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Affine computation
and affine automaton
Alejandro Díaz-Caro
UNIVERSIDAD NACIONAL DE QUILMES
(Argentina)
Abuzer Yakaryılmaz
LAB...
Outline
Motivation
Affine computation
Affine finite Automaton (AfA)
Main results
Alejandro Díaz-Caro and Abuzer Yakaryılmaz Affin...
Probabilistic vs. Quantum
Destructive interference
Probabilistic
+
Alejandro Díaz-Caro and Abuzer Yakaryılmaz Affine computa...
Probabilistic vs. Quantum
Destructive interference
Probabilistic
1
2
+
1
2
1
2 + 1
2 = 1
Alejandro Díaz-Caro and Abuzer Ya...
Probabilistic vs. Quantum
Destructive interference
Probabilistic
1
2
+
1
2
1
2 + 1
2 = 1
Quantum
1
√
2
+
1
√
2
1√
2
2
+ 1√...
Probabilistic vs. Quantum
Destructive interference
Probabilistic
1
2
+
1
2
1
2 + 1
2 = 1
Quantum
1
√
2
−
1
√
2
1√
2
2
+ −1...
Probabilistic vs. Quantum
Destructive interference
Probabilistic
1
2
+
1
2
1
2 + 1
2 = 1
Quantum
1
√
2
−
1
√
2
1√
2
2
+ −1...
Probabilistic vs. Quantum
Destructive interference
Probabilistic
1
2
+
1
2
1
2 + 1
2 = 1
Quantum
1
√
2
−
1
√
2
1√
2
2
+ −1...
Probabilistic vs. Quantum
Destructive interference
Probabilistic
1
2
+
1
2
1
2 + 1
2 = 1
Quantum
1
√
2
−
1
√
2
1√
2
2
+ −1...
Probabilistic vs. Quantum
Destructive interference
Probabilistic
1
2
+
1
2
1
2 + 1
2 = 1
Quantum
1
√
2
−
1
√
2
1√
2
2
+ −1...
Probabilistic vs. Quantum
Destructive interference
Probabilistic
1
2
+
1
2
1
2 + 1
2 = 1
Quantum
1
√
2
−
1
√
2
1√
2
2
+ −1...
Probabilistic vs. Quantum
Destructive interference
Probabilistic
1
2
+
1
2
1
2 + 1
2 = 1
Quantum
1
√
2
−
1
√
2
1√
2
2
+ −1...
Is there any computational power in the
destructive interference?
Alejandro Díaz-Caro and Abuzer Yakaryılmaz Affine computat...
Affine systems
Preliminaries
Probabilistic state: l1-norm-1 vector (defined on R+
0 )
Probabilistic operator: Linear operator...
Affine systems
Preliminaries
Probabilistic state: l1-norm-1 vector (defined on R+
0 )
Probabilistic operator: Linear operator...
Affine systems
Preliminaries
Probabilistic state: l1-norm-1 vector (defined on R+
0 )
Probabilistic operator: Linear operator...
Affine systems
Getting information
Weighting operator
Analogous to quantum measurement
Projects the state into the computati...
Affine systems (AfS)
Formal definition: Affine state
E = {e1, . . . , en} basis states (deterministic states)
Affine state: linea...
Affine systems (AfS)
Formal definition: Affine state
E = {e1, . . . , en} basis states (deterministic states)
Affine state: linea...
Affine systems (AfS)
Formal definition: Affine transformation and weighting operator
Affine transformation
A = (aij )ij is an affin...
Affine systems (AfS)
Formal definition: Affine transformation and weighting operator
Affine transformation
A = (aij )ij is an affin...
Affine finite Automaton (AfA)
Formal definition
An AfA M is a 5-tuple
M = (E, Σ, {Aσ | σ ∈ Σ}, es, Ea)
where
E is the set of d...
Affine finite Automaton (AfA)
Language recognition
Input: w ∈ Σ∗
After reading the whole input, a weighting operator is appli...
Affine finite Automaton (AfA)
Language recognition
Input: w ∈ Σ∗
After reading the whole input, a weighting operator is appli...
