Super-Exponential Size Advantage of Quantum Finite Automata with ...
Super-Exponential Size Advantage of Quantum Finite Automata with Mixed States R¯ usi¸ nˇs Freivalds Department of Computer Science, University of Latvia, Rai¸ na bulv¯ aris 29, R¯ıga, Latvia
Abstract. Quantum finite automata with mixed states are proved to be super-exponentially more concise rather than quantum finite automata with pure states. It was proved earlier by A.Ambainis and R.Freivalds that quantum finite automata with pure states can have exponentially smaller number of states than deterministic finite automata recognizing the same language. There was a never published ”folk theorem” proving that quantum finite automata with mixed states are no more than superexponentially more concise than deterministic finite automata. It was not known whether the super-exponential advantage of quantum automata is really achievable. We use a novel proof technique based on Kolmogorov complexity to prove that there is an infinite sequence of distinct integers n such that there are languages Ln in a 4-letter alphabet such that there are quantum finite automata with mixed states with 2n + 1 states recognizing the language Ln with probability 34 while any deterministic finite automaton recognizing Ln needs to have at least eO(nlnn) states.
1
Introduction
A.Ambainis and R.Freivalds proved in [4] that for recognition of some languages the quantum finite automata can have smaller number of the states than deterministic ones, and this difference can even be exponential. The proof contained a slight non-constructiveness, and the exponent was not shown explicitly. For probabilistic finite automata exponentiality of such a distinction was not yet proved. The best (smaller) gap was proved by Ambainis [2]. The languages recognized by automata in [4] were presented explicitly but the exponent was not. In a very recent paper by R.Freivalds [13] the non-constructiveness is modified, and an explicit (and seemingly much better) exponent is obtained at the expense of having only non-constructive description of the languages used. Moreover, the best estimate proved in this paper is proved under assumption of the well-known Artin’s Conjecture (1927) in Number Theory. [13] contains also a theorem that does not depend on any open conjectures but the estimate is worse, and the description of the languages used is even less constructive. This seems to be the first result in finite automata depending on open conjectures in Number Theory.
This research is supported by Grant No.05.1528 from the Latvian Council of Science.
S.-H. Hong, H. Nagamochi, and T. Fukunaga (Eds.): ISAAC 2008, LNCS 5369, pp. 931–942, 2008. c Springer-Verlag Berlin Heidelberg 2008
932
R. Freivalds
The following two theorems are proved in [13]: Theorem 1. Assume Artin’s Conjecture. There exists an infinite sequence of regular languages L1 , L2 , L3 , . . . in a 2-letter alphabet and an infinite sequence of positive integers z(1), z(2), z(3), . . . such that for arbitrary j: 1. there is a probabilistic reversible automaton with (z(j) states recognizing Lj with the probability 19 36 , 2. any deterministic finite automaton recognizing Lj has at least (21/4 )z(j) = = (1.1892071115 . . .)z(j) states, Theorem 2. There exists an infinite sequence of regular languages L1 , L2 , L3 , . . . in a 2-letter alphabet and an infinite sequence of positive integers z(1), z(2), z(3), . . . such that for arbitrary j: 1. there is a probabilistic reversible automaton with z(j) states recognizing Lj 68 with the probability 135 , 1 2. any deterministic finite automaton recognizing Lj has at least (7 14 )z(j) = = (1.1149116725 . . .)z(j) states, The paper [13] concluded a long research on relative size of probabilistic and deterministic automata [8,9,10,11,16,2,15,13,14]. The two theorems above are formulated in [13] as assertions about reversible probabilistic automata. For probabilistic automata (reversible or not) it was unknown before the paper [13] whether the gap between the size of probabilistic and deterministic automata can be exponential. It is easy to re-write the proofs in order to prove counterparts of Theorems 1 and 2 for quantum finite automata with pure states. The aim of this paper is to prove a counterpart of these theorems for quantum finite automata with mixed states. Quantum algorithms with mixed states were first considered by D.Aharonov, A.Kitaev, N.Nisan [1]. More detailed description of quantum finite automata with mixed states can be found in A.Ambainis, et al.[3]. The automaton is defined by the initial density matrix ρ0 . Every symbol ai in the input alphabet is associated with a unitary matrix Ai . When the automaton reads the symbol ai the current density matrix ρ is transformed into A∗i ρAi . When the reading of the input word is finished and the end-marker $ is read, the current density matrix ρ is transformed into A∗end ρAend and separate measurements of all states are performed. After that the probabilities of all the accepting states are totaled, and the probabilities of all the rejecting states are totaled. Like quantum finite automata with pure states described by A.Kondacs and J.Watrous [19] we allow measurement of the accepting states and rejecting states after every step of the computation. The main result in our paper is: Theorem 3. There is an infinite sequence of distinct integers n such that there are languages Ln in a 4-letter alphabet such that there are quantum finite automata with mixed states with 2n + 1 states recognizing the language Ln with probability 34 while any deterministic finite automaton recognizing Ln needs to have at least eO(nlnn) states.
Super-Exponential Size Advantage of Quantum Finite Automata
933
Proof is delayed till Section 5. Since the numbers of the states for deterministic automata and quantum automata with pure states differ no more than exponentially, we have Theorem 4. There is an infinite sequence of distinct integers n such that there are languages Ln in a 4-letter alphabet such that there are quantum finite automata with mixed states with 2n + 1 states recognizing the language Ln with probability 34 while any quantum finite automaton with pure states recognizing Ln with bounded error needs to have at least eO(nlnn) states.
2
Permutations and Hamming Distance
Permutation of the set Nn is a 1-1 correspondence from Nn onto itself. Let f be such a permutation. The fact that it is onto means that for any k ∈ Nn there exists i ∈ Nn such that f (i) = k. We need a notion similar to Hamming codes for permutations. Since Hamming distance between permutations is already considered in several well-known textbooks (e.g. [6]), it seemed natural that the corresponding theory might be already published. Very far from truth! Hamming distance between two objects is the number of changes one needs to perform to obtain one object from the. The Hamming distance between two binary words (of the same length) is defined to be the number of positions at which they differ. For instance, we consider a set of three binary words {0011, 0110, 1100}. The first word is at Hamming distance 3 from the other two. Additionally, every word in this set is at Hamming distance at least 2 from any other. Such systems of words are called codes. They are important because they allow us to eliminate accidental errors when transmitting the words through noisy information channels. We consider Hamming distance between permutations. Hamming distance between the permutation s of the set {1, 2, 3, · · · n} and the permutation r of the same set is the number of distinct numbers i such that s(i) = r(i). For instance, let s be a permutation of the set {1, 2, 3, · · · n} and the number of it’s fixed points be p. Then the Hamming distance between the permutation s and the identity permutation is the number n − p. Lemma 1. Let d be an arbitrary real number such that 0 ≤ d ≤ 1. No more than 2dnlnn permutations can be on Hamming distance less or equal than dn from the identity permutation. Proof. By Stirling formula, n! = en.lnn−o(nlnn) . Let π be an arbitrary npermutation. How many there are distinct n-permutations differing from the permutation π in no more than dn positions? The differing positions can be chosen in n ≤ < 2n d ways and these ≤ dn positions are permuted. Hence there are no more than 2n .2dnlnn−o(nlnn) ≤ 2dnlnn permutations of this type.
934
R. Freivalds
Theorem 5. For arbitrary constant c < 1 such that for arbitrary n there is a set Gn of n-permutations containing eΩ(n log n) permutations such that the Hamming distance of any two permutations is at least c · n. Proof. Immediately from Lemma 1.
