,z c } A, B A c,b c O = AA c + AB c BA c + BB c

ψ B Z c,x c Z(θ),X(θ) Z(θ) =Z cos θ X sin θ X(θ) =X cos θ + Z sin θ θ O θ = π 4 A = X+Z ; B = X Z A c = Z c ; B c = X c AZ c BZ c O ideal = ( e 8 β ) V O vis = V( e 8 β ) AX c BX c O loss = (1 e 8 β e β γ )

γ = t eff τs τ s t eff O pred = V(1 e 8 β e γ β ) V =0.85 t eff =1.4 µs X, Y X c (α),y c (α) X c (α) =D jα P 0 D jα X c cos α 4β + Y c sin α 4β Y c (α) =D jα P jπ 8β D jα Y c cos α 4β X c sin α 4β α O α =0.15 β =1 A = X; A c = X c+y c B = Y B c = X c Y c O ideal =(cos4α 0 β +sin4α 0 β)e α 0 α 0

O pred =Ve γ β (cos 4α 0 β +sin4α 0 β)e α 0 V =0.85 t eff =1.4 µs β 1 P ±jα0 1 ( ˆX c ± Ŷc) β α 0 β + α 0 =tan4α 0 β α 0 D jα0 P jα0 β β 1 ( ˆX c + Ŷc) P α= jπ 16β W = II c XX c +YY c ZZ c ψ = 1 ( gg + ee )

β W < 0 W F β β =0 1 ( g + e ) 0

W = II ZZ XX + YY F = II + XX YY + ZZ F > 0.5 a) b)

M m ψ m = M m ψ ψ M mm m ψ {X, Y, Z} X : 1 Y : 1 Z : ( ) 1 1 1 1 ( ) 1 j j 1 ( ) 1 0 0 0 c, c, 1 c, 1 ( ) 1 1 1 1 ( ) 1 j j 1 ( ) 0 0 0 1 c c c ψ m = ψ q ψ c ψ cav X : N ( β + β ) N ( β β ) Y : N ( β j β ) N ( β + j β ) Z : β β ψ B

ψ B = 1 ( g, β + e, β ) X Y e m th m β m =3 β = 3 ψ = C m e, m + n m C n g, n C m = m β Ẑ +1

ψ cav = N ( β C m m ) 1 ψ cav = m β β = 3 m th m =3 ψ = C m e, m + n m C n g, n C n n th C n = n β Z +Z ψ c = N n 3 C n n

a a α = α α ap = Pa

0 L = N ( β + β ) 1 L = N ( jβ + jβ ) N 1 β 0 L 0 L, 1 L 1 L P

S n (θ) =e iθ n n S( θ) S( θ)= S n (θ n ) n=0 θ = {θ n } n=0

ψ = N ( β,β + β, β ).

a) 1 P L b) 0 L X Im(α) 0 - c) Readout (mv) 40 0-40 0-0 Re(α) 100 Time (μs) 1 0-1 00 Parity 0 L 1 L N ( 0 L + 1 L ) P

b) Signal (mv) 1 a) c) d) x y 8 x y 4 0 0 x -10-0 -30-40 Qubit drive detuning (MHz) initial y manipulation final 6B;m` RyXk, * pbiv K MBTmH ibqm rbi? i? al S ; ix U V h? T?QiQM@ MmK#` bthbiibm; /m iq i? /BbT`bBp BMi` +ibqm HHQrb QM iq T`7Q`K [m#bi `Qi @ ibqmb +QM/BiBQM/ QM T?QiQM MmK#` bi i. U+? Ti` 8VX U#V h? + pbiv bi i BM i? T?QiQM MmK#` # bbb Bb r`biim b ψc = n cn n r?` cn Bb +QKTHt MmK#`X U+V "v /`BpBM; i? [m#bi +QM/BiBQM/ QM i? T?QiQM MmK#` bi i n bm+? i? i i? [m#bi bi `ib M/ M/b # +F g - bh+ibp MmK#`@/TM/Mi `#Bi` `v T? b ; i Bb TTHB/ UaL SVX U/V h?bb //BiBQM H T? b K MB7bib BibH7 QM +? T?QiQM bi i +QKTQMMi cn X _T`Q/m+/ 7`QK (>`b i HX- kyr8)x Rde

