Beyond Ansätze: Learning Quantum Circuits
as Unitary Operators

While the future of quantum computation is in quantum computational hardware, the currently available hardware is expensive and accessibly limited; this is noted at the core reason for the work of Padilha et al. [2019], which is one of many advances in designing classical software simulators of quantum computational systems. While these quantum simulators allow the research community to experiment on toy problems, even the best simulators pose major problems for many QML applications.

Focusing in on the circuit ansatz challenge, we may consider on the application area of quantum chemistry. Within quantum chemistry, there are some some algorithms wherein “problem-inspired" circuit ansatz exists, but due to the current quantum computational hardware constraints, a “hardware efficient ansatz" is used instead. This is a practical compromise; a circuit ansatz which performs well in simulation but fits no current hardware may be traded off for a circuit ansatz which has a slew of theoretical problems but can be experimented with on NISQ devices.

QML applications in this sense have an extreme disadvantage: unlike quantum chemistry problems which are built on strong foundations, there are no known theoretical ansatz which work well on arbitrary machine learning problems. By their very nature, machine learning applications can have arbitrary datasets with patterns that do not match any clear physical model. In this case, there is no clear answer between “what does the ideal anstaz look like?" vs “what is close to the ideal we can fit on hardware?" but rather “what ansatz is useful on this particular dataset within this particular application?"

The key observation that motivated this paper is that a quantum system of $N$ qubits can be described by a state vector in $\mathbb{C}^{2^{N}}$ and every quantum circuit is a unitary transformation on this state space. This implies that every machine learning algorithm that optimizes a quantum circuit on $N$ wires can be viewed as a optimization process in (a subspace of) the space of $2^{N}$-dimensional unitaries, denoted by $U(2^{N})$. Note that this a fundamentally different from classical machine learning where the space of all possible networks is infinite dimensonal. In the quantum case, the unitary group $U(2^{N})$ has real dimension $2^{2N}$ and allows to fully parametrize the possible quantum circuits on $N$ wires.

The Lie algebra $\mathfrak{u}(d)$ of the unitary group $U(d)$ is the space of $d\times d$ skew-Hermitian matrices

This space is of real dimension $d^{2}$. Moreover, since $U(d)$ is connected and compact, the (matrix-)exponential map $\mathfrak{u}(d)\rightarrow U(d)$ is surjective [Hall, 2015, Corollary 11.10], i.e. every unitary transformation $A\in U(d)$ can be written as the exponential of some skew-Hermitian $X\in\mathfrak{u}(d)$.

Motivated by this result, we propose to parametrise the Lie algebra $\mathfrak{u}(d)$, and use the exponential map to obtain unitary transformations. Since the exponential map is differentiable, we can propagate gradients all the way back to the Lie algebra and do gradient based optimization there.

Note that is significantly eases the optimization process. The reason for this is that the Lie group of unitary transformations has non-trivial geometry and doing gradient descent is cumbersome on such spaces. On the other hand, the Lie algebra is a vector space, where gradient descent can shine.

Since PyTorch [Paszke et al., 2017] has built-in support both for complex-valued tensors and the matrix exponential, random unitary transformations can be generated with just a few lines of Python code:

Parametrising a unitary transformation on $D$ dimensions requires $D^{2}$ scalar parameters, and the dimension of an $N$ qubit system is $D=2^{N}$. It follows that the use our approach on $N$ wires requires $2^{2N}$ trainable parameters.

Since the dimension of the group of unitary transformations scales exponentially as we increase the number of qubits, it is impractical to optimize unitary operators on a large number of wires. Restraining to optimization to subsets of $U(2^{N})$ can ease this computational issue. One interesting subgroup is $U(2^{k})\otimes U(2^{N-k})$, i.e. applying one unitary to the first $k$ wires, and a second one to the last $N-k$ wires. This of course can be generalized by a different partitioning of the $N$ wires. For instance, consider the following alternatives:

• •

  Optimizing directly on $N$ qubits requires $2^{2N}$ parameters to train.

• •

  Optimizing $N/k$ unitary operators on $k$ wires requires $(N/k)2^{2k}$ parameters. This, of course, does not allow any interaction between qubits belonging to different partitions. To overcome this, one could repeat the same procedure with a different partitioning of the wires into groups of size $k$. Repeating this $m$ times results in $(Nm/k)\,2^{2k}$ trainable parameters, which only scales linearly in $N$.

