1. Uncountable Classical and Quantum Complexity Classes
University of Latvia
Faculty of computing
PhD program student
Maksims Dimitrijevs,
Abuzer Yakaryılmaz

2. Introduction
2DFAs, 2NFAs and even 2AFAs can recognize only regular languages. Rūsiņš Freivalds has shown that 2PFAs can recognize nonregular languages with bounded error, but in this case they require exponential expected-time. 2QCFAs can do it in polynomial expected time.

3. Introduction
Deterministic Turing machines can recognize only countable many languages. We can write the program of TM in binary, and then enumerate all possible programs in ascending lexicographical order.
ℵ 0

4. Introduction
Probabilistic or quantum models can be defined with uncomputable transition values and therefore their cardinalities are uncountably many.
ℵ 1

5. Introduction
What are the minimal bounded-error probabilistic and quantum classes that contain uncountable many languages?
ℵ 1

6. Known results
𝑎 4 𝑘 | 𝑎 𝑘 ∈𝐿 can be recognized by PTM with bounded error (𝑙𝑜𝑔𝑛 space is required).
Bounded-error polynomial-time 2QCFA can recognize 𝑃𝑂𝑊𝐸𝑅−𝐸𝑄 𝐼 = 𝑤∈ 𝑎,𝑏 ∗ |𝑤∈𝑃𝑂𝑊𝐸𝑅−𝐸𝑄 𝑎𝑛𝑑 log 8 ( 𝑤 𝑎 )∈𝐼 𝑤∈ 𝑎,𝑏 ∗ |𝑤∈𝑃𝑂𝑊𝐸𝑅−𝐸𝑄 𝑎𝑛𝑑 log 8 ( 𝑤 𝑎 )∈𝐼 , where 𝑃𝑂𝑊𝐸𝑅−𝐸𝑄={𝑎𝑏 𝑎 7 𝑏 𝑎 7∗8 𝑏 𝑎 7∗ 8 2 𝑏…𝑏 𝑎 7∗ 8 𝑛 |𝑛≥ 0}

7. Notion of 𝜮 ∗
We order the elements of Σ ∗ lexicographically and then represent the 𝑖-th element by Σ ∗ (𝑖), where the first value Σ ∗ (1) is the empty string. Σ ∗ ( log 64 𝑛)

8. Probabilistic TM δ:𝑆× Σ × Γ →𝑆× Γ ×{←,↓,→}×{←,↓,→}
Error bound ε (0≤ε <1/2):
if 𝑤∈𝐿, it is accepted with pr. ≥1−ε
if 𝑤∉𝐿, it is accepted with pr. ≤ε
δ:𝑆× Σ × Γ →𝑆× Γ ×{←,↓,→}×{←,↓,→}

9. Lemma for 𝟔𝟒 𝒌 coin flips
Let 𝑥= 𝑥 1 𝑥 2 𝑥 3 … be an infinite binary sequence. If a biased coin lands on head with probability 𝑝= 0. 𝑥 1 01 𝑥 2 01 𝑥 3 01…, then the value 𝑥 𝑘 can be determined with probability after 64 𝑘 coin tosses.

10. Lemma for 𝟔𝟒 𝒌 coin flips
𝐸 𝑋 = 𝑥 1 01 𝑥 2 01… 𝑥 𝑘 01… 𝑥 2𝑘 01=𝑝∗ 64 𝑘 𝑃𝑟 𝑋−𝐸[𝑋]≥ 8 𝑘 ≤ 𝑝∗ 𝑝−1 ∗ 64 𝑘 8 𝑘 2 ≤ 1 4 𝑥 1 01 𝑥 2 01… 𝑥 𝑘 01… 𝑥 2𝑘 01− 8 𝑘 ≥ 𝑥 1 01 𝑥 2 01… 𝒙 𝒌 00∗ 8 𝑘 𝑥 1 01 𝑥 2 01… 𝑥 𝑘 01… 𝑥 2𝑘 𝑘 ≤ 𝑥 1 01 𝑥 2 01… 𝒙 𝒌 10∗ 8 𝑘

