1 Quantum Computation with coupled quantum dots

2 Two sides of a coin Two different polarization of a photon Alignment of a nuclear spin in a uniform magnetic field Two states of an electron orbiting a single atom.

3 Single qubit gate: Hadamard gate H Pauli gates X, Y and Z … Two qubit gate: Swap gate Controlled-Not gate or CNOT, … Qubit Manipulation Gate = qubit operation CNOT + single qubit gates form a universal Set. Quantum Algorithm is a (2^N)x(2^N) unitary CNOT + single qubit gates form a universal Set. Quantum Algorithm is a (2^N)x(2^N) unitary

4 Physical systems actively considered for quantum computer implementation Liquid-state NMR NMR spin lattices Linear ion-trap spectroscopy Neutral-atom optical lattices Cavity QED + atoms Linear optics with single photons Nitrogen vacancies in diamond Topologically ordered materials Electrons on liquid He Small Josephson junctions – “charge” qubits – “flux” qubits Spin spectroscopies, impurities in semiconductors Coupled quantum dots – Qubits: spin,charge,excitons – Exchange coupled, cavity coupled

5 DiVincenzo Criteria A scalable physical system with well charactrized qubits The ability to initialize the state of the qubits to a simple fixed state Long relevant decoherence times, much longer than the gate operation time A universal set of quantum gates. A qubit-specific measurment capability. A qubit must be ‘well characterized’ in the sense that one has a good theoretical description not only of the qubit itself (in terms of an internal Hamiltonian, accurate knowledge of all physical parameters, etc.), but also of all relevant mechanisms that couple qubits among each other and to the environment. The word ‘scalable’ plays an important role: A proposal for a quantum computer must at least in principle be preparable or manufacturable in large numbers of qubits, even if current fundamental experiments are performed with Only few qubits.

6 DiVincenzo Criteria A scalable physical system with well charactrized qubits The ability to initialize the state of the qubits to a simple fixed state Long relevant decoherence times, much longer than the gate operation time A universal set of quantum gates. A qubit-specific measurment capability. every computation needs to be started in an initially known state such as | >. Having also a fast initialization mechanism at hand is crucial for quantum error correction

7 DiVincenzo Criteria A scalable physical system with well charactrized qubits The ability to initialize the state of the qubits to a simple fixed state Long relevant decoherence times, much longer than the gate operation time A universal set of quantum gates. A qubit-specific measurment capability. Quantum states in contact with the outside world ultimately evolve into a fully mixed states. By encoding information not directly into single qubits, but rather into ‘logical qubits’ consisting of several single qubits, a certain amount of errors due to decoherence and imperfect gates maybe corrected, depending on what kind of code is used. There is however still a limit on how faulty elementary gates are allowed to be: The accuracy threshold theorem states that error correction is possible if the error probability per gate is smaller than A certain threshold. The threshold value depends on the error models Studied and on the details of the codes considered. Typical values are in the range of to 10^(-5) to 10^(-3), implying that decoherence times must be a thousand to a hundred thousand times longer than gate operation times.

8 DiVincenzo Criteria A scalable physical system with well charactrized qubits The ability to initialize the state of the qubits to a simple fixed state Long relevant decoherence times, much longer than the gate operation time A universal set of quantum gates. A qubit-specific measurment capability. The generic quantum computing is possible in the standard model if certain one- and two-qubit gates are available. The single qubit gates may be either implemented directly, or can be approximated to arbitrary precision using a finite set of gates. The only necessary two-qubit gate is the controlled-not gate.

9 DiVincenzo Criteria A scalable physical system with well charactrized qubits The ability to initialize the state of the qubits to a simple fixed state Long relevant decoherence times, much longer than the gate operation time A universal set of quantum gates. A qubit-specific measurment capability. Measuring qubits without disturbing the rest of the quantum computer is required in the verification steps of quantum error correction and, not remarkably, in order to reveal the outcome of a computation. A meas- -urement is said to have 100% quantum efficiency if it yields, performed on a state the outcome “0” with probability p and “1” with probability (1 − p) independent of α, the states of neighboring qubits, or any other parameters of the system. Real measure- -ments cannot have perfect quantum efficiency.