The languages and automata zoo
Cutpoint Language Class Automaton
CP > 0 Stochastic lang. SL PFA
CP = 0 Regular lang. REG N...
The languages and automata zoo
Cutpoint Language Class Automaton
CP > 0 Stochastic lang. SL PFA
CP = 0 Regular lang. REG N...
Bounded-error affine languages (BAfL)
Language EQ = {w ∈ {a, b}∗
| |w|a = |w|b} /∈ REG
Alejandro Díaz-Caro and Abuzer Yakary...
Bounded-error affine languages (BAfL)
Language EQ = {w ∈ {a, b}∗
| |w|a = |w|b} /∈ REG
e1 e2
(a, 2)
(b, 1/2)
(a, −1)
(b, 1/2...
Bounded-error affine languages (BAfL)
Language EQ = {w ∈ {a, b}∗
| |w|a = |w|b} /∈ REG
e1 e2
(a, 2)
(b, 1/2)
(a, −1)
(b, 1/2...
Bounded-error affine languages (BAfL)
e1 e2e3
(a, 1)
(a, x)(a, −x)
(b, 1)
(b, −x)(b, x)
(a, 1)
(b, 1)
(a, 1)
(b, 1)
Aa =

...
Bounded-error affine languages (BAfL)
e1 e2e3
(a, 1)
(a, x)(a, −x)
(b, 1)
(b, −x)(b, x)
(a, 1)
(b, 1)
(a, 1)
(b, 1)
Aa =

...
Cutpoint affine languages (AfL)
LAPINS = {w ∈ {a, b, c}∗
| |w|2
a > |w|b and |w|2
b > |w|c } /∈ SL
[J¯anis Lapin¸š, 1974]
PF...
Cutpoint affine languages (AfL)
LAPINS = {w ∈ {a, b, c}∗
| |w|2
a > |w|b and |w|2
b > |w|c } /∈ SL
[J¯anis Lapin¸š, 1974]
PF...
Cutpoint affine languages (AfL)
LAPINS = {w ∈ {a, b, c}∗
| |w|2
a > |w|b and |w|2
b > |w|c } /∈ SL
[J¯anis Lapin¸š, 1974]
PF...
Cutpoint affine languages (AfL)
LAPINS = {w ∈ {a, b, c}∗
| |w|2
a > |w|b and |w|2
b > |w|c } /∈ SL
[J¯anis Lapin¸š, 1974]
PF...
Cutpoint affine languages (AfL)
LAPINS = {w ∈ {a, b, c}∗
| |w|2
a > |w|b and |w|2
b > |w|c } /∈ SL
[J¯anis Lapin¸š, 1974]
PF...
Cutpoint affine languages (AfL)
LAPINS = {w ∈ {a, b, c}∗
| |w|2
a > |w|b and |w|2
b > |w|c } /∈ SL
[J¯anis Lapin¸š, 1974]
PF...
Nondeterministic affine languages (NAfL)
(NQAL contains some famous languages like the complement of EQ)
Theorem
NAfL = NQAL...
Summarising
Bounded and unbounded error:
AfAs more powerful than QFAs and PFAs
Nondeterministic computation:
AfAs equivale...
Summarising
Bounded and unbounded error:
AfAs more powerful than QFAs and PFAs
Nondeterministic computation:
AfAs equivale...
Summarising
Bounded and unbounded error:
AfAs more powerful than QFAs and PFAs
Nondeterministic computation:
AfAs equivale...
Backup slides
Weighting operator
Can we use the weighting operator as a projective measurement?
Answer: No
v =


1
−1
1
...
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Affine computation and affine automaton

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Slides to be used in the 11th International Computer Science Symposium in Russia (CSR 2016) to present the paper at http://dx.doi.org/10.1007/978-3-319-34171-2_11 (and arXiv'ed at http://arxiv.org/abs/1602.04732).

