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

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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

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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

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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)

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

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

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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

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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

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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:

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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

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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

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

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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)

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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

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Deutsch – Algorithm II create superposition: UfUf H H H

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

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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)

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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

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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

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

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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

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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)

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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

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

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} 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

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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

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Single-qubit gates II Lab frame Rotation frame

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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

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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

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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

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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

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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

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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

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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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Figure 12.1: (a) Schematic of a quantum dot embedded in a host. The electron wavefunction is largely confined to the dot material, but does penetrate into.

Figure 12.1: (a) Schematic of a quantum dot embedded in a host. The electron wavefunction is largely confined to the dot material, but does penetrate into.

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