1. Quantum Information Processing with Semiconductors Martin Eberl, TU Munich JASS 2008, St. Petersburg

2. 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

3. Quantum bit (qubit) classical bit: 0 or 1 qubit: 0 or 1 or superposition measurement: either with probability orwith probability (normalization) After measurement: Collapse of the wave function or

4. 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)

5. Hadamard gate pure state mixed state Only 1 classical single-bit gate, but single-qubit gates H² = 1

6. Two qubits Probability for measuring first qubit 0: After measuring 1st qubit 0:

7. Two-qubit states product state: for example Measurement of 1st qubit doesnt 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

8. Two-qubit gates I classical: AND, NAND, OR, NOR, XOR, XNOR NAND is universal 2 bits input 1 bit output not reversible quantum: CNOT controltarget

9. 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:

10. No-Cloning-Theorem its impossible to copy arbitrary quantum states proof: only true for 0 or 1 only pure states can be copied copy with CNOT data space \ / CNOT

11. Function evaluation unitary transformation U f : UfUf By carrying along, it is possible to use a non bijective function as a unitary one picture of a controlled operation f for f(x) = x we get CNOT

12. Quantum parallelism I quantum register of n qubits: create mixed state: for n = 3: === Superposition of 2 n states

13. Quantum parallelism II UfUf H H H …… entangled state for n = 3: simultaneous evaluation of f(x) for 2 n arguments! problem: measurement gives random f(x)

14. 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

15. Deutsch – Algorithm II create superposition: UfUf H H H

16. Deutsch – Algorithm III evaluate f (note that and) _ ___ UfUf ___ UHUH UHUH { constant balanced |

17. Advantages Only for certain problems: exploitation of special properties: e.g. period, correlation Deutsch-Algorithm Shors Algorithm (prime-factoring) Repetition of the same task on large number of input values e.g. search through an unstructured database (Grovers Algorithm)

18. 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 1Dot 2Dot 1Dot 2Dot 1Dot 2Dot 1Dot 2

19. Spin resonance transistor with SiGe heterostructures heterostructure of different Si x Ge 1-x layers Landé g-factor changes spin of weakly bound electron from 31 P 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

20. Donor-based quantum computing Design: B rf 10 -3 Tesla B 2 Tesla T 100 mK AJA

21. Overview Only Si – Isotopes with nuclear spin I n = 0 31 P – Donors have I n = 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 B rf Initialization and measurement is made by gauging electron charges

22. 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 10 18 s at mK temperatures)

23. 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

24. Hyperfine structure I electron nucleus interaction Probability density of electron wave function at nucleus

25. } ΔfΔf Hyperfine structure II Logical operations between electron and nucleus: SWAP-Operation: Transfer of nuclear spin state to electron CNOT: = frequency for B rf to perform SWAP

26. Single-qubit gates I Precession of nuclear spin around B with the Larmor frequency B spin Bring B rf into resonance with spin precession arbitrary rotation possible Problem: B rf is globally applied, not locally

27. Single-qubit gates II Lab frame Rotation frame

28. 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 its possible to address nuclear spin of single donor with B rf

29. 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 B rf can modify the spin states of the system and thus perform logical operations like SWAP or CNOT

30. 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

31. 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

32. Calibration Variation of donor positions and gate sizes its necessary to calibrate each gate set B rf = 0 and measure nuclear spin switch B rf 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

33. 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: its possible to incorporate 10 6 qubits

34. 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