Quantum computing : principles and practical implementations

Transcription

1 Quantum computing : principles and practical implementations D. Vion QUAN UM ELECT RONICS GROUP Scott Adams/Dilbert 4 st ORAP Forum, March 29 th 28, Paris, France

2 Warning: several types of quantum processors Logic gate computing Adiabatic computing DWave more exotic: - one way QC - Quantum Turing machine running sequentially gates from a universal set of gates to implement an algorithm T. Lanting et al, PRX 4 (24) Energy minimization, by quantum annealing this introductory talk see K. Michielson s talk

3 Outline I. Basics of logic gate quantum computing H. X Y Z H II. Practical implementations

4 The adiabatic quantum processor Couplings encoding the problem Adiabatically establish the Hamiltonian B Read the spins - Encoding all problems not easy - Better suited for optimization problems - Role of decoherence and temperature vs speed not fully understood State of the art: Dwave (2 Josephson qubits): - Interesting tests and material simulations - Quantum speedup not demonstrated see K. Michielson s talk

5 Reminder : the classical gate computer n=log 2 (N) T (N states) Turing (937) slowing-down poly(n) U Elementary information: Processed with logic gate arrays bit or NOT and a single 2 bit gate (as XOR) are enough : Universal set Imperfections error detection based on redundancy : Ex: parity bit / byte Do the same with quantum bits to benefit from quantum superposition and entanglement

6 Quantum objects (bit) and superposition of states Mach-Zender interferometer photon

7 Quantum objects (bit) and superposition of states Mach-Zender interferometer A PM A PM B probability /2 /2 photon B

8 Quantum objects (bit) and superposition of states Mach-Zender interferometer A B PM A PM B probability /4 /4 /2 photon

9 Quantum objects (bit) and superposition of states Mach-Zender interferometer A B PM A PM B probability /4 /4 /2 photon

10 Quantum objects (bit) and superposition of states Mach-Zender interferometer A B photon PM A PM B probability!!! The photon goes along both paths at the same time «within the meaning of interference».

11 Quantum objects (bit) and superposition of states photon ϕ The photon goes along both paths at the same time probability photon = α + β which path measurement or 2 α β 2 2 = α

12 The quantum bit and single qubit gates qb = α + β iϕ iϕ θ θ cos e + sin e 2 2 x φ z θ ϕ y projective measurement ou 2 α 2 β important property: no-cloning Duplication of unknown state is impossible Single qubit logic gate : all rotations A few gates: r R x (, φ ) probability ( φ ) i ( φ ) ( φ ) ( φ ) cos / 2 sin / 2 = isin / 2 cos / 2 X, Y, Z, H = In practice, 8, 9 and one 45 rotations are enough But informational content not larger than classical bit: Power will come from having several qubits and entanglement

13 Quantum register and entangling 2-qubit gate n qubit register N = 2 n basis states... = p n Most general register state : reg = N p= a p p Universal set of gates (any unitary evolutions) - single qubit gates - «good» (entangling) 2-qubit gate control target CNOT (quantum XOR) qb qb qb2 qb2 ' qb qb2 qb2 '

14 What entanglement means control target CNOT (quantum XOR) qb qb2 qb2 ' +? 2? + 2 very strong quantum corelations measure qubit projects both! maximum entanglement

15 Entanglement is a ressource (dense coding as a simple proof) EPR pair qba + 2 qbb 2 bits b b 2 encodage transmission qba decoding (disentanglement) system states qb A Z qb A X qb A iy qb A + + qb A qb B H b b 2 2 classical bits transmitted with a single transmitted qubit!

16 Quantum algorithms Ensemble de n qubits N = 2 n états de base n... = p H X. Z H espace temps N p= a p p Y see A. Leverrier s talk

17 Example : find the period of a periodic function f(x) λ =? Shor (994) spectra x N=2 n samples n qubits ΠH x x TF + n qubits U f f ( x)

18 Example : find the period of a periodic function f(x) λ =? Shor (994) spectra x N=2 n samples n qubits ΠH x x TF + U f n qubits f ( x) N x= x

19 Example : find the period of a periodic function f(x) λ =? Shor (994) spectra x N=2 n samples n qubits poly(n) ΠH x x TF + U f n qubits f ( x) N x= x N x= x f ( x) quantum parallelism + entanglement!

