Nondeterministic Quantum Recognition
Updated 21 January 2026
Nondeterministic quantum recognition is a paradigm for quantum automata where acceptance requires a nonzero probability path while non-members have zero acceptance probability.
It employs models such as quantum finite, pushdown, and affine automata to recognize exclusive stochastic languages, surpassing classical nondeterministic and probabilistic systems.
Its framework supports efficient pairwise word separation and complex language class recognition, highlighting advantages in state complexity and error modes.
Nondeterministic quantum recognition refers to the acceptance paradigm for quantum automata and related computational models in which a word or input is accepted if, and only if, there exists at least one accepting computational path with nonzero probability amplitude, while all non-members are accepted with probability zero. This zero-error, or “cutpoint 0,” convention constitutes a quantum analog of classical one-sided or nondeterministic acceptance and yields substantial differences in language recognition power and automata-theoretic properties compared to both classical nondeterministic and probabilistic computation. Quantum models exhibit strict superiority over their classical counterparts under this regime, both in the finite and stack-based contexts.
1. Models of Nondeterministic Quantum Recognition 1.1 Nondeterministic Quantum Finite Automata (NQFA)
The nondeterministic quantum finite automaton is most commonly formalized as a one-way Kondacs–Watrous quantum finite automaton (KWQFA), defined as a tuple
$(Q, \Sigma, \{U_\sigma\}_{\sigma\in\Sigma\cup\{\lrcorner, \$\}}, Q_{acc}, Q_{rej})< / p > < p > w h e r e </p>
<p>where < / p >< p > w h ere i s a f i n i t e s t a t e s e t ( w i t h i n i t i a l s t a t e is a finite state set (with initial state i s a f ini t es t a t ese t ( w i t hini t ia l s t a t e ) , ), ) , a r e are a re u n i t a r y m a t r i c e s a s s o c i a t e d w i t h t a p e s y m b o l s , a n d unitary matrices associated with tape symbols, and u ni t a ry ma t r i ces a ssoc ia t e d w i t h t a p esy mb o l s , an d , , , a r e d i s j o i n t a c c e p t i n g a n d r e j e c t i n g s t a t e s u b s e t s . C o m p u t a t i o n p r o g r e s s e s t h r o u g h u n i t a r y e v o l u t i o n , i n t e r l e a v e d w i t h m e a s u r e m e n t s p r o j e c t i n g o n t o h a l t i n g ( a c c e p t / r e j e c t ) o r n o n − h a l t i n g s u b s p a c e s ; f u r t h e r c o m p u t a t i o n c o n t i n u e s f r o m t h e p r o j e c t e d s t a t e . T h e a c c e p t a n c e p r o b a b i l i t y are disjoint accepting and rejecting state subsets. Computation progresses through unitary evolution, interleaved with measurements projecting onto halting (accept/reject) or non-halting subspaces; further computation continues from the projected state. The acceptance probability a re d i s j o in t a cce pt in g an d re j ec t in g s t a t es u b se t s . C o m p u t a t i o n p ro g resses t h ro ug h u ni t a rye v o l u t i o n , in t er l e a v e d w i t hm e a s u re m e n t s p ro j ec t in g o n t o ha lt in g ( a cce pt / re j ec t ) or n o n − ha lt in g s u b s p a ces ; f u r t h erco m p u t a t i o n co n t in u es f ro m t h e p ro j ec t e d s t a t e . T h e a cce pt an ce p ro babi l i t y i s t h e t o t a l p r o b a b i