Quantum Walks and Quantum Computationjoye/quawagtalks/Barr.pdf · Quantum Walks and Quantum Computation Katie Barr Department of Physics and Astronomy, University of Leeds ... Spatially

INTRODUCTION AUTOMATA LANGUAGE ACCEPTANCE STATE DISCRIMINATION CONCLUSIONS

Quantum Walks and QuantumComputation

Katie BarrDepartment of Physics and Astronomy, University of Leeds

Quantum walks in Grenoble

November 13, 2012

INTRODUCTION AUTOMATA LANGUAGE ACCEPTANCE STATE DISCRIMINATION CONCLUSIONS

OUTLINE

� Introduction to quantum walks1. Definition and basic properties2. Universal quantum computation

� Language acceptance and automata� Quantum walks accepting languages

1. Spatially distributed input2. Sequentially distributed input

� Quantum inputs- state discrimination� Conclusions

INTRODUCTION AUTOMATA LANGUAGE ACCEPTANCE STATE DISCRIMINATION CONCLUSIONS

MOTIVATION

Why look to quantum walks in relation to computation?

Classical random walks used in best known classical algorithms:

� factorisation� k-sat� approximating the permanent of a matrix� graph isomorphism

→ early results concerning quantum walks showed speedupsover classical walks

INTRODUCTION AUTOMATA LANGUAGE ACCEPTANCE STATE DISCRIMINATION CONCLUSIONS

DISCRETE-TIME QUANTUM WALKS ON GENERALGRAPHS

For an arbitrary graph G = {E,V} a quantum walk evolvesaccording to U = SC.

C is the coin operator, which is any unitary. A different coin canbe applied at every node.

S|a, v� = |a, u� if u is the a’th neighbour of v.Aharonov et al, STOC’01, July 2001

INTRODUCTION AUTOMATA LANGUAGE ACCEPTANCE STATE DISCRIMINATION CONCLUSIONS

CONTINUOUS TIME QUANTUM WALKS

Hamiltonian determined by adjacency matrix A.

� H = γA� γ = prob of moving to connected site per unit time� Schrodinger evolution: |ψ(t)� = e−iHt|ψ(0)�

E. Farhi, S. Gutmann. PRA 58, 915-928 (1998)Traverses glued trees exponentially faster than classical walk

INTRODUCTION AUTOMATA LANGUAGE ACCEPTANCE STATE DISCRIMINATION CONCLUSIONS

TRANSPORT PROPERTIES OF QUANTUM WALKS

Algorithms which achieve exponentially faster hitting times:E. Farhi, S. Gutmann. P R A, 58:915-928, 1998A. Childs, E. Farhi, S. Gutmann. Quantum Inf Process, 1:35, 2002A. Childs, R. Cleve, E. Deotto, E. Farhi, S. Gutmann, D. Spielman. STOC03, pp. 59-68J. Kempe. Proceedings of RANDOM03, LNCS, 2764:354-369, 2003

Other applications:

� modelling exciton transport in light harvesting antennacomplexes

M. Mohseni, P. Rebentrost, S. Lloyd, and A. Aspuru-Guzik. J. Chem. Phys. 129 174106

� wires in quantum circuit simulationN. Lovett, S. Cooper, M. Everitt, M. Trevers, and V. Kendon, Phys. Rev. A 81, (2010)

INTRODUCTION AUTOMATA LANGUAGE ACCEPTANCE STATE DISCRIMINATION CONCLUSIONS

GROVERS ALGORITHM- OVERVIEW

Algorithm to search an unsorted database.

Classically: cannot be performed in less than linear time

- checking each item is as good as you can get

Grover performs the search in O(logN)space and O(N1/2) time.

→ Asymptopically fastest time to solve this problem using aquantum computer.

INTRODUCTION AUTOMATA LANGUAGE ACCEPTANCE STATE DISCRIMINATION CONCLUSIONS

GROVERS ALGORITHM- DETAILS

Quantum walk equivalent to Grovers Algorithm:

� Start with uniform superposition over N nodes� Use Grover coin at each node except marked node� At marked node use any unitary- coin acts as oracle� Run for approx π

2

�(N/2) steps

� Measure particle position

Shenvi, Kempe, Whaley, PRA 67, 052307, 2003

INTRODUCTION AUTOMATA LANGUAGE ACCEPTANCE STATE DISCRIMINATION CONCLUSIONS

QUANTUM WALKS ARE COMPUTATIONALLYUNIVERSAL

Both discrete and continuous time quantum walks arecomputationally universal.