20 Example : find the period of a periodic function f(x) λ =? Shor (994) f spectra x N=2 n samples n qubits ΠH x x TF + U f n qubits f ( x) f N x= x N x= x f ( x) j= m l, j= x l + jλ f quantum parallelism + entanglement!

21 Example : find the period of a periodic function f(x) λ =? Shor (994) spectrum x N=2 p samples n qubits ΠH x x TF K N / λ + U f n qubits f ( x) λ / K N x= x N x= x f ( x) j= m l, j= x l + jλ f λ if coprime quantum parallelism + entanglement! x= K N / λ x

22 Exemple : détermination de la période d une fonction Application : Factorisation des grands nombres (*) Tâche longue (base de la cryptographie) M =?? Shor (994) Nombres à 3 chiffres 26 chiffres Durée avec GHz Ordinateur quantique ~ mois ~ million d années 9 portes T=? 8 T x ( ) mod f x = a M a < M M (*) Méthode précédente avec et co-premier avec / 2 a λ m et M partagent un facteur commun déterminé par un algorithme classique.

23 Serious problem : decoherence i x = e x x = reg a ϕ 2 hazards:sudden death disease ϕ drift x... drive U hν (X)? autres d de liberté dj i 2 πν dt α β e + n = 2 ò X t ) dt X ( t ) p d ( relaxation déphasage aléatoire random dephasing sensitivity coef noise

24 Un gros problème: la décohérence Correction d erreur possible (malgré le non clonage) : encodage sur plusieurs qubits avec enchevêtrement Exemple: correction de relaxation (bit flip) ( E, E ) = (, ) ( U, U 2 ) = ( + + +, ) ( V, V 2 ) = ( + + +, ) ( U, U 2 ) ( V, V 2 ) X ψ = α + β E E X X Ne marche que si erreur <,% par porte quantique!

25 Fragility of quantum superpositions: decoherence reg iϕ N = α e,,..., + α e,,..., α e,,..., iϕ iϕ N Sudden death: relaxation Disease: pure dephasing qubit j environnment λ i (t) low-frequency noise j hω a hω ( λ ) i iϕ ( t) + be j j t e iϕ ( t) Γ t e ϕ T = Γ Γ ϕ T Γ = Γ = Γ ϕ

26 Fragility of quantum superpositions: error correction Use redundancy (despite non-cloning) Example: relaxation (bit-flip) correction ( L, L ) = (, ) ( U, U 2 ) = ( + + +, ) ( V, V 2 ) = ( + + +, ) bit flip or not detection ( U, U 2 ) ( V, V 2 ) X correction ψ = a + b L L X X syndroms Generalization to all 3 possible errors: 7-bit Steane code Works if error <.% per gate! Surface code: %!

27 Outline I. Basics of logic gate quantum computing H. X Y Z H II. Practical implementations

28 Outline I. General overview II. Qbits with superconducting circuits

29 The gate quantum processor A quantum Turing machine based on a universal set of gates U U? U 2? - qubit: 2-level-system ψ = a + b resetable to - A few single qubit gates U ψ - a 2-qubit entangling gate U 2 α, α, α 2, α 3,,, - qubit readout with high fidelity universal set

30 The local and network approaches U U 2

31 The local and network approaches U U 2 flying qubit quantum state transfer entanglement between separate small systems

32 Physical implementations NMR non scalable

33 Physical implementations NMR Photons qb gate 2qb gate non scalable single photon sources needed

34 Physical implementations NMR Photons Spins in solid - Implanted - Quantum dots - electronic - nuclear non scalable non deterministic yet (in progress) A. Morello Sydney

35 Physical implementations NMR Photons Spins in solid - Implanted - Quantum dots - electronic - nuclear A. Morello Sydney non scalable non deterministic yet in progress

36 Physical implementations NMR Photons Spins in solid Trapped ions - Implanted (or atoms) - Quantum dots - electronic - nuclear R. Blatt (Insbrück) non scalable non deterministic yet in progress DRF-DRT

37 Physical implementations NMR Photons Spins in solid Trapped ions Superconducting - Implanted (or atoms) circuits - Quantum dots - electronic - nuclear J. Martinis Google non scalable non deterministic yet in progress

