Quantum computation and quantum information

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1 Quantum computation and quantum information Chapter 7 - Physical Realizations - Part 2 First: sign up for the lab! do hand-ins and project!

2 Ch. 7 Physical Realizations Deviate from the book 2 lectures, 4h, 4 parts ) Overview, general info on implementations 2) DiVincenzo criteria + Ion traps 3) Rare-earth impurities 4) Other systems (linear optics/superconducting qubits) Summary and comparisons

3 What do you need? Di Vincenzo criteria ) Qubit 2) Initialize 3) Single qubit gate 4) Two qubit gate 5) Long coherence time 6) Qubit readout 7) Scalability Reformulated from: D. P. di Vincenzo, The physical implementation of quantum computation, Fortschritte der Physik, 48, 77 (2) (

4 Rare-earth-ion quantum computing Rare-earth oxides Rare-earth ions doped in solids

5 Why Rare-earths? - Well shielded system! G. H. Dieke, Spectra and energy levels of Rare earth Ions in Crystals Element 4f electrons La 3+ Ce 3+ Pr 3+ 2 Nd 3+ 3 Pm 3+ 4 Sm 3+ 5 Eu 3+ 6 Gd 3+ 7 Tb 3+ 8 Dy 3+ 9 Ho 3+ Er 3+ Tm 3+ 2 Yb 3+ 3 Lu 3+ 4 Leads to long coherence times!

6 Rare-earths: ) Qubit 4.8 MHz 4.6 MHz Pr 3+ ±5/2 exc ±3/2 ±/2 66 nm = 495 THz Long coherence times: up to 3 s demonstrated Spatially close: strong interactions Ensembles: good photon interaction.2 MHz 7.3 MHz ±/2 ±3/2 ±5/2 But... compensation needed for inhomogeneities

7 Rare-earths: 2) Initialize Conceptual picture of crystal 3 Pr :Y 5 2SiO

8 Absorption Rare-earths: 2) Initialize inhom 5GHz hom khz Frequency 3 Pr :Y 5 2SiO

9 Absorption Absorption Rare-earths: 2) Initialize f 2 f f exc inhom 5GHz hom khz f 2 Frequency 5 MHz peak ~ 2 khz Frequency

10 Absorption Rare-earths: 2) Initialize Pr:Y 2 SiO 5 e 8 MHz Frequency.2 MHz 7.3 MHz aux

11 Absorption Rare-earths: 2) Initialize 4.8 MHz 4.6 MHz e MHz 7 MHz aux MHz 7 MHz aux

12 Absorption (αl) Absorption Rare-earths: 2) Initialize 633 pulses later....6 /2 g /2 e Initialized to the state!.4 /2 g 3/2 e.2 /2 g 5/2 e Frequency (MHz)

13 Frequency stabilized light source (- Hz)

14 Rare-earths: 3) Single qubit gate First, let s try a simple approach: Gaussian pulses e µs long gaussian pulse.2 µs long gaussian pulse Detuning (MH) Detuning (MHz) Problem #: Not the same Rabi frequency everywhere Problem #2: Dephasing due to the inhomogeneous width Problem #3: Wings excite non-initialized ions

15 Rare-earths: 3) Single qubit gate Second, let s try something more complicated: Complex hyperbolic secant pulses (sechyp) Amplitude: sechyp Frequency: tanhyp time (µs)

16 Rare-earths: 3) Single qubit gate Detuning (MHz) Sechyp pulses + Solve the problem with different rabi frequencies + Solve the problem with dephasing due to inhomogeneous broadening + No excitation outside initialized region - Can only handle pole to pole transfers

17 Absorption (al) Absorption (al) Rare-earths: 3) Single qubit gate /2 g /2 e /2 g 3/2 e Preparation of > state /2 Frequency (MHz).4 g /2 3/2 g 3/2 e e Single transfer to > state Frequency (MHz) 97.5% single transfer efficiency!... but only pole to pole transfers. Arbitrary states require a yet more complicated approach.

