Estimation of one-dimensional discrete-time quantum walk parameters by using machine learning algorithms

Quantum walks R58 ; F86 are the quantum counterparts of classical random walks, and are therefore used as the basis to generate relevant models for controlled quantum dynamics of a particle. Much like a classical random walk, the formalism for quantum walks has also developed in two forms - the discrete-time quantum walk (DTQW) and the continuous-time quantum walk (CTQW). Both the formalisms exhibit features that enable them to effectively realise quantum computational tasks DW08 ; C09 ; A07 ; MSS07 ; BS06 ; FGG07 ; K08 . Quantum superposition and interference have the effect of allowing the quantum walker to have a quadratically faster spread in position space in comparison to a classical walker P88 ; ADZ93 ; M96 ; K03 ; CCD03 ; CG04 ; VA12 . This has applications in modelling dynamics of several quantum systems, like quantum percolation CB14 ; KKNJ12 ; CAC19 , energy transport in photosynthesis ECR07 ; MRLA08 , graph isomorphism DW08 , quantum algorithms CMC20 ; PMCM13 , and even in generating a scheme for universal quantum computation SCSC19 ; SCASC20 .

Much like a classical walk, the dynamics of a walker undergoing CTQW can be described only by a position Hilbert space, whereas a walker performing DTQW requires an additional degree of freedom to express its controlled dynamics. This is realised by the coin Hilbert space, which is an internal state of the walker, and provides the relevant additional degree of freedom for the walker. Tuning the parameters and evolution operators of a DTQW enables the walker to simulate several quantum mechanical phenomena, such as topological phases COB15 ; SRFL08 ; KRBD10 ; AO13 , relativistic quantum dynamics S06 ; CBS10 ; MC16 ; C13 ; MBD13 ; MBD14 ; AFF16 ; P16 ; RLBL05 , localization J12 ; C12 ; CB15 , and neutrino oscillations MMC17 ; MP16 . Quantum walks have been experimentally implemented in various physical systems, such as photonic systems SCP10 ; BFL10 ; P10 , nuclear magnetic resonance systems RLBL05 , trapped ion systems SRS09 ; ZKG10 , and cold atoms KFCSWMW09 .

It is known that the coin parameters of a quantum walk play a significant role in determining the overall dynamics of the system. For instance, the coin parameters of a split-step quantum walk determine topological phases COB15 ; SRFL08 ; KRBD10 , neutrino oscillation MMC17 ; MP16 , and the mass of a Dirac particle MC16 . Thus the coin parameter is an important piece of the puzzle while using a quantum walk as a quantum simulation scheme. It thus becomes a crucial problem to be able to estimate the parameters of a quantum walk in order to facilitate better quantum simulations and also for further research into modelling realistic quantum dynamics.

In this regard, the problem partially becomes one of finding a pattern hidden in this complex data, and an effective approach is to use an algorithm that automates its learning process. In this context, the term ’learning’ aptly described in Mitchell97 : A computer program is said to learn from experience E with respect to some class of tasks T and performance measure P, if its performance at tasks in T, as measured by P, improves with experience E. The task T in this particular case is defined as, to output a function $f(\mathbb{R}^{N}\rightarrow\mathbb{R})$, such that the input vector corresponds to the known probability distribution after $N$ steps and the output corresponds to the parameter $\theta$. It may, however, be noted that due to the No Free Lunch theorem W96 , the machine learning strategies for different types of quantum walks are not likely to be the same as for this particular case.

It is known that the Quantum Fisher Information (QFI) P11 ; TP13 ; S13 ; TA14 $H_{w}(\theta)$ of a quantum walker’s position quantifies a bound to the amount of information that can be extracted from the probability distribution P12 ; SCP19 . It has also been shown that it is indeed possible to estimate the coin parameter of a DTQW with a single-parameter coin. However, the approaches that rely on QFI to predict the parameters in a DTQW share a similar constraint, viz, even though $H_{w}(\theta)$ is a continuous, single-valued function over $\theta$, the variations in the plot are small, and thus contribute to the error in determining $\theta$ SCP19 . In this work, we train various machine learning models and demonstrate that such models can indeed estimate the quantum walk parameters to a good accuracy. We also estimate the coin parameter and the number of steps for a DTQW simultaneously with a multilayer perceptron model, and demonstrate that it performs much better than a baseline model that does not learn with experience.

