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25 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

26 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] =

27 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

28 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

29 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

30 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

31 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

32 = 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

33 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

34 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

35 a) b) W(α) Q(α) ψ = N ( β β ) β =

36 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

37 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

38 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).

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

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

41 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)

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

43 3

44 I = I c sin πφ Φ 0 Φ 0 I c ϕ =πφ/φ 0

45 ( ) 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)

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

47 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

48 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

49 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

50 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)

51 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 )

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53 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

54 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 /π 1 W(α ) 0 W(α ) Im(α ) /π 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 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 ibqmb BM+Hm/ i? Mi M;HBM; Q7 T?QiQM MmK#` bi ib rbi? i? [m#bi KQ/ (CQ?MbQM 8j

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56 4

57 100 µs 10 ms

58 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

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

60 6B;m` 9XR, * pbiv Z1. [mbp HMiX hrq@+ pbiv j. +B`+mBi Z1. + M # BHHmb@ i` i/ #v + pbiv Z1. [mbp HMi /B ;` K rbi? 6 #`v@s`qi `bqm iq`b U VX "Qi? bvbikb mb irq + pbibb +QmTH/ iq bbm;h U `ib}+b HV iqkx PM + pbiv Bb /bb;m/ iq # HQM;@HBp/ 7Q` T?QiQM K MBTmH ibqm M/ biq` ; r?bh i? Qi?` + pbiv +QM@ i BMb H Fv KB``Q` UQp`@+QmTH/ QmiTmi +QmTH`V 7Q` iqk bi i /i+ibqmx U#V a?qrm Bb QM? H7 Q7 i? T?vbB+ H /pb+ rbi? irq #Qt@KQ/ `bqm iq`b M/ i` Mb@ KQM [m#bi +QmTH/ iq #Qi? KQ/bX hvtb+ H BKTHKMi ibqmb H p +? + pbiv rbi? 1 GHz bt ` ibqm BM i` MbBiBQM 7`[mM+Bb Q++m``BM; #irm 7 10 GHzX 9XRXk o`ib+ H i` MbKQM [m#bi AM Q`/` iq ` HBx i? irq@+ pbiv BHHmbi` ibqm b?qrm BM 6B;X 9XR- r M/ iq +Qm@ TH #Qi? + pbibb iq bbm;h [m#bix 6Q` TH M ` +B`+mBi /bb;m- i?bb + M T`7Q`K/ #v BKTHKMiBM; +QmTHBM; + T +BiQ`b r?b+? HBMF BM/BpB/m H + pbibb iq bbm;h [m#bi (CQ?MbQM i HX- kyryc J `B MiQMB i HX- kyrrc aizm i HX- kyrj)x 6Q` i?`@/bkmbbqm H `+?Bi+im` i?bb Bb Hbb Q#pBQmbX lmhbf TH M ` ;QKi`v- j. i` MbKQM [m#bi Bb TH +/ T?vbB+ HHv BMbB/ Q7 i? + pbiv M/ MiMM b ` mb/ iq +` i /BTQH +QmTHBM; #irm i? [m#bi M/ + pbiv `bqm iq`x AMbi /- r M/ iq /phqt [m#bi i? i /Qb MQi M/ iq ǵhbpƕ BMbB/ Q7 i? + pbiv #mi + M K`Hv mb M MiMM i? i ` +?b BMiQ i? + pbiv iq +` i bi`qm; [m#bi@+ pbiv +QmTHBM;X h?bb Bb i? # bbb 7Q` i? ǵp`ib+ H i` MbKQMǶ /bb;m r?b+? r rbhh QmiHBM BM KQ` /i BH?`X h? p`ib+ H i` MbKQM Bb /bb;m/ b +Q tb H KQ/ i? i tbbib #irm i? /@ 8N

61 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

62 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).

63 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

64 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

65 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

66 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 ) ( 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 ).

