Implementing arbitrary quantum operations via quantum walks on a cycle graph

Quantum walk Aharonov , the quantum counterpart of classical random walk, has been widely applied in achieving various quantum information processing tasks Portugal . Because of its quadratic enhancement of variances, the quantum walk plays a vital role in many quantum search algorithms and provides possible exponential speedups due to the quantum interference during the walk Kempe . Moreover, various experimental implementations of quantum walks prove its feasibility in real-life circumstances of quantum information processing Kia .

On the other hand, machine learning is a core technology in the age of artificial intelligence. Since machine learning faces the challenge of the lack of computational power and quantum computing has a vast computational potential, the possibility of combining quantum computing and machine learning has been considered. Quantum neural networks (QNNs), a newer class of models in the field of quantum machine learning, operate on quantum computers and perform calculations using quantum effects like superposition, entanglement, and interference. Investigations on QNNs farhi2018classification ; zhao2019building ; mitarai2018quantum ; dallaire2018quantum ; amin2018quantum ; CZoufal ; VDunjko ; MSchuld have revealed their potential advantages, such as training and processing speedups. Despite significant developments in the growing field of quantum machine learning, the trade-offs between quantum and classical models have not been systematically studied. In particular, the question of whether quantum neural networks are more powerful than classical neural networks is still open SAaronson2015nph .

A gate-model QNN is a QNN constructed on a gate-model quantum computer using a sequence of unitaries with associated gate parameters farhi2018classification . Recent developments, such as quantum generative adversarial networks and quantum circuit learning, have more general and diverse QNN structures dallaire2018quantum ; mitarai2018quantum ; gyongyosi2019training . Researchers have already proved that typical quantum walks are universal for quantum computation childs2009universal ; lovett2010universal ; kurzynski2013quantum ; bian2015realization ; zhao2015experimental . However, these works mainly focus on state processing, and many auxiliary systems should be employed in general. In contrast, what we are attempting to achieve in this work is a universal control of the quantum system to implement arbitrary quantum operations, without any auxiliary system. For this purpose, we shall introduce a QNN based on discrete-time quantum walks (DTQW) on a cycle graph with specifically parameterized coin operators. We choose the graph to be a cycle because it is simple for laboratory implementations. We will prove that the DTQW-based QNN is indeed capable of realizing arbitrary unitary evolution of the closed system.

Determining the parameters of the DTQW-based QNN analytically is possible. However, any further adjustments on the network, such as a reduction in the number of circuit depth, will pose extraordinary difficulties for analytical methods. In contrast, we will show that such adjustments can be effectively made with gradient descent, a well-known optimization algorithm frequently employed to train machine learning models, including both classical and quantum neural networks darken1992learning ; bengio2013advances . Another significant advantage of using gradient descent is that explicitly decomposing the desired operator into a sequence of gates from a universal set is no longer necessary. Furthermore, we shall simplify the network in various ways to facilitate laboratory implementations. For example, we shall use only rotations along the x-axis as the gates involved in the DTQW. We can still find decent approximations of the desired quantum operations in this situation using our DTQW-based QNN.

Our work is organized as follows. We first introduce our DTQW-based neural network in Sec. II and then prove its universality for quantum control in Sec. III. We further modify gradient descent and apply it to our DTQW-based QNN in Sec. IV. Finally, we simplify the QNN in Sec. V to facilitate the laboratory implementations.

The quantum neural network based on quantum gates, the gate-model QNN, was first introduced due to its high experimental feasibility farhi2018classification . The gate-model QNNs utilize a series of unitary operations in a certain order to process the quantum state. The unitary operations involve adjustable parameters. By optimizing these parameters and encoding information to the input and output states, the gate-model QNNs are sufficient to solve various learning tasks. In this section, we introduce the DTQW on a cycle graph. We choose the graph to be a cycle because it is simple for laboratory implementation. We will show that such DTQW also involves a series of adjustable unitary gates and is sufficient to learn quantum operations. Thus the DTQW on a cycle graph can be treated as a special type of the gate-model QNN.