l i t y ( s q u a r e d a m p l i t u d e ) o f r e a c h i n g a n y a c c e p t i n g s t a t e a f t e r f u l l p r o c e s s i n g o f is the total probability (squared amplitude) of reaching any accepting state after full processing of i s t h e t o t a lp ro babi l i t y ( s q u a re d am pl i t u d e ) o f re a c hin g an y a cce pt in g s t a t e a f t er f u llp rocess in g o f a n d m a r k e r s . < / p > < p > A K W Q F A i s n o n d e t e r m i n i s t i c i f i t r e c o g n i z e s a l a n g u a g e and markers.</p>
<p>A KWQFA is nondeterministic if it recognizes a language an d ma r k ers . < / p >< p > A K W QF A i s n o n d e t er mini s t i c i f i t reco g ni zes a l an gu a g e a c c o r d i n g t o t h e r u l e : < / p > < p > according to the rule:</p>
<p> a ccor d in g t o t h er u l e :< / p >< p > < / p > < p > w i t h </p>
<p>with < / p >< p > w i t h f o r a l l for all f or a ll . T h i s i s k n o w n a s “ c u t p o i n t 0 ” , o r p o s i t i v e o n e − s i d e d e r r o r ( < a h r e f = " / p a p e r s / 0902.2081 " t i t l e = " " r e l = " n o f o l l o w " d a t a − t u r b o = " f a l s e " c l a s s = " a s s i s t a n t − l i n k " x − d a t a x − t o o l t i p . r a w = " " > 0902.2081 < / a > ) . < / p > < h 3 c l a s s = ′ p a p e r − h e a d i n g ′ i d = ′ n o n d e t e r m i n i s t i c − q u a n t u m − p u s h d o w n − a u t o m a t a ′ > 1.2 N o n d e t e r m i n i s t i c Q u a n t u m P u s h d o w n A u t o m a t a < / h 3 > < p > A Q u a n t u m < a h r e f = " h t t p s : / / w w w . e m e r g e n t m i n d . c o m / t o p i c s / p u s h d o w n − a u t o m a t o n − p d a " t i t l e = " " r e l = " n o f o l l o w " d a t a − t u r b o = " f a l s e " c l a s s = " a s s i s t a n t − l i n k " x − d a t a x − t o o l t i p . r a w = " " > P u s h d o w n A u t o m a t o n < / a > w i t h a G a r b a g e T a p e ( Q P A G ) i s a r e v e r s i b l e e x t e n s i o n o f t h e q u a n t u m p u s h d o w n a u t o m a t o n , f o r m a l i z e d a s a t u p l e < / p > < p > . This is known as “cutpoint 0”, or positive one-sided error (<a href="/papers/0902.2081" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">0902.2081</a>).</p>
<h3 class='paper-heading' id='nondeterministic-quantum-pushdown-automata'>1.2 Nondeterministic Quantum Pushdown Automata</h3>
<p>A Quantum <a href="https://www.emergentmind.com/topics/pushdown-automaton-pda" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">Pushdown Automaton</a> with a Garbage Tape (QPAG) is a reversible extension of the quantum pushdown automaton, formalized as a tuple</p>
<p> . T hi s i s kn o w na s “ c u tp o in t 0” , or p os i t i v eo n e − s i d e d error ( < ah re f = "/ p a p ers /0902.2081" t i tl e = "" re l = " n o f o ll o w " d a t a − t u r b o = " f a l se " c l a ss = " a ss i s t an t − l ink " x − d a t a x − t oo lt i p . r a w = "" > 0902.2081 < / a > ) . < / p >< h 3 c l a ss = ′ p a p er − h e a d in g ′ i d = ′ n o n d e t er mini s t i c − q u an t u m − p u s h d o w n − a u t o ma t a ′ > 1.2 N o n d e t er mini s t i c Q u an t u m P u s h d o w n A u t o ma t a < / h 3 >< p > A Q u an t u m < ah re f = " h ttp s : // www . e m er g e n t min d . co m / t o p i cs / p u s h d o w n − a u t o ma t o n − p d a " t i tl e = "" re l = " n o f o ll o w " d a t a − t u r b o = " f a l se " c l a ss = " a ss i s t