18 Rare-earths: 3) Single qubit gate e i R c e t i c 2 ) ( e i R c e t i c 2 ) ( 2 ) ( 2 ) ( c e t i c e t i c i R i R e e, Ω,, Ω, 2 ) ( c e c e t i c i i R e R R R Simultaneous fields The most complicated scheme: Dark state pulses

19 Rare-earths: 3) Single qubit gate 2 ) ( c e c e t i c i i R e = 2 2 i i e D e B B e B i z y x B D z y x B e 2 2) / cos( 2) / sin( 2) / sin( 2) / cos( 2 / i i i ie ie e U Qubit basis: = Arbitrary rotation around an axis on the equator Dark state pulses:

20 Absorption, αl Absorption, αl Absorption, αl Rare-earths:.4 + 3) Single qubit gate 6) Qubit readout Z measurement.2 /2g 3/2g x z y Frequency (MHz) X measurement /2g 3/2g 5 5 Frequency (MHz) Y measurement F op =.93.2 /2g 3/2g tr( ) I tr( X) X tr( Y) Y tr( Z) Z Frequency (MHz) Eq 8.48 in Nielsen and Chuang

21 Rare-earths:. Consider two ions in the crystal that are spatially close 2. One of the ions is excited on its optical transition 3. The dipole moment is different in the excited state. This leads to a shift of the second ion energy levels 4) Two qubit gate + e 4. Static dipole-dipole interaction scales as r Ion distance frequency shift nm line width nm line widths nm line widths

22 Rare-earths: 4) Two qubit gate Control e Target 2 e 2 Dark state pulse Thus, if q is in state a not gate will be performed on q2

23 Rare-earths: 4) Two qubit gate Control e Target 2 e 2 + Dark state pulse Thus, if q is in state NO gate will be performed on q2. Final state:, +,, Entangled Bell state

24 Rare-earths: 7) Scalability Dipole-dipole interaction Arbitrary volume in the crystal Ions close enough for interaction ion absorbing at frequency ion absorbing at frequency 2

25 Rare-earths: 7) Scalability Single instance! Hard to readout because the long qubit lifetimes give very few photons X Arbitrary volume in the crystal Ions close enough for interaction X X X ion absorbing at frequency ion absorbing at frequency 2

26 Rare-earths: 7) Scalability Minimal laser focus Read out ion of different species

27 Rare-earths: 7) Scalability q3 q q2 q4 q5 Readout ion q6 q7 q-qn: Qubit ions Requirements: Short lifetime Cycleable transition Can interact with qubit ions Current state of the art: Cerium

28 Quantum computing with linear optics Advantage: Photons are good information carriers, little decoherence One of the first proposals for quantum gates: - Non-linear optics by Milburn (989) Described in the book (photonic QC) Disadvantage: weak interaction and almost impossible to implement Two-qubit interactions via an intensity dependent Kerr-nonlinearity Impossibly small phase shift of ~ 8 Can improve strength by cavity coupling, but still difficult Knill, Laflamme, Milburn (KLM) shows in 2 that QC possible with only linear optical elements

29 Quantum computing with linear optics Qubits via a single photon: Single line with or photons is no good n not conserved Perpendicular polarizations ( photon either way) Dual rail ( photon in two spatial modes): Ψ in Ψ out - photon is in the upper path - photon is in the lower path Note: Each mode can have more than photon

30 Quantum computing with linear optics Single qubit gates with dual-rail: Phase shifter: φ Ψ in Ψ out Can be accomplished by e.g. a medium with n> Beam splitter: Ψ in Ψ out Can be made by partial mirror Arbitrary gate: φ φ 2 Ψ in Ψ out Combining the two elements can create any superposition state Very simple components!