11. Polynomial time loglogn space PTM
𝐴𝑀75={ 𝑎 𝑛 |𝑛>0 𝑎𝑛𝑑 𝐹 𝑛 𝑖𝑠 𝑎 𝑝𝑜𝑤𝑒𝑟 𝑜𝑓 2} 𝐴𝑀75′={ 𝑎 𝑛 |𝑛>0 𝑎𝑛𝑑 𝐹 𝑛 𝑖𝑠 𝑎 𝑝𝑜𝑤𝑒𝑟 𝑜𝑓 64} 𝐹 𝑛 = min 𝑖 𝑖 𝑑𝑜𝑒𝑠 𝑛𝑜𝑡 𝑑𝑖𝑣𝑖𝑑𝑒 𝑛} 𝑝 𝐼 =0. 𝑥 1 01 𝑥 2 01 𝑥 3 01… 𝑥 𝑘 01…, 𝑥 𝑖 =1↔𝑖∈𝐼 Ι={𝐼|𝐼∈ 𝑍 + }, cardinality of Ι is ℵ 1 𝐴𝑀75′(𝐼)={ 𝑎 𝑛 |𝑛∈𝐴𝑀 75 ′ 𝑎𝑛𝑑 Σ ∗ ( log 64 (𝐹(𝑛))) ∈𝐼}
Alt and Mehlhorn in 1975.

12. Polynomial time loglogn space PTM
We compute 𝐹(𝑛) deterministically.
If 𝐹 𝑛 = 64 𝑘 for some 𝑘, we proceed with probabilistic check.
We perform 𝐹 𝑛 coin tosses and keep the result on work tape.
To check, whether Σ ∗ ( log 64 (𝐹(𝑛))) ∈𝐼 or not, we need to get the value of 𝑥 𝑘 : (3∗𝑘−2)-th bit after decimal point.

13. Even less space?
For any binary language 𝐿⊆ {0,1} ∗ , we define another language 𝐿𝑂𝐺(𝐿) as follows:
𝐿𝑂𝐺 𝐿 ={0 1 𝑤 𝑤 𝑤 … 0 2 𝑚−1 1 𝑤 𝑚 𝑚 |
𝑤= 𝑤 1 𝑤 2 𝑤 3 … 𝑤 𝑚 ∈𝐿}
Fact. If a binary language 𝐿 is recognized by a bounded-error PTM in space 𝑠(𝑛), then the binary language 𝐿𝑂𝐺(𝐿) is recognized by a bounded- error PTM in space log 𝑠 𝑛 .

14. Arbitrary small non-constant space PTM
Language 𝐿𝑂𝐺 𝑘 (𝐴𝑀75′(𝐼)) for 𝑘>0 can be recognized by a bounded-error PTM in space 𝑂( 𝑙𝑜𝑔 𝑘+2 (𝑛))

15. Linear time linear space 2PCA
𝐷𝐼𝑀𝐴={ … 𝑘+1 𝟏𝟏 𝟎 𝟐 𝟑𝒌+𝟐 𝟏𝟏 0 2 3𝑘+3 1… 𝑘 | 𝑘>0} 𝐷𝐼𝑀𝐴 𝐼 ={𝑤∈ 0,1 ∗ 1 0 𝑚 |𝑚>0, 𝑤∈𝐷𝐼𝑀𝐴, 𝑎𝑛𝑑 Σ ∗ ( log 64 𝑚) ∈𝐼} 𝑥 1 01 𝑥 2 01… 𝒙 𝒌 ∗ 2 3𝑘+2

16. Minimizing counter space
Theorem. For any 𝐼⊆Ι, the language 𝐿𝑂𝐺(𝐷𝐼𝑀𝐴 𝐼 ) can be recognized by a bounded- error 2PCA that uses 𝑂(𝑙𝑜𝑔𝑛) space on the counter.
𝐿𝑂𝐺 𝐷𝐼𝑀𝐴(𝐼) ={0 1 𝑤 𝑤 𝑤 … 0 2 𝑚−1 1 𝑤 𝑚 𝑚 |
𝑤= 𝑤 1 𝑤 2 𝑤 3 … 𝑤 𝑚 ∈𝐷𝐼𝑀𝐴(𝐼)}

17. Arbitrary small non-constant space 2PCA
Language 𝐿𝑂𝐺 𝑘 (𝐷𝐼𝑀𝐴(𝐼)) for 𝑘>0 can be recognized by a bounded-error 2PCA in space 𝑂( 𝑙𝑜𝑔 𝑘+2 (𝑛))

18. log space 1-way unary PTM
UPOWER64= 𝑘 |𝑘>0 - 1DTM can recognize with 𝑙𝑜𝑔𝑛 space in a straightforward way.
𝑈𝑃𝑂𝑊𝐸𝑅64 𝐼 = 0 𝑛
0 𝑛 ∈𝑈𝑃𝑂𝑊𝐸𝑅64 𝑎𝑛𝑑 Σ ∗ ( log 64 𝑛) ∈𝐼}