an t − l ink " x − d a t a x − t oo lt i p . r a w = "" > P u s h d o w n A u t o ma t o n < / a > w i t ha G a r ba g e T a p e ( QP A G ) i s a re v ers ib l ee x t e n s i o n o f t h e q u an t u m p u s h d o w na u t o ma t o n , f or ma l i ze d a s a t u pl e < / p >< p > < / p > < p > w i t h q u a n t u m c o n t r o l , c l a s s i c a l o r q u a n t u m s t a c k , a n d a w r i t e − o n l y g a r b a g e t a p e e n s u r i n g r e v e r s i b i l i t y o f s t a c k o p e r a t i o n s . C o n f i g u r a t i o n s e n c o d e c u r r e n t s t a t e , i n p u t p o s i t i o n , s t a c k c o n t e n t s , a n d g a r b a g e t a p e . E v o l u t i o n p r o c e e d s v i a u n i t a r y o p e r a t o r s s p e c i f i e d b y </p>
<p>with quantum control, classical or quantum stack, and a write-only garbage tape ensuring reversibility of stack operations. Configurations encode current state, input position, stack contents, and garbage tape. Evolution proceeds via unitary operators specified by < / p >< p > w i t h q u an t u m co n t ro l , c l a ss i c a l or q u an t u m s t a c k , an d a w r i t e − o n l y g a r ba g e t a p ee n s u r in g re v ers ibi l i t yo f s t a c k o p er a t i o n s . C o n f i gu r a t i o n se n co d ec u rre n t s t a t e , in p u tp os i t i o n , s t a c k co n t e n t s , an d g a r ba g e t a p e . E v o l u t i o n p rocee d s v ia u ni t a ryo p er a t orss p ec i f i e d b y , a n d p r o j e c t i v e m e a s u r e m e n t p a r t i t i o n s t h e s p a c e i n t o o n g o i n g , a c c e p t i n g , a n d r e j e c t i n g s u b s p a c e s . N o n d e t e r m i n i s t i c r e c o g n i t i o n i s d e f i n e d a s a c c e p t a n c e w i t h n o n z e r o p r o b a b i l i t y f o r m e m b e r s a n d z e r o o t h e r w i s e ( < a h r e f = " / p a p e r s / 1402.3449 " t i t l e = " " r e l = " n o f o l l o w " d a t a − t u r b o = " f a l s e " c l a s s = " a s s i s t a n t − l i n k " x − d a t a x − t o o l t i p . r a w = " " > N a k a n i s h i , 2014 < / a > ) . < / p > < h 3 c l a s s = ′ p a p e r − h e a d i n g ′ i d = ′ m o o r e − c r u t c h f i e l d − q u a n t u m − a u t o m a t a − a n d − a f f i n e − a u t o m a t a ′ > 1.3 M o o r e – C r u t c h f i e l d Q u a n t u m A u t o m a t a a n d A f f i n e A u t o m a t a < / h 3 > < p > M o o r e – C r u t c h f i e l d < a h r e f = " h t t p s : / / w w w . e m e r g e n t m i n d . c o m / t o p i c s / q u a n t u m − f i n i t e − a u t o m a t a − q f a s " t i t l e = " " r e l = " n o f o l l o w " d a t a − t u r b o = " f a l s e " c l a s s = " a s s i s t a n t − l i n k " x − d a t a x − t o o l t i p . r a w = " " > q u a n t u m f i n i t e a u t o m a t a < / a > ( M C Q F A ) e m p l o y a r b i t r a r y u n i t a r y e v o l u t i o n s , i n i t i a l v e c t o r s t a t e s , a n d a c c e p t a n c e v i a p r o j e c t i v e m e a s u r e m e n t o n t o a d e s i g n a t e d a c c e p t i n g s u b s p a c e . A f f i n e f i n i t e a u t o m a t a ( A f A ) g e n e r a l i z e p r o b a b i l i s t i c / q u a n t u m a u t o m a t a b y a l l o w i n g a f f i n e ( ℓ 1 − n o r m p r e s e r v i n g ) t r a n s i t i o n s a n d w e i g h t e d m e a s u r e m e n t s , a d m i t t i n g p o w e r f u l s e p a r a t i o n c a p a b i l i t i e s . < / p > < h 2 c l a s s = ′ p a