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 (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

*H**s*(*t*) = *J*(*t*)*S̄*1 ⋅ *S̄*2, where *J*(*t*) = 4*t*02(*t*)/U is the time dependent exchange constant produced by turning on and off the tunneling matrix element *t*0(*t*), U is the charging energy of a single dot, and *S̄**i* is the spin-1/2 operator for the i-th dot. The Heisenberg Hamiltonian *H**s*(*t*) provides a good description of the quantum-dot system if the following conditions are met:

- Δ
*E*≫*k**T*(where Δ*E*is the level spacing and*T*is the temperature) so that higher-lying single-particle states of the dots can be ignored; -
*τ**s*≫ ℏ/Δ*E*(where*τ**s*is the time scale for pulsing the gate potential "low") in order to prevent transitions to higher orbital levels; - U >
*t*0(*t*) ∀*t*, in order for the Heisenberg-exchange approximation to be accurate; - Γ − 1 ≫
*τ**s*, where Γ − 1 is the decoherence time.

In general the decoherence times of the electron spins are longer than those of the 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 *H**o**r**b* can thus be effectively replaced by the Heisenberg spin Hamiltonian *H**s* where the exchange constant *J*0 = *ε**t* − *ε**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 *U**s*(*t*) = **T***e**x**p*{ − *i*∫0*t**H**s*(*t*ʹ)*d**t*ʹ} to the initial state of the two spins: ∣Ψ(*t*)⟩ = *U**s*∣Ψ(0)⟩. For a specific duration *τ**s* of the spin-spin coupling such that ∫*J*(*t*)*d**t* = *J*0*τ**s* = *π*(*m**o**d* 2*π*), the pulsed Heisenberg Hamiltonian *H**s*(*t*) leads to the *swap operator* *U**s**w**a**p* ≡ *U**s*(*J*0*τ**s* = *π*). 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:

$$\; U_{CNOT} = e^{i\frac{\pi}{2}S_1^z}e^{-i\frac{\pi}{2}S_1^z}U_{swap}^{1/2}
e^{i \pi S_1^z}U_{swap}^{1/2} ,$$ where *e**i**π**S*1*z* etc. are single-qubit operations and *U**s**w**a**p*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 *e**i**π**S*1*z* can be achieved by pulsing a magnetic field *B̄**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 , (*i* = 1, 2), by the same kind of electrical gating which connects the two spins in the Heisenberg Hamiltonian *H**s*. If we want a magnetic field *B̄**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 *τ**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

∫0*τ**s**H**s**B**d**t* = ∑*i* = 12*ω**i**τ**s**S**i**z*, with *ω**i* = *g**μ**B**B**i**z*, where *g* is the effective g-factor, *μ**B* the Bohr magneton and *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)