Quantum Information Processing with Semiconductors

Martin Eberl, TU MunichJASS 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 ⇔

qubit:

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

classical: quantum:

Representation of quantum gates:Unitary matrices:

NOT- gate

Single-qubit gate: NOT- gate

(adjoint = transpose & complex conjugate)

Hadamard gate

pure state → mixed state

Only 1 classical single-bit gate, but ∞ single-qubit gates

H² = 1

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 Iclassical: AND, NAND, OR, NOR, XOR, XNOR

⇒ NAND is universal

2 bits input → 1 bit output ⇒ not reversible

quantum: CNOTcontrol target

Two-qubit gates II

is unitary ⇒ reversible (bijection)

CNOT is universal:every logical operation can be performed by CNOT + single-qubit gates

Operation on state:

No-Cloning-Theoremit‘s impossible to copy arbitrary quantum states

proof:

only true for 0 or 1only pure states can be copied

copy with CNOTdata space \ /

CNOT

CNOT

Function evaluationunitary transformation Uf:

Uf

By carrying along, it is possible to use a non bijective function as a unitary one

picture of a controlled operation

ffor f(x) = x we get CNOT

Quantum parallelism Iquantum register of n qubits:create mixed state:

for n = 3:

== =

Superposition of 2n states

Quantum parallelism II

Uf

H

H

H

……

entangled state

for n = 3:

⇒ simultaneous evaluation of f(x) for 2n arguments! problem: measurement gives random f(x)

Deutsch – Algorithm I4 possible functions

{constantfunctions

{balancedfunctions

Problem: determinate if a function f(x) isbalanced or constant

Classical: 2 function calls needed

Deutsch – Algorithm II

create superposition:

Uf

H

H

H

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 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

Design:

Brf ≅ 10-3 Tesla

B ≅ 2 Tesla

T 100 mK≅

A J A

Overview

Only Si – Isotopes with nuclear spin In = 0

31P – Donors have In = ½ 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 spinlittle 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

electron nucleus interaction

Probability density of electron wave function at nucleus

} Δf

Hyperfine structure II

Logical operations between electron and nucleus:SWAP-Operation:

⇒ Transfer of nuclear spin state to electronCNOT:

= frequency for Brf

to perform SWAP

Single-qubit gates I

Precession of nuclear spin around B with the Larmor frequency B

spinBring Brf into resonance with spin precession

⇒ arbitrary rotation possible

Problem: Brf is globally applied, not locally

Single-qubit gates II

Lab frame Rotation frame

Single-qubit gates IIILarmor 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 gatesApply 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 electronImportant: fast read out, before information decays

Spin measurement possible, but too slow ⇒ charge measurement

Readout 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

CalibrationVariation 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