10 Quantum dot

11 QC with quantum dots the physical system representing a qubit is given by the localized spin state of one electron, and the computational basis states and are identified with the two spin states and, respectively.

Scalability is due to the availability of local gating. Gating operations are realized through the exchange coupling, which can be tuned locally with exponential precision. QC with quantum dots

13 QC with quantum dots In spin qubits, initialization could be achieved by either forcing the spins to align with a strong externally applied magnetic field, or by performing a measurement on the dot followed by a subsequent rotation of the state depending on the measurement outcome.

Initialization of the quantum computer could be realized at low temperature T by applying an external magnetic field B satisfying |gμB| ≫ kT, where g is the g-factor, μ is Bohr’s magneton, and k is the Boltzmann constant. After a sufficiently long time, virtually all spins will have equilibrated to their thermodynamic ground state |0> = |↑>. QC with quantum dots Experiments are usually performed in dilution Refrigerators with base temperature around 20 mK, which is smaller than typical Zeeman splittings ( ∼ 300 mK at B = 1 T and using the bulk value g = −0.44). The initialization time is of the order of a few relaxation times, which in GaAs dots have been reported to be as high as ∼ 1 s.

In quantum dots where electron spins are used as qubits, the most important mechanisms of decoherence are the spin-orbit and the hyperfine interaction QC with quantum dots

Dresselhaus contribution Rashba contribution Hyperfine interaction An electron moving through a solid experiences electric fields, from charged atoms in the lattice. This electric fields lead to a net contribution to the spin- orbit interaction. This effect is known as the Dresselhaus contribution to the spin-orbit interaction and its Hamiltonian for 2DEG with strong confinement along growth direction (001) reads Where depends on the material properties and on

QC with quantum dots Dresselhaus contribution Rashba contribution Hyperfine interaction Electric fields associated with asymmetric confining potentials also give rise to a spin-orbit interaction (SIA, or structural inversion asymmetry). The spin-orbit contribution from SIA is known as the Rashba term. Assuming that the confining electric field is along the z axis, the Hamiltonian for 2DEG of the Rashba contribution reads Where depends on the material specific and on the confining potential

QC with quantum dots Dresselhaus contribution Rashba contribution Hyperfine interaction The spin of an electron in an atom can interact with the spin of “its” atomic nucleus through the hyperfine coupling. An electron spin in a quantum dot, in contrast, may interact with many nuclear spins in the host material. The Hamiltonian for the Fermi Contact hyperfine interaction is then given by Where and are the spin operator for nucleus k and the electron spin Respectively

19 Single spin rotations may be achieved by dragging electrons down (by changing back gate voltages) to a region where the Zeeman splitting in the presence of the external static magnetic field changes due to magnetization or an inhomogeneous g-factor present in that layer. A resonant magnetic ac pulse can then be used to rotate the spin under consideration, while leaving all other qubits unaffected due to the off-resonant Zeeman splitting (ESR). All-electrical single spin manipulation may be realized in the presence of spin-orbit interaction by applying ac electric pulses directly via the gates (EDSR).

QC with quantum dots TWO-QUBIT GATES The interaction of the two spins may be described in terms of the isotropic Heisenberg Hamiltonian Then the corresponding unitary evolution of the state of the double dot is given

For the constant intraction and time s.t performs the so-called ‘square-root of swap’ denoted by This gate together with single-qubit rotations about a fixed (say, the z-) axis can be used to synthesize the cnot operation as QC with quantum dots

Readout of electron spin states. Several methods are available for reading out the spin state of single and double quantum dots and all of them rely on the mechanism of spin-to-charge conversion.

23 References: - D.Loss, D.P. DiVincenzo Phys. Rev. A 57, (1998) - R. Hanson, L. P. Kouwenhoven, J. R. Petta, S. Tarucha, L. M. K. Vandersypen, Rev. Mod. Phys, 79, 1217 (2007)