3
Permutations and Automata
Definition 1. The Hamming distance or simply distance d(r, s) between two n-permutations r and s on the set S is the number of elements x ∈ S such that r(x) = s(x). The similarity e(r, s) is the number of x ∈ S such that r(x) = s(x). Note that d(r, s) + e(r, s) = |S| = n. Theorem 6. Let c be a fixed constant and let there be an infinite sequence of distinct integers n such that for each n there exists a group Gn of permutations of the set {1, 2, . . . , n}, the group has order(Gn ) elements and k generating elements, and the Hamming distance of any two permutations is at least c · n. Then there is an infinite sequence of distinct integers n such that for each n there is a language Ln in a k-letter alphabet that can be recognized with probability 2c by a quantum finite automata with mixed states that has 2n states, while any deterministic finite automaton recognizing Ln must have at least order(Gn ) states. Proof. For each permutation group Gn we define the language Ln as follows: The letters of Ln are the k generators of the group Gn and it consists of words s1 s2 s3 . . . sm such that the product s1 ◦ s2 ◦ s3 ◦ · · · ◦ sm differs from the identity permutation. (A) Any deterministic automaton recognizing Ln is to remember the first input letter by a specific state. (B) We will construct a quantum automaton with mixed states. It has 4n states and the initial density matrix ρ0 is a diagonal block-matrix that consists of n blocks ρ0 : ⎛ ⎞ 1100 ⎟ 1 ⎜ ⎜1 1 0 0⎟ ρ0 = ⎝ 2n 0 0 0 0 ⎠ 0000 For each of k generators gi ∈ Gn we will construct the corresponding unitary matrix Ui as follows – it is a 2n × 2n permutation matrix, that permutes the elements in the even positions according to permutation gi , but leaves the odd positions unpermuted. For example, g = 3241 can be expressed as the following permutation matrix that acts on a column vector: ⎛ ⎞ 0010 ⎜0 1 0 0⎟ ⎟ g=⎜ ⎝0 0 0 1⎠ 1000
Super-Exponential Size Advantage of Quantum Finite Automata
935
The initial density matrix ρ0 for n = 4 and the unitary matrix U that corresponds to the permutation matrix (3) of permutation g are as follows: ⎛ ⎞ ⎛ ⎞ 11000000 10000000 ⎜1 1 0 0 0 0 0 0⎟ ⎜0 0 0 0 0 1 0 0⎟ ⎜ ⎟ ⎜ ⎟ ⎜0 0 1 1 0 0 0 0⎟ ⎜0 0 1 0 0 0 0 0⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 1 ⎜0 0 1 1 0 0 0 0⎟ ⎟,U = ⎜0 0 0 1 0 0 0 0⎟ ρ0 = ⎜ ⎜ ⎟ ⎜ ⎟ 8 ⎜0 0 0 0 1 1 0 0⎟ ⎜0 0 0 0 1 0 0 0⎟ ⎜0 0 0 0 1 1 0 0⎟ ⎜0 0 0 0 0 0 0 1⎟ ⎜ ⎟ ⎜ ⎟ ⎝0 0 0 0 0 0 1 1⎠ ⎝0 0 0 0 0 0 1 0⎠ 00000011 01000000 The unitary matrix U$ for the end-marker is also a diagonal block-matrix. It consists of n blocks that are the Hadamard matrices 1 1 = √1 H 2 1 −1 acts on two specific 2 × 2 density matrices: Notice how the Hadamard matrix H 1 20 1 11 † if ρ = , then HρH = , 2n 1 1 2n 0 0 1 10 H † = 1 1 0 . if ρ = , then Hρ 2n 0 1 2n 0 1 For example, when the letter g is read, the unitary matrix U is applied to the density matrix ρ0 (both are given in equation (3)) and the density matrix ρ1 = U ρ0 U † is obtained. When the end-marker “$” is read, the density matrix becomes ρ$ = U$ ρ1 U$† . Matrices ρ1 and ρ$ are as follows: ⎛ ⎛ ⎞ ⎞ 1 0 0 0 12 12 12 − 21 10000001 ⎜0 1 0 0 1 0 0 0⎟ ⎜ 0 1 0 0 −1 −1 1 −1 ⎟ 2 2 2 2⎟ ⎜ ⎜ ⎟ ⎜0 0 1 1 0 0 0 0⎟ ⎜ 0 0 20 0 0 0 0 ⎟ ⎜ ⎜ ⎟ ⎟ 1⎜ 1⎜ 0 0 00 0 0 0 0 ⎟ 0 0 1 1 0 0 0 0⎟ ⎜ ⎜ ⎟. ⎟ ρ1 = ⎜ , ρ$ = ⎜ 1 8 ⎜0 1 0 0 1 0 0 0⎟ 8 ⎜ 2 − 21 0 0 1 0 12 12 ⎟ ⎟ ⎟ ⎜0 0 0 0 0 1 1 0⎟ ⎜ 1 −1 0 0 0 1 −1 −1 ⎟ 2 2⎟ ⎜ ⎜ 21 12 ⎟ 1 1 ⎝0 0 0 0 0 1 1 0⎠ ⎝ 0 ⎠ 2 2 0 0 2 −2 1 1 1 1 10000001 − 2 − 2 0 0 2 − 21 0 1 Finally, we declare the states in the even positions to be accepting, but the states in the odd positions to be rejecting. Therefore one must sum up the diagonal entries that are in the even positions of the final density matrix to find the probability that a given word is accepted. In our example the final density matrix ρ$ is given in (3). It corresponds to the input word “g$”, which is accepted with probability 18 (1 + 0 + 1 + 1) = 38 and rejected with probability 18 (1 + 2 + 1 + 1) = 58 . Note that the accepting and rejecting probabilities sum up to 1.
936
R. Freivalds
It is easy to see, that the words that do not belong to the language Ln are rejected with certainty, because the matrix U$ ρ0 U$† has all zeros in the even positions on the main diagonal. However, the words that belong to Ln are accepted d cn = 2n = 2c , because all permutations are at least with the probability at least 2n at the distance d from the identity permutation. It is also easy to see that any deterministic automaton that recognizes the language Ln must have at least order(Gn ) states. If the number of states is less than order(Gn ), then there are two distinct words u and v such that the deterministic automaton ends up in the same state no matter which one of the two words it reads. Since Gn is a group, for each word we can find an inverse, that returns the automaton in the initial state (the only rejecting state). Since u and v are different, they have different inverses and u ◦ u−1 is the identity permutation and must be rejected, but v ◦ u−1 is not the identity permutation and must be accepted – a contradiction.
4
Super-Exponential Size Advantage
Now we wish to prove a theorem differing from Theorem 3 only in one way. In this section we allow the alphabets of the languages Ln to grow with n. Consider the following infinite sequence of languages. For every n take the set Gn considered in Theorem 5. The language Ln consists of all the words aa (of the length 2) where a is a symbol for an arbitrary element from Gn . Hence there are eΩ(n log n) letters in the alphabet of the language Ln and equally many words in Ln . Theorem 7. There is an infinite sequence of distinct integers n such that there are languages Ln such that there are quantum finite automata with mixed states with 2n + 1 states recognizing the language Ln with probability 34 while any deterministic finite automaton recognizing Ln needs to have at least eO(nlnn) states. Proof is similar to the proof of Theorem 6. It would be nice to improve our theorem in the natural way by finding algebraic groups Gn of n-permutations such that they are generated by a small (constant) number of generating elements and at the same time the order of Gn (the number of elements) would be superexponentially large with the respect to n. This would give us a natural proof for Theorem 3. Unfortunately, there exists a paper [17] by J.Kempe et al. where it is proven that large Hamming distance between all the n-permutations in a group implies that the order of the group is no more than exponential with the respect to n.