38 Physical implementations (non exhaustive) NMR Photons Spins in solid Trapped ions Superconducting - Implanted (or atoms) circuits - Quantum dots - electronic - nuclear non scalable non deterministic yet in progress Most advanced not easily scalable Most advanced

39 The gate quantum computer roadmap photons spins QD ions superconduc. circuits qubits not really yes 5 7 -qubit gate F yes F>99.8 F>99. F>99.7, 99.9 readout F F>9% F>99.8 F>99. F>96, 99 2-qubit gate F F>95% yes / no 25µs F>97 ~ns F>96.5, 99.4 coherence T 2 /T 2q-gate?? yes / no 2 2, error correction demo no no yes yes logical qubit better than physical no no? no algorithm w/o error correc no no yes yes algorithm w error correc no no no no fault tolerance and scalability no no no no but try with surface code

40 The gate quantum computer roadmap photons spins QD ions superconduc. circuits qubits not really yes qubit gate F yes F>99.8 F>99.9 F>99.7, 99.9 readout F F>9% F>99.8 F>99.9 F>96, 99 2-qubit gate F F>95% yes / no 25µs F>97 ~ns F>96.5, 99.4 coherence T 2 /T 2q-gate?? yes / no > 2 error correction demo no no yes yes logical qubit better than physical no no marginally marginally algorithm w/o error correc no no yes yes algorithm w error correc no no no no fault tolerance and scalability no no no no but try with surface code best results not always obtained all at the same time

41 Actors in quantum computing (not exhaustive) ACADEMIC WORLD NATIONAL LABS: Lincoln Lab, JET, CEA LETI, BIG COMPANIES: - IBM => gate QC (7 qbits, > 5 soon) - GOOGLE + UCSB: gate QC (> 5) - INTEL + QTECH: gate QC (7 qbits) - MICROSOFT + QTECH : soft - ATOS: 4-qubit emulator + compilers see P. Mueller s talk see C. Piechurski s talk SMALLER COMPANIES: - DWAVE: Adiabatic QC, annealing with 2 qubits - RIGETTI Computing: Hard + soft - HYPRES: fabrication.

42 II. Practical implementation with superconducting qubits ) Making qubits with Josephson junctions 2) Circuit-QED architecture for qubit readout 3) Single-qubit gates with microwave and dc pulses 4) Example of the iswap two-qubit gate 5) Example of a simple algorithm: Grover with two qubits

43 . Making qbits with superconducting circuits a) Start from a classical LC electrical harmonic oscillator: L C

44 . Making qubits with superconducting circuits a) Start from a classical LC electrical harmonic oscillator: θ = t V ( t ') dt ' +Q potential energy E = θ 2 / 2L «kinetic» energy p E = Q 2 / 2C c ω = LC 2 2 θ Q H = + 2L 2C θ

45 . Making qubits with superconducting circuits a) Start from a classical LC electrical harmonic oscillator: b) Make it quantum ˆ θ = t Vˆ( t ') dt ' ˆ +Q θ, ˆ ˆ Q = ih potential energy E = θ 2 / 2L «kinetic» energy p Hˆ hω θ E c ˆ2 2 θ Qˆ = + =hω 2L 2C 2 Q = 2C a a

46 . Making qubits with superconducting circuits a) Start from a classical LC electrical harmonic oscillator: b) Make it quantum ˆ θ = t Vˆ( t ') dt ' ˆ +Q θ, ˆ ˆ Q = ih potential energy E = θ 2 / 2L «kinetic» energy p Hˆ hω θ E c ˆ2 2 θ Qˆ = + =hω 2L 2C 2 Q = 2C a a

47 . Making qubits with superconducting circuits a) Start from a classical LC electrical harmonic oscillator: b) Make it quantum by fullfilling 2 conditions: hω >> θ L nh, C pf ν 5GHz hν T = 3 mk 8 P( ) < k T kbt B 4 nκ ω >> κ >> θ Q energy decay rate superconductors

48 . Making qubits with superconducting circuits a) Start from a classical LC electrical harmonic oscillator: hω >> k T Q >> b) Make it quantum by fullfilling and c) Replace the inductor by a nonlinear one B => a Josephson junction 4 ω 3 2 Hˆ θˆ Ep = θ 2 / 2L ˆ2 2 θ Qˆ = + =hω 2L 2C ˆ +Q a a hω θ L ( θ ) J ( e) 2 / 2 / EJ = h cosθ qubit ω Hˆ E p = E J cos( θ ) = E cos( θ ) + J θˆ hω ˆ 2 Q 2C ˆ +Q hω 23 hω 2 θ