σ x = σ y = σ z = 1 σ x σ y = iσ z σ y σ z = iσ x σ z σ x = iσ y e iθσ n = 1 cos θ + iσ n sin θ e i π σn e i π σm = σ n σ m H / = 1 (ξ + ξ )σ x + 1 i (ξ ξ )σ y + 1 σ z ξ ξ σ x σ y δt U =e i δt 0 H(t)dt ξ(t) ξ(t) =0 t<0 t>δt

A x =e i A σ x B y =e i B σ y ξ Ω(t)σ x Ω(t) σ y π/ Uˆx π/ = ( X π/ X π/ ) N Xπ/ = ( e i π 4 σx e i π 4 σx ) N e i π 4 σx

N π π (1 + ϵ) [ U ˆx π/ = e i π 4 (1+ϵ)σ x e i π N 4 x] (1+ϵ)σ e i π 4 (1+ϵ)σ x [ Nπ =e i (1+ϵ)+ π ] 4 (1+ϵ) σ x e iθ/σ x Z =cosθ Uˆx π/ 0 Z =cos [ Nπ(1 + ϵ)+ π (1 + ϵ)] =( 1) N+1 sin [ πϵ + Nπϵ] ϵ 1 Z ϵ Z ( 1) N+1 [ Nπϵ+ π ϵ] π π/m Uˆx π/m = ( X π/m ) mn Xπ/ = (e i π m σ x) mn e i π 4 σ x σ x σ y σ y σ y =cosφσ y sin φσ x X Y

U = Y π/ (X π Y π X π Y π ) N X π/ =e i π 4 σy (e i π σx e i π σy e i π σx e i π σy ) N e i π 4 σx Y σ y X Y π e i π σ x e i π σ y e i π σ x e i π σ y = σ x σ yσ x σ y = σ x [cos φσ y sin φσ x ] σ x [cos φσ y sin φσ x ] = [cos φσ x σ y +sinφ][cosφσ x σ y +sinφ] = 1 i sin(φ)σ z = 1 cos(π +sin(φ)) + iσ z sin(π +sin(φ)) =e iσz(π+sin(φ)) Z π/ X/Y Z Z =( 1) N+1 sin(n sin(φ)) φ/(π) 1 Z ( 1) N+1 Nφ

U =(X π Y π X π Y π ) N X π/ = ( e i π σ x e i π σ y e i π σ x e i π σ y) N e i π 4 σ x σ x σ y σ x + δσ z σ y + δσ z δ e i π σ x e i π σ y e i π σ x e i π σ y = σ xσ yσ xσ y = [σ x + δσ z ][σ y δσ z ][σ x + δσ z ][σ y + δσ z ] = [ σ x σ y + δ(σ z σ y σ x σ z )+δ ][ σ x σ y + δ(σ z σ y + σ x σ z )+δ ] 1 δiσ x = 1 cos( δ)+iσ x sin( δ) =e δiσ x X N Z 4Nδ V out =(1+ϵ)[cos(ω IF t φ)+γ]cos(ω LO t)+(1 ϵ)[sin(ω IF t + φ)+γ]sin(ω LO t) ϵ φ γ

