Quantum Information Processing with Semiconductors Martin Eberl TU

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

Overview ¢ Quantum Computation l Quantum bits l Quantum gates l Quantum parallelism l Deutsch - Algorithm ¢ Semiconductor quantum computer l Self-assembled quantum dots l SRT with Si. Ge heterostructures l Donor-based quantum computing l Quantum bits l Hyperfine structure l Quantum gates l Readout l Calibration

Quantum bit (qubit) classical bit: 0 or 1 ⇔ measurement: either or qubit: 0 or 1 or superposition 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 1 st qubit 0:

Two-qubit states • product state: for example ⇒ Measurement of 1 st qubit doesn‘t affect the 2 nd one • entangled state: not writeable as a product state Bell state: Measurement of 1 st qubit = 0 (with probability 0. 5) then 2 nd qubit must be 0 too

Two-qubit gates I classical: AND, NAND, OR, NOR, 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 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 f for f(x) = x we get CNOT

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

Quantum parallelism II H H … … Uf H entangled state for n = 3: ⇒ simultaneous evaluation of f(x) for 2 n 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 H Uf create superposition:

Deutsch – Algorithm III evaluate f (note that Uf → _ and ___ ___ { ) ___ ___ 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 In. As over Ga. As • 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

Spin resonance transistor with Si. Ge heterostructures • heterostructure of different Six. 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

Donor-based quantum computing B ≅ 10 rf -3 Tesla T ≅ 100 m. K Design: A J A B ≅ 2 Tesla

Overview ¢ ¢ ¢ Only Si – Isotopes with nuclear spin In = 0 31 P – 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 m. K temperatures)

Electron structure Low temperature T ≅ 100 m. K ⇒ 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 ¢ ¢ ¢ Prepare electron spin of 1 st donor in a known state Transfer electron from 2 nd 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 ¢