understanding, controlling, and overcoming decoherence and noise in quantum computation nsf...
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Understanding, controlling, and overcoming decoherence and noise in quantum
computation
NSFSeptember 10, 2007
Kaveh Khodjasteh, D.A.L., PRL 95, 180501 (2005); PRA 75, 062310 (2007)
Quantum Computers are Quantum Computers are OpenOpen Systems: Decoherence Systems: Decoherence
Quantum computers use superposition and entanglement (“massive parallelism”)
Every real quantum system interacts with an environment (“bath”).
Environment is noisy & uncontrollable.
The environment acts as an uncontrollable observer, making random-time measurements, in random basis. Destroys superposition states.
+
Devastating for quantum computation:A sufficiently decohered quantum computer admits efficient simulation on a classical computer.
Is Decoherence a Problem?Is Decoherence a Problem?2 level system
(qubit)Switch time T Rabi flop [sec]
Decoherence time T2 (upper bound)
Quality factor T2/T (no. of ops)
Charge of electron in bulk GaAs
Exciton in GaAs quantum dot
Electron spin in GaAs quantum dot
Trapped ion (In) sdddd
Nuclear spin in liquid state NMR
Atom in microwave cavity
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How can we overcome decoherence?
Example: electron spins in a semiconductor quantum dot
Model:
Dipolar interaction among thenuclei: ¯
Somedata for GaAsJ = O(I § nAn) ¼1MHz¯ =O(I 2§ n<mBnm) ¼10KHz
[Merkulov, Efros, Rosen, PRB. 65, 205309, (2002)]
Hyper¯ne interaction between electrons and nuclei: J
H=HS +HSB +HB
=X
i
iZi +X
i ;n
Ai ;n~¾i ¢~I n +X
n;m
Bnm[~I n~Im]
System-Bath Model of Decoherence and Noise
Besides the quantum system S there is always an environment or bath B.
Hamiltonian:
For a single system qubit
Errors come from faulty control and undesired couplings with the environment. Indirectly also from .BH
Contains our (faulty) control Hc(t)
H =HS (t) I B + I S HB +HSB
He = I S B0+X BX +Y BY +Z BZ
Thereexists a pulse sequence that eliminates any arbitrary HSB
Apulsesequencethateliminatesthesystem-bathinteractionforasinglequbit:
Universal Dynamical Decoupling
exp[ ( )]SB Bi H H f
X Z ZX X
f f f f f'=
exp[ ]Bi H f'
exp[ ] ( )2 Z Zi UZ
exp[ ] ( )2 X Xi UX
on system
f
±;¿ ! 0
Problem: works imperfectlywhen ±;¿ 6= 0.Errors accumulateassequencegrows.
Concatenated Universal Dynamical Decoupling
Tocountererroraccumulation,correcterrorsatalltimescales:NesttheuniversalDDpulsesequenceintoitsownfreeevolution
periodsf:
p(1)= X f Z f X f Z f
p(2)= X p(1)Z p(1)X p(1)Z p(1)
etc.Level Concatenated DD Series after multiplying Pauli matrices
1 XfZfXfZf
2 fZfXfZfYfZfXfZffZfXfZfYfZfXfZf
3 XfZfXfZfYfZfXfZffZfXfZfYfZfXfZfZfZfXfZfYfZfXfZffZfXfZfYfZfXfZfXfZfXfZfYfZfXfZffZfXfZfYfZfXfZfZfZfXfZfYfZfXfZffZfXfZfYfZfXfZf
Length grows exponentially; how about error reduction?
Performance of Concatenated Sequences
[Khodjasteh & Lidar, PRA 75, 062310 (2007) ]
Fix the total sequenceduration so it has 4n¿ pulses.n = concatenation level¿ = pulse intervalAssumezero pulsewidth
¯d(n) ¸ 1¡ 4j a j n
4j bj n 2
Dynamical Decoupling for Quantum Memory: Numerically Exact Simulations for Spin Chain
Simulations for a spin chain system+bath, coupled through Heisenberg for a small number of bath spins.n is the concatenation level or log4N. Themeasure of error is theoneminus purity of the ¯nal tracedout systemstate. Westart thebath at a thermal equilibriumat a temperaturenear 1K.
10log (1 fidelity)
log(1¡ ¯d(n)) · jajn ¡ jbjn2
Spin bath initially in equilibrium at 1K
Theory bound:
Computation
Problem: DD pulses can interfere with logic gates (cancel them too)
How can they be reconciled?
• Need a commuting structure of pulses and computation.
• Use encoded qubits from a DFS.Pick DD pulses to commute with logical gates over DFS, such that DD pulses are still a universal decoupling group.
1
23
j0L i = 12 (j01i ¡ j10i) (j01i ¡ j10i)
j1L i = 12p3(2j0011i +2j1100i ¡ j0110i ¡ j1001i ¡ j1010i ¡ j0101i)
Heisenberg Computation over DFS is Universal
• Heisenberg exchange interaction:
• Universal over collective-decoherence DFS [J. Kempe, D. Bacon, D.A.L., B. Whaley, Phys. Rev. A 63, 042307 (2001)]
• Over 4-qubit DFS:
CNOT involves 14 elementary steps (D. Bacon, Ph.D. thesis)
HHeis =P
i ;j J i j (X iX j +YiYj +ZiZj ) ´P
i ;j J i j E i j
¹X = ¡ 2p3(E13+ 1
2E12) ¹Z = ¡ E12
eiµ ¹X and eiµ ¹Z generatearbitrary singleencoded qubit gates
Universal Decoupling Group Commutes with Heisenberg Exchange
• n levels of concatenation, N=4n pulses
• Universal decoupling group on M (even) system-spins:
p(1)= X U Z U X U Z U
p(2)= X p(1)Z p(1)X p(1)Z p(1) …
X 1¢¢¢XM Z1¢¢¢ZM
[HHeis;X or Z]= 0
e¡ i (µ=N )H gate
t
X Z ZX X
U U U U
Next: demonstrate discrete set of (encoded) single-qubit gates from universal set
4-Qubit DFS π/8 Gate + CDD: ideal pulses, varying internal bath coupling
J=10MHz, T=100nsec
Concatenation Level
System
Bath
A 1;1
B 1;3
¯ = 0:1MHz
¯ = 100MHz
Internal bath coupling
CDD
PDD
β=10MHz, T=100nsec
Concatenation Level
CDD
PDD
System
Bath
A 1;1
B 1;3
system-bath coupling
J = 0:01MHz
J = 10MHz
4-Qubit DFS π/8 Gate + CDD: ideal pulses, varying system-bath coupling
Conclusions
• Decoherence and noise remain the fundamental obstacle to large scale implementation of quantum computers
• A concatenated dynamical decoupling strategy drastically improves fidelity of quantum memory and quantum logic gates
• What next?– Consider Hybrid CDD-QEC strategy– What is the fault-tolerance threshold for this
hybrid setting?– Optimal Decoupling: Can we do better than
CDD?