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Graph identification by quantum walks

J.B. Wang and B.L. Douglas

School of Physics, The University of Western Australia, Perth 6009, Australia.

Quantum random walks display remarkably different properties from their classical

counterparts, most notably their fast spreading characteristics1-3. For example, they were

proven to provide an exponential algorithmic speedup for traversing a randomised glued-tree

graph4. However, despite such potentially superior efficiency in quantum random walks, they

have yet to be applied to problems of practical importance3. Graph isomorphism is a long-

standing open problem in mathematics, which is to decide whether two given structures are

topologically identical5. This has applications in many areas of science and engineering. In this

paper, we present an algorithm using quantum walks to solve the graph isomorphism problem.

In particular, a novel measurement scheme is presented which makes it possible to identify

graph isomorphism in polynomial time.

Graph isomorphism (GI) is of considerable importance in solving a wide range of practical problems.

For example, it is often critically important in chemistry or molecular biology that we know if two

molecules have topologically the same structure, a generalization of which is a graph with specified

nodes and connectivity. Graph isomorphism identification provides an efficient tool for protein

structure comparison and classification. It can be used as well for structural analysis of kinematic

chains, oil pipelines, roads and subways, scheduling problems, network management, communication

systems, and circuit design.

Graph isomorphism is also of considerable theoretical importance due to its computational complexity

and its relationship to the concepts of P vs NP vs NP-completeness. Moreover, a whole set of problems

is referred to as GI-reducible and GI-complete, such as finding the actual isomorphism mapping, graph

isomorphism for directed graphs, graph automorphism, and graph automorphism mapping, which are

proven to be either Turing equivalent to GI or Turing reducible to GI5. This means these and some

other combinatorial problems can be considered as special cases of the graph isomorphism problem.

The two graphs shown in Fig. 1 appear to be very different but are in fact topologically identical. A

naive approach to identify isomorphism is to generate all n! permutations of the n nodes and test each

in turn. The solution then scales in O(n!). More efficient algorithms exist for certain classes of graphs

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such as trees, planar graphs, bipartite graphs, circular-arc graphs, and graphs with bounded degree or

bounded genus6-9. However, the best-known algorithm for general graphs scales exponentially in

!

O(en +O(1)

) , which is based on a canonical labeling scheme after a set of transformations10.

Given a specified graph, we can also learn about its structure by evolving random walks along it. That

is at each node or intersection we throw a coin and move according to the outcome. This process is

stochastic by nature and can be described by a transition matrix:

!

p1'

...

pn '

"

#

$ $ $

%

&

' ' '

=

s11 ... s1n

... ... ...

sn1 ... snn

"

#

$ $ $

%

&

' ' '

p1

...

pn

"

#

$ $ $

%

&

' ' ' ,

which reflects on the actual design of coins and can be either biased or non-biased. Quantum walks are

similar in many ways, except that we are now dealing with a superposition of coin states, involving a

simultaneous walk in all directions. At each time step we apply a coin operator, which mixes the coin

states of each node, and the system evolves accordingly. This is no longer a stochastic process but

rather a unitary process. In fact there is nothing really random about quantum walks. It is deterministic

in the quantum sense and completely reversible. The quantum transition operator is unitary which acts

on the system wavefunction, yielding amplitudes instead of probabilities:

!

"1'

...

"n'

#

$

% % %

&

'

( ( (

=

u11 ... u1n

... ... ...

un1 ... u

nn

#

$

% % %

&

'

( ( (

"1...

"n

#

$

% % %

&

'

( ( ( .

We can also walk on graphs continuously like a flux instead of making discrete time steps. In this case

the classical walk can be described as a continuous time Markov chain, i.e.

!

P(t) = e"A t

P(0) ,

while the quantum walk is described by a unitary evolution, which is a formal solution of the time-

dependent Schrödinger equation, i.e.

!

"(t) = e#i A t"(0) .

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Here

!

A is the graph adjacency or Laplacian matrix with elements

!

Aij = m , where

!

m is the number of

edges connecting nodes

!

i and

!

j . The diagonal elements

!

Aii can be defined in various ways, providing

some freedom in controlling the properties of the walks, such as adding self-loops.