5
4-Letter Input Alphabet
We consider 2 algebraic groups in this Section. The set of all permutations of the set 1, ..., n with algebraic operation ”product of permutations” can be considered as the group G1 . This group has two generating elements, the permutations
Super-Exponential Size Advantage of Quantum Finite Automata
937
(123 · · · n) and (12)(3)(4) · · · (n). The other group G2 is defined as follows. Two binary strings α1 and α2 , each one of the length nlog2 n are taken and two realvalued unitary matrices are constructed. The corresponding unitary operators are rotations around axes depending, correspondingly, on α1 and α2 . The angle 3 of rotation is a multiple of 2π p where p is a prime slightly exceeding n . The inverses of these operators again are unitary, and their matrices are real-valued. The group direct product G1 × G2 is again a group G with 4 generating elements and the elements of this group can be represented by n-dimensional unitary operators. G has the following properties. Property 1. For arbitrary fixed pair (α1 , α2 ) distinct n-permutations generate distinct unitary operators. Property 2. For arbitrary fixed n-permutation distinct pairs (α1 , α2 ) generate distinct unitary operators. Property 3. The n-permutation and the pair (α1 , α2 ) is transformed into the unitary operator described above by an algorithmic process. Property 4. The n-permutation and the unitary operator can be transformed into the pair (α1 , α2 ) by an algorithmic process. These properties will be used below to prove the main result, to prove Theorem 3. At first we construct a quantum finite automaton with mixed states having 2n states. This automaton has the following initial density matrix (we use an 8 × 8-matrix instead of formal description of n × n-matrix for the sake of easier readability). ⎞ ⎛1 1 8 0 0 0 8 0 0 0 1 1 ⎜0 0 0 0 0 0⎟ 8 ⎟ ⎜ 8 1 ⎜0 0 0 0 0 1 0⎟ 8 8 ⎟ ⎜ ⎜0 0 0 1 0 0 0 1⎟ 8 8 ⎟ ⎜ ρ0 = ⎜ 1 1 ⎟ ⎜ 8 01 0 0 8 01 0 0 ⎟ ⎜0 0 0 0 0 0⎟ 8 ⎟ ⎜ 8 1 ⎝0 0 0 0 0 1 0⎠ 8 8 0 0 0 18 0 0 0 18 At every step an input letter ai (one of four possible input letters a1 , a2 , a3 or a4 ) is read from the input. Hence the density matrix ρx is changed into Ui∗ ρx Ui . ∗ · · · U2∗ U1∗ ρ0 U1 U2 · · · Hence if x = x1 x2 x3 · · · xt then the current Ux equals Ut∗ Ut−1 Ut−1 Ut . Let the matrix ⎞ ⎛ u11 u12 · · · u1n ⎜ u21 u22 · · · u2n ⎟ ⎟ ⎜ ⎟ Ux = ⎜ ⎜ u31 u32 · · · u3n ⎟ ⎠ ⎝ ··· un1 un2 · · · unn be the n × n-matrix corresponding to the input word x. We associate a 2n × 2nmatrix Mx
938
R. Freivalds
⎛
1 0 ··· ⎜ 0 1 ··· ⎜ ⎜ 0 0 ··· ⎜ ⎜··· ⎜ ⎜ 0 0 ··· Mx = ⎜ ⎜ 0 0 ··· ⎜ ⎜ 0 0 ··· ⎜ ⎜ 0 0 ··· ⎜ ⎝··· 0 0 ···
0 0 0
··· ··· ···
0 u12 u22 u32
··· ··· ··· ···
0 0 0 0 0 0 1 0 0 0
0 u11 u21 u31
⎞
0 0 0
0 un1 un2 u · · ·
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ 0 ⎟ ⎟ u1n ⎟ ⎟ u2n ⎟ ⎟ u3n ⎟ ⎟ ⎠ unn
to it. We use the initial density matrix and the following property ⎛
1000 0 ⎜0 1 0 0 0 ⎜ ⎜0 0 1 0 0 ⎜ ⎜0 0 0 1 0 ⎜ ⎜ 0 0 0 0 u11 ⎜ ⎜ 0 0 0 0 u21 ⎜ ⎝ 0 0 0 0 u31 0 0 0 0 u41
0 0 0 0 u12 u22 u32 u42
0 0 0 0 u13 u23 u33 u43
⎞⎛ 1 0 8 ⎜ 0 ⎟ ⎟⎜0 ⎜ 0 ⎟ ⎟⎜0 ⎜ 0 ⎟ ⎟ ⎜ 01 ⎜ u14 ⎟ ⎟⎜ 8 ⎜ u24 ⎟ ⎟⎜0 u34 ⎠ ⎝ 0 u44 0 ⎛
1 8
0
0 0 1 8 0 0 18 0 0 0 0 1 8 0 0 18 0 0 0 0
⎜ 0 1 8 ⎜ ⎜ 0 0 1 8 ⎜ ⎜ 0 0 0 =⎜ ⎜ u11 u12 u13 ⎜ u8 u8 u8 ⎜ 21 22 23 ⎜ u8 u8 u8 ⎝ 31 32 33 8 8 8 u41 u42 u43 8 8 8
⎞⎛ 0 0 1000 0 ⎜0 1 0 0 0 0 0⎟ ⎟⎜ 1 ⎟⎜ 8 0 ⎟⎜0 0 1 0 0 ⎜ 0 18 ⎟ ⎟⎜0 0 0 1 0 ⎜ 0 0⎟ ⎟ ⎜ 0 0 0 0 u11 ⎜ 0 0⎟ ⎟ ⎜ 0 0 0 0 u12 1 ⎠ ⎝ 0 0 0 0 u13 0 8 0 0 0 0 u14 0 18
0 18 0 0 0 18 0 0 0 1 8 0 0 0 18 0 0 0 18 0 0 0 1 8 0 0 0 0 0 1 8 u14 8 u24 8 u34 8 u44 8
u11 8 u12 8 u13 8 u14 8 1 8
0 0 0
u21 8 u22 8 u23 8 u24 8
u31 8 u32 8 u33 8 u34 8
0 1 8
0 0
0 0
0
u41 8 u42 8 u43 8 u44 8
0 0 0 0 u21 u22 u23 u24
0 0 0 0 u31 u32 u33 u34
⎞ 0 0 ⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟= u41 ⎟ ⎟ u42 ⎟ ⎟ u43 ⎠ u44
⎞
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ 0 ⎠
1 8
1 8
to justify the correspondence between the n × n-matrices Ux and the 2n × 2nmatrices Mx at every moment. When all the input word is read the end-marker $ comes in, the unitary matrix Mx corresponding to the end-marker $ is ⎛
√1 2
⎜ 0 ⎜ ⎜ 0 ⎜ ⎜ ⎜ 0 M$ = ⎜ ⎜ √1 ⎜ 2 ⎜ 0 ⎜ ⎜ ⎝ 0 0
0 √1 2
0 0 0
0 0 √1 2
0 0 0
0 0 0 √1 2
√1 2
0 0 0
0 √1 2
0 0 0
0 0 √1 2
0 0 − √12 0 √1 0 0 − √12 0 2 1 0 √2 0 0 0 − √12 0 0 0 0 √12 0
0 0 0
⎞
⎟ ⎟ ⎟ ⎟ ⎟ √1 ⎟ 2 ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ ⎟ 0 ⎠ − √12
Super-Exponential Size Advantage of Quantum Finite Automata
939
This matrix can be regarded as a block matrix with blocks of size 2 × 2. It is worth to remember that
√1 2 √1 2
√1 2 − √12
a11 a12 a21 a22
√1
√1 2 2 1 √ − √1 2 2
a11 +a21 +a12 +a22 =
a11 +a21 −a12 −a22 2 2 a11 −a21 +a12 −a22 a11 −a21 −a12 +a22 2 2
.
If the input word is such that the corresponding member of the free group G1 equals the empty word then the final density matrix becomes ⎛1 4
ρf inal
⎜0 ⎜ ⎜0 ⎜ ⎜0 =⎜ ⎜0 ⎜ ⎜0 ⎜ ⎝0 0
0 0 1 4 0 0 14 0 0 0 0 0 0 0 0 0 0
⎞ 00000 0 0 0 0 0⎟ ⎟ 0 0 0 0 0⎟ ⎟ 1 ⎟ 4 0 0 0 0⎟. 0 0 0 0 0⎟ ⎟ 0 0 0 0 0⎟ ⎟ 0 0 0 0 0⎠ 00000
Otherwise the total of the first n elements on the main diagonal is less than 1. At the end separate measurements of all states are performed. After that the probabilities of all the accepting states are totaled, and the probabilities of all the rejecting states are totaled. What is the probability of the acceptance in the case when the member of the free group G1 does not equal the empty word? It depends on the values of (α1 , α2 ). We will prove in the subsequent sections that if the Kolmogorov complexity of (α1 , α2 ) is maximal then this probability is MUCH less than 1.
6
Kolmogorov Complexity
The theorems in this section are well-known results in spite of the fact that it is not easy to find exact references for all of them. Definition 2. We say that the numbering Ψ = {Ψ0 (x), Ψ1 (x), Ψ2 (x), . . .} of 1argument partial recursive functions is computable if the 2-argument function U (n, x) = Ψn (x) is partial recursive. Definition 3. We say that a numbering Ψ is reducible to the numbering η if there exists a total recursive function f (n) such that, for all n and x, Ψn (x) = ηf (n) (x). Definition 4. We say that a computable numbering ϕ of all 1-argument partial recursive functions is a G¨ odel numbering if every computable numbering (of any class of 1-argument partial recursive functions) is reducible to ϕ. Theorem. [7] There exists a G¨ odel numbering.
940
R. Freivalds
Definition 5. We say that a G¨ odel numbering ϑ is a Kolmogorov numbering if for arbitrary computable numbering Ψ (of any class of 1-argument partial recursive functions) there exist constants c > 0, d > 0, and a total recursive function f (n) such that: 1. for all n and x, Ψn (x) = ϑf (n) (x), 2. for all n, f (n) ≤ c · n + d. Kolmogorov Theorem. [18] There exists a Kolmogorov numbering.