49 . Making qubits with superconducting circuits a) Start from a classical LC electrical harmonic oscillator: hω >> k T Q >> b) Make it quantum by fullfilling and c) Replace the inductor by a nonlinear one B => a Josephson junction insulator S S 2e ω L ( ) J θ θˆ ˆ +Q E p = E J cos( θ ) Cooper pair box qubit hω θ ˆ ω ˆ σ / 2 H = h z

50 . Making qubits with superconducting circuits a) b) c) d) Split the Josephson junction (SQUID) to make the qubit magnetically tunable insulator S S Φ 2e L (, ) J θ Φ θˆ Φ ˆ +Q E p = E J cos( θ ) qubit ( Φ) H = hω σ z hω ˆ ˆ / 2 θ

51 . Making qubits with superconducting circuits: summary The transmon qubit is a kind of superconducting LC oscillator ω - at microwave frequency - cold enough at ω - nonlinear at the single photon level - thus behaving as a 2LS - electric dipole - magnetically tunable - driven resonantly at ω Φ ω ( Φ) ˆ +Q H ˆ = h ˆ σ / 2 z 4µm 2µm

52 . Making qubits with superconducting circuits: summary The (Cooper pair box) transmon qubit is a kind of superconducting LC oscillator ω - at microwave frequency - cold enough at ω - nonlinear at the single photon level - thus behaving as a 2LS - electric dipole - magnetically tunable - driven resonantly at ω Φ ω ( Φ) ˆ +Q H ˆ = h ˆ σ / 2 z 2 nm Al/Al 2 O 3 /Al

53 . Making qubits with superconducting circuits Ex: transmon qubit = nonlinear superconducting LC oscillator at 5- GHz Φ artificial two-level atom hν ( Φ) Al/Al 2 O 3 /Al 2 nm 2µm or effective spin /2 ˆ ˆ / 2 H = hν σ z 4µm

54 Fabrication technique ) e-beam patterning 2) development 3) first evaporation 4) oxidation 5) second evap. 6) lift-off 7) electrical test small junctions Al/Al 2 O 3 /Al junctions O 2 e - e-beam lithography PMMA PMMA-MAA SiO 2 small junctions Multi angle shadow evaporation

55 2. Qubit readout: introduction α >+β > a b +? t meas << T a b + a b +?? To measure β 2 : repeat same experimetal sequence ON OFF ON OFF Cool down Q. state preparation measure Cool down Q. state preparation measure time

56 2. Circuit-QED architecture to readout a Josephson qubit Couple qubit to a resonator like atoms in quantum electrodynamics Cooper pair box qubit = artificial atom D CPW resonator A. Blais et al., Phys. Rev. A 69, 6232 (24) A. Walraff et al., Nature 43, 62 (24) I. Chiorescu et al., Nature 43, 59 (24)

57 2. Circuit-QED architecture to readout a Josephson qubit Couple qubit to a resonator like atoms in quantum electrodynamics A. Blais et al., Phys. Rev. A 69, 6232 (24) A. Walraff et al., Nature 43, 62 (24) Jaynes-Cummings Hamiltonian (dispersive) hωσ ω + c σ + σ - + Ĥ = - ˆ ˆ ˆ Z + h a a + hg( ˆ aˆ + ˆ a ˆ ) 2 >> G ω c ω r E resonator (ω c ) d qubit ν h + Ĥ eff = - ( ωge + χ)ˆ σz + h( ωc - χσˆ Z ) â â 2 χ = G 2 / cavity frequency = ω + c χ if qubit projected on ω - c χ

58 2. Circuit-QED architecture to readout a Josephson qubit Vcos( 2π f t) f ' = f -χσ Z ( π φ ) Vcos 2 f t + ν a b + reflected phase φ φ φ frequency f/f

59 2. Circuit-QED architecture to readout a Josephson qubit Homodyne detection of reflected signal iωt Ve C c c iωt + φ(t) Ve A(t) φ ( t) CPB π φ - φ = 8 χ L.O κ φ π frequency ω/ω c