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Affine computation and affine automaton

  1. 1. Affine computation and affine automaton Alejandro Díaz-Caro UNIVERSIDAD NACIONAL DE QUILMES (Argentina) Abuzer Yakaryılmaz LABORATÓRIO NACIONAL DE COMPUTAÇÃO CIENTÍFICA (Brazil) Computer Science Symposium in Russia St. Petersburg, June 9–13, 2016
  2. 2. Outline Motivation Affine computation Affine finite Automaton (AfA) Main results Alejandro Díaz-Caro and Abuzer Yakaryılmaz Affine computation and affine automaton 2 / 16
  3. 3. Probabilistic vs. Quantum Destructive interference Probabilistic + Alejandro Díaz-Caro and Abuzer Yakaryılmaz Affine computation and affine automaton 3 / 16
  4. 4. Probabilistic vs. Quantum Destructive interference Probabilistic 1 2 + 1 2 1 2 + 1 2 = 1 Alejandro Díaz-Caro and Abuzer Yakaryılmaz Affine computation and affine automaton 3 / 16
  5. 5. Probabilistic vs. Quantum Destructive interference Probabilistic 1 2 + 1 2 1 2 + 1 2 = 1 Quantum 1 √ 2 + 1 √ 2 1√ 2 2 + 1√ 2 2 = 1 Alejandro Díaz-Caro and Abuzer Yakaryılmaz Affine computation and affine automaton 3 / 16
  6. 6. Probabilistic vs. Quantum Destructive interference Probabilistic 1 2 + 1 2 1 2 + 1 2 = 1 Quantum 1 √ 2 − 1 √ 2 1√ 2 2 + −1√ 2 2 = 1 Alejandro Díaz-Caro and Abuzer Yakaryılmaz Affine computation and affine automaton 3 / 16
  7. 7. Probabilistic vs. Quantum Destructive interference Probabilistic 1 2 + 1 2 1 2 + 1 2 = 1 Quantum 1 √ 2 − 1 √ 2 1√ 2 2 + −1√ 2 2 = 1 1 2 1 2 + 1 2 + 1 2 Alejandro Díaz-Caro and Abuzer Yakaryılmaz Affine computation and affine automaton 3 / 16
  8. 8. Probabilistic vs. Quantum Destructive interference Probabilistic 1 2 + 1 2 1 2 + 1 2 = 1 Quantum 1 √ 2 − 1 √ 2 1√ 2 2 + −1√ 2 2 = 1 1 2 1 2 + 1 2 + 1 2 α + β Alejandro Díaz-Caro and Abuzer Yakaryılmaz Affine computation and affine automaton 3 / 16
  9. 9. Probabilistic vs. Quantum Destructive interference Probabilistic 1 2 + 1 2 1 2 + 1 2 = 1 Quantum 1 √ 2 − 1 √ 2 1√ 2 2 + −1√ 2 2 = 1 1 2 1 2 + 1 2 + 1 2 3 4 + 1 4 Alejandro Díaz-Caro and Abuzer Yakaryılmaz Affine computation and affine automaton 3 / 16
  10. 10. Probabilistic vs. Quantum Destructive interference Probabilistic 1 2 + 1 2 1 2 + 1 2 = 1 Quantum 1 √ 2 − 1 √ 2 1√ 2 2 + −1√ 2 2 = 1 1 2 1 2 + 1 2 + 1 2 3 4 + 1 4 5 8 + 3 8 Alejandro Díaz-Caro and Abuzer Yakaryılmaz Affine computation and affine automaton 3 / 16
  11. 11. Probabilistic vs. Quantum Destructive interference Probabilistic 1 2 + 1 2 1 2 + 1 2 = 1 Quantum 1 √ 2 − 1 √ 2 1√ 2 2 + −1√ 2 2 = 1 1 2 1 2 + 1 2 + 1 2 3 4 + 1 4 5 8 + 3 8 1 √ 2 1 √ 2 + 1 √ 2 + 1 √ 2 1 √ 2 − 1 √ 2 Alejandro Díaz-Caro and Abuzer Yakaryılmaz Affine computation and affine automaton 3 / 16
  12. 12. Probabilistic vs. Quantum Destructive interference Probabilistic 1 2 + 1 2 1 2 + 1 2 = 1 Quantum 1 √ 2 − 1 √ 2 1√ 2 2 + −1√ 2 2 = 1 1 2 1 2 + 1 2 + 1 2 3 4 + 1 4 5 8 + 3 8 1 √ 2 1 √ 2 + 1 √ 2 + 1 √ 2 1 √ 2 − 1 √ 2 Alejandro Díaz-Caro and Abuzer Yakaryılmaz Affine computation and affine automaton 3 / 16
  13. 13. Is there any computational power in the destructive interference? Alejandro Díaz-Caro and Abuzer Yakaryılmaz Affine computation and affine automaton 4 / 16