V out =cos(ω IF t)cos(ω LO t)+sin(ω IF t)sin(ω LO t) =cos([ω LO ω IF ]t) V out =(1+ϵ)cos(ω IF t)cos(ω LO t)+(1 ϵ)sin(ω IF t)sin(ω LO t) =cos([ω LO ω IF ]t)+ϵ cos([ω LO + ω IF ]t) ϵ ϵ =10 P dbc /0 P dbc V out =cos(ω IF t + φ)cos(ω LO t)+sin(ω IF t + φ)sin(ω LO t) =cos(ω LO t)[cos(ω IF t)cos(φ) sin(ω IF t)sin(φ)] +sin(ω LO t)[cos(ω IF t)cos(φ) sin(ω IF t)sin(φ)] =cos(φ)cos([ω LO ω IF ]t) sin(φ)sin([ω LO + ω IF ]t) tan(φ) tan(φ) =10 P dbc /0 P dbc

V out =[cos(ω IF )+γ]cos(ω LO t)+[sin(ω IF t)+γ]sin(ω LO t) =cos([ω LO ω IF ]t)+γ [cos(ω LO t)+sin(ω LO t)] =cos([ω LO ω IF ]t)+γ sin(ω LO t + π/4) F = [χ χ ]

α / π = 50MHz

α q 8.453 ξ π φ π 5.5 10 3 ϵ 0.0 σ τ X Y π/ /π

a) b) (1 3.5e 3) (1 1.0e 3)

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10 4.63e 05 ±3.4e 05 7.9e 06 ±3.7e 05.0e 06 ±6.3e 05.6e 05 ±3.4e 05 4.9e 05 ±9.7e 05

1 1e 04.1e 06% 50 > 60 σ z

.8e 5 8.9e 5 X π/ π X Y

(1 8.e 4) (1 1.4e 3)

1 8.e 04 1 1.4e 3 (0IF) 1.1e 05 ±.7e 05 1.4e 06 ±8.5e 05 8.1e 05 ±1.3e 04

ρ Q(α) =F{C a (λ)} ] C a (λ) =Tr [ρe λ a e λa F {} = 1 π d λe αλ α λ Q(α) = 1 π [ d λe αλ α λ Tr ρe λ a e λa ] 1 π d β β = Q(α) = 1 [ρ π Tr 3 ] d λd βe λ (α β) λ(α β ) β β λ e λ µ λµ = π δ(µ) Q(α) = 1 [ρ π Tr ] d βδ(α β) β β = 1 Tr [ρ α α ] π = 1 π α ρ α. α

ρ W (α) =F{C s (λ)} C s (λ) =Tr[ρD(λ)] F {} = 1 π d λe αλ α λ W (α) = 1 π d λe αλ α λ Tr [ρd(λ)]. α, λ α + iα,λ + iλ e αλ α λ = e i(α λ α λ ) D(λ) =e λa λ a = e iλ ( a ) ( +a iλ a ) a i = e iλ λ T P =λ T X=λ x C s (λ) =Tr[ρD(λ)] = dx x ρd(λ) x.

W (α) = 1 π d λdxe i(α λ α λ ) x ρd(λ) x. D(λ) x = e iλ λ T P =λ T X=λ x = e iλ λ T P =λ x + λ = e iλ λ e iλ (x+λ ) x + λ. W (α) = 1 π = 1 π d λdxe i(α λ α λ ) e iλ λ e iλ (x+λ ) x ρ x + λ d λdxe iλ (λ +x α ) e iα λ x ρ x + λ. dµe iµν =πδ(ν) W (α) = π = π = π dλ dxδ(λ +x α )e iα λ x ρ x + λ dxe iα (α x) x ρ x +α x dxe 4iα (α x) x ρ α x u =(x α ) D(α) u = eiα α e iα u α u u D (α) = α + u e iα α e iα u W (α) = 1 π = 1 π due iα u e iα α e iα α e iα u u D (α)ρd(α) u du u D (α)ρd(α) u.

