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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