Coupled Superconducting Qubits Daniel Esteve, lecture 3

What would be the ideal two-qubit experiment ? 1) Individual qubits initialized in their ground state 2) Preparation of wanted two-qubit state by switching ON and OFF microwaves + two-qubit coupling 3) Measure each qubit state and reconstruct joint probabilities a) Measure « longitudinal correlations » b) Measure « transverse correlations » i.e. perform individual qubit rotations before measurement.

What would be the ideal two-qubit experiment ? Q1 Q2 Initialization State preparationReadout/Analysis Correlations 1) « Longitudinal » correlations 2) « Transverse » correlations == ???

Measuring longitudinal correlations  z1  z2 All superconducting qubit detectors up to now measure along the energy basis i.e along  z 4 possible outputs N(i,j) Remark : only one detector could be OK even for measuring correlations although 2 are of course much better … + it is a bit of a cheat since scaled to N qubits it would represent an exponential slow down of the experiments !

What would be the ideal two-qubit experiment ? Q1 Q2 Initialization State preparationReadout/Analysis  Single-qubit rotations after preparation of two-qubit states

Measuring transverse correlations Use single-qubit rotations around axis n, angle  prior to the detection along  z 4 possible outputs N(i,j) Detection along axis Z  n Z z

Z versus X coupling Z coupling always ONOK (interaction does not change populations) X coupling always ON Need to measure very fast compared to two-qubit evolution time (includes single-qubit rotations to measure Transverse correlations) 1) Coupling efficiency 2) Detection Different properties Interaction picture : X coupling effective only when Z coupling effective whatever the relative qubit frequencies. (energy conservation)

Two-qubit experiments  … frozen detection g Non-adiabatic

Gate protocol for X coupling: resonant vs non-resonant qubits   2 -  1  R1 -resonant qubits: -tuning difficult (requires moving away from optimal point) Z I1 * > I0 * > drive Z I1 * > I0 * > drive qubit 1 qubit 2  non-resonant qubits: -NMR like coupling in the doubly rotating frame qubit 1 Yale FLICFORQ protocol: (team M. Devoret)

Possible implementations for flux and charge qubits (tunable inductance/capacitance) B. Plourde et al., PRB 70, (2004) A. Niskanen et al., cond-mat/ (2006) SQUID Tunable inductance D. Averin and C. Bruder, Phys. Rev. Lett. 91, (2003) Cooper-pair box Tunable capacitance VgVg

Gate protocol : resonant vs non-resonant qubits   2 -  1  R1 -resonant qubits: -tuning difficult (requires moving away from optimal point) Z I1 * > I0 * > drive Z I1 * > I0 * > drive qubit 1 qubit 2  non-resonant qubits: -NMR like coupling in the doubly rotating frame qubit 1 Yale FLICFORQ protocol: (team M. Devoret)

NMR like coupling in the doubly doubly rotating frame at optimal points Z I1 * > I0 * > resonant drive Z I1 * > I0 * > Protocol: - stay at P 0 and apply: - remove z rotations with single qubit gates 8t 0 gate 1 gate 2 Rigetti et al., PRL, 94 (2005)

Coupled Qubits CcCc C On Resonance: 1 qubit fidelity vs. coupling Typically: easier harder Phase qubit: harder easier Why is coupling easier? Wires |Z qubit | ~ 30  cQED Plus: fast readout, simultaneous measurement, tunable

B A CcCc CcCc C AB Coupled Qubits: Spectroscopy Flux Bias A  A /2  (GHz) Flux Bias B  B /2  (GHz) B qubit unbiased:A qubit unbiased: Off Resonant:

B A CcCc CcCc C AB Coupled Qubits: Spectroscopy Flux Bias B  B /2  (GHz) A qubit unbiased: Flux Bias A  A /2  (GHz) B qubit biased: Moves with bias on B S= 74 MHz Resonant:

Cross Coupling when Measurement is Delayed  When measure 1 state: pumps energy into 2 nd qubit, producing 0 -> 1 transition  P10P10 P11P11 P 01 same P01P01 P11P11 P 10 same A B Qubit gate easy to make, During measurement coupling still on ! Measurement of 1 state dissipates energy Fixed Coupling:

Time Scale of Measurement Crosstalk 16 GHz V(t) I(t)=C x dV/dt I(t) E/E 10 t [ns] on resonance 0 1 Small crosstalk for misalignment <1 ns theory: experiment: V(t)

Simultaneous Measure of Coupled Qubits: i-SWAP gate  S i-SWAP gate  PABPAB A B  osc  z-gate  /2 z-gate P 10 P 01 P 11 Eigenstate, Bell singlet

