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]

Robust Hadamard Gate for Spin Bath

System

Bath

A 1;1

B 1;3

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

4-Qubit DFS Logic Gate + CDD, Finite Width Pulses

t

System

Bath

A 1;1

B 1;3

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?