A. M. Childs, Phys. Rev. Lett. 102, (2009)N. Lovett, S. Cooper, M. Everitt, M. Trevers, and V. Kendon, Phys. Rev. A 81, (2010)

They can simulate the elementary gate set required foruniversal computation in the quantum circuit model.

An exceptionally simple quantum circuit

INTRODUCTION AUTOMATA LANGUAGE ACCEPTANCE STATE DISCRIMINATION CONCLUSIONS

QUANTUM CIRCUIT AS QUANTUM WALK-PROPAGATION

Deterministic evolution provided by the d dimensional Groveroperator:

Gd =

2−dd

2d · · · 2

d

2d

2−dd · · · 2

d

...... . . . ...

2d

2d · · · 2−d

d

Evolution of amplitude using 4d Grover coin:

INTRODUCTION AUTOMATA LANGUAGE ACCEPTANCE STATE DISCRIMINATION CONCLUSIONS

QUANTUM CIRCUIT AS QUANTUM WALK- EXAMPLEGATE

The CNOT gate:

1 0 0 00 1 0 00 0 0 10 0 1 0

Discrete-time walkversion of the CNOTgate

An n-qubitcomputation requires2n wires +exponentially moregate structures...

... but 2n wires canbe encoded in nqubits

INTRODUCTION AUTOMATA LANGUAGE ACCEPTANCE STATE DISCRIMINATION CONCLUSIONS

REMARKS ON GATE SET SIMULATION

� Continuous time walk requires graph of maximum degree 3� Discrete time walk requires graph with maximum degree 8� Both require 2n wires for an n-qubit input� Walk on N vertex graph can be efficiently simulated on a

quantum computer with poly(log2)N gates� Childs more recent work proves universal computation

with multiple walkers, which does not require 2n wiresChilds et al arXiv:1205.3782

INTRODUCTION AUTOMATA LANGUAGE ACCEPTANCE STATE DISCRIMINATION CONCLUSIONS

SCHEMA FOR THEORETICAL COMPUTERS

Turing machines compute bymoving a tape head along an in-put tape, which determines theevolution of the computationalstates

INTRODUCTION AUTOMATA LANGUAGE ACCEPTANCE STATE DISCRIMINATION CONCLUSIONS

FORMAL LANGUAGES

Quick summary:

� A formal language is a set of words from some alphabet Σ� Example Leq = {ambm|m ∈ N} = {ab, aabb, aaabbb, ...}� Want to find machines which can distinguish words in a

given language� Some, such as Leq can be recognised by models of

computation which are not Turing universal

INTRODUCTION AUTOMATA LANGUAGE ACCEPTANCE STATE DISCRIMINATION CONCLUSIONS

PROBABILISTIC ACCEPTANCE

Probabilistic automata (PFAs) have probabilistic outputs,which are accepted either with bounded or unbounded error:

� Language recognised with cut-point λ ∈ [0, 1) by PFA M isL = {w|w ∈ Σ∗ fM(w) > λ}

� fM is the probability of acceptance� M accepts L with bounded error if there is some � > 0 such

that for all w ∈ L, M accepts w with probability greaterthan λ+ � and accepts all words w /∈ L with probabilityless than λ− �

� Acceptance is with unbounded error if there is no such �

INTRODUCTION AUTOMATA LANGUAGE ACCEPTANCE STATE DISCRIMINATION CONCLUSIONS

LANGUAGE RECOGNISERS: FINITE STATE AUTOMATA

An Finite state automaton, FSA, is a 5-tuple:

M = (Q, qi, F, δ,Σ)

� Q is a finite set of states� Σ is the finite input alphabet� δ:Q × Σ → Q is the transition function� qi is the initial state of the automaton� F ⊂ Q is the set of accepting states of the automaton

Accept regular languages.

INTRODUCTION AUTOMATA LANGUAGE ACCEPTANCE STATE DISCRIMINATION CONCLUSIONS

COMPUTATION AS MATRIX MULTIPLICATION

Take Q as a basis for a vector space. Then, for example, thestate of the FSA M in state qi might be represented by

10...0

.