Although the mathematical formulation for the classical and quantum walks is very similar, they

display remarkably different walking characteristics. For non-biased classical random walks on any

graph, the system diffuses into an even population at all (connected) nodes. Introducing some

difference in the graph makes little difference in the overall appearance of the probability distribution

as shown in Fig. 2 (middle panel). On the other hand, quantum walks are a unitary process and

completely reversible. They will not diffuse into a steady state, but rather the wavefunction amplitude

at each node oscillates in such a way which reflects upon the topological structure of the graph. The

amplitude distribution is significantly different for even slightly modified graphs, as illustrated in Fig.

2 (bottom panel).

An important question arises: can we identify graph isomorphism from the amplitude distributions

resulting from quantum walks on graphs? There have been a number of suggestions in the literature to

study graph isomorphism using quantum walks4,11. In particular, two recent papers have actually

touched upon this topic. Shiau et. al. performed single-particle quantum walks on closed graphs, but

they concluded that such walks fail to identify non-isomorphic strongly regular graphs (srg)12. They

introduced quantum walks of two interacting particles, which distinguished a set of srg’s with up to 29

nodes. However, the number of matrix overlaps required in their algorithm grows exponentially as the

graph size increases. One class of classical GI algorithms relies on the eigenvalues of Laplacian

matrices, but many non-isomorphic graphs are co-spectral using the Laplacian matrices or some other

modified matrix representation13,14*. Emms et. al. introduced a new matrix representation inspired by

quantum walks15. This new matrix displays different spectra for a set of non-isomorphic srg’s, but it

fails to distinguish fairly simple general graphs.

Strongly regular graphs can be represented by a set of parameters srg(n, d, λ, µ). Each of the n nodes in

such a graph has d neighbours, every two adjacent nodes have λ common neighbours, and every two

non-adjacent nodes have µ common neighbours16. Srg’s with the same parameters have a high degree

of symmetry and similarity, and are consequently difficult to distinguish. Indeed, groups of srg’s are

often used to test proposed GI algorithms. Not surprisingly, there is no difference between the

probability distributions resulting from classical walks along two srg’s with the same parameters. For * It is readily verifiable that the improved AAM scheme14 cannot distinguish non-isomorphic srg’s.

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example, two srg’s with the parameter set (16,6,2,2) are seen to yield identical probability distributions

in Fig. 3 (middle panel). For the same pair of srg(16,6,2,2), quantum walks show regular oscillation

due to the symmetry of these graphs, but unfortunately a straight-forward implementation of quantum

walks yields exactly the same pattern at many nodes as demonstrated in Fig. 3 (bottom panel),

although the two graphs are intrinsically different.

We were facing two rather formidable difficulties. Firstly, non-isomorphic graphs with a high degree

of symmetry and similarity can display exactly the same amplitude distribution, making them

indistinguishable with respect to quantum walks. Secondly, even when the two amplitude distributions

are different, to build up such a distribution would still be a formidable task, since every projective

quantum measurement causes a collapse of the wave function and the walks need to start all over

again.

We circumvented the symmetry problem by adding inhomogeneity into the graphs, which in effect

breaks the symmetry with respect to the walks that might exist between two non-isomorphic graphs

and consequently makes them distinguishable. This has been successfully done through the addition of

phases, which can be added to nodes (or equivalently to the coin operator) and to edges (both directed

and undirected). It is important to note that we add the inhomogeneity symmetrically about certain

reference points and we also apply symmetric coin operators at all nodes, so that we do not

differentiate the two graphs with respect to labeling.

Shown in Fig. 4 are two randomly generated graphs. Suppose we do not know whether graph

!

Y is

topologically different from

!

X . The proposed quantum algorithm is as follows. We chose a pair of

measurement and phase nodes in

!

X and

!

Y denoted as

!

{Xm,X"} and

!

{Ym,Y"} , start quantum walks

with the addition of phases to

!

X" and

!

Y" at each time step, and then compare the amplitude

distributions measured at

!

Xm

and

!

Ym

. If they are the same at all times we write “0”, otherwise “1” on

a comparison table, which is built up by cycling through all possible pairs of

!

{Ym,Y"} for fixed

!

{Xm,X"} . Typically this table has some “0”s and some “1”s. We then change to another pair of

!