7
Back to Automata
We denote by U (τ, α) the pair of unitary matrices generated from the npermutation τ and the binary string α of the length 2|τ |log2 |τ | (later divided into the pair (α1 , α2 )) each one of the length |τ |log2 |τ | . There exist many distinct Kolmogorov numberings. We now fix one of them and denote it by η. Since Kolmogorov numberings give indices for all partial recursive functions, for arbitrary τ and n there is an i such that ηi (τ, n) = x. Let i(τ, n) be the minimal i such that ηi (τ, n) = α. It is easy to see that if x1 = x2 , then i(α1 , n) = i(α2 , n). We consider all binary words α corresponding to n and denote by α(n) the word α such i(α, n) exceeds i(β, n) for all binary words β corresponding to n different from α. It is obvious that i ≥ 22n(log2 n)−1 . Until now we considered generating matrices U (τ, α) for independently chosen τ and α. From now on we consider only α corresponding to i(τ, n). We wish to prove that if n is sufficiently large, then Hamming distances between the vectors 1 1 1 1 1 n , n , n , · · · , n and a11 , a22 , · · · , ann in the matrix ρf inal exceeds 18 . We introduce a partial recursive function μ(z, n) defined as follows. Above when defining U (τ, α) we considered auxiliary function U (τ, α). To define μ(z, n) we consider all binary words α of the length 2|τ |log2 |τ |. If z is not a binary word of such a length, then μ(z, n) is not defined. If z is a binary word of length 2|τ |log2 |τ |, then we consider all α ∈ {0, 1}2|τ |log2|τ | such that U (τ, α) = z. If there are no such α, then μ(z, n) is not defined. If there is only one such α, then μ(z, n) = α. If there are more than two such x, then μ(z, n) is not defined. Now we introduce a computable numbering of some partial recursive functions. This numbering is independent of n. For each n (independently from other values of n) we order the set of all the 22|τ |log2 |τ | binary words z of the length 2|τ |log2 |τ | : z0 , z1 , z2 , . . . , z22|τ |log2 |τ | −1 . We define z0 as the word 000 . . . 0. The words z1 , z2 , . . . , z22p are words with exactly one symbol 1. We strictly follow a rule ”if the word zi contains less symbols 1 than the word zj , then i < j”. Words with equal number of the symbol 1 are ordered lexicographically. Hence z22p −1 = 111 . . . 1.
Super-Exponential Size Advantage of Quantum Finite Automata
941
For each n, we define w = |τ |log2 |τ | and Ψ0 (n) = μ(z0 , n) Ψ1 (n) = μ(z0 , n) Ψ2 (n) = μ(z1 , n) ... Ψ22w+1 −2 (p) = μ(z22w −1 , n) Ψ22w+1 −1 (p) = μ(z22w −1 , n) For j ≥ 22w+1 , Ψj (w) is undefined. We have fixed a Kolmogorov numbering η and we have just constructed a computable numbering Ψ of some partial recursive functions. Lemma 2. There exist constants c > 0 and d > 0 (independent of p) such that for arbitrary i there is a j such that 1. Ψi (t) = ηj (t) for all t, and 2. j ≤ ci + d. Proof. Immediately from Kolmogorov Theorem. The rest of the proof of Theorem 3 is from the contrary. Suppose that n is sufficiently large (because Kolmogorov numbering is optimal only for large lengths of the words) and for α(n) the value of a11 + a22 + · · · + ann had exceeded 17 18 in the unitary matrix U (τ, α). Then our α could be uniquely and algorithmically be described in more concise way than our obvious assertion i(τ ) ≥ 2nlog2n − 1 allows. Contradiction. This way we have proved that the languages Ln can be recognized by 2n-state quantum finite automata with mixed states so that every word in the language is accepted with the probability 1 and every word not in Ln is accepted with probability less than 17 18 . If we add one more state to the automata we can get automata with a cut-point 12 and a uniformly bounded error. It remains only to find the complexity (the number of states) of deterministic finite automata recognizing these languages. Theorem 5 above shows that there are eO(nlnn) n-permutations τj with pairwise Hamming distances no less than const.n. These n-permutations can be constructed using no more than n3 generating elements (123 · · · n) and (12)(3)(4) · · · (n). Input words corresponding to these eO(nlnn) n-permutations τj should be remembered by distinct states of any deterministic finite automaton recognizing the language.
Acknowledgments I would like to thank Andris Ambainis for pointing at a possibility to use random unitary operators instead of permutations. In the final version of my proof the unitary operators are far from being random but this idea helped to move my proof off a deadlock.
942
R. Freivalds
References 1. Aharonov, D., Kitaev, A., Nisan, N.: Quantum circuits with mixed states. In: Proc. STOC 1998, pp. 20–30 (1998) 2. Ambainis, A.: The complexity of probabilistic versus deterministic finite automata. LNCS, vol. 1178, pp. 233–237. Springer, Heidelberg (1996) 3. Ambainis, A., Beaudry, M., Golovkins, M., K ¸ ikusts, A., Mercer, M., Th´erien, D.: Algebraic Results on Quantum Automata. Theory Comput. Syst. 39(1), 165–188 (2006) 4. Ambainis, A., Freivalds, R.: 1-way quantum finite automata: strengths, weaknesses and generalizations. In: Proc. IEEE FOCS 1998, pp. 332–341 (1998) 5. Artin, E.: Beweis des allgemeinen Reziprozit¨ atsgesetzes. Mat. Sem. Univ. Hamburg B.5, 353–363 (1927) 6. Cameron, P.: Permutation groups. London Mathematical Society Student Texts series. Cambridge University Press, Cambridge (1999) 7. Ershov, Y.L.: Theory of numberings. In: Griffor, E.R. (ed.) Handbook of computability theory, pp. 473–503. North-Holland, Amsterdam (1999) 8. Freivalds, R.: Recognition of languages with high probability on different classes of automata. Doklady Akademii Nauk SSSR 239(1), 60–62 (1978) (Russian) 9. Freivalds, R.: Projections of languages recognizable by probabilistic and alternating finite multi-tape automata. Information Processing Letters 13(4–5), 195–198 (1981) 10. Freivalds, R.: On the growth of the number of states in result of the determinization of probabilistic finite automata. Avtomatika i Vichislitel’naya Tekhnika 3, 39–42 (1982) (Russian) 11. Freivalds, R.: Complexity of probabilistic versus deterministic automata. In: Barzdins, J., Bjorner, D. (eds.) Baltic Computer Science. LNCS, vol. 502, pp. 565–613. Springer, Heidelberg (1991) 12. Freivalds, R.: Languages Recognizable by Quantum Finite Automata. In: Farr´e, J., Litovsky, I., Schmitz, S. (eds.) CIAA 2005. LNCS, vol. 3845, pp. 1–14. Springer, Heidelberg (2006) 13. Freivalds, R.: Non-constructive methods for finite probabilistic automata. In: Harju, T., Karhum¨ aki, J., Lepist¨ o, A. (eds.) DLT 2007. LNCS, vol. 4588, pp. 169– 180. Springer, Heidelberg (2007) 14. Freivalds, R.: Hamming, Permutations and Automata. In: Hromkoviˇc, J., Kr´ aloviˇc, R., Nunkesser, M., Widmayer, P. (eds.) SAGA 2007. LNCS, vol. 4665, pp. 18–29. Springer, Heidelberg (2007) 15. Freivalds, R., Ozols, M., Manˇcinska, L.: Permutation Groups and the Strength of Quantum Finite Automata with Mixed States. In: Proceedings of the Workshop Probabilistic and Quantum Algorithms colocated with the 11th International Conference Developments in Language Theory DLT 2007, Turku, Finland, July 3-5, 2007, vol. 45, pp. 13–27. TUCS General Publication, No 45 (June 2007) 16. Ka¸ neps, J., Freivalds, R.: Running time to recognize nonregular languages by 2way probabilistic automata. In: Leach Albert, J., Monien, B., Rodr´ıguez-Artalejo, M. (eds.) ICALP 1991. LNCS, vol. 510, pp. 174–185. Springer, Heidelberg (1991) 17. Kempe, J., Pyber, L., Shalev, A.: Permutation groups, minimal degrees and quantum computing. Groups, Geometry, and Dynamics 1(4), 553–584 (2007) 18. Kolmogorov, A.N.: Three approaches to the quantitative definition of information. Problems in Information Transmission 1, 1–7 (1965) 19. Kondacs, A., Watrous, J.: On the power of quantum finite state automata. In: Proc. IEEE FOCS 1997, pp. 66–75 (1997)
Abstract. Quantum finite automata with mixed states are proved to be super-exponentially more concise rather than quantum finite automata with pure states. It was proved earlier by A.Ambainis and R.Freivalds that quantum finite automata with pure states can have exponentially smaller number of states than deterministic finite automata recognizing the same language. There was a never published ”folk theorem” proving that quantum finite automata with mixed states are no more than superexponentially more concise than deterministic finite automata. It was not known whether the super-exponential advantage of quantum automata is really achievable. We use a novel proof technique based on Kolmogorov complexity to prove that there is an infinite sequence of distinct integers n such that there are languages Ln in a 4-letter alphabet such that there are quantum finite automata with mixed states with 2n + 1 states recognizing the language Ln with probability 34 while any deterministic finite automaton recognizing Ln needs to have at least eO(nlnn) states.