  14. 14. Affine systems Preliminaries Probabilistic state: l1-norm-1 vector (defined on R+ 0 ) Probabilistic operator: Linear operator (stochastic matrix) Quantum state: l2-norm-1 vector (defined on C) Quantum operator: Linear operator (unitary matrix) Alejandro Díaz-Caro and Abuzer Yakaryılmaz Affine computation and affine automaton 5 / 16
  15. 15. Affine systems Preliminaries Probabilistic state: l1-norm-1 vector (defined on R+ 0 ) Probabilistic operator: Linear operator (stochastic matrix) Quantum state: l2-norm-1 vector (defined on C) Quantum operator: Linear operator (unitary matrix) Aim Generalization of probabilistic system Allowing negative values Linear operator Defined in a simple way Alejandro Díaz-Caro and Abuzer Yakaryılmaz Affine computation and affine automaton 5 / 16
  16. 16. Affine systems Preliminaries Probabilistic state: l1-norm-1 vector (defined on R+ 0 ) Probabilistic operator: Linear operator (stochastic matrix) Quantum state: l2-norm-1 vector (defined on C) Quantum operator: Linear operator (unitary matrix) Aim Generalization of probabilistic system Allowing negative values Linear operator Defined in a simple way Affine state: Barycentric vector (defined on R) Affine operator: Linear operator (affine transformation) Alejandro Díaz-Caro and Abuzer Yakaryılmaz Affine computation and affine automaton 5 / 16
  17. 17. Affine systems Getting information Weighting operator Analogous to quantum measurement Projects the state into the computational basis The weight is the absolut value Normalization after measurement (l1-norm can be > 1) Normalized magnitude = probability of observation 10 − 9 Probability of : 10 19 Probability of : 9 19 Alejandro Díaz-Caro and Abuzer Yakaryılmaz Affine computation and affine automaton 6 / 16
  18. 18. Affine systems (AfS) Formal definition: Affine state E = {e1, . . . , en} basis states (deterministic states) Affine state: linear combination a1e1 + · · · + anen with n i=1 ai = 1 ai ∈ R Alejandro Díaz-Caro and Abuzer Yakaryılmaz Affine computation and affine automaton 7 / 16
  19. 19. Affine systems (AfS) Formal definition: Affine state E = {e1, . . . , en} basis states (deterministic states) Affine state: linear combination a1e1 + · · · + anen with n i=1 ai = 1 ai ∈ R In R2 Probabilistic state Set of states: A segment 1 1 Quantum state Set of states: A circle 1 Affine state Set of states: A line 1 1 Alejandro Díaz-Caro and Abuzer Yakaryılmaz Affine computation and affine automaton 7 / 16
  20. 20. Affine systems (AfS) Formal definition: Affine transformation and weighting operator Affine transformation A = (aij )ij is an affine transformation ⇔ ∀j, i aij = 1 Alejandro Díaz-Caro and Abuzer Yakaryılmaz Affine computation and affine automaton 8 / 16
  21. 21. Affine systems (AfS) Formal definition: Affine transformation and weighting operator Affine transformation A = (aij )ij is an affine transformation ⇔ ∀j, i aij = 1 Weighting operator In QC, sign of amplitudes does not matter for measurement We follow the same idea Magnitude of an affine state: |v| = i |ai | + |a2| + · · · + |an| ≥ 1 (l1 norm) Probability of observing the j−th state: |aj | |v| Alejandro Díaz-Caro and Abuzer Yakaryılmaz Affine computation and affine automaton 8 / 16