{Xm,X"} , which will also be cycled through all possible combinations. If for a chosen pair of

!

{Xm,X"} we record a whole table of “1”s whilst cycling through all pairs of

!

{Ym,Y"} , graphs

!

X and

!

Y must be different. This is a sufficient condition. We then compare one of the graphs with its own

permutation, in which case there are many more “0”s in the comparison tables than comparing two

non-isomorphic graphs. Since the comparison cycles through all permutations and, by symmetry

arguments, a graph cannot be more similar to graph X than X’s own permutation, the total number of

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“0”s summed over the comparison tables is minimal if graph Y is actually a permutation of graph X.

We conjecture that this is a necessary and sufficient condition.

To solve the measurement problem we propose the following physical implementation scheme, but the

same principle also applies to other implementations. In this scheme, the nodes of a graph are

represented by energy levels of a quantum system, which can be trapped ions or quantum dots or any

other quantum systems with discrete energy levels. The quantum walker starts at an equal

superposition of both

!

Xm

and

!

Ym

, and walks quantum mechanically through both graphs with a

constant phase added to

!

X" and

!

Y" at each time step. After certain number of time steps, which is

linearly dependent on the diameter of the graphs, a probe node is introduced which is simply an

ancillary energy level in the system as illustrated in Fig. 4. Two coherent lasers of the same intensity

are then applied with frequencies matching the energy gaps and with a

!

" -phase shift added to one of

the lasers. If the amplitudes at the two measurement nodes

!

Xm

and

!

Ym

are exactly the same at all

times, no transition will ever occur and the probe node will remain empty. This is the so called

“coherent population trapping” or “EIT, electromagnetically induced transparency”, which has been

studied extensively in the past few years17,18. The lasers can be chosen from a coherent laser comb,

which typically has over one million frequency components with ultra stable phase relations19-21. In

most coherent laser combs there is around

!

1012 photons per comb line (~

!

1µW ), which is sufficient to

excite the required transitions. The laser comb should still be useable even if there is only

!

1nW per

comb line. One can record a population at the probe node only if the two amplitudes are different. In

this way, the “0”s and “1”s in the comparison tables are determined.

This measurement scheme allows continuous monitoring of the quantum walk process without

disturbing the system in ways that could change the measured results at the probe node (i.e. the “0”s

and “1”s) until useful information is extracted. If the chosen pairs

!

{Xm,X"} and

!

{Ym,Y"} are

equivalent with respect to quantum walks, the proposed measurement scheme will affect the sub-

system representing graph X and the sub-system representing graph Y in exactly the same way so

that the wavefunction amplitudes at the two measurement nodes

!

Xm

and

!

Ym

will remain the same to

each other and a "0" will be recorded. Otherwise the two amplitudes would be different, in which case

a “1” will be recorded regardless of measurement disturbance. To cycle through all possible pairs in

both graph X and Y requires

!

O(n4) measurements. Considering this plus other polynomial overheads

required for the actual walking and error corrections, we conjecture that the proposed quantum

algorithm solves the graph isomorphism problem in polynomial time.

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For all graphs that we have tested, including trees, planar graphs, randomly generated graphs,

connected and disconnected graphs, regular graphs, strongly regular graphs, and strongly regular

graphs that are also distance-transitive, this scheme has successfully identified isomorphic and

nonisomorphic graphs. Furthermore, the proposed scheme has a very different flavour from the well-

known Shor, Grover, and Deutsch-Jozsa algorithms. It may inspire further development of entirely

new quantum algorithms. It may also provide a breakthrough or insights in the search of efficient

classical algorithms for the GI problem.

Acknowledgment

We thank A. Woods, C. H. Li, K. Manouchehri, A. Luiten, J. J. McFerran, T. Fortier, and J. L. Hall for

discussions.

Figure captions

Fig. 1 Two isomorphic graphs.

Fig. 2 Top panel: randomly generated graph and its modification; middle and bottom panels: probability at node M for classical and quantum walks, respectively.

Fig. 3 Top panel: non-isomorphic srg(16,6,2,2); middle and bottom panels: probability at node M for classical and quantum walks, respectively.

Fig. 4 Measurement scheme for graph isomorphism identification using quantum walks, with an inset showing the coherent frequency comb [after Udem et. al.19].

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