1
Introduction
A.Ambainis and R.Freivalds proved in [4] that for recognition of some languages the quantum finite automata can have smaller number of the states than deterministic ones, and this difference can even be exponential. The proof contained a slight non-constructiveness, and the exponent was not shown explicitly. For probabilistic finite automata exponentiality of such a distinction was not yet proved. The best (smaller) gap was proved by Ambainis [2]. The languages recognized by automata in [4] were presented explicitly but the exponent was not. In a very recent paper by R.Freivalds [13] the non-constructiveness is modified, and an explicit (and seemingly much better) exponent is obtained at the expense of having only non-constructive description of the languages used. Moreover, the best estimate proved in this paper is proved under assumption of the well-known Artin’s Conjecture (1927) in Number Theory. [13] contains also a theorem that does not depend on any open conjectures but the estimate is worse, and the description of the languages used is even less constructive. This seems to be the first result in finite automata depending on open conjectures in Number Theory.
This research is supported by Grant No.05.1528 from the Latvian Council of Science.
S.-H. Hong, H. Nagamochi, and T. Fukunaga (Eds.): ISAAC 2008, LNCS 5369, pp. 931–942, 2008. c Springer-Verlag Berlin Heidelberg 2008
932
R. Freivalds
The following two theorems are proved in [13]: Theorem 1. Assume Artin’s Conjecture. There exists an infinite sequence of regular languages L1 , L2 , L3 , . . . in a 2-letter alphabet and an infinite sequence of positive integers z(1), z(2), z(3), . . . such that for arbitrary j: 1. there is a probabilistic reversible automaton with (z(j) states recognizing Lj with the probability 19 36 , 2. any deterministic finite automaton recognizing Lj has at least (21/4 )z(j) = = (1.1892071115 . . .)z(j) states, Theorem 2. There exists an infinite sequence of regular languages L1 , L2 , L3 , . . . in a 2-letter alphabet and an infinite sequence of positive integers z(1), z(2), z(3), . . . such that for arbitrary j: 1. there is a probabilistic reversible automaton with z(j) states recognizing Lj 68 with the probability 135 , 1 2. any deterministic finite automaton recognizing Lj has at least (7 14 )z(j) = = (1.1149116725 . . .)z(j) states, The paper [13] concluded a long research on relative size of probabilistic and deterministic automata [8,9,10,11,16,2,15,13,14]. The two theorems above are formulated in [13] as assertions about reversible probabilistic automata. For probabilistic automata (reversible or not) it was unknown before the paper [13] whether the gap between the size of probabilistic and deterministic automata can be exponential. It is easy to re-write the proofs in order to prove counterparts of Theorems 1 and 2 for quantum finite automata with pure states. The aim of this paper is to prove a counterpart of these theorems for quantum finite automata with mixed states. Quantum algorithms with mixed states were first considered by D.Aharonov, A.Kitaev, N.Nisan [1]. More detailed description of quantum finite automata with mixed states can be found in A.Ambainis, et al.[3]. The automaton is defined by the initial density matrix ρ0 . Every symbol ai in the input alphabet is associated with a unitary matrix Ai . When the automaton reads the symbol ai the current density matrix ρ is transformed into A∗i ρAi . When the reading of the input word is finished and the end-marker $ is read, the current density matrix ρ is transformed into A∗end ρAend and separate measurements of all states are performed. After that the probabilities of all the accepting states are totaled, and the probabilities of all the rejecting states are totaled. Like quantum finite automata with pure states described by A.Kondacs and J.Watrous [19] we allow measurement of the accepting states and rejecting states after every step of the computation. The main result in our paper is: Theorem 3. There is an infinite sequence of distinct integers n such that there are languages Ln in a 4-letter alphabet such that there are quantum finite automata with mixed states with 2n + 1 states recognizing the language Ln with probability 34 while any deterministic finite automaton recognizing Ln needs to have at least eO(nlnn) states.
Super-Exponential Size Advantage of Quantum Finite Automata
933
Proof is delayed till Section 5. Since the numbers of the states for deterministic automata and quantum automata with pure states differ no more than exponentially, we have Theorem 4. There is an infinite sequence of distinct integers n such that there are languages Ln in a 4-letter alphabet such that there are quantum finite automata with mixed states with 2n + 1 states recognizing the language Ln with probability 34 while any quantum finite automaton with pure states recognizing Ln with bounded error needs to have at least eO(nlnn) states.
2
Permutations and Hamming Distance
Permutation of the set Nn is a 1-1 correspondence from Nn onto itself. Let f be such a permutation. The fact that it is onto means that for any k ∈ Nn there exists i ∈ Nn such that f (i) = k. We need a notion similar to Hamming codes for permutations. Since Hamming distance between permutations is already considered in several well-known textbooks (e.g. [6]), it seemed natural that the corresponding theory might be already published. Very far from truth! Hamming distance between two objects is the number of changes one needs to perform to obtain one object from the. The Hamming distance between two binary words (of the same length) is defined to be the number of positions at which they differ. For instance, we consider a set of three binary words {0011, 0110, 1100}. The first word is at Hamming distance 3 from the other two. Additionally, every word in this set is at Hamming distance at least 2 from any other. Such systems of words are called codes. They are important because they allow us to eliminate accidental errors when transmitting the words through noisy information channels. We consider Hamming distance between permutations. Hamming distance between the permutation s of the set {1, 2, 3, · · · n} and the permutation r of the same set is the number of distinct numbers i such that s(i) = r(i). For instance, let s be a permutation of the set {1, 2, 3, · · · n} and the number of it’s fixed points be p. Then the Hamming distance between the permutation s and the identity permutation is the number n − p. Lemma 1. Let d be an arbitrary real number such that 0 ≤ d ≤ 1. No more than 2dnlnn permutations can be on Hamming distance less or equal than dn from the identity permutation. Proof. By Stirling formula, n! = en.lnn−o(nlnn) . Let π be an arbitrary npermutation. How many there are distinct n-permutations differing from the permutation π in no more than dn positions? The differing positions can be chosen in n ≤ < 2n d ways and these ≤ dn positions are permuted. Hence there are no more than 2n .2dnlnn−o(nlnn) ≤ 2dnlnn permutations of this type.
934
R. Freivalds
Theorem 5. For arbitrary constant c < 1 such that for arbitrary n there is a set Gn of n-permutations containing eΩ(n log n) permutations such that the Hamming distance of any two permutations is at least c · n. Proof. Immediately from Lemma 1.
