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Optimal lower bounds for quantum automata and random access codes
"... Consider the finite regular ¢¤£¦¥¨§�©�����©�� language ©������� �. In [3] it was shown that while this language is accepted by a deterministic finite automaton of ������ � size, any oneway quantum finite automaton (QFA) for it has ���¤ � £��� � ����£� � size. This was based on the fact that the e ..."
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Cited by 120 (9 self)
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Consider the finite regular ¢¤£¦¥¨§�©�����©�� language ©������� �. In [3] it was shown that while this language is accepted by a deterministic finite automaton of ������ � size, any oneway quantum finite automaton (QFA) for it has ���¤ � £��� � ����£� � size. This was based on the fact that the evolution of a QFA is required to be reversible. When arbitrary intermediate measurements are allowed, this intuition breaks down. Nonetheless, we show ���� � £�� a lower bound for such QFA ¢ £ for, thus also improving the previous bound. The improved bound is obtained from simple entropy arguments based on Holevo’s theorem [8]. This method also allows us to obtain an asymptotically op���������������� � timal bound for the dense quantum codes (random access codes) introduced in [3]. We then turn to Holevo’s theorem, and show that in typical situations, it may be replaced by a tighter and more transparent inprobability bound.
Twoway finite automata with quantum and classical states
"... We introduce 2way finite automata with quantum and classical states (2qcfa's). This is a variant on the 2way quantum finite automata (2qfa) model which may be simpler to implement than unrestricted 2qfa's; the internal state of a 2qcfa may include a quantum part that may be in a (mixed ..."
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Cited by 60 (0 self)
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We introduce 2way finite automata with quantum and classical states (2qcfa's). This is a variant on the 2way quantum finite automata (2qfa) model which may be simpler to implement than unrestricted 2qfa's; the internal state of a 2qcfa may include a quantum part that may be in a (mixed) quantum state, but the tape head position is required to be classical. We show two languages for which 2qcfa's are better than classical 2way automata. First, 2qcfa's can recognize palindromes, a language that cannot be recognized by 2way deterministic or probabilistic finite automata. Second, in polynomial time 2qcfa's can recognize fa n b n
Characterizations of 1Way Quantum Finite Automata
 SIAM Journal on Computing
"... The 2way quantum finite automaton introduced by Kondacs and Watrous[KW97] can accept nonregular languages with bounded error in polynomial time. If we restrict the head of the automaton to moving classically and to moving only in one direction, the acceptance power of this 1way quantum finite aut ..."
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Cited by 51 (0 self)
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The 2way quantum finite automaton introduced by Kondacs and Watrous[KW97] can accept nonregular languages with bounded error in polynomial time. If we restrict the head of the automaton to moving classically and to moving only in one direction, the acceptance power of this 1way quantum finite automaton is reduced to a proper subset of the regular languages. In this paper we study two different models of 1way quantum finite automata. The first model, termed measureonce quantum finite automata, was introduced by Moore and Crutchfield[MCar], and the second model, termed measuremany quantum finite automata, was introduced by Kondacs and Watrous[KW97]. We characterize the measureonce model when it is restricted to accepting with bounded error and show that, without that restriction, it can solve the word problem over the free group. We also show that it can be simulated by a probabilistic finite automaton and describe an algorithm that determines if two measureonce automata are equiv...
Dense Quantum Coding and Quantum Finite Automata
, 2002
"... We consider the possibility of encoding m classical bits into many fewer n quantum bits (qubits) so that an arbitrary bit from the original m bits can be recovered with good probability. We show that nontrivial quantum codes exist that have no classical counterparts. On the other hand, we show that ..."
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Cited by 37 (9 self)
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We consider the possibility of encoding m classical bits into many fewer n quantum bits (qubits) so that an arbitrary bit from the original m bits can be recovered with good probability. We show that nontrivial quantum codes exist that have no classical counterparts. On the other hand, we show that quantum encoding cannot save more than a logarithmic additive factor over the best classical encoding. The proof is based on an entropy coalescence principle that is obtained by viewing Holevo's theorem from a new perspective. In the
Dense quantum coding and a lower bound for 1way quantum automata
 Proceedings of the ThirtyFirst Annual ACM Symposium on the Theory of Computing
, 1999
"... We consider the possibility of encoding m classical bits into much fewer n quantum bits so that an arbitrary bit from the original m bits can be recovered with a good probability, and we show that nontrivial quantum encodings exist that have no classical counterparts. On the other hand, we show that ..."
