Transcript Quantum Information Processing with Semiconductors

Quantum Information
Processing with
Semiconductors
Martin Eberl, TU Munich
JASS 2008, St. Petersburg
Overview

Quantum Computation
 Quantum bits
 Quantum gates
 Quantum parallelism
 Deutsch - Algorithm

Semiconductor quantum computer
 Self-assembled quantum dots
 SRT with SiGe heterostructures
 Donor-based quantum computing
 Quantum bits
 Hyperfine structure
 Quantum gates
 Readout
 Calibration
Quantum bit (qubit)
classical bit:
0 or 1
⇔
measurement: either
or
qubit:
0 or 1 or superposition
with probability
with probability
(normalization)
After measurement: Collapse of the wave function
or
Quantum gates
= logical operation on qubits
Single-qubit gate: NOT- gate
classical:
quantum:
Representation of quantum gates:
(adjoint = transpose &
Unitary matrices:
complex conjugate)
NOT- gate
Hadamard gate
H² = 1
pure state → mixed state
Only 1 classical single-bit gate, but ∞ single-qubit gates
Two qubits
Probability for measuring first qubit 0:
After measuring 1st qubit 0:
Two-qubit states
• product state:
for example
⇒ Measurement
of 1st qubit doesn‘t affect the 2nd one
• entangled state:
not writeable as a product state
Bell state:
Measurement of 1st qubit = 0 (with probability 0.5)
then 2nd qubit must be 0 too
Two-qubit gates I
classical: AND, NAND, OR, NOR, XOR, XNOR
⇒ NAND is universal
2 bits input → 1 bit output ⇒ not reversible
quantum: CNOT
control
target
Two-qubit gates II
Operation on state:
is unitary ⇒ reversible (bijection)
CNOT is universal:
every logical operation can be performed by
CNOT + single-qubit gates
No-Cloning-Theorem
it‘s impossible to copy arbitrary quantum states
proof:
copy with CNOT
data space
\
/
CNOT
CNOT
only true for 0 or 1
only pure states can be copied
Function evaluation
unitary transformation Uf:
Uf
By carrying
along, it is possible to use a non
bijective function as a unitary one
picture of a controlled operation
for f(x) = x we get CNOT
f
Quantum parallelism I
quantum register of n qubits:
create mixed state:
for n = 3:
=
=
=
Superposition of 2n states
Quantum parallelism II
H
H
…
…
Uf
H
entangled state
for n = 3:
⇒ simultaneous evaluation of f(x) for 2n arguments!
problem: measurement gives random f(x)
Deutsch – Algorithm I
4 possible functions
constant
functions
balanced
functions
{
{
Problem:
determinate if a function f(x) is
balanced or constant
Classical:
2 function calls needed
Deutsch – Algorithm II
H
H
Uf
H
create superposition:
Deutsch – Algorithm III
evaluate f (note that
_
and
)
Uf
→
___
___
___
___
___
{
___
UH
UH
|
___
constant
___
balanced
Advantages
Only for certain problems:
 exploitation of special properties:
e.g. period, correlation
⇒ Deutsch-Algorithm
⇒ Shor‘s Algorithm (prime-factoring)
 Repetition of the same task on large
number of input values
e.g. search through an unstructured
database (Grover‘s Algorithm)
Self-assembled quantum
dots
• quantum dots self-assembled by
growing InAs over GaAs
• Excitons (electron-hole pairs)
used as qubits
⇒ created by light absorption
⇒ confined in quantum dots
• 4-8 nm distance
⇒ overlap of wave functions
⇒ tunneling
Dot 1 Dot 2
Dot 1 Dot 2
Dot 1 Dot 2
Dot 1 Dot 2
Spin resonance transistor
with SiGe heterostructures
• heterostructure of different SixGe1-x layers
⇒ Landé g-factor changes
• spin of weakly bound electron from
the qubit
• Voltage at gate
pulls wave function
away from donor
• different g-factor
⇒ resonance
frequency changes
• magnetic field in
resonance performs
logical operations
31P
represents
Donor-based quantum
computing
B ≅ 10
rf
-3
Tesla
T ≅ 100 mK
Design:
A
J
A
B ≅ 2 Tesla
Overview

