Implementing impossible functions via tomography

A long time ago I had a conversation with Daniel Nagaj about the following problem: given access to an unknown unitary gate $U$, is it possible to build the gate control-$U$ out of it? Nagaj argued that it was impossible, because $U$ and $e^{i\alpha}U$ are physically equivalent for any phase $\alpha$, but control-$U$ and control-$e^{i\alpha}U$ are physically distinct. This is true, but it felt to me too cheap: the global phase $\alpha$ is not even defined, so of course it is not possible to make it appear in the control. Moreover, why would anyone want to have this ghostly phase in the controlled unitary? The more meaningful question is whether is it possible to implement control-$e^{i\phi(U)}U$ for any phase $\phi(U)$.

A quick search through the literature revealed that nobody had answered this question, so I proved it was impossible and wrote a short and simple paper about it. There’s also the plot twist that implementing control-$U$ is often easy in practice, as any realistic implementation of an unknown unitary will have some extra information that makes it possible, but that’s another story.

In any case, at that time I believed that if one is given a large number of copies of $U$, one could just do process tomography on it, and thus easily implement control-$e^{i\phi(U)}U$. Luckily I didn’t write that down anywhere, because a couple of weeks ago Gavorová et al. proved that it is not possible to implement control-$e^{i\phi(U)}U$ for any number of copies of $U$, so in particular even tomography cannot do it.

Now how could this possibly be true? One can always do tomography, right? And if you did tomography, you can implement any function you want of the thing you tomographed, right? It turns out it is not so simple. If you successfully tomograph $U$, what you end up with is with a unitary $U’$ such that $\|\mathfrak{C}(U’) – \mathfrak{C}(U)\| < \epsilon$, where $\mathfrak{C}(U)$ is the Choi-Jamiołkowski representation of $U$. This complication is unfortunately necessary, as $U$ is not physically observable, only $\mathfrak{C}(U)$ is, for the very reason that $\mathfrak{C}(e^{i\alpha}U)$ does not depend on the global phase $\alpha$. It is just not true that $\|U’-U\| < \epsilon$, because we would need to tomograph the global phase to make it happen.

Ok, great, we have tomography, how about the function then? We want that $\|f\big(\mathfrak{C}(U’)\big)-f\big(\mathfrak{C}(U)\big)\| < \delta$, but this only follows from $\|\mathfrak{C}(U’) – \mathfrak{C}(U)\| < \epsilon$ if $f$ is continuous! If $f$ is not continuous then by definition there will exist a pair $U,U’$ such that $\mathfrak{C}(U’),\mathfrak{C}(U)$ are arbitrary close but $f\big(\mathfrak{C}(U’)\big),f\big(\mathfrak{C}(U)\big)$ are not. And Gavorová et al. showed precisely that if $f$ implements the controlled unitary it must be discontinuous.

The source of the discontinuity is not really the controlling, but just the killing off the global phase, so the seemingly simpler task of producing a unitary that does not depend on the global phase is equally impossible.

To be more precise, let $f$ be a function that takes $\mathfrak{C}(U)$ to a unitary $V$ such that $\mathfrak{C}(U) = \mathfrak{C}(V)$. Or equivalently, let $g$ be a function such that $g(U) = g(e^{i\alpha}U)$ for all $\alpha$ and $\mathfrak{C}\big(g(U)\big) = \mathfrak{C}(U)$. Then it is impossible to tomograph $U$ and produce $U’$ such that $\|f\big(\mathfrak{C}(U’)\big)-f\big(\mathfrak{C}(U)\big)\| < \delta$, or that $\|g(U’)-g(U)\| < \delta$.

To see why, let’s consider a specific example of such a function, $g(U) = \frac{u^*}{|u|}U$, where $u$ is the first nonzero element of the first column of $U$. Let
\[U = \begin{pmatrix} \sqrt{p} & -\sqrt{1-p}\ e^{i\beta} \\ \sqrt{1-p}\ e^{i\alpha} & \sqrt{p}\ e^{i(\alpha+\beta)} \end{pmatrix} \] be an arbitrary unitary of dimension $2$ with a global phase already removed, so that for $p > 0$ we have $g(U) = U$, and for $p=0$ we have
\[g(U) = \begin{pmatrix} 0 & -e^{i(\beta-\alpha)} \\ 1 & 0 \end{pmatrix}. \]Suppose that $p >0$, $\alpha=\pi$, and $\beta=0$, so that
\[U = \begin{pmatrix} \sqrt{p} & -\sqrt{1-p} \\ -\sqrt{1-p} & -\sqrt{p}\end{pmatrix}. \] Now if $p$ is small enough, this is very close to
\[U’ = \begin{pmatrix} 0 & -1 \\ -1 & 0\end{pmatrix}, \] so tomography might very well return us that instead. The problem is that
\[g(U’) = \begin{pmatrix} 0 & 1 \\ 1 & 0\end{pmatrix}, \] which is as far from $g(U)$ as it gets.

You might object that the probability of obtaining such $U’$ is extremely small; it is given roughly1 by the probability of measuring the state $U|0\rangle$ in the computational basis and always obtaining outcome 1, which is $(1-p)^n$. It is way too high for small $p$! In particular for fixed $n$, there is always a $p$ such that this probability is higher than any failure probability $f$ that we accept.

This breaks the fundamental promise of tomograhy: that for any error tolerance $\epsilon > 0$ and failure probability $f > 0$ you just need to measure $n$ copies of $U$ in order to obtain an estimate $U’$ that with probability $1-f$ is $\epsilon$-close to $U$. Sure, for any $U$ there will be an $n$ such that tomography works, but this doesn’t help us since we don’t know what $U$ is before we tomographed it!

Ok, ok, so tomography cannot converge to $g(U)$; it does converge instead to $g(U)$ modulo some random phase. Doesn’t seem so dramatic, does it? Since you didn’t care about the global phase to start with, why should you worry about whether it is random or a function of $U$? Well, yes, if you just do tomography it doesn’t matter, but perhaps you have some other, efficient protocol that does the control modulo a random phase. Now in every iteration of the protocol you might get a different phase, and if you just ignore this you’re going to lose coherence.

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