Approaching graph problems with continuous variable quantum computing

TL;DR

This paper presents a novel quantum computing method using continuous variables and variational algorithms to solve the Max-Cut problem, analyzing the impact of non-Gaussian gates and framing the circuit as a machine learning model.

Contribution

It introduces a new continuous-variable quantum approach for Max-Cut, combining graph embedding and variational circuits, with analysis of non-Gaussian gate effects.

Findings

Non-Gaussian gates influence optimization performance

Proposed a machine learning perspective for quantum circuits

Numerical simulations demonstrate potential effectiveness

Abstract

We introduce a method for solving the Max-Cut problem using a variational algorithm and a continuous-variables quantum computing approach. The quantum circuit consists of two parts: the first one embeds a graph into a circuit using the Takagi decomposition and the second is a variational circuit which solves the Max-Cut problem. We analyze how the presence of different types of non-Gaussian gates influences the optimization process by performing numerical simulations. We also propose how to treat the circuit as a machine learning model.

Equations

$$\mathbb{X}=\left[\begin{array}{cc}0& {\mathbb{I}}_n\\ {\mathbb{I}}_n& 0\end{array}\right]$$

$${\sigma }_A=({\mathbb{I}}_{2n}-\mathbb{X}A{)}^{-1}-{\mathbb{I}}_{2n}/2$$

$$A^{\prime }=\left[\begin{array}{cc}A& 0\\ 0& A\end{array}\right]$$

$${\sigma }_{c,d,A}^{\prime }=({\mathbb{I}}_{4n}-\mathbb{X}d\cdot (c{\mathbb{I}}_{4n}+A^{\prime }){)}^{-1}-{\mathbb{I}}_{4n}/2$$

$$B=UD{U}^T$$

$$w(S,V\backslash S)=\sum _{i\in S,j\in V\backslash S}w(i,j)$$

$$mc(G,w)=ma{x}_{S\subset V}w(S,V\backslash S)$$

$$w(S)=\sum _{i,j\in E}{w}_{i,j}(1-s_i\ast s_j),$$

$$\hat{H}=\sum _{k=1}^N{\hat{H}}_k$$

$${\hat{H}}_k=\frac{1}{2}{\omega }_k^2{\hat{x}}_k^2+\frac{{\hat{p}}_k}{2}$$

$${\hat{X}}_k=\sqrt{\frac{{\omega }_k}{\hslash }}{\hat{x}}_k$$

$${\hat{P}}_k=\sqrt{\frac{1}{\hslash {\omega }_k}{\hat{p}}_k}$$

$${\hat{H}}_k=\frac{\hslash \omega }{2}\left[{\hat{X}}_k^2+{\hat{P}}_k^2\right]$$

$$G(x)=C\times exp\left\{-\frac{1}{2}{x}^{T}Ax+b^{T}x\right\}$$

$$W(x,y)=\frac{2}{\pi }{\int }_{-\infty }^{+\infty }dy\times exp\left\{+4iyp\right\}\times ⟨x-y|\hat{\rho }|x+y⟩$$

$${\hat{D}}^{†}(\alpha )\hat{x}\hat{D}(\alpha )=\hat{x}+\sqrt{2\hslash }Re(\alpha )I$$

$${\hat{D}}^{†}(\alpha )\hat{p}\hat{D}(\alpha )=\hat{p}+\sqrt{2\hslash }Im(\alpha )I$$

$$\left[\begin{array}{c}{x}^{\prime }\\ p^{\prime }\end{array}\right]=\left[\begin{array}{c}x+Re(\alpha )\\ p+Im(\alpha )\end{array}\right]$$

$${\hat{S}}^{†}(r)\hat{x}\widehat{S(r)}=e^{-r}\hat{x}$$

$${\hat{S}}^{†}(r)\hat{p}\widehat{S(r)}=e^{+r}\hat{p}$$

$$\left[\begin{array}{c}{x}^{\prime }\\ p^{\prime }\end{array}\right]=\left[\begin{array}{cc}{e}^{-r}& 0\\ 0& e^{+r}\end{array}\right]\left[\begin{array}{c}x\\ p\end{array}\right]$$

$${\hat{R}}^{†}(\varphi )\hat{x}\hat{R}(\varphi )=\hat{x}\cos \varphi -\hat{p}\sin \varphi$$

$${\hat{R}}^{†}(\varphi )\hat{p}\hat{R}(\varphi )=\hat{x}\sin \varphi +\hat{p}\cos \varphi$$