3
Permutations and Automata
Definition 1. The Hamming distance or simply distance d(r, s) between two n-permutations r and s on the set S is the number of elements x ∈ S such that r(x) = s(x). The similarity e(r, s) is the number of x ∈ S such that r(x) = s(x). Note that d(r, s) + e(r, s) = |S| = n. Theorem 6. Let c be a fixed constant and let there be an infinite sequence of distinct integers n such that for each n there exists a group Gn of permutations of the set {1, 2, . . . , n}, the group has order(Gn ) elements and k generating elements, and the Hamming distance of any two permutations is at least c · n. Then there is an infinite sequence of distinct integers n such that for each n there is a language Ln in a k-letter alphabet that can be recognized with probability 2c by a quantum finite automata with mixed states that has 2n states, while any deterministic finite automaton recognizing Ln must have at least order(Gn ) states. Proof. For each permutation group Gn we define the language Ln as follows: The letters of Ln are the k generators of the group Gn and it consists of words s1 s2 s3 . . . sm such that the product s1 ◦ s2 ◦ s3 ◦ · · · ◦ sm differs from the identity permutation. (A) Any deterministic automaton recognizing Ln is to remember the first input letter by a specific state. (B) We will construct a quantum automaton with mixed states. It has 4n states and the initial density matrix ρ0 is a diagonal block-matrix that consists of n blocks ρ0 : ⎛ ⎞ 1100 ⎟ 1 ⎜ ⎜1 1 0 0⎟ ρ0 = ⎝ 2n 0 0 0 0 ⎠ 0000 For each of k generators gi ∈ Gn we will construct the corresponding unitary matrix Ui as follows – it is a 2n × 2n permutation matrix, that permutes the elements in the even positions according to permutation gi , but leaves the odd positions unpermuted. For example, g = 3241 can be expressed as the following permutation matrix that acts on a column vector: ⎛ ⎞ 0010 ⎜0 1 0 0⎟ ⎟ g=⎜ ⎝0 0 0 1⎠ 1000
Super-Exponential Size Advantage of Quantum Finite Automata
935
The initial density matrix ρ0 for n = 4 and the unitary matrix U that corresponds to the permutation matrix (3) of permutation g are as follows: ⎛ ⎞ ⎛ ⎞ 11000000 10000000 ⎜1 1 0 0 0 0 0 0⎟ ⎜0 0 0 0 0 1 0 0⎟ ⎜ ⎟ ⎜ ⎟ ⎜0 0 1 1 0 0 0 0⎟ ⎜0 0 1 0 0 0 0 0⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 1 ⎜0 0 1 1 0 0 0 0⎟ ⎟,U = ⎜0 0 0 1 0 0 0 0⎟ ρ0 = ⎜ ⎜ ⎟ ⎜ ⎟ 8 ⎜0 0 0 0 1 1 0 0⎟ ⎜0 0 0 0 1 0 0 0⎟ ⎜0 0 0 0 1 1 0 0⎟ ⎜0 0 0 0 0 0 0 1⎟ ⎜ ⎟ ⎜ ⎟ ⎝0 0 0 0 0 0 1 1⎠ ⎝0 0 0 0 0 0 1 0⎠ 00000011 01000000 The unitary matrix U$ for the end-marker is also a diagonal block-matrix. It consists of n blocks that are the Hadamard matrices 1 1 = √1 H 2 1 −1 acts on two specific 2 × 2 density matrices: Notice how the Hadamard matrix H 1 20 1 11 † if ρ = , then HρH = , 2n 1 1 2n 0 0 1 10 H † = 1 1 0 . if ρ = , then Hρ 2n 0 1 2n 0 1 For example, when the letter g is read, the unitary matrix U is applied to the density matrix ρ0 (both are given in equation (3)) and the density matrix ρ1 = U ρ0 U † is obtained. When the end-marker “$” is read, the density matrix becomes ρ$ = U$ ρ1 U$† . Matrices ρ1 and ρ$ are as follows: ⎛ ⎛ ⎞ ⎞ 1 0 0 0 12 12 12 − 21 10000001 ⎜0 1 0 0 1 0 0 0⎟ ⎜ 0 1 0 0 −1 −1 1 −1 ⎟ 2 2 2 2⎟ ⎜ ⎜ ⎟ ⎜0 0 1 1 0 0 0 0⎟ ⎜ 0 0 20 0 0 0 0 ⎟ ⎜ ⎜ ⎟ ⎟ 1⎜ 1⎜ 0 0 00 0 0 0 0 ⎟ 0 0 1 1 0 0 0 0⎟ ⎜ ⎜ ⎟. ⎟ ρ1 = ⎜ , ρ$ = ⎜ 1 8 ⎜0 1 0 0 1 0 0 0⎟ 8 ⎜ 2 − 21 0 0 1 0 12 12 ⎟ ⎟ ⎟ ⎜0 0 0 0 0 1 1 0⎟ ⎜ 1 −1 0 0 0 1 −1 −1 ⎟ 2 2⎟ ⎜ ⎜ 21 12 ⎟ 1 1 ⎝0 0 0 0 0 1 1 0⎠ ⎝ 0 ⎠ 2 2 0 0 2 −2 1 1 1 1 10000001 − 2 − 2 0 0 2 − 21 0 1 Finally, we declare the states in the even positions to be accepting, but the states in the odd positions to be rejecting. Therefore one must sum up the diagonal entries that are in the even positions of the final density matrix to find the probability that a given word is accepted. In our example the final density matrix ρ$ is given in (3). It corresponds to the input word “g$”, which is accepted with probability 18 (1 + 0 + 1 + 1) = 38 and rejected with probability 18 (1 + 2 + 1 + 1) = 58 . Note that the accepting and rejecting probabilities sum up to 1.
936
R. Freivalds
It is easy to see, that the words that do not belong to the language Ln are rejected with certainty, because the matrix U$ ρ0 U$† has all zeros in the even positions on the main diagonal. However, the words that belong to Ln are accepted d cn = 2n = 2c , because all permutations are at least with the probability at least 2n at the distance d from the identity permutation. It is also easy to see that any deterministic automaton that recognizes the language Ln must have at least order(Gn ) states. If the number of states is less than order(Gn ), then there are two distinct words u and v such that the deterministic automaton ends up in the same state no matter which one of the two words it reads. Since Gn is a group, for each word we can find an inverse, that returns the automaton in the initial state (the only rejecting state). Since u and v are different, they have different inverses and u ◦ u−1 is the identity permutation and must be rejected, but v ◦ u−1 is not the identity permutation and must be accepted – a contradiction.
4
Super-Exponential Size Advantage
Now we wish to prove a theorem differing from Theorem 3 only in one way. In this section we allow the alphabets of the languages Ln to grow with n. Consider the following infinite sequence of languages. For every n take the set Gn considered in Theorem 5. The language Ln consists of all the words aa (of the length 2) where a is a symbol for an arbitrary element from Gn . Hence there are eΩ(n log n) letters in the alphabet of the language Ln and equally many words in Ln . Theorem 7. There is an infinite sequence of distinct integers n such that there are languages Ln such that there are quantum finite automata with mixed states with 2n + 1 states recognizing the language Ln with probability 34 while any deterministic finite automaton recognizing Ln needs to have at least eO(nlnn) states. Proof is similar to the proof of Theorem 6. It would be nice to improve our theorem in the natural way by finding algebraic groups Gn of n-permutations such that they are generated by a small (constant) number of generating elements and at the same time the order of Gn (the number of elements) would be superexponentially large with the respect to n. This would give us a natural proof for Theorem 3. Unfortunately, there exists a paper [17] by J.Kempe et al. where it is proven that large Hamming distance between all the n-permutations in a group implies that the order of the group is no more than exponential with the respect to n.