67 L J

68 cavity 1 Y in (ω) cavity Z 1 (ω) Z (ω) 4 mm substrate stripline 0.4 mm Z 1 (ω) Admittance (ms) Y in (ω) cavity Frequency (GHz) LC Z 1 (ω), Z (ω) Y in (ω)

69 O (ϕ 6 )

70 Ω

71 6B;m` 9Xd, * pbiv T`T ` ibqmx hq KBMBKBx i? z+ib Q7 bm`7 + HQbbb- r rbb? iq `KQp HH `bb/m 7`QK i? bm`7 + Q7 i? + pbiv r HHbX a?qrm Bb ` QTiB+ H BK ;b Q7 + pbiv r HH, U V mbbm; M +iqm- Ki? MQH +H MBM; T`Q+bb M/ U#V i? b K T`Q+bb rbi? M //BiBQM H H+QMQt /i`;mi +H MBM; bitx 9XkXk 6`B/; M/ rb`bm; /bb;m HH tt`bkmib b?qrm?` r` T`7Q`K/ rbi? +`vq;m@7` /BHmiBQM iq` QT` ibm; i ky KEX S`QT` bb;m H M/ i?`k H }Hi`BM; Kmbi # T`7Q`K/ iq Mbm` +QH/- T`Qi+i/ MpB`QMKMi 7Q` +? tt`bkmi Ub (a `b- kyrj)vx 1 +? irq@+ pbiv tt`bkmi ivtb+ HHv +QMi BMb irq BMTmi TQ`ib- QM 7Q` +? + pbivx //B@ ibqm HHv- QM Q7 i?b TQ`ib rbhh HbQ b`p b M BMTmi 7Q` i? [m#bi KQ/ UivTB+ HHv i?`qm;? i? /bb;m i/ ` /Qmi + pbivvx h? QmiTmi 7Q` i? ` /Qmi + pbiv Bb ivtb@ + HHv Qp`+QmTH/ iq HHQr bb;m H iq tbi i? + pbivx.tm/bm; QM i? ibqm- Bi?` /BbT`bBp Q`?B;?@TQr` ` /Qmi Bb mb/(: K#ii i HX- kyyec _/ i HX- kyry)x 6Q` H i` tt`bkmib U*?X NV- [m MimK KTHB}` Bb HbQ KTHQv/ 7Q`?B;?@}/HBiv ZL. [m#bi ` /QmiX JQbi tt`bkmib rbhh MQi +QMi BM M QmiTmi TQ`i QM i? biq` ; + pbivx h?bb Bb /m i? 7 +i i? i HH BM7Q`K ibqm Q7 i? biq` ; + pbiv KQ/ rbhh # Q#i BM/ #v T`Q#BM; i? [m#bi bi i /BbT`bBpHv +QmTH/ iq i? KQ/X a 6B;X 9X3 7Q` /i BHb QM 7`B/; rb`bm; 7Q` irq@+ pbiv BKTHKMi ibqmb BM@ +Hm/BM; }Hi`BM; M/ i?`k HBx ibqmx dy

72 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 )

73

74 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

75 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

76 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

77 6B;m` 9XRR, _ /Qmi +QM};m` ibqmx q mb irq /Bz`Mi [m#bi bi i ` /Qmi +QM};m` ibqmb- Bi?` rbi? Q` rbi?qmi M `@[m MimK HBKBi/ KTHB}`X U V qbi?@ Qmi KTHB}+ ibqm- bb;m Hb 7`QK i? ` /Qmi `bqm iq` ` bmi /B`+iHv iq i? >1Jh KTHB}+ ibqm +? BM i 9 EX AM H i` tt`bkmib- r BMi;` i/ [m MimK KTHB}`b U#V iq T`@ KTHB7v i? ` /Qmi bb;m HX q? p BKTHKMi/ #Qi? CQbT?bQM T Ki`B+ KTHB}`b UCS*V M/ CQbT?bQM #B7m`+ ibqm KTHB}`b UC" V 7Q` irq@+ pbiv tt`bkmibx de

