Quantum walks: Definition and applications

Ashley Montanaro

Talk structure

Introduction to quantum walks

Defining a quantum walk ...on the line ...on undirected graphs ...on directed graphs

Applications of quantum walks

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What are quantum walks?

A random walk is the simulation of the random movement of a particle around a graph

A quantum walk is the same – but with a quantum particle not the same as running a normal random walk algorithm on a

quantum computer

Random walks are a useful model for developing classical algorithms; quantum walks provide a new way of developing quantum algorithms which is particularly important because producing new quantum

algorithms is so hard

Physical intuition behind a classical random walk on a graph

1 3

2 4

After 3 steps we are in position “5” or “6” with equal probability.

Time Probability at vertex

1 2 3 4 5 6

0 1

1

2

3

6

5

½ ½

1

½ ½

Physical intuition behind a quantum walk on a graph

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1

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Light detectors

Mirror

Half-silveredmirror

5

6

Physical intuition behind a quantum walk on a graph

After 3 steps we are guaranteed to be in detector “6” – this is caused by quantum interference.

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1

3

4

Time Amplitude at point

1 2 3 4 5 6

0 1

1

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5

6

1

Mathematical definition of a random walk Express a classical random walk as a

matrix W of transition probabilities where the entries in each column sum to 1

Express a position as a column vector v

Performing a step of the walk corresponds to pre-multiplying v by W

Performing n steps of the walk corresponds to pre-multiplying v by Wn

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2 4

=

Mathematical definition of a quantum walk Very similar, but:

probabilities combine differently (sum of the amplitudes squared must be 1)

the transition matrix must be unitary (ie. send unit vectors to unit vectors)

This will not in general be the case, so we may need to modify the structure of the graph – for example, by adding a coin space

This can be considered as a quantum analogue of flipping a coin to decide which direction to go at each step of the walk

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= (e.g.)

Classical random walk on the line

Versions of this walk are useful models for many random processes

When the walker has equal probability to move left or right, it’s well-known that the average distance from the start position after time n is sqrt(n)

But we can define a quantum walk on the same graph with different behaviour: an average distance of n

Consider a walk on the following simple infinite graph:

Quantum walk on the line

We have two quantum registers: a coin register holding |L or |R, and a position register |p

Our walk operation is a coin flip followed by a shift coin flip: send |L |L + i|R,

|R i|L + |R shift: send |L|p |L|p-1

|R|p |R|p+1

These are both unitary operations, and hence their combination is too so, together, they provide a way of defining a quantum walk on

the line there are other ways – e.g. the continuous-time formulation of

quantum walks

A few iterations of the walk on the line 1. start |R|0

2. coin (i|L + |R)|0shift i|L|-1 + |R|1

3. coin (i|L - |R)|-1 + (i|L + |R)|1 shift i|L|-2 - |R|0 + i|L|0 + |R|2

4. coin (i|L - |R)|-2 + (i|L + |R)|2 shift i|L|-3 - |R|-1 + i|L|1 + |R|3

Equal probability to be at |-3, |-1, |1 or |3 - whereas classical random walk favours |-1, |1

Classical vs. quantum walk on the line

Running a classical walk on the lineresults in a probability distribution like:

Whereas running this quantum walk for thesame number of steps gives:

The peaks and troughs in this graph are caused byquantum interference.

position

Quantum walks on undirected graphs Consider a d-regular graph G (each vertex has d arcs

leaving it)

We can label each arc and choose between them using a d-dimensional “coin” A variety of coin operators can be used: we usually pick one to

mix between all arcs equally

As before, one step of the walk consists of a coin flip followed by a shift

An irregular graph can be handled using a different coin for each vertex of a different degree or other methods...

Behaviour of quantum walks on undirected graphs We can define quantum equivalents of the mixing time

and hitting time of a walk

The mixing time of a random walk is the time it takes to converge to a limiting distribution Quantum walks have quadratically faster mixing time for any

undirected graph

The hitting time is the time it takes to reach a given vertex On certain graphs, quantum walks have exponentially faster

hitting time Open question: for which graphs is this true?