p e r − h e a d i n g ′ i d = ′ c h a r a c t e r i z a t i o n s − a n d − l a n g u a g e − r e c o g n i t i o n − p o w e r ′ > 2. C h a r a c t e r i z a t i o n s a n d L a n g u a g e R e c o g n i t i o n P o w e r < / h 2 > < p > T h e c e n t r a l r e s u l t f o r n o n d e t e r m i n i s t i c q u a n t u m f i n i t e a u t o m a t a e s t a b l i s h e s t h a t t h e i r l a n g u a g e c l a s s c o i n c i d e s w i t h t h e e x c l u s i v e s t o c h a s t i c l a n g u a g e s : , and projective measurement partitions the space into ongoing, accepting, and rejecting subspaces. Nondeterministic recognition is defined as acceptance with nonzero probability for members and zero otherwise (<a href="/papers/1402.3449" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">Nakanishi, 2014</a>).</p>
<h3 class='paper-heading' id='moore-crutchfield-quantum-automata-and-affine-automata'>1.3 Moore–Crutchfield Quantum Automata and Affine Automata</h3>
<p>Moore–Crutchfield <a href="https://www.emergentmind.com/topics/quantum-finite-automata-qfas" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">quantum finite automata</a> (MCQFA) employ arbitrary unitary evolutions, initial vector states, and acceptance via projective measurement onto a designated accepting subspace. Affine finite automata (AfA) generalize probabilistic/quantum automata by allowing affine (ℓ₁-norm preserving) transitions and weighted measurements, admitting powerful separation capabilities.</p>
<h2 class='paper-heading' id='characterizations-and-language-recognition-power'>2. Characterizations and Language Recognition Power</h2>
<p>The central result for nondeterministic quantum finite automata establishes that their language class coincides with the exclusive stochastic languages: , an d p ro j ec t i v e m e a s u re m e n tp a r t i t i o n s t h es p a ce in t oo n g o in g , a cce pt in g , an d re j ec t in g s u b s p a ces . N o n d e t er mini s t i creco g ni t i o ni s d e f in e d a s a cce pt an ce w i t hn o n zero p ro babi l i t y f or m e mb ers an d zeroo t h er w i se ( < ah re f = "/ p a p ers /1402.3449" t i tl e = "" re l = " n o f o ll o w " d a t a − t u r b o = " f a l se " c l a ss = " a ss i s t an t − l ink " x − d a t a x − t oo lt i p . r a w = "" > N akani s hi , 2014 < / a > ) . < / p >< h 3 c l a ss = ′ p a p er − h e a d in g ′ i d = ′ m oore − cr u t c h f i e l d − q u an t u m − a u t o ma t a − an d − a ff in e − a u t o ma t a ′ > 1.3 M oore – C r u t c h f i e l d Q u an t u m A u t o ma t aan d A ff in e A u t o ma t a < / h 3 >< p > M oore – C r u t c h f i e l d < ah re f = " h ttp s : // www . e m er g e n t min d . co m / t o p i cs / q u an t u m − f ini t e − a u t o ma t a − q f a s " t i tl e = "" re l = " n o f o ll o w " d a t a − t u r b o = " f a l se " c l a ss = " a ss i s t an t − l ink " x − d a t a x − t oo lt i p . r a w = "" > q u an t u m f ini t e a u t o ma t a < / a > ( MCQF A ) e m pl oy a r bi t r a ry u ni t a rye v o l u t i o n s , ini t ia l v ec t ors t a t es , an d a cce pt an ce v ia p ro j ec t i v e m e a s u re m e n t o n t o a d es i g na t e d a cce pt in g s u b s p a ce . A ff in e f ini t e a u t o ma t a ( A f A ) g e n er a l i ze p ro babi l i s t i c / q u an t u ma u t o ma t ab y a ll o w in g a ff in e ( ℓ 1 − n or m p reser v in g ) t r an s i t i o n s an d w e i g h t e d m e a