A Quantum Information Theoretic View On A Deep Quantum Neural Network

The two-qubit output state of the minimal network of “two-qubits–two-qubits–two-qubits” is given by

where $\rho_{in}^{z}$ is a given input state, the initial states of the hidden and output layers have been chosen to be in the state $|0\rangle$ (w.o.l.g.) and the unitaries only address the subspaces described before.

The cost function for our problem is then defined by

In Ref. [5] a composite parametrization was introduced which will allow us to compute the derivative of the unitaries, for the optimization of the network, analytically. For any unitary operation $U$ acting on a Hilbert space $\mathcal{H}=\mathbb{C}^{d}$ with $d\geq 2$ spanned by the orthonormal basis $\{|1\rangle,\ldots,|d\rangle\}$ there exist $d^{2}$ real values $\lambda_{m,n}$ with $m,n\in\left\{1,\ldots,d\right\}$ and ${\color[rgb]{1,0,0}\lambda_{n,n}}\in\left[0,2\pi\right]$ and ${\color[rgb]{0,0,1}\lambda_{n,m}}\in\left[0,2\pi\right]$ for $m<n$ and ${\color[rgb]{0,1,0}\lambda_{m,n}}\in\left[0,\frac{\pi}{2}\right]$ for $m<n$ such that any $U\equiv U_{C}$ with

The sequence of the product is defined by $\prod_{i=1}^{N}A_{i}=A_{1}\cdot A_{2}\cdots A_{N}$. Here, the $P_{l}$ are one-dimensional projectors $P_{l}=|l-1\rangle\langle l-1|$ and $Y_{m,n}$ are the generalized anti-symmetric Pauli-matrices $Y_{m,n}=-i|m-1\rangle\langle n-1|+i|n-1\rangle\langle m-1|$ with $1\leq m<n\leq d$.

The parameter $\lambda_{m,n}$ can be gathered in a $d\times d$ “parameterization matrix”

where the diagonal entries ${\color[rgb]{1,0,0}{\lambda_{n,n}}}$ represent global phase transformations, the upper right entries ${\color[rgb]{0,1,0}{\lambda_{m,n}}}$ are related to rotations in the subspaces spanned by $|n\rangle$ and $|m\rangle$, while the lower left entries ${\color[rgb]{0,0,1}{\lambda_{n,m}}}$ are relative phases in these subspaces (with respect to the basis $\{|0\rangle,\ldots,|d-1\rangle\}$). Note that for optimization one does not need to restrict the parameter $\lambda_{n,m}$ to the intervals given above.

Now we want to change the unitaries of the neural network in order to maximize the cost function $C$ and this for each parameter $\lambda_{xy}$, i.e. we can consider a Taylor expansion:

with

and

where we have used the results of Ref. [5]. Note that $\tilde{Y}$ are hermitian, thus the unitarity condition holds for every order in the expansion. Moreover, note that $\tilde{Y}$ depends for $x\not=y$ on all other $\lambda$ parameters except $\lambda_{x,y}$.

Thus the parameters of the unitaries are changed by $\lambda_{x,y}\longrightarrow\lambda_{x,y}+\;\delta\lambda_{x,y}$ which is given for the first perceptron by the term

and for the second perceptron by the term

and so on. Here $\varepsilon$ can be chosen arbitrarily and in principle different for each unitary and plays the role of a learning parameter in a classical network, i.e. chosen too low the cost function will only increase slowly but chosen too high we may miss the optimum. It is a hyper parameter in the learning process. We chose it for all four unitaries the same in our applications.

Let us emphasize here that each equation forms a Heisenberg uncertainty, in the so called Robertson version [8], i.e.

where the $\Delta$ is the standard deviation of the operator with respect to $\Psi$. Clearly, there are only two ways how the lower bound on a Heisenberg uncertainty can vanish, either the two observables are commuting or the state $\Psi$ has a spectrum of zero. The first way is the general foundational limit provided by Nature, if two observables are not commuting, for instance the famous position operator $\hat{x}$ and momentum operator $\hat{p}$, we have $[\hat{x},\hat{p}]=i\hbar\mathbbm{1}$ and therefore, for all possible states $\Psi$ the lower bound is $\frac{\hbar}{2}$. Differently stated, there exists no states for which the standard deviation of the position and momentum can be smaller than this value.

On the other hand if we consider e.g. Pauli operators $\sigma_{i}$, then the commutator is $[\sigma_{i},\sigma_{j}]=2i\varepsilon_{ijk}\sigma_{k}$ and thus the lower bound gives

which may be a non-zero value for a general $\Psi$, but choosing an appropriate $\Psi$ it may still vanish though the Pauli operators do not commute. This property of the Robertson version of the Heisenberg uncertainty relation was criticized and an entropic version was found overcoming this issue, which we discuss in the conclusions. The question we want to discuss first is if in the optimization process of the quantum net, these fundamental limits are utilized.