31 Quantum computing with linear optics Initialization of photonic quantum computing: Single photons! p p 2 2 Spontaneous Parametric Down Conversion (SPDC) Non-linear, intensity dependent frequency conversion Tune power to obtain single events Wait until detector clicks heralded single photon Gives a probabilistic source of single photons

32 Quantum computing with linear optics The difficult part: a Two-qubit gate Despite name, a non-linear component is needed: Detectors! Non-linear Shift (NS): used to construct C-phase gate One rail gets a phase shift Ancilla qubits How can detection increase entanglement? Ancillas are detected, and for some answers, the main qubit is used. otherwise thrown away (probabilistic gate) Consider the state : + + +, no entanglement Add one qubit: + + +, partially entangled Post-select q3=: +, Maximally entangled! Trick is to add the ancilla qubits in a meaningful way

QC with superconducting qubits Introduction Motivation: QC scheme based on electronics might integrate better with conventional technology Basic mechanism: superconductivity

33 QC with superconducting qubits Introduction Motivation: QC scheme based on electronics might integrate better with conventional technology Basic mechanism: superconductivity via Cooper pairing + Josephsson junction Fully explained by BCS theory (Nobel prize 972 to Bardeen, Cooper and Schrieffer) Tunneling barrier Josephsson et al. 962, Nobel prize 973

34 QC with superconducting qubits Superconductivity a hand-waving explanation Consider a metallic structure: Net forced towards ion/other electron forms bound Cooper pair - F - Loose electrons allowed to move freely Core positive ions in a lattice Paired electrons are Bosons and can thus be in the same state Electron scattering vanishes No resistance, superconductivity Simple picture, a proper description requires full many body interactions Weak effect, requires very low temperatures to form bound

QC with superconducting qubits Qubits based on superconducting patches: superconducting Barrier Controlling the bias allows different number of Cooper pairs, n, on island Josephsson effect

35 QC with superconducting qubits Qubits based on superconducting patches: superconducting Barrier Controlling the bias allows different number of Cooper pairs, n, on island Josephsson effect quantization of charges Cooper pairs can coherently tunnel between the patches Superposition state of n and n + Cooper pairs Requires very low thermal noise ~ -3 mk Can be readout by a Single Electron Transistor coupled to the island ΔE~.5 K

currents: Flux qubit i - Single qubit gates Flux can be used to control the

36 QC with superconducting qubits Different qubit types possible i + Charge qubit Conjugate variable to the number, n: Phase qubit By left and right circulating currents: Flux qubit i - Single qubit gates Flux can be used to control the Josephsson junction interaction on charge qubits Magnetic fields can be used to affect Flux qubits

37 QC with superconducting qubits Two-qubit gates Multiple qubits in series can be coupled together capacitively All qubits driven unless one can switch off coupling to specific qubits LC-oscillation depends on state of qubit Other way: Use an inductor in resonantor mode: LC-circuit Conditional oscillations can then mediate qubit-qubit interactions ) Induce LC-oscillation conditioned on state of single qubit 2) Perform 2nd qubit gate conditioned on state of LCoscillator Flux qubit-qubit interaction via coupled magnetic fields Challenge to switched it off

Summary No protocol is fully scalable Ion traps Rare-earth systems Superconducting qubits Linear optics photonics NV-centers in diamond Nuclear Magnetic Resonance (NMR) Quantum dots Cold atoms (in

38 Summary No protocol is fully scalable Ion traps Rare-earth systems Superconducting qubits Linear optics photonics NV-centers in diamond Nuclear Magnetic Resonance (NMR) Quantum dots Cold atoms (in optical traps) Most qubits so far (8-4) Good fidelity Drawbacks: resource heavy Simpler system + solid state Strong signals from ensembles (QM) Drawbacks: multi-qubit gates not done Conclusion: We don t yet know which protocol will be the best so we have to look into many different directions! Solid system/electronics Fast operations Drawbacks: Difficult to get good large systems Simple components Photons are good qubits (Long T 2 and flying) Drawbacks: probabilistic makes it very slow Simple system Some room temperature operation Drawbacks: Limited scalability around NV Early success but no real quantum scaling

Quantum gates in rare-earth-ion doped crystals

Quantum gates in rare-earth-ion doped crystals Atia Amari, Brian Julsgaard Stefan Kröll, Lars Rippe Andreas Walther, Yan Ying Knut och Alice Wallenbergs Stiftelse Outline Rare-earth-ion doped crystals