s u re m e n t s , a d mi tt in g p o w er f u l se p a r a t i o n c a p abi l i t i es . < / p >< h 2 c l a ss = ′ p a p er − h e a d in g ′ i d = ′ c ha r a c t er i z a t i o n s − an d − l an gu a g e − reco g ni t i o n − p o w e r ′ > 2. C ha r a c t er i z a t i o n s an d L an gu a g e R eco g ni t i o n P o w er < / h 2 >< p > T h ece n t r a l res u lt f or n o n d e t er mini s t i c q u an t u m f ini t e a u t o ma t a es t ab l i s h es t ha tt h e i r l an gu a g ec l a ssco in c i d es w i t h t h ee x c l u s i v es t oc ha s t i c l an gu a g es : w h e r e
where w h ere i s e x c l u s i v e s t o c h a s t i c i f t h e r e e x i s t s a o n e − w a y p r o b a b i l i s t i c f i n i t e a u t o m a t o n is exclusive stochastic if there exists a one-way probabilistic finite automaton i se x c l u s i v es t oc ha s t i c i f t h eree x i s t s a o n e − w a y p ro babi l i s t i c f ini t e a u t o ma t o n a n d c u t p o i n t and cutpoint an d c u tp o in t s u c h t h a t < / p > < p > such that</p>
<p> s u c h t ha t < / p >< p > < / p > < p > T h i s c a n b e r e a l i z e d i n p a r t i c u l a r b y </p>
<p>This can be realized in particular by < / p >< p > T hi sc anb ere a l i ze d in p a r t i c u l a r b y . T h e N Q F A w i t h c u t p o i n t 0 r e c o g n i z e s a l l a n d o n l y s u c h l a n g u a g e s , w i t h m e m b e r s h i p i n d i c a t e d b y n o n z e r o ( s t r i c t l y p o s i t i v e ) a c c e p t a n c e p r o b a b i l i t y ( < a h r e f = " / p a p e r s / 0902.2081 " t i t l e = " " r e l = " n o f o l l o w " d a t a − t u r b o = " f a l s e " c l a s s = " a s s i s t a n t − l i n k " x − d a t a x − t o o l t i p . r a w = " " > 0902.2081 < / a > ) . < / p > < p > C l a s s i c a l l y , P F A s w i t h c u t p o i n t 0 r e c o g n i z e p r e c i s e l y t h e r e g u l a r l a n g u a g e s , b u t N Q F A s t r i c t l y e x t e n d t h i s c l a s s . F o r e x a m p l e , t h e l a n g u a g e < / p > < p > . The NQFA with cutpoint 0 recognizes all and only such languages, with membership indicated by nonzero (strictly positive) acceptance probability (<a href="/papers/0902.2081" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">0902.2081</a>).</p>
<p>Classically, PFAs with cutpoint 0 recognize precisely the regular languages, but NQFA strictly extend this class. For example, the language</p>
<p> . T h e NQF A w i t h c u tp o in t 0 reco g ni zes a ll an d o n l ys u c h l an gu a g es , w i t hm e mb ers hi p in d i c a t e d b y n o n zero ( s t r i c tl y p os i t i v e ) a cce pt an ce p ro babi l i t y ( < ah re f = "/ p a p ers /0902.2081" t i tl e = "" re l = " n o f o ll o w " d a t a − t u r b o = " f a l se " c l a ss = " a ss i s t an t − l ink " x − d a t a x − t oo lt i p . r a w = "" > 0902.2081 < / a > ) . < / p >< p > Cl a ss i c a ll y , PF A s w i t h c u tp o in t 0 reco g ni ze p rec i se l y t h ere gu l a r l an gu a g es , b u tNQF A s t r i c tl ye x t e n d t hi sc l a ss . F ore x am pl e , t h e l an gu a g e < / p >< p > < / p > < p > i s i n N Q F A b u t n o t r e g u l a r . T h i s d e m o n s t r a t e s t h a t q u a n t u m n o n d e t e r m i n i s m a t t h e o n e − s i d e d u n b o u n d e d − e r r o r l e v e l i s s t r i c t l y m o r e e x p r e s s i v e t h a n c l a s s i c a l n o n d e t e r m i n i s m ( < a h r e f = " / p a p e r s / 0902.2081 " t i t l e = " " r e l = " n o f o l l o w " d a t a − t u r b o = " f a l s e " c l a s s = " a s s i s t a n t − l i n k " x − d a t a x − t o o l t i p . r a w = " " > 0902.2081 < / a > ) . < / p > < p > F o r < a h r e f = " h t t p s : / / w w w . e m e r g e n t m i n d . c o m / t o p i c s / q u a n t u m − p u s h d o w n − a u t o m a t a " t i t l e = " " r e l = " n o f o l l o w " d a t a − t u r b o = " f a l s e " c l a s s = " a s s i s t a n t − l i n k " x − d a t a x − t o o l t i p . r a w = " " > q u a n t u m p u s h d o w n a u t o m a t a < / a > w i t h a g a r b a g e t a p e , i t i s p r o v e n t h a t a n y c l a s s i c a l p r o b a b i l i s t i c o r n o n d e t e r m i n i s t i c < a h r e f = " h t t p s : / / w w w . e m e r g e n t m i n d . c o m / t o p i c s / p a n e l − d a t a − a p p r o a c h − p d a " t i t l e = " " r e l = " n o f o l l o w " d a t a − t u r b o = " f a l s e " c l a s s = " a s s i s t a n t − l i n k " x − d a t a x − t o o l t i p . r a w = " " > P D A < / a > c a n b e s i m u l a t e d e x a c t l y ( a c c e p t i n g w i t h n o n z e r o p r o b a b i l i t y o n a c c e p t e d i n p u t s a n d z e r o o t h e r w i s e ) , a n d , c r u c i a l l y , t h a t Q P A G s t r i c t l y o u t p e r f o r m s c l a s s i c a l P D A o n c e r t a i n p r o m i s e p r o b l e m s ( < a h r e f = " / p a p e r s / 1402.3449 " t i t l e = " " r e l = " n o f o l l o w " d a t a − t u r b o = " f a l s e " c l a s s = " a s s i s t a n t − l i n k " x − d a t a x − t o o l t i p . r a w = " " > N a k a n i s h i , 2014 < / a > ) . < / p > < h 2 c l a s s = ′ p a p e r − h e a d i n g ′ i d = ′ s e p a r a b i l i t y − s t a t e − c o m p l e x i t y − a n d − s t r u c t u r a l − r e s u l t s ′ > 3. S e p a r a b i l i t y , S t a t e C o m p l e x i t y , a n d S t r u c t u r a l R e s u l t s < / h 2 > < p > A c e n t r a l s e p a r a b i l i t y r e s u l t s t a t e s t h a t f o r e v e r y p a i r o f d i s t i n c t w o r d s </p>
<p>is in NQFA but not regular. This demonstrates that quantum nondeterminism at the one-sided unbounded-error level is strictly more expressive than classical nondeterminism (<a href="/papers/0902.2081" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">0902.2081</a>).</p>
<p>For <a href="https://www.emergentmind.com/topics/quantum-pushdown-automata" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">quantum pushdown automata</a> with a garbage tape, it is proven that any classical probabilistic or nondeterministic <a href="https://www.emergentmind.com/topics/panel-data-approach-pda" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">PDA</a> can be simulated exactly (accepting with nonzero probability on accepted inputs and zero otherwise), and, crucially, that QPAG strictly outperforms classical PDA on certain promise problems (<a href="/papers/1402.3449" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">Nakanishi, 2014</a>).</p>
<h2 class='paper-heading' id='separability-state-complexity-and-structural-results'>3. Separability, State Complexity, and Structural Results</h2>