This paper is structured as follows. In section II we introduce a standard DTQW, an SSQW, and describe the evolution operators that determine their dynamics. We give a brief overview of the machine learning models used and their parameters in Section III. Section IV details our results of training the machine learning algorithms and their performance. We wrap up the paper in section V with a small discussion on our results and conclusions drawn.

The evolution of a walker executing DTQW is defined in a Hilbert space $\mathcal{H}=\mathcal{H}_{C}\otimes\mathcal{H}_{P}$, where $\mathcal{H}_{C}$ and $\mathcal{H}_{P}$ are the walker’s coin and position Hilbert spaces, respectively. The coin Hilbert space has the basis states $\left\{\ket{\uparrow},\ket{\downarrow}\right\}$, and the position Hilbert space is defined by the basis states $\ket{x}$, where $x\in\mathbb{Z}$. The evolution is described by two unitary operators, known as the coin and shift operations. The coin operator affects the coin Hilbert space, and the shift operator evolves the walker in a superposition of position states, the amplitudes of which are determined by the coin operation. In case of the one-dimensional DTQW, the most general unitary coin operator is an $SU(2)$ matrix, which has three independent parameters. That said, even one- and two-parameter coins are very useful while simulating various systems. As an example, the single-parameter split step DTQW is very effective to simulate neutrino oscillations MMC17 , topological phases COB15 , and the Dirac equation MC16 .

In a one-dimensional DTQW, the coin operation is an $SU(2)$ matrix, defined as,

To obtain the the one-parameter coin form from Eq. (1), we fix the values $\xi=0$ and $\zeta=-\frac{\pi}{2}$. This is the convention adopted in the remainder of this text. The one-parameter coin operator is thus given as,

Fig. 1 shows the probability distributions in position space of a walker performing a DTQW with different values of $\theta$ as in Eq. (5).

Fig. 2 shows the probability distributions obtained by a walker executing a SSQW with multiple combinations of $\theta_{1}$ and $\theta_{2}$. Unlike the two peak probability distribution from the single coin DTQW, SSQW can results in four peak probability distribution.

Machine learning was originally created to try and create solutions to problems which were hard to define formally, such as recognizing faces in images, or cognition of spoken words. This necessitated the creation of algorithms that could learn from experience of examples supplied to them, so the programmer was absolved of the requirement to formally specify all knowledge needed to solve the problem beforehand.

While the hardcoded knowledge methods worked very well for small, relatively sterile environments such as the world of chess CHH02 , such programs face difficulties in subjective and intuitive tasks such as understanding speech or object recognition. Inference-based methods have been suggested and implemented as a possible solution to this type of problem, but have not had much success LG89 ; D15 . Modern approaches circumvent this difficulty by implementing algorithms that have the capability of recognising patterns in raw data on their own GBC16 - a trait now known as machine learning.

In this work, we have used machine learning algorithms and trained an MLP model in an attempt to estimate the parameters of a one-dimensional DTQW. The models were trained on DTQW with $N=500$, and $\theta$ varying from $\frac{\pi}{180}$ to $\frac{89\pi}{180}$, at an interval of $\frac{\pi}{1800}$. The dataset consists of DTQW distributions that resulted after using these parameters in conjunction with a symmetric initial state as,

We have chosen the ratio of training to testing data as $3:1$ ($75\%:25\%$) for this dataset. As a result, $668$ randomly selected probability distributions are used to train our algorithm, and the rest are used to test its performance. Three regression models, namely, K-nearest neighbours, linear regression and ridge regression, were trained on this data, and their performance was evaluated by the mean square error (MSE) of their predictions. The results are shown as in Table 1. It is observed that while all the models return a very low MSE, but the performance of the linear regression model is a few orders of magnitude better than the next best performing model (k-nearest neighbours).

