# Quantum Information Processing with Semiconductors

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```					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
 Calibration
Quantum bit (qubit)
classical bit:            qubit:
0 or 1                ⇔   0 or 1 or superposition

measurement: either           with probability
or       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:
Unitary matrices:          (adjoint = transpose &
complex conjugate)

NOT- 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            only true for 0 or 1
data space
\   /
CNOT
only pure states can be copied
CNOT
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           ___
constant

{                    ___    UH
|
___
balanced

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   31P   represents
the qubit
• Voltage at gate
pulls wave function
away from donor
• different g-factor
⇒ resonance
frequency changes
• magnetic field in
resonance performs
logical operations
Donor-based quantum
computing        B ≅ 10       rf
-3   Tesla
Design:       T ≅ 100 mK

B ≅ 2 Tesla
A     J    A
Overview

   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
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
Larmor frequency                     B

Bring Brf into resonance with
spin
spin precession
⇒ arbitrary rotation possible

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
Qubit stored in nucleus spin
⇒ little interaction with the environment
SWAP between nucleus and electron
Important: fast read out, before information
decays
Spin measurement possible, but too slow
⇒ charge measurement
   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 A-
Gate 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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