The DTQW on a cycle graph involves two Hilbert spaces, namely the coin space $\mathcal{H}_{c}$ and the position space $\mathcal{H}_{p}$, which are spanned by orthonormal basis $\{|{0}\rangle_{c},|{1}\rangle_{c}\}$ and $\{|{x}\rangle_{p}\}_{x=0}^{n-1}$ respectively, where $n$ is the number of sites in the cycle. The walker state $|{\Psi}\rangle$ is then in the space $\mathcal{H}=\mathcal{H}_{c}\otimes\mathcal{H}_{p}$. A schematic representation of the DTQW on a cycle graph is shown in Fig. 1. The process of the DTQW is an iteration of applying coin operators $\hat{C}^{(t)}$ and shift operators

to the walker state, i.e.,

where $t=0,1,2,...$ denotes the ordinal of iterations, integer $\delta_{c}$ represents how far the walker is shifted if its coin is in the state $|{c}\rangle$. For simplicity, we choose $\delta_{c}=c$ throughout this letter. To make sure that the DTQW is flexible enough to implement various quantum operations, the coin operator $\hat{C}^{(t)}$ need to be site-dependent, i.e.,

where $\hat{c}_{x}^{(t)}\in\mathrm{U}(2)$ flips the coin of the walker during the $t$-th iteration if the walker is at the site $x$. Since the operators $\hat{c}_{x}^{(t)}$ are applied to the coin only if the walker is at certain sites $x$, they are called single-site coin operators.

Since the operations during every iteration are unitary, the total effect of a $T$-step DTQW

is also unitary, where $\mathcal{T}\prod$ denotes the time-ordered product. We define $\hat{U}_{t_{1},t_{0}}=\mathcal{T}\prod_{t=t_{0}}^{t_{1}-1}\hat{S}\hat{C}^{(t)}$ so that it is the time evolution operator, i.e., $|{\Psi^{(t_{1})}}\rangle=\hat{U}_{t_{1},t_{0}}|{\Psi^{(t_{0})}}\rangle$. One can notice that our version of the DTQW on a cycle graph is a straightforward generalization of the conventional Hadamard walk of which $\hat{c}_{x}^{(t)}=\hat{H}$ and $\delta_{c}=1-2c$.

Every step of DTQW is unitary and is parameterized by $\hat{c}_{x}^{(t)}\in\mathrm{U}(2)$. These operators $\hat{c}_{x}^{(t)}$ can be treated as the adjustable gates in a gate-model quantum neural network. By adjusting these gates $\hat{c}_{x}^{(t)}$, we can use the DTQW to implement various quantum operations. Therefore the DTQW can be seen as a special type of gate-model quantum neural network. A schematic representations of the quantum neural network based on the DTQW on a cycle graph is shown in Fig. 2. The circuit depth of this network is the number of walking steps $T$ of the DTQW. In this work, we will denote the quantum neural network based on the DTQW on a cycle graph simply as the DTQW-QNN. We call the system of the quantum walker the underlying system of the DTQW-QNN.

The implementation of quantum operations via the DTQW on a cycle graph is one of the primary motivations of our work. The universality of DTQW for quantum computation has been shown in general childs2009universal ; lovett2010universal . While the previous work mainly focuses on the mapping from the initial state to the final state in a certain small subspace of the total system, in this work we take the overall effect on the total system into account. In this section, we investigate the capacity and universality of the DTQW on a cycle graph in implementing quantum operations, and show that it is universal for unitary operations, which is the main theorem of this section.

By saying that the DTQW on a cycle graph is universal, we mean that any unitary operation on the overall Hilbert space $\mathcal{H}=\mathcal{H}_{c}\otimes\mathcal{H}_{p}$ can be realized by a DTQW. Hence it is not only universal for computation but also universal for controlling the whole quantum system. To be more formal and specific, the following theorem is provided.

