Does quantum control allow to implement any gate?

Question

Using quantum control techniques it is possible to control quantum systems in a wide range of different scenarios (e.g. 0910.2350 and 1406.5260).

In particular, it was shown that using these techniques it is possible to implement gates like the (quantum) Toffoli gate (1501.04676) with good fidelities. More precisely, they show that given the Toffoli gate $\newcommand{\on}[1]{\operatorname{#1}}\mathcal{U}_{\on{Toff}}$, defined as the C-CNOT gate $$ \newcommand{\ketbra}[2]{\lvert#1\rangle\langle#2\rvert} \newcommand{\ket}[1]{\lvert#1\rangle} \newcommand{\bra}[1]{\langle#1\rvert} \mathcal{U}_{\on{Toff}}\equiv \ket{0}_1\!\bra{0}\otimes \on{CNOT} + \ket{1}_1\!\bra{1}\otimes I, $$ and a time-dependent Hamiltonian $H(t)$ containing a specific set of interactions, one can find a set of (time-dependent) parameters of $H(t)$ such that $$ \mathcal T \exp\left(-i \int_0^\Theta H(\tau)d\tau\right) \simeq \mathcal U_{\on{Toff}}. $$

Are there known results on the universality of such an approach? In other words, do the tools provided by quantum control theory allow to say when, given set of constraints on the allowed Hamiltonian parameters, a given target gate can be realized? (1)

More precisely, the problem is the following: fix a target gate $\mathcal U$ acting over a set of qubits (or more generally qudits), and a parametrised Hamiltonian of the form $H(t)=\sum_k c_k(t) \sigma_k$, where $\{\sigma_k\}_k$ is a fixed set of (Hermitian) operators, and $c_k(t)$ are time-dependent parameters to be determined. Is there a way to tell whether there are coefficients $\{c_k(t)\}_k$ such that $$ \mathcal T\exp\left(-i\int_0^{\Theta} H(\tau)d\tau\right) \stackrel{?}{=} \mathcal U. $$

---

(1) Note that I here talk of quantum control only because that is the term used in the paper. If this is not the most suitable term to use to refer to this kind of problems please let me know.

Moreover, note also that the problem solved in the paper is slightly different than the one I stated here. In particular, the Hamiltonian they consider actually acts in the space of three four-dimensional qudits, and the Toffoli is only implemented as an effective dynamics in the lower levels of each ququart. I'm also ok with results of this sort of course.

Answer 1

There is the concept of controllability of a quantum system, i.e. do the given set of controls permit you to create any state or unitary? Usually this is computed by looking at the Lie Algebra of the system, and can be quite messy; you need to take the individual Hamiltonian terms that you can control, and calculate all their commutators to arbitrary orders. If you can take linear combinations of those and make any arbitrary Hamiltonian, then your full Hilbert space is controllable; you can make any unitary you want, and any quantum state is said to be reachable from any other. See Complete controllability of quantum systems (PRA 2001) for an example.

One important point to emphasise, however, is that this tells you nothing about efficiency, i.e. how long it takes you to reach a given state (as a function of the system size). There is an explicit construction that you can make based on the above decomposition in terms of commutators, but the time required scales exponentially in the order of the commutator required. Numerical techniques of control theory are methods that attempt to find the required control fields (as a function of time) in a more efficient manner, but (to my knowledge) rarely give you any guarantees. So, if you have fixed $\Theta$ and bounded $c_k(t)$, the controllability concept may not be sufficient.

Answer 2

Quantum control does not necessarily allow implementing just any gate. Imagine your control of the system is a time-dependent energy. That corresponds to the Hamiltonian $\hat{H}(t) = c(t) \hat{Z}$. Then you can only ever rotate about one axis of the Bloch sphere and your only choice is how fast to rotate when. This is obviously insufficient to even generate arbitrary single-qubit gates because then you would need to be able to effect rotations about any (arbitrary) axis.

I cannot answer the second part of your question, about known results for the universality. Note, however, that I picked a very special case to illustrate that simple quantum control is not enough. Imagine I had picked the Hamiltonian $\hat{H}(t) = c(t) \hat{X} + \frac{E_0}{2} \hat{Z}$. This is a constant rotation about one axis (due to an energy difference $E_0$ between the two states of the single qubit involved) plus a rotation you completely control about an orthogonal axis. Since you can produce arbitrary rotation with a suitable combination of such rotations, this is universal (for a single qubit system). This is my attempt at illustrating that not having universal control if you have any control could be seen as a special case rather than the rule.