Using the same training data as specified above, the model was tasked to predict the value of $\theta$, and given the testing data corresponding to $\theta=\frac{\pi}{2}$, which is outside the range used to generate the training data. Table 2 details the predictions by each of these models.

We also implemented an MLP model (detailed in section III) to try and predict two parameters ($\theta$ and $N$) of a 1D-DTQW simultaneously via deep learning. The model contains $3$ layers, out of which there is $1$ hidden layer. The model uses rectified linear units as the activation function for the first two layers (i.e. the input and hidden layers, respectively), and an exponential activation for the output layer. The neural net uses the Nadam optimizer. Other possible optimizers that can be used here are AdaGrad, AdaDelta and Adam. AdaGrad offers the benefit of eliminating the need of manually tuning the learning rate, but has the weakness that it accumulates a sum of squared gradients in its denominator over time, causing the learning rate to shrink and eventually become infinitesimally small, and the algorithm stop learning at this point. AdaDelta aims to reduce the aggressive decay of the learning rate in AdaGrad by fixing a window size of the accumulated past gradients. Adam tries to improve on both AdaGrad and AdaDelta by considering an exponentially decaying average of past gradients to adjust its learning rates. Nadam, however, improves on the performance of Adam by using Nesterov-accelerated gradients, and was chosen as the best fit for this case.

To train this network, we generated a new dataset, which contained all possible combinations of the parameters $\theta$ and $N$, varying between $[\frac{\pi}{180},\frac{\pi}{2}]$ and $[1,499]$ respectively, with a step size of $1$ for $N$, and $\frac{\pi}{180}$ for $\theta$. The dataset thus comprised of $44,910$ different probability distributions. The training-testing split was chosen to be the same as the earlier case ($75\%:25\%$), and the training set thus consisted of a randomly selected $33,682$ probability distributions of the total.

In order to judge the effectiveness of this model, we also designed a baseline model, which would always predict the mean of all supplied training values of $\theta$ and $N$ ($\frac{89\pi}{360}$ and $249$ respectively, for this case), for any input distribution. The MSE from this baseline model was found to be ${26077.4287}$.

Our neural network gave an output MSE of $916.0458$, which is a reduction of $96.488\%$ on the baseline error. This proves that the neural network is able to effectively learn and reproduce the patterns found in the DTQW probability distributions in order to reasonably estimate both the parameters of a 1D-DTQW.

We also tried to estimate selected parameters of an SSQW. A one-dimensional SSQW with two single-parameter coins as defined in Eq. (5) which typically has 3 parameters, $\theta_{1}$, $\theta_{2}$, and $N$. For the purposes of this task, we varied $\theta_{1}$ in the range $[\frac{\pi}{1800},\frac{\pi}{2}]$ at an interval of $\frac{0.04\pi}{180}$, while keeping $\theta_{2}=\frac{\pi}{4}$, and $N=100$ fixed. We trained a linear regression model on this data, as well as a K-nearest neighbour model. The parameter was estimated with a good accuracy, as is seen from the contents of Table 3.

In this work, we have demonstrated the effectiveness of machine learning models while estimating the coin parameter in a DTQW and an SSQW. We have applied these models on the one-dimensional DTQW and SSQW, and attempted to estimate the parameters from these walks.

In the case of DTQW, we applied three different models in order to estimate the coin parameter $\theta$, and conclude that a machine learning-based approach is indeed able to estimate the parameter very well. We also attempt to use a neural network in order to try and predict the two parameters at once, and show that its prediction error reduces by a significant amount from the baseline error, implying that the network is able to distinguish and replicate patterns found in this data. We also use two different models in order to estimate one of the coin parameters of a SSQW.

It is important to keep in mind that the accuracy of the predictions will vary with the amount of training data it is given. Typically, larger datasets improve the accuracy. The accuracy is also very dependent on the model itself under consideration. There is no single known machine learning algorithm that may outperform all others, and it is thus important to choose the algorithm for implementation with care.