$$\left[\begin{array}{c}{x}^{\prime }\\ p^{\prime }\end{array}\right]=\left[\begin{array}{cc}\cos \varphi & \sin \varphi \\ -\sin \varphi & \cos \varphi \end{array}\right]\left[\begin{array}{c}x\\ p\end{array}\right]$$

$$\{\begin{array}{ll}{\widehat{BS}}^{†}(\theta ,\varphi ){\hat{x}}_1\widehat{BS}(\theta ,\varphi )& ={\hat{x}}_1\cos \theta -\sin \theta \left[{\hat{x}}_2\cos \varphi +{\hat{p}}_2\sin \varphi \right]\\ {\widehat{BS}}^{†}(\theta ,\varphi ){\hat{x}}_2\widehat{BS}(\theta ,\varphi )& ={\hat{x}}_2\cos \theta +\sin \theta \left[{\hat{x}}_1\cos \varphi -{\hat{p}}_1\sin \varphi \right]\end{array}$$

$$\{\begin{array}{ll}{\widehat{BS}}^{†}(\theta ,\varphi ){\hat{p}}_1\widehat{BS}(\theta ,\varphi )& ={\hat{p}}_1\cos \theta -\sin \theta \left[{\hat{p}}_2\cos \varphi +{\hat{x}}_2\sin \varphi \right]\\ {\widehat{BS}}^{†}(\theta ,\varphi ){\hat{p}}_2\widehat{BS}(\theta ,\varphi )& ={\hat{p}}_2\cos \theta +\sin \theta \left[{\hat{p}}_1\cos \varphi -{\hat{x}}_1\sin \varphi \right]\end{array}$$

$$\left[\begin{array}{c}{x}_1^{\prime }\\ x_2^{\prime }\\ p_1^{\prime }\\ p_2^{\prime }\end{array}\right]=\left[\begin{array}{cccc}\cos \theta & -\sin \theta \cos \varphi & 0& \sin \theta \sin \varphi \\ \sin \theta \cos \varphi & \cos \varphi & -\sin \varphi \cos \theta & 0\\ 0& \sin \varphi \cos \theta & -\cos \theta & -\sin \theta \cos \varphi \\ -\sin \varphi \cos \theta & 0& \sin \theta \cos \varphi & \cos \theta \end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\\ p_1\\ p_2\end{array}\right]$$

$$\hat{V}(\gamma )\hat{x}\hat{V}(\gamma )=\hat{x}$$

$$\hat{V}(\gamma )\hat{p}\hat{V}(\gamma )=\hat{p}+\gamma \hat{x}$$

$$K=exp\left\{i\kappa {\hat{n}}^2\right\}$$

Full text

1 Introduction

In recent years there has been a boom in quantum computing. Different models of hardware have been proposed, with the most dominant being based on superconducting circuits or trapped ions (cf. IBM, Rigetti, IonQ). All these architectures use discrete values (qubits) as the basis for computation. However, this is not the only possible approach — one might also create a quantum computer based on qudits or even continuous variables [1]. The later is the approach that the Canadian company Xanadu is pursuing, building a photonic quantum computer utilizing the continuous variable paradigm. They have built the open-source libraries Strawberry Fields and PennyLane which provide tools for simulating photonic circuits and perfoming machine learning experiments.

Our goal in this paper is to take a hands-on approach towards optimization problems on continuous-variable quantum computers. This type of problem has already been researched both for discrete quantum computing and quantum annealing in [2], [3], and [4]. In particular, we consider a solution to the Max-Cut problem on weighted graphs. We embed a graph into a quantum state, and then we optimize the parametrizable part of the circuit using a well-chosen cost function. For readers new to continuous-variable quantum computing, the appendices can serve as a quick introduction to the terminology and foundations of the field.

1.1 Related work

In recent years, researchers started to use a new approach for creating quantum algorithms, namely variational circuits [5]. Algorithms like Variational Quantum Eigensolver (VQE) [6] or Quantum Approximate Optimization Algorithm (QAOA) [2] have been succesfully employed to graph problems like the Max-Cut [2] and the traveling salesman problems [7]. However, these algorithms use the discrete variable quantum computing paradigm and there has not been much work done to solve this type of problems with continuous-variable quantum computing; with the notable recent exception of [8]. Among all the graph problems, the Max-Cut problem has thus far gotten much more attention than others as can be found in [2], [3], and [9]. Since many researchers use it as the first problem for testing and benchmarking their variational optimization algorithms, we decided to follow this trend.

We have based our work on the methods described in [10] and [11] and developed them further to solve the Max-Cut problem on a simulator of a photonic quantum computer.