60 2. Circuit-QED architecture to readout a Josephson qubit Typical implementation use distributed coplanar waveguide resonator CPW ~ 6 GHz Nb E j Nb

61 2. Circuit-QED architecture to readout a Josephson qubit Typical implementation (Quantronics) with λ/2 coplanar waveguide resonator 5 mm (f =6.5GHz) Q=7 8µm 4µm (optical+e-beam lithography) =GHz => φ - φ

62 2. Circuit-QED architecture to readout a Josephson qubit A key figure of merit is the single-shot readout error rates (or readout fidelity ) > > P()= -ε Fidelity = - ε P()= -ε - ε Fidelity limited by - qubit relaxation during measurement time - noise of first amplifier p() Quantronics 24 (Josephson bifurcation amplifier) ε =2.4% Today : ε, ε <.2% in 5 ns with quantum limited parametric amplifiers.2 ε =%. 5 5 equivalent Rabi pulse duration (ns) 2

63 3. Single qubit gates: rotation around X a(t) cos(ω t + ) ω > Z α = a ( t ) dt Y X >

64 3. Single qubit gates: rotation around X ϕ in equator b(t) cos(ω t + ϕ) ω > Z β = b ( t ) dt X ϕ X ϕ Y >

65 3. Single qubit gates: rotation around Z Φ ω Φ ( t) > γ = δω ( t ) dt Z X ϕ Y Combine X, Z or X, X ϕ => generate all U Today: fidelity > 99.9 % R. Barends et al., Nature (24) >

66 4. Decoherence in quantum circuits Relaxation (Spontaneous emission) hω ( ) λ i Pure dephasing λ i (t) hω i ( t) + α βe ϕ t D λ, H = h λ Γ = π D, S () 2 ϕ λ z λ π T = Γ = D S 2 ( ω ) 2 λ, λ environmental density of modes at qubit frequency D λ,z ω = λ iϕ ( t ) Γ2t fϕ ( t) e e T Γ = Γ = Γ ϕ Low-frequency noise

67 4. Decoherence in quantum circuits Macroscopic sources : real part of the impedance of the circuits Microscopic sources : moving charges and flipping spins R e - N g +dn g (t) Φ+dΦ(t) Spin flips R ULTIMATE LIMITS ON COHERENCE TIMES UNKNOWN YET

68 4. Decoherence in quantum circuits Best results with transmon R. Barends et al., PRL (23) =3 = 2

69 5. Simple example of the iswap /2 two-qubit gate fixed capacitive coupling Φ H / h ν ν = A B A B A B A B σ z σ z g σ σ σ σ ( ) I II qubits coupled only on resonance ν = ν ( π U ) SWAP int = i 2g Fastest entangling gate + i 2

70 5. Simple example of the iswap /2 two-qubit gate fixed capacitive coupling Φ 2 µm

71 4. Simple example of the iswap /2 two-qubit gate fixed capacitive coupling Φ (Quantronics 2) readout resonator qubits mm coupling capacitor frequency control 2 µm

72 5. Simple example of the iswap /2 two-qubit gate 5 µm frequency control

73 5. Simple example of the iswap /2 two-qubit gate A. Dewes et al., PRL (2) frequencies (GHz) ν ΙΙ ν c Ι ν c ΙΙ ν Ι 2g/π = 9 MHz φ I /φ φ I,II /φ

74 4. Simple example of the iswap /2 two-qubit gate Drive QB II QB I X π 6.82 GHz 6.67 GHz 6.42 GHz f QB I QB II 5.32GHz 5.3 GHz Swap Duration 6.3GHz Occupation proba,,8,6,4,2 Data corrected from readout errors, iswap 2 2 Swap 3 duration (ns) swap duration (ns)

75 5. Characterizing the iswap /2 two-qubit gate readouts I II X,Y iswap Z tomo. I,X,Y I II ns 3*3 rotations*3 independent probabilities (P,P,P ) = 27 measured numbers Fit experimental density matrix ρ exp Compute fidelity ( / 2 / 2 F = Tr ρ ) th ρexpρth

76 4. Characterizing the iswap /2 two-qubit gate occupation proba, > >,8,6 >,4,2, > swap duration (ns) gate tomography F=92% Todays gate fidelity F>98%

77 5. The grover search algorithm: a benchmark for quantum speedup 77 x, y {,,,} f y (x)=, x = y, x y f ()= f ()= f ()= f ()= CLASSICALLY Random trial: probability of success /4 QUANTUM-MECHANICALLY Grover s search algorithm : probability can reach!

