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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