78 5

79 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

80 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

81 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

82 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

83 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

84 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

85 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%

86 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

87 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

88 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

89 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

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

91 f 01 = ωq f π 0/ = (ωq α) π K Spectroscopy Frequency (GHz) 7.36 f 0 / f 01 K

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

93 K s n n =, 3 K Readout Signal (mv) Spectroscopy Frequency (GHz) π n n = 1,, K/π =163

94 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)

95 a) cavity qubit b) c) Wait time ( ) Displacement ( ) Wait time ( ) β β =0 β =0.5 β =1.0 β =1.5

96 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 = χ qs 3 MHz

97 a) b) Displacment amplitude ( ) Wait time ( ) Wait time ( ) Mean photon number ( ) a a

98 6

99 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 δα/α 0.0

100 a) cavity qubit m or b) Readout signal (mv) c) Drive amplitude (DAC value) 6000 Photon probability Displacement ( ) 4 R0ˆn,π P n P n α χ sr P 0 P n n = P n

101 a) cavity qubit or b) Readout signal (mv) c) Drive amplitude (DAC value) 1 3 Displacement ( ) C π P α P α α P α P α P α

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

103 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) Re(α) Q(α) = 1 π 0 D αρd α 0 β

104 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)

105 a) b) Im(α) 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

106 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

107 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

108 a) state cavity b) prep tomography cavity W(α) qubit Im(α) 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}

109 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

110 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

111 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

112 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

113 7

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

115 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

116 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 =

117 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

118 Re(α) - 0 Experiment time Im(α) 0-15 ns 65 ns 440 ns 815 ns Theory - Re(α) 0 Experiment time Im(α) 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

119 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

120 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

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

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

123 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

124 8

125

126 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

127 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.

128 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

129 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

130 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

131 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

132 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

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

134 [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

135 a) Im( ) Re( ) b) Z Re( ) Im( ) Z Y 0.5 Y 0.5 X X Z Y X Z Y X Z 1.0 Y 0.5 X Z 1.0 Y 0.5 X Z Y X Z Y X ψ = N ( β + β ) β, β ψ = { cos( θ ) g +sin(θ ) e } ψ = 1 { g + e iφ e }

136 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 d

137 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

138 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

139 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 =

140 σ z F F recov F F recov

141

142 a) b) Spectroscopy frequency (GHz) Normalized spectroscopy signal Photon number Im( ) β β + β β β β = d Ae Im(α) cos(d Im(α) +δ) S A δ

143 Im( ) Re( ) a) b) c) -4 d) 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

144 a) cavity 1 qubit or or Readout b) Re( ) 0 c) Re( ) 0 Im( ) Im( ) (radians) d) f) Phase (radians) Energy e) (photons) Phase (radians) δφ n Φ P e 1/ n 1/ n δφ C e nκτ / n κ τ nκτ > 1

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

146 9

147 β β

148 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 }

149 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

150 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

151 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

152 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 = V = IPα d α =(85± 1)% π V F V

153 (a) Im( ) 0 - Re( ) - 0 (b) g e g e Re( ) Mean Value (c) Fock state basis Re( ) 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

154 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

155 (a) Re( ) Im( ) (b) Mean Value Re( ) Im( ) 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 }

156 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

157 (a) 3 ideal photon loss visibility 0 Rotation ( ) Cat amplitude ( ) (b) 3 ideal photon loss visibility Displacement ( ) 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 =

158 ± ±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 V V pred =(1 p c )V =8% V 85%

159 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

160 ψ B

161 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 )

162 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

163 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 )

164 β W < 0 W F β β =0 1 ( g + e ) 0

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

166 M m ψ m = M m ψ ψ M mm m ψ {X, Y, Z} X : 1 Y : 1 Z : ( ) ( ) 1 j j 1 ( ) c, c, 1 c, 1 ( ) ( ) 1 j j 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

167 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

168 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

169

170 10

171 a a α = α α ap = Pa

172 0 L = N ( β + β ) 1 L = N ( jβ + jβ ) N 1 β 0 L 0 L, 1 L 1 L P

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

174 ψ = N ( β,β + β, β ).