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Cited by 37 (6 self)
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We consider the possibility of encoding m classical bits into much fewer n quantum bits so that an arbitrary bit from the original m bits can be recovered with a good probability, and we show that nontrivial quantum encodings exist that have no classical counterparts. On the other hand, we show that quantum encodings cannot be much more succint as compared to classical encodings, and we provide a lower bound on such quantum encodings. Finally, using this lower bound, we prove an exponential lower bound on the size of 1way quantum finite automata for a family of languages accepted by linear sized deterministic finite automata. 1
SpaceBounded Quantum Complexity
 Journal of Computer and System Sciences
, 1999
"... This paper investigates the computational power of spacebounded quantum Turing machines. The following facts are proved for spaceconstructible space bounds s satisfying s(n) = Ω(log n). 1. Any quantum Turing machine (QTM) running in space s can be simulated by an unbounded error probabilistic Tur ..."
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Cited by 27 (5 self)
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This paper investigates the computational power of spacebounded quantum Turing machines. The following facts are proved for spaceconstructible space bounds s satisfying s(n) = Ω(log n). 1. Any quantum Turing machine (QTM) running in space s can be simulated by an unbounded error probabilistic Turing machine (PTM) running in space O(s). No assumptions on the probability of error or running time for the QTM are required, although it is assumed that all transition amplitudes of the QTM are rational. 2. Any PTM that runs in space s and halts absolutely (i.e., has finite worstcase running time) can be simulated by a QTM running in space O(s). If the PTM operates with bounded error, then the QTM may be taken to operate with bounded error as well, although the QTM may not halt absolutely in this case. In the case of unbounded error, the QTM may be taken to halt absolutely. We therefore have that unbounded error, space O(s) bounded quantum Turing machines and probabilistic Turing machines are equivalent in power, and furthermore that any QTM running in space s can be simulated deterministically in NC 2 (2 s) ⊆ DSPACE(s 2) ∩ DTIME ( 2 O(s)). We also consider quantum analogues of nondeterministic and onesided error probabilistic spacebounded classes, and prove some simple facts regarding these classes. 1 1
On the hardness of distinguishing mixedstate quantum computations
, 2004
"... This paper considers the following problem. Two mixedstate quantum circuits Q0 and Q1 are given, and the goal is to determine which of two possibilities holds: (i) Q0 and Q1 act nearly identically on all possible quantum state inputs, or (ii) there exists some input state ρ that Q0 and Q1 transform ..."
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Cited by 24 (10 self)
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This paper considers the following problem. Two mixedstate quantum circuits Q0 and Q1 are given, and the goal is to determine which of two possibilities holds: (i) Q0 and Q1 act nearly identically on all possible quantum state inputs, or (ii) there exists some input state ρ that Q0 and Q1 transform into almost perfectly distinguishable outputs. This may be viewed as an abstraction of the problem that asks, given two discrete quantum mechanical processes described by sequences of local interactions, are the processes effectively the same or are they different? We prove that this promise problem is complete for the class QIP of problems having quantum interactive proof systems, and is therefore PSPACEhard. This is in contrast to the fact that the analogous problem for classical (probabilistic) circuits is in AM, and for unitary quantum circuits is in QMA.
Decidable and Undecidable Problems about Quantum Automata
 SIAM Journal on Computing
, 2005
"... We study the following decision problem: is the language recognized by a quantum finite automaton empty or nonempty? We prove that this problem is decidable or undecidable depending on whether recognition is defined by strict or nonstrict thresholds. This result is in contrast with the correspondi ..."
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Cited by 23 (0 self)
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We study the following decision problem: is the language recognized by a quantum finite automaton empty or nonempty? We prove that this problem is decidable or undecidable depending on whether recognition is defined by strict or nonstrict thresholds. This result is in contrast with the corresponding situation for probabilisticfinite automata for which it is known that strict and nonstrict thresholds both lead to undecidable problems.
Quantum communication complexity
 In Proc. Intl. Colloquium on Automata, Languages, and Programming (ICALP
, 2000
"... This paper surveys the field of quantum communication complexity. Some interesting recent results are collected concerning relations to classical communication, lower bound methods, oneway communication, and applications of quantum communication complexity. 1 ..."
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Cited by 19 (3 self)
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This paper surveys the field of quantum communication complexity. Some interesting recent results are collected concerning relations to classical communication, lower bound methods, oneway communication, and applications of quantum communication complexity. 1