Tomography: Direct Proof of Entanglement  A B  state tomography I,X,Y

Fixed coupling of flux qubits … natural inductive coupling is XX at the optimal point E 1 -E 0  x1 -  0 /2 J.B. Majer et al., Phys. Rev. Lett. 94, (2005)  x1  x2 O1 O2 D1diffD2 No coupling at optimal point

Hannes Majer Floor Pauw Alexander ter Haar

A two-qubit molecule

H = h 1  z 1 + t 1  x 1 + h 2  z 2 + t 2  x 2 +j  z 1  z 1 asymmetry in h: E J and area (shift of f=0.5) asymmetry t: E J, junction ratio        symmetric qubitsasymmetric qubits      -   + 

results of fit I p1 = 512 nA ± 6nAt 1 = 0.45 GHz ± 0.2 GHz I p2 = 392nA ± 5nAt 2 = 1.9 GHz ± 0.1 GHz sigma = % ± 0.004% (difference areas) j = 0.50 GHz ± 0.03 GHz (coupling strength) no transitions to 3rd excited state: low transition probability

10.25GHz, -14dBm Rabi oscillations

Two Coupled Flux Qubits with Controllable Interaction Supported by AFOSR, ARO, NSFMS+S2006 Atsugi 28 February 2006 Configuration, measurement and characterization Single qubit spectroscopy Coupled qubit spectroscopy Coupled qubit manipulation Travis Hime Britton Plourde Paul Reichardt Tim Robertson Alexey Ustinov Cheng-En Wu Concluding remarks Introduction Controllable coupling of two flux qubits: theory Experiments on two flux qubits

Two Flux Qubits and a SQUID 35  m SQUID Qubits have interaction of the form      where  is a Pauli spin matrix Qubits coupled to each other via M qq : K 0 = -2M qq | I qA | | I qB | where I qA and I qB are qubit circulating currents Qubits are also coupled via the SQUID: this coupling depends on the SQUID current and flux biases Thus, one can use the SQUID to control the total coupling between the qubits SQUID Qubit B Qubit A

Circulating Current in dc SQUID vs. Applied Flux Plourde et al. Phys. Rev. B 70, (2004)

Variable Qubit Coupling Using dc SQUID B (B)(B) (A)(A) (A)(A) A QUBIT A QUBIT B

QUBIT A QUBIT B Numerical Values

Experiments on Two Flux Qubits Configuration, measurement and characterization

Two Flux Qubits, a SQUID and Flux Lines 180x205 nm 2  C 0 ≈ 6.5 fF 1  m Qubit junctions 215 x 250 nm 2, C ≈ 8.5 fF SQUID junctions 250 nm 30  m Qubit B Qubit A Flux line 1 Flux line 1 Flux line 0 Flux line 0 L q ~ 200 pH L J ~ 600 pH Loop inductance not negligible SQUID Two on-chip flux lines enable one to apply independent fluxes to any two of the three devices Large inductances to keep currents in flux lines small Need to measure the six mutual inductances: M foqA, M fiqA, M foqB, M f1qB, M f0s, M f1s Predictions require theory that includes loop inductance (Robertson et al., submitted) 35  m

Microwave Spectroscopy of Qubits A and B Qubit A Qubit B   = ± GHz d  / d  = 875 ± 9 MHz / m      = ± GHz d  / d  = 923 ± 6 MHz / m    Fit data to = (     ) 1/2,  is splitting at degeneracy point,  = 2I q (    1 photon 2 photon

“Intersecting” Anticrossing of |1> and |2> Qubit intersection frequency GHz Spectra measured at constant SQUID flux, 0.35    and hence at constant qubit-qubit coupling strength Zero SQUID bias current Minimum splitting ± 0.8 MHz Note absence of data for |1> near anticrossing |1> |2> Calculated |T f0 | 2 (a.u.)  A -  0 /2 (m  0 ) |T 20 | 2 |T 10 | 2 Matrix elements | | 2

Anticrossing of |1> and |2>: Experiment and Theory Calculated spectrum (GHz)  A -  0 /2 (m  0 ) |2> |1> Experiment Calculation For SQUID flux of 0.35    Minimum spitting 119 ± 2 MHz Minimum splitting ± 0.8 MHz |2> | 1>

Measurement Configuration

SQUID Readout Pulse bias current: detect switching events Repeat (say) 1000 times to determine probability Increment bias current and repeat  QA = 0.48  0  QA = 0.52  0  S = constant Determine current I S 50% for 50% switching probability