Working with subspaces of the unitary group in order to shrink the optimization space is an effective way of cutting the computational costs. It is important to note that every such reduction of the number of parameters (and in turn the search space) trades off generality for a computational speedup. Moreover, training speed is not a monotonically decreasing function of the parameter count, a large number of parameters may be faster if the resulting map is simple to compute (such as the exponential of a single skew-symmetric matrix) than a mapping with a lower parameter count but with a long chain of simple ingredients, that need to be composed every time the parameters change (such as a long ansatz).

Taking the idea of the previous section to the extreme, we can recover the traditional approach of first choosing an ansatz of predefined quantum gates and optimizing their parameters. As an example, consider a two-qubit quantum system. Our approach would optimize the parameters $(\theta_{1},\,...\,,\theta_{16})\in\mathbb{R}^{16}$ generating the full unitary group $U(4)$:

where $i$ denotes the complex unit.

Now, let us restrict our attention to the ansatz with an RX-gate on the first wire and an RY-gate on the second one. Optimizing this circuit amounts to training $(\theta_{1},\theta_{2})\in\mathbb{R}^{2}$ resulting in the following family of unitaries:

Note that family of unitaries in (2) is a strict subset of the unitaries in (1), i.e. our method can find quantum circuits that the approach with a given ansatz would not be able to find. The price to pay for that is an increased number of parameters, a bigger search space to optimize in.

As with all ansätze, if there is a good reason to believe that the optimal circuit belongs to a certain subclass of unitaries, then it makes a lot of sense to to only consider functions that satisfy the ansatz. Otherwise, keeping the search space as big as possible (and practically still feasible) is a reasonable thing to do.

In the previous section we argued that an ansatz is equivalent to a particular restriction of the search space. Conversely, optimization over all unitaries is at least as general as any choice of ansatz, which in turn implies that the optimum of full unitary optimization represents the best possible quantum circuit that any ansatz could have found¹¹1Note that in practice life is more complicated. Local minima, gradient step size, etc. can have an effect on the optimization process. The statement here is simply that if a search space is contained in an other one, then the global optimum of the latter one cannot be worse than that of the first one.. To put it differently:

This is particularly useful when trying to answer questions such as

Whether $\mathcal{P}$ is the performance of some classical approach (looking for quantum advantage) or not, our method separates these issues from the choice of an ansatz and provides a way to answer them easily. If the answer to a question such as Question 2 is affirmative, there is hope that a (simple) ansatz can also perform well on $\mathcal{T}$. Otherwise one doesn’t have to worry about ansätze as none of them will reach performance $\mathcal{P}$.

We compare our approach to PennyLane [Bergholm et al., 2020] by running the same training procedure (number of qubits, datapoints and training epochs) in both frameworks. The space of unitary transformations on $N$ qubits is $2^{2N}$ dimensional. It follows that our PyTorch model contains $2^{2N}$ trainable parameters. Accordingly, we initizalize a PennyLane model containing a single RandomLayer with $2^{2N}$ trainable parameters.

In our experiments, we generate a classical dataset and use $R_{x}$ rotations to encode the classical data into a quantum representation, apply the quantum circuit and perform $Z-$measurements on the wires to decode from the quantum to a classical representation. We are not interested in accuracy of the trained networks, only in training speed, so we just train the networks to learn the identity function.

In this section we run experiments along the lines of §3.4 and test quantum circuits on two image classification tasks, MNIST and SAT-6²²2The SAT-6 dataset [Basu et al., 2015] consists of satellite images of 4-channels (red, green, blue, near infrared) each belonging to one of 6 categories (barren land, trees, grassland, roads, buildings and water bodies).. Quanvolutional networks [Henderson et al., 2020] generalize the concept of convolutions by having a quantum circuit sliding through the input image and evaluating it at every patch of the input. By construction they inherit the translation equivariance property of convolutional networks, therefore are a fitting architecture for image processing tasks.

In both experiments we compare two networks, one classical and one classical-quantum hybrid. The overall number of parameters of the networks are comparable, the architectures are shown in Figure LABEL:fig:architecture. The quanvolutional layer maps from 16 to 8 channels using 2-by-2 filters, resulting in a total of 32 quantum circuits on 4 qubits.

The results are summarized in Figure 2. We see that the quanvolutional architecture trained by full unitary optimization does not yield an accuracy benefit over the classical network and conclude that no ansatz exists for the 4 qubit circuits that would outperform the classical network given that the classical part of the hybrid architecture is not changed, only retrained with the ansatz.