78 5. The grover search algorithm: principle 78 78gg State Preparation Oracle (O) Decoding (D) Readout > > H ( ) x f y ( x) x x D x x y + x y x 2 y Grover et. al., PRL 79, 997

79 6. The grover search algorithm: encoding 79 State Preparation Oracle (among 4) Decoding (D) > > Y π/2 Y π/2 iswap Z +/-π/2 Z +/-π/2 iswap X π/2 X π/2 x x y + x y x y

80 Implementation of the Algorithm 8 State Preparation Oracle (O) Decoding (D) > > Y π/2 Y π/2 iswap Z -π/2 Z -π/2 iswap X π/2 X π/2 f [φ(t)], a(t) Y(π/2) iswap Z(±π/2) iswap f X(π/2) ns Similar to: DiCarlo et.al., Nature 46(29) F=98% F=87% F=7%

81 5. The grover search algorithm: pulse implementation 8 State Preparation Oracle (O ) Decoding (D) Readout > > Y π/2 Y π/2 iswap Z -π/2 Z -π/2 iswap X π/2 X π/2 readout f [φ(t)], a(t) Y(π/2) iswap Z(±π/2) iswap X(π/2) X 2 (π) ns

82 5. The grover search algorithm: success probability 82 State Preparation Oracle Function (R) Diffusion Operator (D) Readout > > Y π/2 Y π/2 iswap Z ±π/2 Z ±π/2 iswap X π/2 X π/2 67 % f f f f 62 % 55 % 52 % query and check F i > 25 % for all oracles Quantum speedupachieved! Deweset. al.,prb RapidComm85 (22)

83 Results for Different Oracle Functions π 67 % 55 % 62 % 52 % F i > 5 % for all four oracles Demonstration of quantum speed-up Dewes et.al. PRB (22)

84 What we have seen: 84 - A microwave circuit can be quantum during up to µs - Single qubit gate error rate <.%? - Readout with error rate <.3 % - Two qubit gates with error rate ~ -3% - Quantum speedup demonstrated but algorithm error rate above -2% Recent demonstration - Error correction demonstrated with limited improvement - Operational 7 qubit processor demonstrated - «calculation» of very simple diatomic molecules Current research - Surface code: an architecture robust against decoherence at the price of a huge overhead - Stabilization of quantum states by dissipation engineering - Hybrid architectures combining circuits for fast gates and spins for long memory

85 The surface (stabilizer) code architecture data qubit X measure qubit Z measure qubit - Repeat measurements of all X,Z - Memorize errors - don t need to correct But

86 The surface (stabilizer) code architecture data qubit X measure qubit Z measure qubit - Repeat measurements of all X,Z - Memorize errors - don t need to correct - Define plaquette logical qubit - Define logical qubits by holes - Single qubit gate X L = product X - Two qubit gates by braiding But huge overhead: >36 qubit/logical error/gate!

87 V. Schmitt A. Dewes Thank you! QUAN UM ELECT RONICS GROUP A. Dewes, V. Schmitt, P. Bertet, D. Vion, D. Esteve Technical Support: P. Senat, P. Orfila

88

89 A few demos with ion and/or circuit quantum processors Grover algorithm with 8% success Factor 5 with 5 % success Implement Rabi model Hamiltonian by trotterization Other simple simulations Recompute ground state of simple diatomic molecules (quantum chemistry) /

90 Research strategies Demonstrate potentiality of adiabatic QC (Dwave) Complete roadmap to surface code => 7 qubit-plaquette (IBM, Google, QuTech-Intel, DRT) Develop spin qubit in semiconductors QuTech, Princeton, Sydney, DRT, Develop optical flying qubits - single qubit sources - conversion to optical qubits Develop alternative qubits immune to decoherence ( back to roadmap starting point) - Yale encoding in Microwave Schrödinger cat states with autonomous stabilization - Interface spins with minute long coherence (Sydney, DRF) - Topologically protected (Majorana) qubits /