175

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

177 b) Signal (mv) 1 a) c) d) x y 8 x y x 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 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

178 A

179

180 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

181 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

182 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

183 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

184 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

185 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

186 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 ]

187 α / π = 50MHz

188 α q ξ π φ π ϵ 0.0 σ τ X Y π/ /π

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

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192 a) b) 6B;m` Xd, S? b._ : 1``Q`X S? b M/._ : rbhh K MB7bi i?kbhpb 7`QK i? b K ivt Q7 TmHb i` BM Ur + HH Bi " JyVX S? b Bb M+Q// QM i? t@ tbb Q7 i? "HQ+? bt?` r?bh._ : Bb QM i? x@ tbbx 6B;m` U V b?qrb #7Q` imm@ BM; M/ };m` U#V 7i`X U = (Xπ Y π Xπ Yπ )N Xπ/ U XkkV hmm@mt Q`/` M/ +QMp`;M+ +`Bi`B h? i?q`v T`bMi/ BM i? T`pBQmb b+ibqm bbmkb i? i 7Q` +? TmHb@i` BM i?` Bb QMHv bbm;h ``Q` Ui? QM r ` i`vbm; iq /i+i M/ +Q``+i 7Q`VX h?bb K Mb i? i Qi?` mm+q``+i/ ``Q`b +QmH/ bfr i? }ib M/ BM im`m /BbiQ`i i? +Q`@ `+i/ T ` Ki`bX aqk Q7 i?b TmHb i` BMb ` KQ` bmbbibp iq ti` ``Q`b i? M Qi?`bX hq K/B i i?bb- r rbhh ǵ#qqibi` TǶ #v TB+FBM; T `ib+mh ` Q`/` Q7 i?b TmHb@imMBM;bX h? +m``mi T`7``/ Q`/` Bb b 7QHHQrb, _ #B- _ Kbv- snyuny- sr3y- S? b-._ :- sny- sr3yx A U"`B MV? p +?QbM i?bb Q`/` #+ mb sr3y Bb bmbbibp iq sny KTHBim/ ``Q`bc S? b M/._ : ` bmbbibp iq snyuny- M/ sr3y ``Q`bc M/ }M HHv +? M;b BM._ : T ` Ki`b rbhh z+i sny M/ sr3y /`Bp KTHBim/b Ui?Bb Bb /m iq Qm` +m``mi TmHb T ` Ki` /}MBiBQMb 7Q` RNR

193 e 05 ±3.4e e 06 ±3.7e 05.0e 06 ±6.3e 05.6e 05 ±3.4e e 05 ±9.7e 05

194 1 1e 04.1e 06% 50 > 60 σ z

195 .8e 5 8.9e 5 X π/ π X Y

196 (1 8.e 4) (1 1.4e 3)

197 1 8.e e 3 (0IF) 1.1e 05 ±.7e e 06 ±8.5e e 05 ±1.3e 04

198

199 B

200 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

201 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.

202 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.

203 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

204 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

205 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

206 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

207 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 )

208 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)

209 + 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

210 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

211 C

212 d X = Z = ω ω 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

213 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

214 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

215 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 X = Z = 0 ω ω ω 3 ω = e iπ S = Z Z Z = S P V S

216 P = 1 (I + Z )= P = 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 X = X P = 1 (I + X )=

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

218 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 ( ) c.c. + 1 ( + 6 ) c.c. 0 L = 1 ( ) 1 L = 1 ( + 6 ). C(S) 1 ( ), 1 ( + 6 ) S = X 4,Z 4 X, Z d =18 9

219 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

220 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

221 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)

222 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

223 d =18

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