1.2 Acknowledgements

We would like to thank Nathan Killoran and Josh Izaac for their help and guidance, Maria Schuld for help with the QMLT package and Nicolás Quesada for help with understanding the Takagi decomposition. We would also like to acknowledge Wayne Nixalo and Witold Kowalczyk for their contribution at the initial stage of this research.

1.3 Organization

This paper is organized as follows: in section 2 we present the theoretical framework we used in this research. We introduce two ways of representing a graph as a quantum state using either a Gaussian covariance matrix or the Takagi decomposition.

In section 3, we present results of the simulation made on graphs of different sizes and offer preliminary results for extending the research to a machine learning model.

In section 4, we conclude the paper and give directions for further research.

Appendices A, B, C and D introduce the reader to the continuous-variables quantum computing paradigm.

2 Theoretical framework

2.1 Representation of a graph as a quantum state

Let $(G,w)$ be a weighted graph where $G=(V,E)$ is a graph with vertices $V$ and edges $E$, and $w$ is a weight function which attaches to each edge between vertices $i$ and $j$ a real number $w(i,j)$. We set $w(i,j)=0$ if there is no edge connecting $i$ and $j$.

We write $A=(a_{ij})$ for the weighted adjacency matrix, that is we set $a_{ij}=w(i,j)$ for each $i,j\in V$. Inspired by [12] and [10], we match $A$ with a Gaussian covariance matrix (i.e. a matrix describing a state created by a Gaussian quantum circuit). We can either calculate it directly or perform the Takagi decomposition, which allows us to omit direct calculations. We describe both methods, but we use only the second one for the numerical simulations.

2.1.1 Gaussian covariance matrix

We assume that $G$ has $n$ vertices. Let $\mathbb{I}_{n}$ be the identity matrix. We define

[TABLE]

The Gaussian covariance matrix associated with $A$ is defined to be

[TABLE]

Let $c$ be an auxilary real number which will be our parameter to be determined for each graph separately. We choose $c$ such that $\sigma_{c\mathbb{I}_{2n}+A}$ is symplectic and positive definite, if that is possible. The reason we need to scale $A$ is because $\sigma_{A}$ does not always give a proper Gaussian covariance matrix222That is, it cannot be always represented by a gaussian quantum state (cf. [10], especially Appendix A). For arbitrary $A$, the above might not work. That is why we introduce $d$ to be another auxilary positive real, which we use as a parameter. Adopting the method in [10] we define

[TABLE]

and set

[TABLE]

where $\mathbb{X}$ here is defined with $\mathbb{I}_{2n}$. Pursuant to Appendix A in [10] for any $A$ there are always $c,d>0$ such that $\sigma^{\prime}_{c,d,A}$ is symplectic and positive definite.

Now, using Strawberry Fields we are able to associate a quantum circuit with $\sigma_{d(c\mathbb{I}_{2n}+A)}$ if that is possible, or with $\sigma^{\prime}_{c,d,A^{\prime}}$ if not. Note that we try to avoid using $\sigma^{\prime}_{c,d,A^{\prime}}$ as it doubles the dimensions and the number of qumodes we need to use.

The result is a quantum circuit for which the output probability distribution depends on matrix $A$.

This method introduces additional parameters and requires choosing them in such a way that all the matrices meet the required conditions. However, the same result can be achieved by using the Takagi decomposition, as described in the section below. In this research we have used the later approach.

2.1.2 Takagi decomposition

Let’s take a set of $N$ squeezed states, with a squeezing parameter $r_{i}$, followed by an interferometer described by a matrix $U$ (see fig. 2.1.2). If the matrices meet the condition:

[TABLE]

where $D$ is a diagonal matrix with elements $d_{i}$ (which are the eigenvalues of $B$) on the diagonal and $r_{i}=\operatorname{arctanh}(d_{i})$, then the probability distribution of such a state depends on matrix $B$ [12].

Therefore, if we want to embed a weighted graph described by a distance matrix $B$ in a circuit, we do not need to calculate the covariance matrix; it’s enough to perform the Takagi decomposition which is given by the equation 1 and to set the parameters of the gates accordingly.

There are two restrictions on matrix $B$: it has to be symmetrical and its eigenvalues must be from the interval $[-1,1]$ so that it fits the $\operatorname{arctanh}$ function. Matrix $A$ is always symmetrical, but it can have arbitrary eigenvalues. Therefore in order to embed it, we need to rescale it by multipling it by a constant, so that it meets the second condition.

This method has several advantages over the previous one: it does not require calculating the covariance matrix explicitly, it does not introduce any parameters and it is much simpler. Those reasons make it an attractive method for state preparation as we do in this paper.