Represent the transition on symbol σ as a matrix, with nonzeroentries at (qx, qy) if qx is updated to qy when M reads σ.

A stochastic matrix gives rise to a probabilistic automaton.

INTRODUCTION AUTOMATA LANGUAGE ACCEPTANCE STATE DISCRIMINATION CONCLUSIONS

QUANTUM FINITE AUTOMATA

A quantum finite automaton (QFA) is a tuple:

M = (Q, qi, qf , δ,Σ)

The transition matrices induced by δ must be unitary.

Unitary transition matrices offer no advantage over stochastictransition matrices.

INTRODUCTION AUTOMATA LANGUAGE ACCEPTANCE STATE DISCRIMINATION CONCLUSIONS

NEW MODEL OF LANGUAGE ACCEPTANCE

Why?

→ Potentially gain new insights into both quantum walks andformal language acceptance

Quantum circuits have not been experimentally realisedbeyond a few qubits

→ Discrete time quantum walks have been realisedexperimentally, with a large variety of coins

INTRODUCTION AUTOMATA LANGUAGE ACCEPTANCE STATE DISCRIMINATION CONCLUSIONS

SPATIALLY DISTRIBUTED INPUT- SETUP

Designate encoding for input:

Graph structure then directs amplitude encoding words in thelanguage accepted by the graph to an accepting node.

INTRODUCTION AUTOMATA LANGUAGE ACCEPTANCE STATE DISCRIMINATION CONCLUSIONS

WALK RECOGNISING Leq

Graph and coins accepting Leq:

→ Accepts words in Leq with certainty.

Probability of accepting a word w /∈ Leq for strings length n > 1≤ 2/n2.

INTRODUCTION AUTOMATA LANGUAGE ACCEPTANCE STATE DISCRIMINATION CONCLUSIONS

WALK RECOGNISING Leq

Red curve shows Jaro dis-tance between input wordand the appropriately sizedstring from Leq

Black curve shows proba-bility of accepting the stringfor the first 200 strings

Both curves peak at the indices representing strings ab, aabb and aaabbb.

The Jaro distance of strings w1 and w2: dj =

�0 if m = 0

13(

m|w1|

+ m|w2|

+ m−tm ) otherwise

INTRODUCTION AUTOMATA LANGUAGE ACCEPTANCE STATE DISCRIMINATION CONCLUSIONS

DIFFERENT APPROACH- WALK ACCEPTING La

La = {a, b}∗a can’t be accepted by most basic QFAs.

Probability of accepting w ∈ La = (n − 1)( 12n2(n−1) +

1n(1 − 1

2n))

Probability of accepting w /∈ La =n−1

n (1 − 12n)

INTRODUCTION AUTOMATA LANGUAGE ACCEPTANCE STATE DISCRIMINATION CONCLUSIONS

WALK RECOGNISING La

Uses biased Hadamard:

�1

2n

�1 − 1

2n�1 − 1

2n −�

12n

Red curve shows Jaro distancebetween input word and closestword in La

Black curve is probability of walkaccepting the string

INTRODUCTION AUTOMATA LANGUAGE ACCEPTANCE STATE DISCRIMINATION CONCLUSIONS

ADVANTAGES AND DRAWBACKS

Well suited to recognising languages with at most one word ofeach length.

Spatially distributed input allows arbitrarily long strings to beprocessed in a fixed, small number of steps . . .

. . . but increase in input length requires an increase in the number ofnodes.

Number of nodes required to accept a given language can beheld constant regardless of input length if each input symbol

is fed into the structure in turn

INTRODUCTION AUTOMATA LANGUAGE ACCEPTANCE STATE DISCRIMINATION CONCLUSIONS

SEQUENTIALLY DISTRIBUTED INPUT- SETUP

The input can be treated sequentially if we start with it along a chainwith two links between each node. The two symbols are represented:

a =

α000

b =

0α00

Coin on this part of the graph is σx ⊗ I2.