Let us start with a simple example, namely learning a particular unitary $V$, that was first considered in Ref. [6]. In Ref [13] this quantum neural network was applied to the real data of the Iris flower and the performance of this quantum network was compared with other networks. The ground truth is then given by choosing $z$ arbitrary states $\rho^{z}$ and computing the desired output by $\rho_{desired}^{z}=V\rho_{in}^{z}$. The goal is that the network learns this unitary (generally only $16$ parameters) by optimizing the $4\cdot 64$ parameters of the network by utilizing the cost function. One can fix $\varepsilon$, which may be interpreted as a learning parameter, in each round but we optimize $\varepsilon$ by taking the maximum of the cost function for the computed corrections of all $4$ unitaries. We chose randomly $100$ pairs and used $10$ for the optimization and $90$ for the validation. The software for implementation was Mathematica Wolfram.

In Fig. 3 we plotted the cost function and the validation function for different training rounds. At each training round the cost function is plotted as a function of $\varepsilon$ and its maximum is taken. As can be seen the curves are monotonically increasing at each round but the convergence is slow.

A typical update for the $\lambda_{x,y}$ looks like (example for $U_{1}$, round $55$)

or for (example for $U_{1}$, round $60$)

which is the sum of all training states defining the lower bound in the Heisenberg relation and does not vanish. The zeros are due to the fact that we have chosen the hidden layer and output layer states to be $|0\rangle$. In Fig. 4 we show how the cost function typically changes with the learning parameter $\varepsilon$. It is quite constrained if all four unitaries are included, but for a single one the parameter space is quite flat. This suggests that the interplay of all four unitaries is relevant for the problem, but the constraint due to each single unitary does not do the job.

Even though we are close to the maximum cost function value it seems that the derivatives do not vanish. To see if this is due the average over in this case $10$ pairs, we picked one out and optimized it to a cost function value of $0.999003$, the corrections terms are still of same order as above. This suggests that the neural network does not optimize the uncertainty relation but the parameters of the unitaries which are not unique. Let us choose now a non-trivial problem.

Now we create pairs such that the desired output state encodes the quantum properties, i.e. its purity $\mathsf{P}(\rho)=\frac{4}{3}(Tr(\rho^{2})-\frac{1}{4})$ and its entanglement property. For that purpose we choose the concurrence [7], a computable measure for bipartite qubits. The concurrence $\mathsf{Con}(\rho)$ is defined as maximal between zero and the maximal eigenvalue minus the other three eigenvalues of the quantity $\sqrt{\sqrt{\rho}\sigma_{2}\otimes\sigma_{2}\rho^{*}\sigma_{2}\otimes\sigma_{2}\sqrt{\rho}}$ with $\sigma_{2}$ being the $y$-Pauli matrix. For pure states it simplifies to $\mathsf{Con}(\psi)\;=\;|\langle\psi|\;\sigma_{y}\otimes\sigma_{y}|\psi^{*}\rangle|$.

Our desired output states are chosen to be

This means that each pair is again connected by a unitary (if we assume only pure input states), $|\Phi^{z}_{\mathsf{desired}}\rangle=U_{\psi^{z}_{in}}|\psi^{z}_{in}\rangle$, but it is chosen according to the quantum properties of the input states. Thus the net needs to learn a set of unitaries defined by the quantum properties (entanglement&purity) of the arbitrary input state. Consequently, the question is whether the neural net processes also the properties of the state itself or if only the information of the training pair is exploited as it would be the case in a classical neural network.

We tried different sets for the training and here we discuss the result for $70$ pairs for the training. The convergence is even slower than for the problem of single unitaries. We find typically a cost function value of $0.91$ with a standard deviation of $0.08$. If we use for the validation only $30$ pairs the cost function value was found to be higher, i.e. for our set $0.93$. This shows a high statistic fluctuation with the randomly chosen set, meaning the general problem is not (yet) fully learnt. As a further test we can interchange output with the input, this gave a cost function of $0.30$ with a standard deviation of $0.26$. We also tried random inputs, which gave in general very low cost function values. Consequently, the net is indeed learning some features of the training set, which also applies to an arbitrary set.

In Fig. 5 we visualize how well the quantum properties are learnt per se. The first graphs correspond to an early time in the optimization process, here the cost function gave the value $0.84$ with a standard deviation of $0.09$. We see that in the optimization process the net learns e.g. the symmetry between the $|01\rangle$ and $|10\rangle$ states (Fig. 5(b)), but the range of the errors does not get smaller when compared to a later stage in the optimization. From Fig. 5(a) we can deduce that the error in purity is significantly reduced (having only pure states in the training), but the system also predicts values greater than $1$, which is of course unphysical. This could be compensated by adding a Lagrange multiplier to the cost function. In general we observe that the training and validation pairs distribute quite similarly. The range in the error of the concurrence (Fig. 5(a)), however, is not reduced.

The correction terms obtained by back propagation are always of similar size similar to the trivial example discussed in the previous section and have been visualized in Fig. 6. The dependence on the learning parameter $\varepsilon$ is depicted in Fig. 7. In conclusion, the net learns partial properties of the set but the cost function does converge slowly. In the next section we discuss if the learning exploits the limits by the Heisenberg uncertainty relation.