5
4-Letter Input Alphabet
We consider 2 algebraic groups in this Section. The set of all permutations of the set 1, ..., n with algebraic operation ”product of permutations” can be considered as the group G1 . This group has two generating elements, the permutations
Super-Exponential Size Advantage of Quantum Finite Automata
937
(123 · · · n) and (12)(3)(4) · · · (n). The other group G2 is defined as follows. Two binary strings α1 and α2 , each one of the length nlog2 n are taken and two realvalued unitary matrices are constructed. The corresponding unitary operators are rotations around axes depending, correspondingly, on α1 and α2 . The angle 3 of rotation is a multiple of 2π p where p is a prime slightly exceeding n . The inverses of these operators again are unitary, and their matrices are real-valued. The group direct product G1 × G2 is again a group G with 4 generating elements and the elements of this group can be represented by n-dimensional unitary operators. G has the following properties. Property 1. For arbitrary fixed pair (α1 , α2 ) distinct n-permutations generate distinct unitary operators. Property 2. For arbitrary fixed n-permutation distinct pairs (α1 , α2 ) generate distinct unitary operators. Property 3. The n-permutation and the pair (α1 , α2 ) is transformed into the unitary operator described above by an algorithmic process. Property 4. The n-permutation and the unitary operator can be transformed into the pair (α1 , α2 ) by an algorithmic process. These properties will be used below to prove the main result, to prove Theorem 3. At first we construct a quantum finite automaton with mixed states having 2n states. This automaton has the following initial density matrix (we use an 8 × 8-matrix instead of formal description of n × n-matrix for the sake of easier readability). ⎞ ⎛1 1 8 0 0 0 8 0 0 0 1 1 ⎜0 0 0 0 0 0⎟ 8 ⎟ ⎜ 8 1 ⎜0 0 0 0 0 1 0⎟ 8 8 ⎟ ⎜ ⎜0 0 0 1 0 0 0 1⎟ 8 8 ⎟ ⎜ ρ0 = ⎜ 1 1 ⎟ ⎜ 8 01 0 0 8 01 0 0 ⎟ ⎜0 0 0 0 0 0⎟ 8 ⎟ ⎜ 8 1 ⎝0 0 0 0 0 1 0⎠ 8 8 0 0 0 18 0 0 0 18 At every step an input letter ai (one of four possible input letters a1 , a2 , a3 or a4 ) is read from the input. Hence the density matrix ρx is changed into Ui∗ ρx Ui . ∗ · · · U2∗ U1∗ ρ0 U1 U2 · · · Hence if x = x1 x2 x3 · · · xt then the current Ux equals Ut∗ Ut−1 Ut−1 Ut . Let the matrix ⎞ ⎛ u11 u12 · · · u1n ⎜ u21 u22 · · · u2n ⎟ ⎟ ⎜ ⎟ Ux = ⎜ ⎜ u31 u32 · · · u3n ⎟ ⎠ ⎝ ··· un1 un2 · · · unn be the n × n-matrix corresponding to the input word x. We associate a 2n × 2nmatrix Mx
938
R. Freivalds
⎛
1 0 ··· ⎜ 0 1 ··· ⎜ ⎜ 0 0 ··· ⎜ ⎜··· ⎜ ⎜ 0 0 ··· Mx = ⎜ ⎜ 0 0 ··· ⎜ ⎜ 0 0 ··· ⎜ ⎜ 0 0 ··· ⎜ ⎝··· 0 0 ···
0 0 0
··· ··· ···
0 u12 u22 u32
··· ··· ··· ···
0 0 0 0 0 0 1 0 0 0
0 u11 u21 u31
⎞
0 0 0
0 un1 un2 u · · ·
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ 0 ⎟ ⎟ u1n ⎟ ⎟ u2n ⎟ ⎟ u3n ⎟ ⎟ ⎠ unn
to it. We use the initial density matrix and the following property ⎛
1000 0 ⎜0 1 0 0 0 ⎜ ⎜0 0 1 0 0 ⎜ ⎜0 0 0 1 0 ⎜ ⎜ 0 0 0 0 u11 ⎜ ⎜ 0 0 0 0 u21 ⎜ ⎝ 0 0 0 0 u31 0 0 0 0 u41
0 0 0 0 u12 u22 u32 u42
0 0 0 0 u13 u23 u33 u43
⎞⎛ 1 0 8 ⎜ 0 ⎟ ⎟⎜0 ⎜ 0 ⎟ ⎟⎜0 ⎜ 0 ⎟ ⎟ ⎜ 01 ⎜ u14 ⎟ ⎟⎜ 8 ⎜ u24 ⎟ ⎟⎜0 u34 ⎠ ⎝ 0 u44 0 ⎛
1 8
0
0 0 1 8 0 0 18 0 0 0 0 1 8 0 0 18 0 0 0 0
⎜ 0 1 8 ⎜ ⎜ 0 0 1 8 ⎜ ⎜ 0 0 0 =⎜ ⎜ u11 u12 u13 ⎜ u8 u8 u8 ⎜ 21 22 23 ⎜ u8 u8 u8 ⎝ 31 32 33 8 8 8 u41 u42 u43 8 8 8
⎞⎛ 0 0 1000 0 ⎜0 1 0 0 0 0 0⎟ ⎟⎜ 1 ⎟⎜ 8 0 ⎟⎜0 0 1 0 0 ⎜ 0 18 ⎟ ⎟⎜0 0 0 1 0 ⎜ 0 0⎟ ⎟ ⎜ 0 0 0 0 u11 ⎜ 0 0⎟ ⎟ ⎜ 0 0 0 0 u12 1 ⎠ ⎝ 0 0 0 0 u13 0 8 0 0 0 0 u14 0 18
0 18 0 0 0 18 0 0 0 1 8 0 0 0 18 0 0 0 18 0 0 0 1 8 0 0 0 0 0 1 8 u14 8 u24 8 u34 8 u44 8
u11 8 u12 8 u13 8 u14 8 1 8
0 0 0
u21 8 u22 8 u23 8 u24 8
u31 8 u32 8 u33 8 u34 8
0 1 8
0 0
0 0
0
u41 8 u42 8 u43 8 u44 8
0 0 0 0 u21 u22 u23 u24
0 0 0 0 u31 u32 u33 u34
⎞ 0 0 ⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟= u41 ⎟ ⎟ u42 ⎟ ⎟ u43 ⎠ u44
⎞
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ 0 ⎠
1 8
1 8
to justify the correspondence between the n × n-matrices Ux and the 2n × 2nmatrices Mx at every moment. When all the input word is read the end-marker $ comes in, the unitary matrix Mx corresponding to the end-marker $ is ⎛
√1 2
⎜ 0 ⎜ ⎜ 0 ⎜ ⎜ ⎜ 0 M$ = ⎜ ⎜ √1 ⎜ 2 ⎜ 0 ⎜ ⎜ ⎝ 0 0
0 √1 2
0 0 0
0 0 √1 2
0 0 0
0 0 0 √1 2
√1 2
0 0 0
0 √1 2
0 0 0
0 0 √1 2
0 0 − √12 0 √1 0 0 − √12 0 2 1 0 √2 0 0 0 − √12 0 0 0 0 √12 0
0 0 0
⎞
⎟ ⎟ ⎟ ⎟ ⎟ √1 ⎟ 2 ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ ⎟ 0 ⎠ − √12
Super-Exponential Size Advantage of Quantum Finite Automata
939
This matrix can be regarded as a block matrix with blocks of size 2 × 2. It is worth to remember that
√1 2 √1 2
√1 2 − √12
a11 a12 a21 a22
√1
√1 2 2 1 √ − √1 2 2
a11 +a21 +a12 +a22 =
a11 +a21 −a12 −a22 2 2 a11 −a21 +a12 −a22 a11 −a21 −a12 +a22 2 2
.
If the input word is such that the corresponding member of the free group G1 equals the empty word then the final density matrix becomes ⎛1 4
ρf inal
⎜0 ⎜ ⎜0 ⎜ ⎜0 =⎜ ⎜0 ⎜ ⎜0 ⎜ ⎝0 0
0 0 1 4 0 0 14 0 0 0 0 0 0 0 0 0 0
⎞ 00000 0 0 0 0 0⎟ ⎟ 0 0 0 0 0⎟ ⎟ 1 ⎟ 4 0 0 0 0⎟. 0 0 0 0 0⎟ ⎟ 0 0 0 0 0⎟ ⎟ 0 0 0 0 0⎠ 00000
Otherwise the total of the first n elements on the main diagonal is less than 1. At the end separate measurements of all states are performed. After that the probabilities of all the accepting states are totaled, and the probabilities of all the rejecting states are totaled. What is the probability of the acceptance in the case when the member of the free group G1 does not equal the empty word? It depends on the values of (α1 , α2 ). We will prove in the subsequent sections that if the Kolmogorov complexity of (α1 , α2 ) is maximal then this probability is MUCH less than 1.
6
Kolmogorov Complexity
The theorems in this section are well-known results in spite of the fact that it is not easy to find exact references for all of them. Definition 2. We say that the numbering Ψ = {Ψ0 (x), Ψ1 (x), Ψ2 (x), . . .} of 1argument partial recursive functions is computable if the 2-argument function U (n, x) = Ψn (x) is partial recursive. Definition 3. We say that a numbering Ψ is reducible to the numbering η if there exists a total recursive function f (n) such that, for all n and x, Ψn (x) = ηf (n) (x). Definition 4. We say that a computable numbering ϕ of all 1-argument partial recursive functions is a G¨ odel numbering if every computable numbering (of any class of 1-argument partial recursive functions) is reducible to ϕ. Theorem. [7] There exists a G¨ odel numbering.
940
R. Freivalds
Definition 5. We say that a G¨ odel numbering ϑ is a Kolmogorov numbering if for arbitrary computable numbering Ψ (of any class of 1-argument partial recursive functions) there exist constants c > 0, d > 0, and a total recursive function f (n) such that: 1. for all n and x, Ψn (x) = ϑf (n) (x), 2. for all n, f (n) ≤ c · n + d. Kolmogorov Theorem. [18] There exists a Kolmogorov numbering.
