Quantum Information Processing with Semiconductors

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

4. 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

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

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 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

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 control target

9. 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

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

11. 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

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

13. 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)

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 H H Uf H create superposition:

16. Deutsch – Algorithm III _ evaluate f (note that and ) Uf → ___ ___ ___ ___ ___ ___ UH { constant ___ ___ UH balanced |

17. 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)

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 1 Dot 2 Dot 1 Dot 2 Dot 1 Dot 2 Dot 1 Dot 2

19. 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

20. Donor-based quantum computing Brf≅ 10-3 Tesla Design: T ≅ 100 mK B ≅ 2 Tesla A J A

21. 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

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 1018 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 Probability density of electron wave function at nucleus electron nucleus interaction

25. 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:

26. Single-qubit gates I Precession of nuclear spin around B with the Larmor frequency B Bring Brf into resonance with spin precession ⇒ arbitrary rotation possible spin Problem: Brf 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 ⇒ it’s possible to address nuclear spin of single donor with Brf

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 Brf 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 ⇒ 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

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: it’s possible to incorporate 106 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