Spin qubits

Semiconductor [[quantum dots]] are small devices in which charge carriers are confined in all three dimensions: this can be achieved by electrical gating and/or etching techniques applied e. g. to a two-dimensional electron gas. In the quantum-dot scenario either the electron spin or the [[Charge qubits|charge]] (orbital) degrees of freedom can be chosen as the qubit. Here we will describe the '''spin dot''' which has two immediate advantages: #the qubit represented by a real spin-1/2 is always well-defined as the two-dimensional Hilbert space is the entire space available and therefore there are no extra dimensions into which the qubit state could "leak"; #real spins have quite long dephasing times (order of microseconds in GaAs). In order to be able to perform quantum computation, in addition to a well-defined qubit, we also need a controllable ''source of entanglement'', i. e. a mechanism by which two specific qubits at a time can be entangled so as to produce the fundamental [[CNOT]] gate operation. This can be achieved by temporarily coupling two spins via the Heisenberg Hamiltonian :$\backslash ;\; H\_s(t)\; =\; J(t)\backslash bar\{S\}\_1\; \backslash cdot\; \backslash bar\{S\}\_2\; ,$ where $\backslash ;\; J(t)=4t\_0^2(t)/\backslash mathcal\{U\}$ is the time dependent exchange constant produced by turning on and off the tunneling matrix element $\backslash ;\; t\_0(t)$, $\backslash ;\; \backslash mathcal\{U\}$ is the charging energy of a single dot, and $\backslash ;\; \backslash bar\{S\}\_i$ is the spin-1/2 operator for the i-th dot. The Heisenberg Hamiltonian $\backslash ;\; H\_s(t)$ provides a good description of the quantum-dot system if the following conditions are met: #$\backslash ;\; \backslash Delta\; E\; \backslash gg\; kT$ (where $\backslash ;\; \backslash Delta\; E$ is the level spacing and $\backslash ;\; T$ is the temperature) so that higher-lying single-particle states of the dots can be ignored; #$\backslash ;\; \backslash tau\_s\; \backslash gg\; \backslash hbar/\backslash Delta\; E$ (where $\backslash ;\; \backslash tau\_s$ is the time scale for pulsing the gate potential "low") in order to prevent transitions to higher orbital levels; #$\backslash ;\; \backslash mathcal\{U\}\; >\; t\_0(t)\; \backslash quad\; \backslash forall\; t$, in order for the Heisenberg-exchange approximation to be accurate; #$\backslash ;\; \backslash Gamma^\{-1\}\; \backslash gg\; \backslash tau\_s$, where $\backslash ;\; \backslash Gamma^\{-1\}$ is the decoherence time. In general the decoherence times of the electron spins are longer than those of the [[Charge qubits|charge degrees of freedom]], since the former are insensitive to environmental fluctuations of the electric potential. Condition 1. above ensures that we can focus on the two lowest orbital eigenstates which are the spin ''singlet'' and the spin ''triplet''. The general Hamiltonian $\backslash ;\; H\_\{orb\}$ can thus be effectively replaced by the Heisenberg spin Hamiltonian $\backslash ;\; H\_s$ where the exchange constant $\backslash ;\; J\_0\; =\; \backslash epsilon\_t\; -\; \backslash epsilon\_s$ is the difference between the triplet and singlet energy. In this setting, quantum computing can be achieved by applying the unitary time evolution operator $\backslash ;\; U\_s(t)\; =\; \backslash mathbf\{T\}\; exp\backslash \{-i\backslash int\_0^t\; H\_s(t\text{'})dt\text{'}\backslash \}$ to the initial state of the two spins: $\backslash ;\; |\backslash Psi(t)\backslash rangle\; =\; U\_s|\backslash Psi(0)\backslash rangle$. For a specific duration $\backslash ;\; \backslash tau\_s$ of the spin-spin coupling such that $\backslash ;\; \backslash int\; J(t)dt\; =\; J\_0\backslash tau\_s\; =\; \backslash pi(mod\backslash ,2\backslash pi)$, the pulsed Heisenberg Hamiltonian $\backslash ;\; H\_s(t)$ leads to the ''swap operator'' $\backslash ;\; U\_\{swap\}\; \backslash equiv\; U\_s(J\_0\backslash tau\_s\; =\; \backslash pi)$. This operation conserves the total angular momentum of the system and therefore is not sufficient by itself to perform useful quantum computing. However, we can pulse the interaction for just half the duration and thus obtain the ''square root of swap'' which is a fundamental quantum gate in the implementation, for instance, of the [[CNOT]]: :$\backslash ;\; U\_\{CNOT\}\; =\; e^\{i\backslash frac\{\backslash pi\}\{2\}S\_1^z\}e^\{-i\backslash frac\{\backslash pi\}\{2\}S\_1^z\}U\_\{swap\}^\{1/2\}\; e^\{i\; \backslash pi\; S\_1^z\}U\_\{swap\}^\{1/2\}\; ,$ where $\backslash ;\; e^\{i\; \backslash pi\; S\_1^z\}$ etc. are single-qubit operations and $\backslash ;\; U\_\{swap\}^\{1/2\}$ is the ''square root of swap''. === One-qubit gates === We have seen how to obtain the two-qubit operations ''swap gate'' and ''root of swap gate'' but we still have to see how to implement one-qubit gates. Single-qubit rotations such as $\backslash ;\; e^\{i\; \backslash pi\; S\_1^z\}$ can be achieved by pulsing a magnetic field $\backslash ;\; \backslash bar\{B\}\_i$ exclusively onto the i-th spin. This could be done with: #a scanning-probe tip; #an auxiliary dot. In the latter case the auxiliary dot (FM) is made of an insulating, ferromagnetically-ordered material that can be connected to the i-ht dot of interest ,$\backslash ;\; (i=1,2)$, by the same kind of electrical gating which connects the two spins in the Heisenberg Hamiltonian $\backslash ;\; H\_s$. If we want a magnetic field $\backslash ;\; \backslash bar\{B\}\_i$ to be pulsed exclusively onto the i-th spin, we just have to lower the barrier between the FM dot and the i-th dot so that the electron's wavefunction overlaps with the magnetized region for a fixed time $\backslash ;\; \backslash tau\_s$. Thus, during this time, the Hamiltonian for the qubit on the i-th dot will contain a Zeeman term which leads to the single-qubit rotation. Therefore the i-th spin is rotated and the corresponding Hamiltonian is :$\backslash ;\; \backslash int\_0^\{\backslash tau\_s\}H\_s^B\; dt\; =\; \backslash sum\_\{i=1\}^2\; \backslash omega\_i\; \backslash tau\_s\; S\_i^z,$ with $\backslash ;\; \backslash omega\_i\; =\; g\backslash mu\_B\; B\_i^z$, where $\backslash ;\; g$ is the effective g-factor, $\backslash ;\; \backslash mu\_B$ the Bohr magneton and $\backslash ;\; B\_i^z$ the magnetic field acting on the i-th spin in the z direction. == Related papers == *D. Loss, D. P. DiVincenzo, ''Phys. Rev. A'' '''57''', 120 (1998) *G. Burkard, D. Loss, D. P. DiVincenzo, ''Phys. Rev. B'' '''59''', 2070 (1999)