19. loglog space sweeping unary PTM
𝐴𝑀75′={ 𝑎 𝑛 |𝑛>0 𝑎𝑛𝑑 𝐹 𝑛 𝑖𝑠 𝑎 𝑝𝑜𝑤𝑒𝑟 𝑜𝑓 64} 𝐴𝑀75′(𝐼)={ 𝑎 𝑛 |𝑛∈𝐴𝑀 75 ′ 𝑎𝑛𝑑 Σ ∗ ( log 64 (𝐹(𝑛))) ∈𝐼} Polynomial time

20. Linear space sweeping PCA
𝐷𝐼𝑀𝐴={ … 𝑘+1 𝟏𝟏 𝟎 𝟐 𝟑𝒌+𝟐 𝟏𝟏 𝑘+3 1… 𝑘 |
𝑘>0}
𝐷𝐼𝑀𝐴 can be recognized by sweeping PCA in three passes.
𝐷𝐼𝑀𝐴 𝐼 ={𝑤∈ 0,1 ∗ 1 0 𝑚 |𝑚>0,
𝑤∈𝐷𝐼𝑀𝐴, 𝑎𝑛𝑑 Σ ∗ ( log 64 𝑚) ∈𝐼}
Subquadratic time: 𝑂 2 3𝑘 𝑂 2 6𝑘 =𝑂 2 9𝑘 = 𝑂(𝑛 𝑛 ).

21. 2QCFA
Two-way finite automaton with quantum and classical states, which can use unitary operators and projective measurements on the quantum part.
A superoperator, determined by the current classical state and the symbol being scanned on the input tape, is applied to the quantum register, yielding an outcome.
Then, the next classical state and tape head movement direction is determined by the current classical state, the symbol being scanned on the input tape, and the observed outcome.

22. POWER-EQ continued
𝑃𝑂𝑊𝐸𝑅−𝐸𝑄={𝑎𝑏 𝑎 7 𝑏 𝑎 7∗8 𝑏 𝑎 7∗ 8 2 𝑏…𝑏 𝑎 7∗ 8 𝑛 |𝑛≥0} This language can be recognized by a restarting rtQCFA with bounded error.

23. Quantum trick
θ 𝐼 =2π 𝑖=1 ∞ 𝑥 𝑖 8 𝑖+1 , 𝑥 𝑖 =1, if 𝑖∈𝐼; 𝑥 𝑖 =−1, if 𝑖∉𝐼. 8 𝑗 ∗2π 𝑖=1 ∞ 𝑥 𝑖 8 𝑖+1 =π 𝑥 𝑗 4 + 𝑖=𝑗+1 ∞ 𝑥 𝑖 8 𝑖+1 After an additional rotation by π 4 , the final angle from | 𝑞 1 > is π 2 +δ if 𝑥 𝑖 =1 (𝑖∈𝐼) and it is δ if 𝑥 𝑖 = −1 (𝑖∉𝐼), where δ is sufficiently small such that the probability of the qubit being in | 𝑞 2 > (| 𝑞 1 > ) is bigger than 0.98 if 𝑖∈𝐼 (𝑖∉𝐼).

24. Restarting rtQCFA
𝑃𝑂𝑊𝐸𝑅−𝐸𝑄 𝐼 = 𝑤∈ 𝑎,𝑏 ∗ |𝑤∈𝑃𝑂𝑊𝐸𝑅−𝐸𝑄 𝑎𝑛𝑑 log 8 ( 𝑤 𝑎 )∈𝐼 Exponential expected time.

25. Middle log space 2QCCA
𝑈𝑃𝑂𝑊𝐸𝑅2= 0 2 𝑛 |𝑛≥0 2QCCA can recognize in middle log space. 𝑈𝑃𝑂𝑊𝐸𝑅8= 0 8 𝑛 |𝑛≥0 𝑈𝑃𝑂𝑊𝐸𝑅8(𝐼)= 0 8 𝑛 |𝑛−1∈𝐼

26. Unary languages
Polynomial-time 𝑂(𝑙𝑜𝑔𝑙𝑜𝑔𝑛)-space sweeping PTMs (open for 𝑜(𝑙𝑜𝑔𝑙𝑜𝑔𝑛)-space)
Linearithmic-time 𝑂(𝑙𝑜𝑔𝑛)-space 1PTMs
Middle 𝑂(𝑙𝑜𝑔𝑛)-space 2QCCAs (open for better space bounds and/or polynomial-time; open for PCAs)

27. Binary languages
ω(1)-space 2PCAs (open for 𝑂(1)-space (or equivalently 2PFAs)
Polynomial-time 𝑂(𝑛)-space 2PCAs (open for polynomial-time 𝑜(𝑛)-space)
Restarting rtQCFAs (open for polynomial-time)

28. Thank you for your attention!