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

Only Si – Isotopes with nuclear spin In = 0
31P – Donors have I = ½
n
Nuclear spin of donors is used for qubits
Logical operations are performed with different
voltages on the gates above the donors in
combination with the magnetic field Brf
Initialization and measurement is made by
gauging electron charges
Nuclear spin as qubit
Problem in general:
Interaction of quantum system with environment
⇒ decay of information (decoherence time)
⇒ computation must be completed before the
information has significantly decayed
Solution: nuclear spin
little interaction ⇒ large decoherence time
(estimated to be in the order of 1018 s at mK
temperatures)
Electron structure
Low temperature T ≅ 100 mK
⇒ no electrons in the conduction band
⇒ isolator
Phosphorus is a group V element
⇒ one additional electron, which is very
weakly bound, close to the conduction band
⇒ Similar to a Hydrogen atom with bigger
radius and smaller energy
Hyperfine structure I
Probability density of
electron wave function
at nucleus
electron
nucleus
interaction
Hyperfine structure II
}
Δf =
frequency for Brf
to perform SWAP
Logical operations between electron and nucleus:
SWAP-Operation:
⇒ Transfer of nuclear spin state to electron
CNOT:
Single-qubit gates I
Precession of nuclear spin around B with the
B
Larmor frequency
Bring Brf into resonance with
spin precession
⇒ arbitrary rotation possible
spin
Problem: Brf is globally applied, not locally
Single-qubit gates II
Lab frame
Rotation frame
Single-qubit gates III
Larmor frequency is dependent on the
hyperfine interaction of the electron
with the nucleus
Apply voltage at the A-Gate:
⇒ electron is drawn away from the
nucleus
⇒ Larmor frequency for single donor
changes
⇒ it’s possible to address nuclear spin
of single donor with Brf
Two-qubit gates
Apply positive electric field
on J-Gate ⇒ turn electron
mediated interaction between
nuclei on or off
New hyperfine structure for
the system of both nuclei
and their electrons
Magnetic field Brf can modify
the spin states of the system
and thus perform logical
operations like SWAP or CNOT
Readout
Qubit stored in nucleus spin
⇒ little interaction with the environment
⇒ hard to read out
SWAP between nucleus and electron
Important: fast read out, before information
decays
Spin measurement possible, but too slow
⇒ charge measurement
Readout
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Prepare electron spin of 1st donor in a known state
Transfer electron from 2nd donor using A-Gate
voltage
⇒ only possible, if spin is pointing in different
direction
Perform charge measurement
Calibration
Variation of donor positions and gate sizes
⇒ it’s necessary to calibrate each gate
• set Brf = 0 and measure nuclear spin
• switch Brf on and sweep through small voltage
interval at A-Gate
• measure nuclear spin again
⇒ it will only flip, if resonance occurred in the AGate voltage range
• After A-Gates have been calibrated, use same
procedure with the J-Gates
• Calibration can be performed parallel on many
Gates, resonance voltages can be stored on
capacitors
Challenges for building the
computer
Silicon completely free of spin &
charge impurities
 Donors in an ordered array ~ 25 nm
beneath the surface
 Very small gates must be placed on
the surface right above the donors

Advantage to other quantum computer concepts:
it’s possible to incorporate 106 qubits
Quantum Information
Processing with
Semiconductors
Nielsen, Chuan, Quantum computation and quantum information, 2001
 Stolze, Suter, Quantum computing, 2004
 Chen et. al., Optically induced entanglement of excitons in a
single quantum dot, 2000
 Rutger Vrijen et. al., Electron spin resonance transistors for quantum
computing in silicon-germanium heterostructures, 2000
 B.E. Kane, A silicon-based nuclear spin quantum computer, Nature
393: 133-137, 1998.
 B.E. Kane, Silicon-based quantum computation, 2008
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