7
Back to Automata
We denote by U (τ, α) the pair of unitary matrices generated from the npermutation τ and the binary string α of the length 2|τ |log2 |τ | (later divided into the pair (α1 , α2 )) each one of the length |τ |log2 |τ | . There exist many distinct Kolmogorov numberings. We now fix one of them and denote it by η. Since Kolmogorov numberings give indices for all partial recursive functions, for arbitrary τ and n there is an i such that ηi (τ, n) = x. Let i(τ, n) be the minimal i such that ηi (τ, n) = α. It is easy to see that if x1 = x2 , then i(α1 , n) = i(α2 , n). We consider all binary words α corresponding to n and denote by α(n) the word α such i(α, n) exceeds i(β, n) for all binary words β corresponding to n different from α. It is obvious that i ≥ 22n(log2 n)−1 . Until now we considered generating matrices U (τ, α) for independently chosen τ and α. From now on we consider only α corresponding to i(τ, n). We wish to prove that if n is sufficiently large, then Hamming distances between the vectors 1 1 1 1 1 n , n , n , · · · , n and a11 , a22 , · · · , ann in the matrix ρf inal exceeds 18 . We introduce a partial recursive function μ(z, n) defined as follows. Above when defining U (τ, α) we considered auxiliary function U (τ, α). To define μ(z, n) we consider all binary words α of the length 2|τ |log2 |τ |. If z is not a binary word of such a length, then μ(z, n) is not defined. If z is a binary word of length 2|τ |log2 |τ |, then we consider all α ∈ {0, 1}2|τ |log2|τ | such that U (τ, α) = z. If there are no such α, then μ(z, n) is not defined. If there is only one such α, then μ(z, n) = α. If there are more than two such x, then μ(z, n) is not defined. Now we introduce a computable numbering of some partial recursive functions. This numbering is independent of n. For each n (independently from other values of n) we order the set of all the 22|τ |log2 |τ | binary words z of the length 2|τ |log2 |τ | : z0 , z1 , z2 , . . . , z22|τ |log2 |τ | −1 . We define z0 as the word 000 . . . 0. The words z1 , z2 , . . . , z22p are words with exactly one symbol 1. We strictly follow a rule ”if the word zi contains less symbols 1 than the word zj , then i < j”. Words with equal number of the symbol 1 are ordered lexicographically. Hence z22p −1 = 111 . . . 1.
Super-Exponential Size Advantage of Quantum Finite Automata
941
For each n, we define w = |τ |log2 |τ | and Ψ0 (n) = μ(z0 , n) Ψ1 (n) = μ(z0 , n) Ψ2 (n) = μ(z1 , n) ... Ψ22w+1 −2 (p) = μ(z22w −1 , n) Ψ22w+1 −1 (p) = μ(z22w −1 , n) For j ≥ 22w+1 , Ψj (w) is undefined. We have fixed a Kolmogorov numbering η and we have just constructed a computable numbering Ψ of some partial recursive functions. Lemma 2. There exist constants c > 0 and d > 0 (independent of p) such that for arbitrary i there is a j such that 1. Ψi (t) = ηj (t) for all t, and 2. j ≤ ci + d. Proof. Immediately from Kolmogorov Theorem. The rest of the proof of Theorem 3 is from the contrary. Suppose that n is sufficiently large (because Kolmogorov numbering is optimal only for large lengths of the words) and for α(n) the value of a11 + a22 + · · · + ann had exceeded 17 18 in the unitary matrix U (τ, α). Then our α could be uniquely and algorithmically be described in more concise way than our obvious assertion i(τ ) ≥ 2nlog2n − 1 allows. Contradiction. This way we have proved that the languages Ln can be recognized by 2n-state quantum finite automata with mixed states so that every word in the language is accepted with the probability 1 and every word not in Ln is accepted with probability less than 17 18 . If we add one more state to the automata we can get automata with a cut-point 12 and a uniformly bounded error. It remains only to find the complexity (the number of states) of deterministic finite automata recognizing these languages. Theorem 5 above shows that there are eO(nlnn) n-permutations τj with pairwise Hamming distances no less than const.n. These n-permutations can be constructed using no more than n3 generating elements (123 · · · n) and (12)(3)(4) · · · (n). Input words corresponding to these eO(nlnn) n-permutations τj should be remembered by distinct states of any deterministic finite automaton recognizing the language.
Acknowledgments I would like to thank Andris Ambainis for pointing at a possibility to use random unitary operators instead of permutations. In the final version of my proof the unitary operators are far from being random but this idea helped to move my proof off a deadlock.
942
R. Freivalds
References 1. Aharonov, D., Kitaev, A., Nisan, N.: Quantum circuits with mixed states. In: Proc. STOC 1998, pp. 20–30 (1998) 2. Ambainis, A.: The complexity of probabilistic versus deterministic finite automata. LNCS, vol. 1178, pp. 233–237. Springer, Heidelberg (1996) 3. Ambainis, A., Beaudry, M., Golovkins, M., K ¸ ikusts, A., Mercer, M., Th´erien, D.: Algebraic Results on Quantum Automata. Theory Comput. Syst. 39(1), 165–188 (2006) 4. Ambainis, A., Freivalds, R.: 1-way quantum finite automata: strengths, weaknesses and generalizations. In: Proc. IEEE FOCS 1998, pp. 332–341 (1998) 5. Artin, E.: Beweis des allgemeinen Reziprozit¨ atsgesetzes. Mat. Sem. Univ. Hamburg B.5, 353–363 (1927) 6. Cameron, P.: Permutation groups. London Mathematical Society Student Texts series. Cambridge University Press, Cambridge (1999) 7. Ershov, Y.L.: Theory of numberings. In: Griffor, E.R. (ed.) Handbook of computability theory, pp. 473–503. North-Holland, Amsterdam (1999) 8. Freivalds, R.: Recognition of languages with high probability on different classes of automata. Doklady Akademii Nauk SSSR 239(1), 60–62 (1978) (Russian) 9. Freivalds, R.: Projections of languages recognizable by probabilistic and alternating finite multi-tape automata. Information Processing Letters 13(4–5), 195–198 (1981) 10. Freivalds, R.: On the growth of the number of states in result of the determinization of probabilistic finite automata. Avtomatika i Vichislitel’naya Tekhnika 3, 39–42 (1982) (Russian) 11. Freivalds, R.: Complexity of probabilistic versus deterministic automata. In: Barzdins, J., Bjorner, D. (eds.) Baltic Computer Science. LNCS, vol. 502, pp. 565–613. Springer, Heidelberg (1991) 12. Freivalds, R.: Languages Recognizable by Quantum Finite Automata. In: Farr´e, J., Litovsky, I., Schmitz, S. (eds.) CIAA 2005. LNCS, vol. 3845, pp. 1–14. Springer, Heidelberg (2006) 13. Freivalds, R.: Non-constructive methods for finite probabilistic automata. In: Harju, T., Karhum¨ aki, J., Lepist¨ o, A. (eds.) DLT 2007. LNCS, vol. 4588, pp. 169– 180. Springer, Heidelberg (2007) 14. Freivalds, R.: Hamming, Permutations and Automata. In: Hromkoviˇc, J., Kr´ aloviˇc, R., Nunkesser, M., Widmayer, P. (eds.) SAGA 2007. LNCS, vol. 4665, pp. 18–29. Springer, Heidelberg (2007) 15. Freivalds, R., Ozols, M., Manˇcinska, L.: Permutation Groups and the Strength of Quantum Finite Automata with Mixed States. In: Proceedings of the Workshop Probabilistic and Quantum Algorithms colocated with the 11th International Conference Developments in Language Theory DLT 2007, Turku, Finland, July 3-5, 2007, vol. 45, pp. 13–27. TUCS General Publication, No 45 (June 2007) 16. Ka¸ neps, J., Freivalds, R.: Running time to recognize nonregular languages by 2way probabilistic automata. In: Leach Albert, J., Monien, B., Rodr´ıguez-Artalejo, M. (eds.) ICALP 1991. LNCS, vol. 510, pp. 174–185. Springer, Heidelberg (1991) 17. Kempe, J., Pyber, L., Shalev, A.: Permutation groups, minimal degrees and quantum computing. Groups, Geometry, and Dynamics 1(4), 553–584 (2007) 18. Kolmogorov, A.N.: Three approaches to the quantitative definition of information. Problems in Information Transmission 1, 1–7 (1965) 19. Kondacs, A., Watrous, J.: On the power of quantum finite state automata. In: Proc. IEEE FOCS 1997, pp. 66–75 (1997)