  22. 22. Affine finite Automaton (AfA) Formal definition An AfA M is a 5-tuple M = (E, Σ, {Aσ | σ ∈ Σ}, es, Ea) where E is the set of deterministic states es ∈ E is the starting state Ea ⊆ E set of accepting states Σ is the input alphabet Aσ is the affine transformation matrix for the symbol σ. Idem PFA except for the transition matrices (and a PFA with matrices consisting only of 0s and 1s is a DFA) Alejandro Díaz-Caro and Abuzer Yakaryılmaz Affine computation and affine automaton 9 / 16
  23. 23. Affine finite Automaton (AfA) Language recognition Input: w ∈ Σ∗ After reading the whole input, a weighting operator is applied Accepting probability of M in w: fM (w) = ek ∈Ea |vf [k]| |vf | ∈ [0, 1] Alejandro Díaz-Caro and Abuzer Yakaryılmaz Affine computation and affine automaton 10 / 16
  24. 24. Affine finite Automaton (AfA) Language recognition Input: w ∈ Σ∗ After reading the whole input, a weighting operator is applied Accepting probability of M in w: fM (w) = ek ∈Ea |vf [k]| |vf | ∈ [0, 1] A language is recognized by an AfA M with cutpoint λ ∈ [0, 1) iff L = {w ∈ Σ∗ | fM (w) > λ} Nondeterministic AfA: cutpoint 0. A language is recognized by an AfA M with bound error iff ∃δ such that ∀w ∈ L, fM (w) ≥ λ + δ ∀w /∈ L, fM (w) ≤ λ − δ Alejandro Díaz-Caro and Abuzer Yakaryılmaz Affine computation and affine automaton 10 / 16
  25. 25. The languages and automata zoo Cutpoint Language Class Automaton CP > 0 Stochastic lang. SL PFA CP = 0 Regular lang. REG NFA Bound error Regular lang. REG BPFA CP > 0 Stochastic lang. SL QFA CP = 0 Nondeterministic quantum lang. NQAL NQFA Bound error Regular lang. REG BQFA REG NQAL SL Alejandro Díaz-Caro and Abuzer Yakaryılmaz Affine computation and affine automaton 11 / 16
  26. 26. The languages and automata zoo Cutpoint Language Class Automaton CP > 0 Stochastic lang. SL PFA CP = 0 Regular lang. REG NFA Bound error Regular lang. REG BPFA CP > 0 Stochastic lang. SL QFA CP = 0 Nondeterministic quantum lang. NQAL NQFA Bound error Regular lang. REG BQFA REG NQAL SL Cutpoint Language Class Automaton CP > 0 Affine lang. AfL AfA CP = 0 Nondeterministic affine lang. NAfL NAfA Bound error Bounded-error affine lang. BAfL BAfA BAfL0 : All non-members are accepted with value 0 BAfL1 : All members are accepted with value 1. Alejandro Díaz-Caro and Abuzer Yakaryılmaz Affine computation and affine automaton 11 / 16
  27. 27. Bounded-error affine languages (BAfL) Language EQ = {w ∈ {a, b}∗ | |w|a = |w|b} /∈ REG Alejandro Díaz-Caro and Abuzer Yakaryılmaz Affine computation and affine automaton 12 / 16
  28. 28. Bounded-error affine languages (BAfL) Language EQ = {w ∈ {a, b}∗ | |w|a = |w|b} /∈ REG e1 e2 (a, 2) (b, 1/2) (a, −1) (b, 1/2) (a, 1) (b, 1) Aa = 2 0 −1 1 Ab = 1/2 0 1/2 1 After reading m as and n bs the state is 2m−n 1 − 2m−n ∀w ∈ EQ, vf = 1 0 (accepting value 1) ∀w /∈ EQ, max accepting value: vf = 2 −1 (accepting value 2/3) Alejandro Díaz-Caro and Abuzer Yakaryılmaz Affine computation and affine automaton 12 / 16