We prove the universality of the DTQW on a cycle by decomposing arbitrary unitary operators $\hat{V}$ into a product of two-level unitary operators $\hat{V}=\hat{u}_{m},\dots,\hat{u}_{2}\hat{u}_{1}$ and construct a DTQW to implement every $\hat{u}_{i}$ for $i=1,2,\dots,m$. A detailed proof is provided in Appendix A.

As a demonstration of Theorem 1, we first implement the controlled NOT (CNOT) gate with a DTQW on a cycle with two sites. We can find that according to Eq. (1), the shift operator

is just the CNOT gate we need. Hence a simple one-step DTQW is equivalent to the CNOT gate if we choose all the single-site coin operators $\hat{c}_{x}^{(t)}$ to be the identity operator.

Next, let us consider a more complicated two-level unitary operator, a unitary $\hat{U}$ controlled by two qubits

where $a,b,c,d$ are four matrix elements of $\hat{U}$. Comparing this operator $\hat{V}$ with the general form of two-level operators in Eq. (17), we can find that $c_{0}=c_{1}=1$, $x_{0}=3$ and $x_{1}=4$. By substituting $c_{0},c_{1},x_{0},x_{1}$ in Eqs.(26) and (27) with their respective values, we get

where $\hat{\sigma}_{x}$ is the Pauli $x$ matrix. By choosing the single-site coin operators $\hat{c}_{x}^{(t)}$ according to Eq.(7), we can realize the unitary operator $\hat{V}$ with an eight-step DTQW on a cycle with four sites.

For the most general two-level unitary operators $\hat{V}$, the calculation is essentially the same as the above example, i.e., find the values of $c_{0},c_{1},x_{0},x_{1}$ by comparing $\hat{V}$ with Eq. (17) and then substitute them in Eqs. (18) and (19) if $c_{0}=c_{1}$ or Eqs. (26) and (27) if otherwise. For unitary operators $\hat{V}$ which are not two-level, we decompose them into a product of two-level unitary operators $\hat{V}=\hat{u}_{m},\dots,\hat{u}_{2}\hat{u}_{1}$ Nielsen2007Quantum . By combining the DTQWs for $\hat{u}_{i}$ one after one, we can realize $\hat{V}$ with the final combined DTQW. As an example, the calculation to implement the Fourier transformation is provided in Appendix B.

Implementing a unitary operation with the construction in the proof of Theorem 1 as above involves numerous steps of the walk. To reduce the number of steps, we provide in Appendix C a further optimized scheme for implementations. With this scheme, no more than $2n^{2}-2n+1$ steps of walk is needed for the DTQW-QNN to be universal, where $n$ is the number of sites in the cycle.

It is sometimes cumbersome to find exact realizations of desired quantum operations in analytical ways. However, fair approximations to desired operations are often acceptable for practical purposes. In this section, we introduce an algorithm in a machine learning fashion to find the approximations by applying gradient descent to the DTQW-QNN. With this algorithm, the required number of depth can be further reduced when approximations are allowed.

In order to apply gradient descent to the DTQW-QNN, we have to do the following three things in advance.

1. 1.

   Parameterize the single-site coin operators with a four dimensional real vector $\vec{\alpha}^{(x,t)}$:

   $$\hat{c}_{x}^{(t)}=e^{i\alpha_{3}^{(x,t)}\hat{\sigma}_{3}}e^{i\alpha_{2}^{(x,t)}\hat{\sigma}_{2}}e^{i\alpha_{1}^{(x,t)}\hat{\sigma}_{1}}e^{i\alpha_{0}^{(x,t)}\hat{\sigma}_{0}},$$

   (8)

   in which $\hat{\sigma}_{j}$ is the $j$th Pauli matrix, $\hat{\sigma}_{0}=\hat{I}$.

2. 2.

   Introduce a state-wise loss function $L_{|{\Psi}\rangle}$:

   $$L_{|{\Psi}\rangle}=\frac{1}{2}\left\lVert|{\Psi^{(T)}}\rangle-|{\Phi^{(T)}}\rangle\right\rVert^{2},$$

   (9)

   where $|{\Psi^{(T)}}\rangle=\hat{U}_{T,0}|{\Psi}\rangle$ and $|{\Phi^{(T)}}\rangle=\hat{V}|{\Psi}\rangle$ are the final state and the desired final state respectively.