INTRODUCTION AUTOMATA LANGUAGE ACCEPTANCE STATE DISCRIMINATION CONCLUSIONS

WALKS ACCEPTING PARTICULAR STRINGS

To accept specific strings,can specify walks bygraph structure only

Coin is just permutationoperator

Example: graph accept-ing abab

INTRODUCTION AUTOMATA LANGUAGE ACCEPTANCE STATE DISCRIMINATION CONCLUSIONS

WALK ACCEPTING Lab

Lab = {(ab)m|m ∈ N} ={ab}∗

Can use graphs of constantsize to accept words of arbi-trary length

Number of computationalsteps = n (length of input) +3

INTRODUCTION AUTOMATA LANGUAGE ACCEPTANCE STATE DISCRIMINATION CONCLUSIONS

WALK ACCEPTING Lab

Red curve shows Jaro distancebetween input word and theappropriately sized string fromLab

Black curve shows probabilityof accepting the input for first200 strings

Both curves peak at the indices representing strings ab, abab andababab.

INTRODUCTION AUTOMATA LANGUAGE ACCEPTANCE STATE DISCRIMINATION CONCLUSIONS

QUANTUM INPUTS

We have so far examined classical inputs. Our setup allows forquantum inputs: superpositions of words.

� Each symbol can be in superposition of a or b, xa + yb suchthat |x|2 + |y|2 = α2

� Where symbols match, amplitude is allocated to thatsymbol as for the classical encoding

� Otherwise amplitude is distributed between the a and bstates accordingly

Using quantum inputs, language acceptance becomes quantumstate discrimination.

INTRODUCTION AUTOMATA LANGUAGE ACCEPTANCE STATE DISCRIMINATION CONCLUSIONS

EXAMPLE STATE DISCRIMINATION SCHEME

PD1

|! >

" (#/4+$/2)

PD3

PBS6

PD2

A

WP6

WP7

NPBS

PBS3PBS5

B

C

A

C

B

PBS4PD4

" (#/4)

" (#/4)

input state

WP5

j

2

4

|h>

|v>

|h> |v>

2

4

|ψ41� =

1√3

�−|h�+

√2e−2πi/3|v�

�

|ψ42� =

1√3

�−|h�+

√2e+2πi/3|v�

�

|ψ43� =

1√3

�−|h�+

√2|v�

�

|ψ44� = |h�

Clarke, Kendon, Chefles, Barnett, Riis,Sasaki. PRA 64 012303 2001

INTRODUCTION AUTOMATA LANGUAGE ACCEPTANCE STATE DISCRIMINATION CONCLUSIONS

EFFECTS OF QUANTUM INPUTS

Fidelity of final state to ac-cepting state for quantum in-puts in a superposition of aabband:

� string with no matchingsymbols, bbaa (blue)

� strings with 1 matchingsymbol, abaa, baaa, bbaband bbba (green)

� strings with 2 matchingsymbols aaaa, abab...(black)

� strings with 3 matchingsymbols aaab, aaba....(red)

INTRODUCTION AUTOMATA LANGUAGE ACCEPTANCE STATE DISCRIMINATION CONCLUSIONS

ALTERNATIVE QUANTUM WALK PERFORMING STATEDISCRIMINATION

Suppose we know our state has equal probability of being

|ψ+� = cos θ2 |l�+ sin θ2 |r�

or |ψ−� = cos θ2 |l� − sin θ2 |r�.

- can implement a three step walk, which after 3 steps using atime dependent coin, can tell us:

� It is definitely |ψ−� if the particle is at position -1� It is definitely |ψ+� if the particle is at position +1� Particle at position 3 means ’I don’t know’

Ref: Kurzynski/Wojcik arXiv:1208.1800

INTRODUCTION AUTOMATA LANGUAGE ACCEPTANCE STATE DISCRIMINATION CONCLUSIONS

SUMMARY

We have:� Given an overview of quantum walks in relation to

quantum computation� Introduced a new way to implement computation using

the discrete time quantum walk� Shown two ways to do this and discussed examples of

each� Developed the concept of a quantum input...� ... and applied the results to the problem of quantum state

discrimination

Ref: Barr/Kendon arXiv:1209.5238

INTRODUCTION AUTOMATA LANGUAGE ACCEPTANCE STATE DISCRIMINATION CONCLUSIONS

FUTURE WORK

� Find more examples of quantum walks accepting formallanguages

� Develop tools to make complexity considerations- start byexamining classical versions of these walks

� Explore the application to state discrimination further� Further develop concept of quantum input, extend to

quantum languages?

Thank you very much for listening!

Quantum dotting by Aidan Illsley