  29. 29. Bounded-error affine languages (BAfL) Language EQ = {w ∈ {a, b}∗ | |w|a = |w|b} /∈ REG e1 e2 (a, 2) (b, 1/2) (a, −1) (b, 1/2) (a, 1) (b, 1) Aa = 2 0 −1 1 Ab = 1/2 0 1/2 1 After reading m as and n bs the state is 2m−n 1 − 2m−n ∀w ∈ EQ, vf = 1 0 (accepting value 1) ∀w /∈ EQ, max accepting value: vf = 2 −1 (accepting value 2/3) Theorem REG BAfL1 Alejandro Díaz-Caro and Abuzer Yakaryılmaz Affine computation and affine automaton 12 / 16
  30. 30. Bounded-error affine languages (BAfL) e1 e2e3 (a, 1) (a, x)(a, −x) (b, 1) (b, −x)(b, x) (a, 1) (b, 1) (a, 1) (b, 1) Aa =   1 0 0 x 1 0 −x 0 1   Ab =   1 0 0 −x 1 0 x 0 1   After reading m as and n bs the state is   1 (m − n)x (n − m)x   Accepting value:    1 if m = n 1 2x|m − n| + 1 if m = n Taking x larger we get smaller error Alejandro Díaz-Caro and Abuzer Yakaryılmaz Affine computation and affine automaton 13 / 16
  31. 31. Bounded-error affine languages (BAfL) e1 e2e3 (a, 1) (a, x)(a, −x) (b, 1) (b, −x)(b, x) (a, 1) (b, 1) (a, 1) (b, 1) Aa =   1 0 0 x 1 0 −x 0 1   Ab =   1 0 0 −x 1 0 x 0 1   After reading m as and n bs the state is   1 (m − n)x (n − m)x   Accepting value:    1 if m = n 1 2x|m − n| + 1 if m = n Taking x larger we get smaller error Theorem REG BAfL0 Alejandro Díaz-Caro and Abuzer Yakaryılmaz Affine computation and affine automaton 13 / 16
  32. 32. Cutpoint affine languages (AfL) LAPINS = {w ∈ {a, b, c}∗ | |w|2 a > |w|b and |w|2 b > |w|c } /∈ SL [J¯anis Lapin¸š, 1974] PFAs and QFAs can check one of the conditions, with cutpoint 1 2 , but not both Alejandro Díaz-Caro and Abuzer Yakaryılmaz Affine computation and affine automaton 14 / 16
  33. 33. Cutpoint affine languages (AfL) LAPINS = {w ∈ {a, b, c}∗ | |w|2 a > |w|b and |w|2 b > |w|c } /∈ SL [J¯anis Lapin¸š, 1974] PFAs and QFAs can check one of the conditions, with cutpoint 1 2 , but not both Theorem LAPINS is recognized by an AfA with cutpoint 1 2 Proof (sketch). Alejandro Díaz-Caro and Abuzer Yakaryılmaz Affine computation and affine automaton 14 / 16
  34. 34. Cutpoint affine languages (AfL) LAPINS = {w ∈ {a, b, c}∗ | |w|2 a > |w|b and |w|2 b > |w|c } /∈ SL [J¯anis Lapin¸š, 1974] PFAs and QFAs can check one of the conditions, with cutpoint 1 2 , but not both Theorem LAPINS is recognized by an AfA with cutpoint 1 2 Proof (sketch). 1. Check |w|2 a > |w|b with an AfA simulating the PFA |w|2 a |w|b Alejandro Díaz-Caro and Abuzer Yakaryılmaz Affine computation and affine automaton 14 / 16
  35. 35. Cutpoint affine languages (AfL) LAPINS = {w ∈ {a, b, c}∗ | |w|2 a > |w|b and |w|2 b > |w|c } /∈ SL [J¯anis Lapin¸š, 1974] PFAs and QFAs can check one of the conditions, with cutpoint 1 2 , but not both Theorem LAPINS is recognized by an AfA with cutpoint 1 2 Proof (sketch). 1. Check |w|2 a > |w|b with an AfA simulating the PFA |w|2 a |w|b 2. Check |w|2 b > |w|c by producing the state |w|2 b − |w|c 1 − (|w|2 b − |w|c ) Alejandro Díaz-Caro and Abuzer Yakaryılmaz Affine computation and affine automaton 14 / 16