3. 3.

   Derive the partial derivative:

   $$\frac{\partial{L_{|{\Psi}\rangle}}}{\partial{\alpha^{(x,t)}_{j}}}=\operatorname{Im}\left(\langle{\Phi^{(t)}}|{\hat{\Sigma}_{j}^{(x,t)}}|{\Psi^{(t)}}\rangle\right),$$

   (10)

   where $|{\Psi^{(t)}}\rangle=\hat{U}_{t,0}|{\Psi}\rangle$ and $|{\Phi^{(t)}}\rangle=\hat{U}_{T,t}^{\dagger}|{\Phi^{(T)}}\rangle$ are the forward-propagation and back-propagation states respectively, $\hat{\Sigma}_{j}^{(x,t)}=(\hat{n}_{j}^{(x,t)}\cdot{\vec{\sigma}})\otimes\left|{x}\middle\rangle\middle\langle{x}\right|+\sum_{\xi\neq x}\hat{I}_{c}\otimes\left|{\xi}\middle\rangle\middle\langle{\xi}\right|$, ${\vec{\sigma}}=\sum_{j=0}^{3}\hat{\sigma}_{j}\vec{e}_{j}$, and $\hat{n}_{0}^{(x,t)}$, $\hat{n}_{1}^{(x,t)}$, $\hat{n}_{2}^{(x,t)}$, $\hat{n}_{3}^{(x,t)}$ equals

   $$\begin{bmatrix}1\\ 0\\ 0\\ 0\end{bmatrix},\begin{bmatrix}0\\ 1\\ 0\\ 0\end{bmatrix},\begin{bmatrix}0\\ 0\\ \cos{2\alpha_{1}^{(x,t)}}\\ \sin{2\alpha_{1}^{(x,t)}}\end{bmatrix},\begin{bmatrix}0\\ \sin{2\alpha_{2}^{(x,t)}}\\ -\cos{2\alpha_{2}^{(x,t)}}\sin{2\alpha_{1}^{(x,t)}}\\ \cos{2\alpha_{2}^{(x,t)}}\cos{2\alpha_{1}^{(x,t)}}\end{bmatrix}$$

   respectively.

Gradient descent iteratively moves the parameters in the opposite direction of the gradient, i.e.,

where $\eta$ is a positive real number called learning rate. Hence, the loss gradually drops during the iteration and the approximation to $\hat{V}$ by $\hat{U}_{T,0}$ becomes better and better.

The details of the algorithm to find the parameters of the DTQW-QNN $\left\{\vec{\alpha}^{(x,t)}:0\leq x<n\mbox{ and }0\leq t<T\right\}$ to approximate a desired unitary operator $\hat{V}$ are as the following.

1. 1.

   Set the total number of depth $T$ and the learning rate $\eta$ to be an appropriate positive integer and real respectively.

2. 2.

   Randomly initialize all the parameters $\alpha^{(x,t)}_{j}$.

3. 3.

   Randomly sample a state $|{\Psi}\rangle$ from the total Hilbert space $\mathcal{H}$.

4. 4.

   Calculate the partial derivatives $\frac{\partial{L_{|{\Psi}\rangle}}}{\partial{\alpha^{(x,t)}_{j}}}$ for all $t$, $x$, $j$ according to Eq. (10).

5. 5.

   Update all the parameters according to Eq. (11).

6. 6.

   Repeat Steps 3 to 5 until an acceptable approximation is reached.