P P x = x W (α) = 1 π = π du u D (α)ρd(α)p u dv v D (α)ρd(α)p v = π Tr [ D (α)ρd(α)p ] = π Tr [ D(α)PD (α)ρ ] P α = D(α)PD (α) W (α) =Tr[D α PD αρ] Q n (α) =Tr[D α n n D αρ] W (α) = i,j W(α) i,j ρ i,j Q n (α) = i,j Q(α) i,j ρ i,j W(α) =D α PD α, Q(α) =D α n n D α

W(α) W i,j (α) = j D α PD α i D α a =(a α)d α Pa = ap D αa =(a + α)d α Pa = a P. ad α PD α =αd α PD α D α PD αa D α PD αa =α D α PD α a D α PD α W(α) W 0,0 (α) = 0 D α PD α 0 = 0 α = e α

W k,0 (α) = 0 D α PD α k = 1 k 0 D α PD αa k 1 = α k W k 1,0 (α). W(α) W T (α) =W (α) W 0,k (α) = α k W 0,k 1 (α) =W k,0(α). W k,l (α) = l D α PD α k = 1 k l D α PD αa k 1 = 1 (α W k 1,l (α) ) lw k 1,l 1 (α). k W l,k (α) = k D α PD α l = W k,l(α). n max (n max 1) α n max 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 given density matrix. Parameters ---------- basis : integer The truncation number of the density matrix which will be used to determine the Wigner function. alpha : complex matrix An array of complex values which represent the displacement amplitude for a set of measurements Returns ------- Wmat : complex 4-dim array Values representing the design matrix to create a Wigner function given 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(-.0 * np.abs(alpha) ** ) for n in range(1,basis): # calculate 0><n and n><0 Wmat[0][n] = (.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] = (.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 Q n (α) Q n (α) =Tr[Q n (α)ρ] Q n i,j(α) = j D α n n D α i ad α 0 0 D α = αd α 0 0 D α

D α n n D α = 1 n Da 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 D n 1 n 1 D α). Q n i,j(α) Q 0 0,0(α) = 0 D α 0 0 D α 0 = e α Q 0 k,l(α) = l D α 0 0 D α k = 1 l l 1 ad α 0 0 D α k = α l l 1 D α 0 0 D α k = α l Q 0 k,l 1(α) Q T (α) =Q (α) Q n l,k(α) =Q n k,l (α). Q n k,l = 1 n ( lkq n 1 k 1,l 1 (α) α kq n 1 k 1,l (α) α lq n 1 k,l 1 (α)+ α Q n 1 k,l )

n th Q n (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 given density matrix. Parameters ---------- basis : integer The truncation number of the density matrix which will be used to determine the generalized Q function. alpha : complex matrix An array of complex values which represent the displacement amplitude for a set of measurements Returns ------- Qmat : complex 5-dim array Values representing the design matrix to create a generalized Q-function given 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 photon Qmat[0][0][0] = np.exp( -np.abs(alpha) ** ) for k in np.arange(1,basis): # calculate k><0 for 0 photon Qmat[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 photon Qmat[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 photon Qmat[0][0][n] = np.abs(alpha)** * Qmat[0][0][n-1] / n for k in np.arange(1, basis): # calculate k><0 for n photon Qmat[0][k][n] = ( (1./n) * (np.abs(alpha)** * 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 photon Qmat[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)** * Qmat[l][k][n-1] ) ) Qmat[k][l][n] = np.conj(qmat[l][k][n]) return Qmat β ψ(t) = U(t) β = e ikt (a a) β = n e iktn e β β n n! t q = π qk q ψ(t q ) = n F n e β β n! n F n = e iπn q F q q F n+q = e iπ q (n+q) = e iπn q e 4πni e 4πqi = e iπn q = F n F n F n = q 1 p f p e iπpn q

f p = 1 q q 1 k F k e iπkp q = 1 q q 1 k e iπk q e iπkp q = 1 q q 1 k e iπ q k(k p) ψ(t q ) = = q 1 p q 1 p = 1 q f p ( n f p βe ipπ q q 1 p=0 q 1 k=0 e β β n e iπkn q n! n e iπ q k(k p) βe ipπ q ) q = ( ψ(t ) = 1 e iπ 4 ) iπ β + e 4 β