  36. 36. Cutpoint affine languages (AfL) LAPINS = {w ∈ {a, b, c}∗ | |w|2 a > |w|b and |w|2 b > |w|c } /∈ SL [J¯anis Lapin¸š, 1974] PFAs and QFAs can check one of the conditions, with cutpoint 1 2 , but not both Theorem LAPINS is recognized by an AfA with cutpoint 1 2 Proof (sketch). 1. Check |w|2 a > |w|b with an AfA simulating the PFA |w|2 a |w|b 2. Check |w|2 b > |w|c by producing the state |w|2 b − |w|c 1 − (|w|2 b − |w|c ) 3. Tensor both atomata PFA and QFA cannot do steps 1 and 2 at the same time! Alejandro Díaz-Caro and Abuzer Yakaryılmaz Affine computation and affine automaton 14 / 16
  37. 37. Cutpoint affine languages (AfL) LAPINS = {w ∈ {a, b, c}∗ | |w|2 a > |w|b and |w|2 b > |w|c } /∈ SL [J¯anis Lapin¸š, 1974] PFAs and QFAs can check one of the conditions, with cutpoint 1 2 , but not both Theorem LAPINS is recognized by an AfA with cutpoint 1 2 Proof (sketch). 1. Check |w|2 a > |w|b with an AfA simulating the PFA |w|2 a |w|b 2. Check |w|2 b > |w|c by producing the state |w|2 b − |w|c 1 − (|w|2 b − |w|c ) 3. Tensor both atomata PFA and QFA cannot do steps 1 and 2 at the same time! Corollary SL AfL AFAs is more powerful than PFAs and QFAs with cutpoint Alejandro Díaz-Caro and Abuzer Yakaryılmaz Affine computation and affine automaton 14 / 16
  38. 38. Nondeterministic affine languages (NAfL) (NQAL contains some famous languages like the complement of EQ) Theorem NAfL = NQAL Proof. We prove the double inclusion by showing how to simulate one with the other. NAfAs have the same power as NQFAs Alejandro Díaz-Caro and Abuzer Yakaryılmaz Affine computation and affine automaton 15 / 16
  39. 39. Summarising Bounded and unbounded error: AfAs more powerful than QFAs and PFAs Nondeterministic computation: AfAs equivalent to QFAs (and are more powerful than PFAs) Alejandro Díaz-Caro and Abuzer Yakaryılmaz Affine computation and affine automaton 16 / 16
  40. 40. Summarising Bounded and unbounded error: AfAs more powerful than QFAs and PFAs Nondeterministic computation: AfAs equivalent to QFAs (and are more powerful than PFAs) Corroboration of the thesis: The destructive interference plays a role in the computational power of QC Alejandro Díaz-Caro and Abuzer Yakaryılmaz Affine computation and affine automaton 16 / 16
  41. 41. Summarising Bounded and unbounded error: AfAs more powerful than QFAs and PFAs Nondeterministic computation: AfAs equivalent to QFAs (and are more powerful than PFAs) Corroboration of the thesis: The destructive interference plays a role in the computational power of QC Further results Language recognition power and succinctness of affine automata (M. Villagra and A. Yakaryılmaz) To appear in UCNC’16 arXiv:1602.05432 Can one quantum bit separate any pair of words with zero-error? (A. Belovs, J. A. Montoya, A. Yakaryılmaz) arXiv:1602.07967 Alejandro Díaz-Caro and Abuzer Yakaryılmaz Affine computation and affine automaton 16 / 16
  42. 42. Backup slides Weighting operator Can we use the weighting operator as a projective measurement? Answer: No v =   1 −1 1   Weighting based on separation {e1} and {e2, e3}: Probability 1/3 of {e1} Probability 2/3 of {e2, e3} But v =   0 −1 1   Not affine! (not even after normalization) Conclusion: After weighting, the system must collapse to a single state Alejandro Díaz-Caro and Abuzer Yakaryılmaz Affine computation and affine automaton 16 / 16

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