スピントロニクス量子操作研究会「量子コンピュータの基礎」 第7回

3/24/2005

Error correction and fault-tolerant
quantum computation on standard QC
z Error correction and fault-tolerant quantum

Physical systems with inherent
error-correcting properties

computationz Requires tremendous number of qubits and non-local entanglementz Example: Factorize a 130-digit (432-bit)number by Shor’s algorithmz Total number of qubits ∼ 106
z including additional ancillary qubits to implement gatesfault-tolerantlyz Non-local entanglementz over 73 qubits to encode one qubit!

z Need for better design
z with inherent error-correcting and/or fault-tolerant

properties

Good qubit design for syndrome
measurement

Example (1): Non-linear optics
z Error correcting storage for photonic qubitz Active error correction requires

z 3-qubit bit-flip (phase) flip code

syndrome measurement involving ancillaqubitsz Arrangement of qubits/ancilla qubitsmatters!

Ancillary
qubits|ψi

Z 1Z 2z Store qubit information using three micro-spheres
z Syndrome measurement by sending coherentstates

RecoveryZ 2Z 3

micro-spheres

ErrorsEncodedqubits

Encoding|0i

Coherentstate|αi3-qubit bit
flip code|0i|0i

Syndrome measurement
Code subspace {|000i, |111i}
Phase shift

3Homodyne
detection

Encoding
1.

Micro-spheresz No errors (|000i, |111i): |αiz Bit flip X1 (|100i, |011i): |α e± iθiz Bit flip X2 (|010i, |101i): |α e± 3iθiz Bit flip X3 (|001i, |110i): |α e± 2iθiz

24z Syndrome measurement (3-qubit bit flip code)
z1

3|0iz

2

123|0i0θ

−θ|1i

θ

−2θθ

2.
3.

No need for fault-tolerant ancilla preparation(α 0 + β 1 )( 0 + 1 )( 0 + 1 )= α( 0 0 0 + 0 0 1 + 0 1 0 + 0 1 1 )+ β (1 0 0 + 1 0 1 + 1 1 0 + 1 1 1 )Í Use of robust coherent state as an ancilla

X2
X3

micro-spheres

X1
No error

|αi

1

2

3

Transfer qubit information from a photon to
micro-sphere #1
Prepare micro-spheres #1 and #2 in |0i+|1istate
Run error correcting procedure

Error Correction⎯⎯⎯⎯⎯⎯→ α 0 0 0 + β 1 1 1Homodynedetection
5

Fumiko Yamaguchi <yamaguchi@stanford.edu>|αimicro-spheres
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スピントロニクス量子操作研究会「量子コンピュータの基礎」 第7回

3/24/2005

Example (2): 2D nuclear spin networkz Topological (planar) code

Naturally fault tolerant
quantum computationz Check operators = XXXX or ZZZZz Stack of 2D nuclear spin networkz Planes of encoded qubits
z Planes of ancilla qubits

ZZ

X

X

Z

M

X

X

Qubit #1
Ancilla #1

ZX
X

X

Qubit #2

ZZAncilla #2
MZside view

7

Example (1): 3-qubit bit flip code oncoupled quantum dots

Natural error suppressionz Physical system with Hamiltonianz 3-qubit bit flip code

H = – J (g1 + g2 + g3 + g4 + L)

z Stabilizer generators: Z1Z2, Z2Z3

z gi (i = 1, 2, L, n-k) : Stabilizer generators ofz Hamiltonian: H = – J(Z1Z2+Z2Z3)

error correcting codez No errors: Ground statez An error raises energy by 2J[Ferromagnetic interaction]z Natural error suppressionz No error: Ground state E = – 2Jz Natural error suppression
z 2J ¿ kBT

z Bit flip error X1, X2, or X3: E = 0→ No-error states energeticallyfavoredz Timescale À 1/J → Probability amplitude ofQD1

state with an error e-iJt ∼ 0

QD2–JQD3–J

9Example (2): Topological code10

Physical system: Nuclear spin networkz Consider the Hamiltonianz

H = − J ∑ Z a Zb Zc Z d − J ∑ X a X b X c X dP

Physical system
Stack of 2D nuclear spin network

zSzz Ground state of the system formsz

topological code subspace+ constantly simulated 4-body interactions(using ancilla qubits)
Logic operations between encodedqubitsMzz Errors would increase the energy and be

suppressed by e-2J/kTzz Physical realization

Planes of encoded qubits
Planes of ancilla qubits

ZZ

z Intrinsically fault-tolerant

Xz Much less overhead for error correction
z Likely to be scalable!

X
11

Fumiko Yamaguchi <yamaguchi@stanford.edu>

ZX

X
X

X

Qubit #1
Ancilla #1

X

Qubit #2

Z
Z

Ancilla #2

Z

M

122

スピントロニクス量子操作研究会「量子コンピュータの基礎」 第7回

Physical system: AlxGa1-xAs/GaAs
fluctuating quantum dots

3/24/2005

Photoluminescence spectra from QWs(a)(b)z AlxGa1-xAs/GaAs wellstructuresz 10-20 monolayer (ML)thickz Fluctuating quantum

100 - 1000Å

dots at interface (100 AlxGa1-xAs1000Å) by growthinterruption
Excitonz An exciton (Bohr10-20 MLGaAs QW
radius ~ 300Å) isAlGaAsx1-x
trapped in a quantum
dotz

Low-temperature (6K)
photoluminescence (PL)spectra of four single
quantum wells growncontinuously (bottom) andwith interruption at heterointerfaces (top)z

6 K PL spectrum of free
excitons of a single
quantum well containing 21and 22 monolayers. D
denotes defect-related
bound excitons.

13

Proposed systemz GaP monolayer

14

Summary
100 - 1000Å

z Error correction and fault-tolerance are

AlxGa1-xAs
Grown in the quantum
well
z P nuclear spins =qubits
Exciton
10-20 MLGaAs QWz Optical initialization andreadout via hyperfine
AlxGa1-xAs
interaction
z Four body interactions
simulated by applyingz Dipole coupling among
constant NMR pulse
qubitssequencez Akin to arrangement ofz Code subspace
qubits for topological= ground statecode.
Æ Inherent error
suppression
z15

Fumiko Yamaguchi <yamaguchi@stanford.edu>

inevitable for quantum computationz In existing theoriesz Large number of qubitsz Non-local operationsz Non-local entanglementz All correction by codeare requiredz Good design of qubits is required at qubit
element level for scalabilityz Suitable design for syndrome measurement; ORz System with natural fault tolerance
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