Quantum Computers Algorithms and Chaos Varenna 5 15

Quantum Computers, Algorithms and Chaos, Varenna 5 -15 July 2005 Quantum computation with solid state devices “Theoretical aspects of superconducting qubits” Rosario Fazio

“Di. Vincenzo list” • • • Two-state system Preparation of the state Controlled time evolution Low decoherence Read-out Geometric quantum computation Applications (Esteve) (Averin)

Outline Lecture 1 - Quantum effects in Josephson junctions - Josephson qubits (charge, flux and phase) - qubit-qubit coupling - mechanisms of decoherence - Leakage Lecture 2 - Geometric phases - Geometric quantum computation with Josephson qubits - Errors and decoherence Lecture 3 - Few qubits applications - Quantum state transfer - Quantum cloning

Solid state qubits Advantages - Scalability - Flexibility in the design Disadvantages - Static errors - Environment

Qubit = two state system How to go from N-dimensional Hilbert space (N >> 1) to a two-dimensional one?

All Cooper pairs are ``locked'' into the same quantum state

Quasi-particle spectrum There is a gap in the excitation spectrum D D T/Tc

Josephson junction j 1 I j 2 Energy of the ground state • Cooper pairs also tunnel through a tunnel barrier ~ -EJcosj • a dc current can flow when no voltage is applied • A small applied voltage results in an alternating current

SQUID Loop j. L F j. R

Dynamics of a Josephson junction j 1 +++++++ _______ X = j 2

Mechanical analogy

Washboard potential U(f)

Quantum mechanical behaviour The charge and the phase are Canonically conjugated variable From a many-body wavefunction to a one (continous) quantum mechanical degree of freedom Two state system

Josephson qubits are realized by a proper embedding of the Josephson junction in a superconducting nanocircuit Charge qubit 1 Charge-Phase qubit Flux qubit 104 Phase qubit Major difference is in the form of the non-linearity

Phase qubit U(f) Current-biased Josephson junction The qubit is manipulated by varying the current

Flux qubit X j 1 (t) j 2 The qubit is manipulated by varying the flux through the loop f and the potential landscape (by changing EJ)

Cooper pair box tunable: - external (continuous) gate charge nx - EJ by means of a SQUID loop

Cooper pair box Cooper pair number, phase difference voltage across junction current through junction

Cooper pair box

Cooper pair box CHARGE BASIS V EC å (n - n x) 2 N Charging EJ n n 2 å( n n IJ Cx Cj n +1 + n + 1 n ) n Josephson tunneling

From the CPB to a spin-1/2 In the |0>, |1> subspace Hamiltonian of a spin In a magnetic field H= Magnetic field in the xz plane

Coherent dynamics - experiments Chiorescu et al 2003 Nakamura et al 1999 NIST Schoelkopf et al, Yale See also exps by • Chalmers group • NTT group • … Vion et al 2002

Charge qubit coupling - 1 EJ 1 C Vx F EJ 2 C nx Cx EJ 1 C F EJ 2 C L nx Cx Vx Inductance

Charge qubit coupling - 2 EJ 1 C F EJ 2 C nx Cx Capacitance

Charge qubit coupling - 3 EJ 1 C F EJ C F F EJ 2 C nx Cx Josephson Junction

Tunable coupling Variable electrostatic transformer Untunable couplings = more complicated gating The effective coupling is due to the (non-linear) Josephson element The coupling can be switched off even in the presence of parasitic capacitances Averin & Bruder 03

Leakage The Hilbert space is larger than the computational space |m> |m+1> Consequences: a) gate operations differ from ideal ones (fidelity) ~Ec |0> b) the system can leak out from the computational Ej space (leakage) qubit |1> One qubit gate Leakage Fidelity Two qubit gate Fidelity

Sources of decoherence in charge qubits electromagnetic fluctuations of the circuit (gaussian) Z discrete noise due to fluctuating background charges (BC) trapped in the substrate or in the junction Quasi-particle tunneling

Reduced dynamics – weak coupling Full density matrix TRACE OUT the environment RDM for the qubit: populations and coherences

Reduced dynamics – weak coupling Ø q=0 ”Charge degeneracy” (e = 0 , W = EJ) no adiabatic term optimal point Ø q=p/2 ”Pure dephasing” (EJ =0 , W = e) no relaxation

Background charges in charge qubits z Fluctuations due to the environment HQ E E is a stray voltage or current or charge polarizing the qubit x Charged switching impurities close to a solid state qubit d i+d i E electrostatic coupling charged impurities Electronic band

g=v/g weak vs strongly coupled charges “Weakly coupled” charge Decoherence only depends on = oscillator environment “Strongly coupled” charge • large correlation times of environment • discrete nature • keeps memory of initial conditions • saturation effects for g >>1 • information beyond needed

EJ=0 – exact solution Constant of motion

EJ=0 – exact solution In the long time behavior for a single Background Charge ~ ~ The contribution to dephasing due to “strongly coupled” charges (slow charges) saturates in favour of an almost static energy shift

Background charges and 1/f noise Experiments: BCs are responsibe for 1/f noise in SET devices. Standard model: BCs distributed according to with yield the 1/f power spectrum from experiments Warning: an environment with strong memory effects due to the presence of MANY slow BCs

Slow vs fast noise Split “Fast” noise Slow noise ≈ classical noise • slow 1/f noise Two-stage elimination in general quantum noise • fast gaussian noise • fast or resonant impurities

Initial defocusing due to 1/f noise Paladino et al. 04 • Slow noise: x(t) random adiabatic drive g. M <W → adiabatic approximation • Retain fluctuations of the length of the Hamiltonian → longitudinal noise Large Path Nfl central limit theorem → • Static Approximation (SPA) variance • gaussian distributed z expand to second order H in x → quadratic noise Q x Optimal point see also Shnirman Makhlin, 04 Rabenstein et al 04 s 2

Initial defocusing due to 1/f noise Falci, D’Arrigo, Mastellone, Paladino, PRL 2005, cond-mat/0409522 z HQ x with recalibration Optimal point Initial suppression of the signal due essentially to Standard measurements no recalibration SPA inhomogeneuos broadening (no recalibration)