Do multi-parameter unitary subgroups exist?

Question

I'm working with an $N$-dimensional quantum system that is defined by the following Hamiltonian

$$ H = H_{\text{drift}} + \sum_{j=1}^n a^j H_{\text{drive}}^j $$

Where $n \ll N$ (in my case $n = 4$ and $N = 42$)

The unitary matrix in $SU(N)$ generated from $\text{span}\{H_{\text{drift}}, H^j_{\text{drive}}\}$ is presumably in an $(n + 1)$-parameter subgroup of $SU(N)$. I know about one-parameter subgroups so I'm basically asking if multi-parameter subgroups are a thing? And also, if they are, if there are any known parameterizations of them?

Answer 1

Correct me if I'm wrong but your Hamiltonian looks a lot like the basic formulation of quantum control theory: there one extends the usual time-dependent Schrödinger equation $\dot\psi(t)=-\frac{i}\hbar H_{\sf drift}\psi(t)$ by control terms to arrive at a bilinear control system: $$ \dot\psi(t)=-\frac{i}\hbar\Big(H_{\sf drift}+\sum_ju_j(t)H^j_{\sf drive}\Big)\psi(t) $$ Here the $u_j(t)$ are control functions, i.e. locally integrable functions which an experimenter can choose freely. If one works with general mixed states (and not just pure states) one analogously turns the Liouville-von Neumann equation $\dot\rho(t)=-\frac{i}\hbar[H_{\sf drift},\rho(t)]$ into a controlled version: $$ \dot\rho(t)=-\frac{i}\hbar\Big[H_{\sf drift}+\sum_ju_j(t)H^j_{\sf drive},\rho(t)\Big] $$ The most basic question one asks in this setting is the question of "reachability": given some initial state $\psi_0$ ($\rho_0$), which states $\psi_F$ ($\rho_F$) can one generate using the control Hamiltonians at hand? Shifting perspective one can also ask about gate reachability: the set of all solutions to $$ \dot U(t)=-\frac{i}\hbar\Big(H_{\sf drift}+\sum_ju_j(t)H^j_{\sf drive}\Big)\quad \text{ with }\quad U(0)={\bf 1}\tag{1} $$ is the set of all quantum gates (unitary operations) one can implement via the drift & controls at hand. While there are most relevant follow up questions like generating a reachable state or gate using controls which are optimal in some sense -- a central tool for which Pontryagin's maximum principle -- let us focus on reachability. The reason for this is that Lie theory turns out to be of immense use here:

As long as one has access to at least all control functions $u_j(t)$ which are piecewise constant (NB: which is quite reasonable with modern digital equipment) then solutions to the above equation are products of operators of the simple form $e^{-\frac{i}\hbar t(H_{\sf drift}+\sum_ju_jH^j_{\sf drive})}$, that is, time-independent operators which are also featured in your question. In this case an essential tool for analyzing reachability is the system Lie algebra $\mathcal A:=\langle H_{\sf drift},H^j_{\sf drive}:j\rangle_{\sf Lie}$, i.e. $\mathcal A$ is the smallest Lie subalgebra of $\mathfrak u(n)$ (=the Lie algebra of the unitary group ${\bf U}(n)$) which contains $H_{\sf drift}$ as well as all $H^j_{\sf drive}$. Side note: this may sound a bit abstract at first but this object is quite simple to handle on a computer. Either way one has the following fundamental result:

Theorem. Let $G := \langle \exp(\mathcal A)\rangle$ denote the system group of control system (1), that is, $G$ is the smallest subgroup of ${\bf U}(n)$ that contains $\exp(\mathcal A)$). If $G$ is a closed subgroup of ${\bf U}(n)$, then then the set of all unitary matrices that can be generated via (1) (called "reachable set") coincides with the system group $G$.

A direct consequence of this is a very simple criterion for "controllability" (="one can generate all unitaries") via dimension counting:

Corollary. If all Hamiltonians are chosen traceless, then system (1) is controllable if and only if $\operatorname{dim}(\mathcal A)=\operatorname{dim}(\mathfrak{su}(n))=n^2-1$. Otherwise (1) is controllable iff $\operatorname{dim}(\mathcal A)=\operatorname{dim}(\mathfrak u(n))=n^2$.

I'll keep it at that for now, but I'll gladly add some more detail if this is what your question aimed at. As for further literature:

• G. Dirr and U. Helmke, "Lie Theory for Quantum Control", GAMM-Mitteilungen 31 (2008), pp. 59–93; the above theorem is Theorem 3.3 in said paper
• D. D’Alessandro. Introduction to Quantum Control and Dynamics. Chapman & Hall/CRC, Boca Raton, 2008
• or Chapter 3.3 in my PhD thesis "Reachability in Controlled Markovian Quantum Systems"