Topologicaly protected abelian Josephson qubits

28 Pages·2007·2.31 MB·English

by  Lev Ioffe

Topologicaly protected abelian
Josephson qubits: theory and
experiment.
B. Doucot (Jussieu)
M.V. Feigelman (Landau)
L. Ioffe (Rutgers)
M. Gershenson (Rutgers)
Plan
• Honest (pessimistic) review of the state of
the field of superconducting qubits.
• Theoretical models of protected qubits and
their implementations in Josephson
junctions.
• Experimental results
• Conclusions
State of the Art in Superconducting
Qubits
Charge Charge/Phase Flux Phase
Saclay/Yale TU Delft/SUNY NIST/UCSB
NEC/Chalmers
E = E
Junction size
J C
Charging versus Josephson energy
Ideal Hamiltonian : H = 4E (nˆ −n )2 + E cos(ϕ+Φ)− Iϕ
C 0 J
∂
nˆ = i −number of Cooper pairs
∂ϕ
I − external current
Two states of the qubit differ by
charge nˆ if E E
(cid:0)
C J
phase ϕ if E E
(cid:0)
J C
State of the Art in Superconducting
Qubits
Charge Charge/Phase Flux Phase
Saclay/Yale TU Delft/SUNY NIST/UCSB
NEC/Chalmers
E = E
Junction size # of Cooper pairs
J C
In real life H = 4E [nˆ −n +δn(t)]2 +[E +δE (t)]cos(ϕ+δΦ(t))− Iϕ
C 0 J J
δn(t) charge noise, δE (t) critical current noise, δΦ(t) flux noise
J
Partial remedy:
Sweet spots
(linear protection)
Energy as a function of the most
dangerous parameter (charge, flux, etc)
State of the Art in Superconducting
Qubits
Charge Charge/Phase Flux Phase
Saclay/Yale TU Delft/SUNY NIST/UCSB
NEC/Chalmers
E = E
Junction size # of Cooper pairs
J C
Charging versus Josephson energy
History
• 1st qubit demonstrated in 1998 (NEC Labs, Japan)
• “Long” coherence shown 2002 (Saclay/Yale)
• Several experiments with two degrees of freedom
• C-NOT gate (2003 NEC, 2006 Delft and UCSB )
• Bell inequality tests being attempted (2006, UCSB) [failure due to low readout visibility!]
• 1st time domain tunable coupling of two flux qubits (2007, NEC Labs, Japan)
• Coupling superconducting qubits via a cavity bus (2007 Yale and NIST)
Relaxation and dephasing times
T ≡ Relaxation time
1 α0 +β1 T ≡ Dephasing time
2
ω 2π = 5 ÷10 GHz
0
Operation Logical quality
Design Group T T Visibility
2 1 time factor
Phase qubit UCSB ~ 85 nsec 110 nsec > 90%
Flux qubit NEC ~ 0.8 μsec 1 μsec ~ 20/30% ~ 0.02 μsec 10 – 100*
Transmon
Yale ~ 2 μsec ~ 1.5 μsec > 95 % ~ 0.05 μsec 10 - 200*
(CPB in a cavity)
*Projected values
Q ≅νT ≈103 −104
Physical 0 2
Q < 0.1 Q = 102 - 103
Logical Physical
What do we need?
Error rates that we need for
quantum computation.
Log (Program length)
10
Steane PRA (2003)
-Log Q
10 Logical
-Log R R=N/K - redundancy
10
Q < 102 - 103
Logical
Need Q >> 104 for many qubit system
Logical
Advantage of protection
Quantronium, (charge qubit at the
Charge qubit: Q<100
optimal point): Q=103-104
Flux qubit: away from At optimal point Q=103-104
the optimal point Q~10 T ~4μs T ~3μs T echo~4μs
1 2 2
Devices decoupled from the leading
source of noise in the linear order.
Devices decoupled in
? ?
higher orders
Protected Qubit (General)
Protected Doublet:
Special Spin Hamiltonians H with a large
P
k
number of (non-local) integrals of motion P, Q:
[H,P ]=0, [H,Q ]=0, [P ,Q ]≠0
k m k m
Any physical (local) noise term δH(t) commutes with
all P and Q except a O(1) number of each.
k m
Effect of noise appears in N order of the perturbation
Q theory:
m
δE~(δH(t) /∆)N-1 δH(t)
Simplest Spin Hamiltonian
H=Σ J x σx σ x + Σ J z σz σ z
kl kl k l kl kl k l
Rows Columns
P =∏ σz Q =∏ σx
k l l k l l
Crucial issues:
1. Which spin model has a large gap ∆?
2. Which spin model is easiest to realize in Josephson junction arrays?
Numerics for short range model
(nearest neighbor interactions)
z
H=Σ J x σx σ x + Σ J z σz σ
l
(kl) kl k l (kl) kl k
Low energy band
Splitting of two lowest
(degenerate) levels by
Lowest excited state in
random field applied to
the same sector as the
each spin and distributed
ground state
in interval (-0.05, 0.05).
Conclusion: relatively small arrays
Low energy states for 4*4 and 5*5 spin provide very good protection,
array. Low energy band contains 24 and 25 states especially in one channel!