d X = 0 1 0 0 0 0 1 0 0 0 1 0 1 0 0 0 Z = 1 0 0 0 0 ω 0 0 0 0 ω 0 0 0 0 ω (d 1) ω = e πi d X, Z d d = d X Z ZX = ωxz Z d = X d = I. j X j = (j +1)modd Z j = ω j j Y Y = ωxz d

d = G {±I,±X, ±Y,±Z, } g 1,...,g k G G G g 1,...,g k G = g 1,...,g k G = X, Z, I. d G d = X, Z, ωi d X,Z 3, G d S V S S = g 1,...,g l V S S S V S

V S S V S P ψ S P = N l (I + g l ). N 1 S P V S G d S C(S) {E j } E j g l = g l E j C(S) d g l S g l g k = g k g l

P V S g l {E i } S C(S) {E i } V S d =4,S = Z C(S) d =4 G 4 = X, Z, ωi 0 1 0 0 1 0 0 0 X = 0 0 1 0 0 0 0 1 Z = 0 ω 0 0 0 0 ω 0 1 0 0 0 0 0 0 ω 3 ω = e iπ S = Z Z 1 0 0 0 Z = 0 1 0 0 0 0 1 0 0 0 0 1 S P V S

1 0 0 0 P = 1 (I + Z )= 0 0 0 0 0 0 1 0. 0 0 0 0 P = 0 0 + 0 L = 0 1 L =. V S Z G 4 XZ = ω Z X = Z X C(S) 0, Z X d =4,S = X S = X 0 0 1 0 X = 0 0 0 1 1 0 0 0 0 1 0 0 X 1 0 1 0 P = 1 (I + X )= 1 0 1 0 1 1 0 1 0. 0 1 0 1

P = 1 ( 0 + ) c.c. + 1 ( 1 + 3 ) c.c. 0 L = 1 ( 0 + ) 1 L = 1 ( 1 + 3 ). ZX = ω X Z = X Z C(S) 1 ( 0 + ) 1 ( 1 + 3 ) X Z d =4,S = X, Z Z X S Z X = ω 4 X Z = X Z V S 1 0 1 0 P = 1 (I + Z )(I + X )= 1 0 0 0 0 1 0 1 0. 0 0 0 0 P = 1 ( 0 + ) c.c. ψ = 1 ( 0 + ) S = X,Z

d =8,S = X 4, Z 4 Z 4 X 4 (X 4 ) = X 4, (Z 4 ) = Z 4 Z 4 X 4 = ω 16 X 4 Z 4 = X 4 Z 4 ω = e iπ 4 V S P = 1 (I + Z4 )(I + X 4 ) P = 1 ( 0 + 4 ) c.c. + 1 ( + 6 ) c.c. 0 L = 1 ( 0 + 4 ) 1 L = 1 ( + 6 ). C(S) 1 ( 0 + 4 ), 1 ( + 6 ) S = X 4,Z 4 X, Z d =18 9

j = βω j ω = e πi d d βω j β j k δ j,k X j (j +1)modd X = e πi d a a a,a X j

j +1 Z j ω j j d =4,S = Z d =4 Z Z =( β β + β β ) ( iβ iβ + iβ iβ ) 0 L = β 1 L = β. X X = e πi a a d =4,S = X d =4 S = X

X =( β β + β β )+( iβ iβ + iβ iβ ) ( ) 1 = ( β + β ) c.c. + 1 ( iβ + iβ ) c.c ( ) 1 ( β β ) c.c. + 1 ( iβ iβ ) c.c X P = e iπa a 0 L = 1 ( β + β ) 1 L = 1 ( iβ + iβ ). Z Z a ax = ae iπa a = ap = Pa = X a. d =4 S = X C(S)

d =8,S = X 4,Z 4 d =8 X Z X 4 = e iπa a = P Z 4 Z 4 0 L = 1 ( β + β ) 1 L = 1 ( iβ + iβ ) X Z X 4 a