One can notice that our choice of the loss function leads to a friendly form of gradients Eq. (10) for numerical calculation. The states $\left|{\Psi^{(t)}}\right\rangle$ and $\left|{\Phi^{(t)}}\right\rangle$ can be calculated by a forward-propagation and a back-propagation efficiently. Moreover, the gradients can be calculated by implementing a circuit with the help of an ancillary qubit as shown in Fig. 3. At the last of the circuit, the average value $\langle\hat{\sigma}_{3}\rangle$ of the ancillary qubit is measured. The result $\langle\hat{\sigma}_{3}\rangle$ can be used to update the parameters of the DTQW-QNN since $\langle\hat{\sigma}_{3}\rangle$ always coincides with the partial derivative $\partial{L_{|{\Psi}\rangle}}/\partial{\alpha^{(x,t)}_{j}}$ in Eq. (10). This might enable us to implement simultaneous tomography and cloning of an unknown unitary operation.

Besides, the position space $\mathcal{H}_{p}$ is commonly much larger than the coin space $\mathcal{H}_{c}$. Theorem 1 thus indicates that one can indirectly control a large system by controlling a small two-level coin system via DTQW on a cycle graph. For example, unitary operations and general two-outcome measurements described by positive-operator-valued measures (POVMs) can be applied to the position space in this way straightforwardly according to Theorem 1. If we are only interested in the unitary operators that act on the position space $\mathcal{H}_{p}$, we only need one arbitrary site to be allowed to assign nonidentity coin operators. The detailed content is provided in Appendix D.

In all previous parts of this work, we have assumed that the single-site coin operators $\hat{c}_{x}^{(t)}$ can take values from $\mathrm{U}(2)$ arbitrarily. This means the single-site coin operator can have arbitrary phase and arbitrary rotational axis. However, it would be much easier to implement rotations along a fixed axis with fixed phases in laboratories. Hence, in this section, we simplify the DTQW so that it becomes easier to implement. Also, there are always noises when DTQW-QNNs are implemented in laboratories. We test it under the situation where noises are presented in the single-site coin operators $\hat{c}_{x}^{(t)}$. Throughout this section, the numerical demonstrations are all based on DTQWs on a cycle with two sites.

In conclusion, we have proposed a quantum neural network based on a simple DTQW on a cycle graph, and used the network to implement arbitrary quantum computation tasks, i.e., unitary operations on an arbitrary $N$-dimensional Hilbert space.

In order to implement an arbitrary unitary operation via a circuit model, one needs to decompose the unitary into a sequence of smaller unitary operators. However, via our DTQW-QNN, we only need to update the parameters by a learning algorithm. In other words, our model is adaptive to new tasks. With a new computational task given, our network can simply evolve according to the learning algorithm, and there is no need to decompose the desired operation into a sequence of smaller gates.

Regarding the universality of our model, we presented a specific construction of realizing arbitrary two-level unitary operations on the computational basis, and proved that the DTQW-QNN is universal for all unitary operations on the overall Hilbert space of the involved quantum systems. The DTQW-QNN is not only universal for quantum computation but also universal for controlling the whole quantum system. We also provided an optimization so that the circuit depth of the DTQW-QNN does not need to exceed $2n^{2}-2n+1$ to realize an arbitrary unitary operator on a $2n$-dimensional Hilbert space. However, this is only a theoretical limit of the network size in the worst case for the purpose of analytical proof. The appropriate number of nodes for each task may vary, and it is an open question to find this number for a given task.

Our network evolves according to a learning algorithm based on gradient descent, with the loss function carefully chosen so that the parameter updates can be efficiently calculated in a back-propagation fashion and can be, in principle, directly read out from a measurement. The algorithm performs well in updating the parameters of the neural network. We have shown good approximations of unitary operations on a Hilbert space up to $40$ dimensions, as well as arbitrary two-outcome POVMs. Finally, we have also simplified the DTQW-QNN in various aspects. For example, the rotation gates involved in the DTQW are all limited to be along the $x$-axis. Such simplifications make the DTQW-QNN more friendly for laboratory implementations while its capability of implementing desired operations is maintained.

We have shown the capability of the DTQW-QNNs in both analytical and numerical ways. Further studies might reveal their total capacity in completing various quantum computation tasks as well as solving machine learning problems, and further experimental implementations would make them more practically useful and closer to real-life applications.