Quantum walks on directed graphs

A quantum walk can be defined on any undirected graph, with the use of a suitable coin

But it turns out that not all directed graphs support the idea of a quantum walk: only reversible ones do a reversible graph is a graph where, if you can get from a to b,

you can get from b to a each component of such graphs is strongly connected compare the idea that quantum computers have to be reversible

Quantum walks defined on irreversible graphs will not respect the structure of the graph: there will be some possibility to traverse arcs in the “wrong direction”

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Reversible and irreversible graphs

These graphs are irreversible:

These graphs are reversible:

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Implications for translation of classical algorithms Many classical algorithms can be represented as a

random walk on a directed graphs with sinks – the idea is to find a sink, which represents a solution to a problem e.g. Schöning’s random walk algorithm for SAT

A quantum walk cannot be defined on these graphs; this suggests that there is no easy translation of these algorithms into a quantum walk form

However, it is possible to produce a quantum walk which is “like” the original random walk in the sense that, after a long period of time, it has a high probability of ending up in a sink

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Applications of quantum walks

Quantum network routing Kempe, 2002

Quantum walk search algorithm Shenvi, Kempe, Whaley, 2002

Element distinctness Ambainis, 2004

Applications of element distinctness Magniez, Santha, Szegedy, 2003 Buhrmann, Spalek, 2004

Quantum network routing Consider a network whose topology is

a d-dimensional hypercube

We want to route a packet from onecorner of the hypercube to the other(eg. from 000 to 111)

Algorithm: perform ~d steps of a quantum walk. Then measure to see where the packet is.

This has advantages over a classical routing algorithm: it’s noise resistant: deleting intermediate links will not affect the

walk much intermediate nodes need minimal routing “hardware”

000 001

010 011

110

100 101

111

Quantum walk search algorithm

Consider the unstructured search problem: given a function

f(x) = { 1 if x = a, 0 otherwise }find the “marked” element a, where 0 a 2n-1.

Grover’s algorithm can solve this in O(2n/2) queries on a quantum computer, whereas a classical computer needs at least (2n) queries

Can we produce a quantum walk algorithm that requires the same (optimal) number of queries? this may be easier to implement, or provide a better model for

searching a “real” database

Quantum walk search algorithm (2)

We perform a quantum walk on the hypercube of dimension n each vertex, labelled by an n-bit string, corresponds to a

possible input to the oracle each vertex has n neighbours

Our walk consists of a combination of a coin flip and a shift, as before Identify each of the n coin states with each neighbour of a vertex

Use a “marking” coin operator When at an unmarked vertex, pick a coin state randomly When at the marked vertex, stay in the same coin state

Quantum walk search algorithm (3)

Start with a superposition over all vertices

If we run the walk for O(2n/2) steps, can prove that there is a high probability it will “home in” on the marked vertex in fact, there’s a general result stating that “perturbed”

walks like this will always find one of the marked elements

We then simply measure the position and we’ve found the marked item

Element distinctness

Problem: does a (multi-)set S of N elements contain any duplicate elements?

Call reading an element from the set a query

Clearly, classically we need N queries to answer the question with certainty

It turns out that a quantum walk algorithm can solve the problem in O(N2/3) queries which has been proven to be optimal

Quantum walk algorithm for element distinctness We use a quantum walk on a

graph where the vertices are subsets of S containing either M or M + 1 elements for some M < N

Two vertices are connected if they differ in exactly one element

The graph on the right encodes the set {1, 1, 2, 3} for M = 2

11,12

11,2

11,3

12,2

12,3

2,3

11,12,2

12,2,3

11,2,3

11,12,3

{1, 1, 2, 3}

Quantum walk algorithm for element distinctness (2) Basic walk algorithm:

1. start with some subset S’ S (where |S’| = M)2. check whether S’ contains any duplicates (needs

O(M) queries)3. if not, change to a different subset S’’ that differs in

exactly one element4. check S’’ for duplicates (needs 1 query)5. repeat steps 3 and 4 until a duplicate is found

Because this is a quantum walk, we can start with a superposition of all M-subsets

Analysis of quantum walk

In total, we need (M + r) queries, where M is the number of elements in the initial subset r is the number of steps of the quantum walk

It turns out that if we pick M = N2/3, then a solution can be found with high probability in r = N1/3 steps of the walk resulting in O(N2/3) queries in total it also turns out that the number of non-query operations

required is small, so the query complexity is a good measure of the time complexity

Note that this algorithm requires a significant amount of space – enough to store O(N2/3) elements

Applications of element distinctness

Using element distinctness as a subroutine, quantum walk algorithms have been developed to solve other problems: finding a triangle in a graph with n vertices in time

O(n1.3) verifying matrix multiplication (testing if A*B = C for

some n*n matrices A, B, C) in time O(n1.67)

The algorithm has also been generalised to solve the problem of finding any subset that has a given property i.e.: find (a, b) such that (f(a), f(b)) P, where P is

some property

Conclusions and further reading

Quantum walks can be defined on any undirected graph, and on reversible directed graphs.

Quantum walks are a way to develop quantum algorithms that outperform their classical counterparts.

Further reading (on www.arxiv.org): “Quantum walks and their algorithmic applications”, A. Ambainis,

quant-ph/0311001 “Quantum random walks – an introductory overview”, J. Kempe,

quant-ph/0303081 “Quantum walks on directed graphs”, A. Montanaro,

quant-ph/0504116