Two-Qubit Flux Map I S 50% vs flux 0 and flux 1 SQUID contribution has been subtracted, leaving only the contributions of the two flux qubits Contains 10,000 flux values Yields values of: M foqA, M fiqA, M foqB, M f1qB, M f0s, M f1s Typical double degeneracy point Lines of constant flux in qubit A Lines of constant flux in qubit B

Single Qubit Spectroscopy Three microwave frequencies

Coupled Qubit Spectroscopy Examples of previous coupled qubits: Pashkin et al. (2003) Charge qubit Berkley et al. (2003) Phase qubits McDermott et al. (2005) Phase qubits Majer et al. (2005) Flux qubits

Basis states: Symmetric triplet |11>, |S> ≡ (|01> + |10>)/2 1/2, |00> Antisymmetric singlet |A> ≡ (|01> − |10>)/2 1/2 The Coupled-Qubit Hamiltonian H 2qb = (−½  A  Az − ½  A  Ax ) + (−½  B  Bz − ½  B  Bx ) − ½K  Az  Bz |0> |1> |2> |3> Antiferromagnetic coupling: E 1 < E 2 Eigenstates (11) (S) (00) (A) Storcz and Wilhelm, Phys. Rev. A 67, (2003 (11) (S) (00) (A)

qubit 1 qubit 1 qubit 2 qubit 2  Harmonic oscillator・・・ ・・・ : Josephson junction readout SQUID for qubit 2 LC-resonator as q-bit coupler readout SQUID for qubit 1 multi-qubit control scheme  ・・・ Control signal ： RF line f

inner ring ： a flux qubit inner ring ： a flux qubit outer ring ： SQUID detector outer ring ： SQUID detector LC-resonator for qubit-qubit coupling LC-resonator for qubit-qubit coupling material ： Aluminum ( Tc ~ 1.2 K ) |０>|０>|０>|０> |１>|１>|１>|１> |０>|０>|０>|０> |１>|１>|１>|１> Qubit control by resonant MW pulses ( RF-line ) mesoscopic superconductingcircuit technology → prototype Q-computation mesoscopic superconducting circuit technology → prototype Q-computation Two-qubit entanglement control scheme LC-plasma mode qubit coupling LC-plasma mode qubit coupling

Capacitively coupled Cooper pair boxes CcCc  capacitive coupling C C swapping frequency: frequency scale for all protocols Case of resonant qubits : -modulable coupling ?

What has been done: - Easy manipulation using NMR techniques - Single shot measurement design failed: 40% contrast only - Optimal point leading to long coherence time limited by charged TLFs What we try to do: a two qubit swap gate

A iSWAP gate with resonant qubits: uncoupled eigenbasis coupled gate (universal) Protocol: Move to resonance with I b2 But not optimal for coherence P0P0 time T2T2 21 t0t0

NMR like coupling in the doubly doubly rotating frame at optimal points Z I1 * > I0 * > resonant drive Z I1 * > I0 * > Protocol: - stay at P 0 and apply: - remove z rotations with single qubit gates 8t 0 gate 1 gate 2 Rigetti et al., PRL, 94 (2005)

Chip design: electrostatic coupling left island energized Cc=50aF, Cg=40 aF, Cct=4.5 aF (~10%) C2D code  ~ MHz Cct Cg

Implementation Requirements: 50  4k  50  E J /E C ~2 =17GHz T 2 >0.5 µs <25aF T 1 > 5µs coupling and decoupling (protocol 1) ~ 2ns low pass2 ns rise time bias ~ 30aF T 1qubit gate << t swap << T 2 t swap ~ ns  ~ 70 MHz 3pF0.5µA p ~ 4 GHz << single shot sensitivity Q ~ 5 on-chip µw capa Qubit isolation 30pF

100µm 5 µm Implementation: Chip C g =15 aF C c =25 aF On chip µw capacitors 30 nm Al 500 eV O 2 Plasma 70 nm Al 3nF – 12 nF/mm 2 Leakage 1 mm G Breakdown 6V-2V 3 pF

4 cm coil inside Arlon 1000 PCB 8 true coplanar lines Rosenberger mini SMP (S 11 = -18 dB up to 20 GHz) 50  – 13 pF RC decoupler with SMDs Implementation: Sample holder 100µm

VgVg 13 pF F F 30mK 4K 300K -30 dB -20dB 4k  readoutgate 1k  3 pF T 50  Implementation: Setup

The work on SPEC G. ITHIER E. COLLIN N. BOULANT D. VION P. ORFILA P. SENAT P. JOYEZ P. MEESON D. ESTEVE A. SHNIRMAN G. SCHOEN Y. MAKHLIN F. CHIARELLO 1 0 Fluctuating environment A -meter theQuantronium 1µm box qp trap dcgatedcgate µw readoutjunction 2004