<p>A central separability result states that for every pair of distinct words < / p >< p > i s in NQF A b u t n o t re gu l a r . T hi s d e m o n s t r a t es t ha tq u an t u mn o n d e t er mini s ma tt h eo n e − s i d e d u nb o u n d e d − error l e v e l i ss t r i c tl y m oree x p ress i v e t han c l a ss i c a l n o n d e t er mini s m ( < ah re f = "/ p a p ers /0902.2081" t i tl e = "" re l = " n o f o ll o w " d a t a − t u r b o = " f a l se " c l a ss = " a ss i s t an t − l ink " x − d a t a x − t oo lt i p . r a w = "" > 0902.2081 < / a > ) . < / p >< p > F or < ah re f = " h ttp s : // www . e m er g e n t min d . co m / t o p i cs / q u an t u m − p u s h d o w n − a u t o ma t a " t i tl e = "" re l = " n o f o ll o w " d a t a − t u r b o = " f a l se " c l a ss = " a ss i s t an t − l ink " x − d a t a x − t oo lt i p . r a w = "" > q u an t u m p u s h d o w na u t o ma t a < / a > w i t ha g a r ba g e t a p e , i t i s p ro v e n t ha t an yc l a ss i c a lp ro babi l i s t i cor n o n d e t er mini s t i c < ah re f = " h ttp s : // www . e m er g e n t min d . co m / t o p i cs / p an e l − d a t a − a pp ro a c h − p d a " t i tl e = "" re l = " n o f o ll o w " d a t a − t u r b o = " f a l se " c l a ss = " a ss i s t an t − l ink " x − d a t a x − t oo lt i p . r a w = "" > P D A < / a > c anb es im u l a t e d e x a c tl y ( a cce pt in g w i t hn o n zero p ro babi l i t yo na cce pt e d in p u t s an d zeroo t h er w i se ) , an d , cr u c ia ll y , t ha tQP A G s t r i c tl yo u tp er f or m sc l a ss i c a lP D A o n cer t ain p ro mi se p ro b l e m s ( < ah re f = "/ p a p ers /1402.3449" t i tl e = "" re l = " n o f o ll o w " d a t a − t u r b o = " f a l se " c l a ss = " a ss i s t an t − l ink " x − d a t a x − t oo lt i p . r a w = "" > N akani s hi , 2014 < / a > ) . < / p >< h 2 c l a ss = ′ p a p er − h e a d in g ′ i d = ′ se p a r abi l i t y − s t a t e − co m pl e x i t y − an d − s t r u c t u r a l − res u lt s ′ > 3. S e p a r abi l i t y , St a t e C o m pl e x i t y , an d St r u c t u r a lR es u lt s < / h 2 >< p > A ce n t r a l se p a r abi l i t yres u lt s t a t es t ha t f ore v ery p ai ro fd i s t in c tw or d s , t h e r e e x i s t s a 2 − s t a t e M C Q F A o p e r a t i n g i n n o n d e t e r m i n i s t i c a c c e p t a n c e m o d e s u c h t h a t < / p > < p > , there exists a 2-state MCQFA operating in nondeterministic acceptance mode such that</p>
<p> , t h eree x i s t s a 2 − s t a t e MCQF A o p er a t in g inn o n d e t er mini s t i c a cce pt an ce m o d es u c h t ha t < / p >< p > < / p > < p > T h i s m i n i m a l − s t a t e s e p a r a t i o n e x t e n d s e x p o n e n t i a l l y : f o r a n y t w o d i s j o i n t f i n i t e s u b s e t s </p>
<p>This minimal-state separation extends exponentially: for any two disjoint finite subsets < / p >< p > T hi s minima l − s t a t ese p a r a t i o n e x t e n d se x p o n e n t ia ll y : f or an y tw o d i s j o in t f ini t es u b se t s w i t h with w i t h , a , a , a − s t a t e M C Q F A s u f f i c e s t o s e p a r a t e ( a c c e p t -state MCQFA suffices to separate (accept − s t a t e MCQF A s u ff i ces t ose p a r a t e ( a cce pt , r e j e c t , reject , re j ec t i n c u t p o i n t − 0 m o d e ) . T h e c o n s t r u c t i o n e m p l o y s t e n s o r p r o d u c t s o f s i n g l e t o n s ’ 2 − s t a t e r e c o g n i z e r s ( < a h r e f = " / p a p e r s / 1602.07967 " t i t l e = " " r e l = " n o f o l l o w " d a t a − t u r b o = " f a l s e " c l a s s = " a s s i s t a n t − l i n k " x − d a t a x − t o o l t i p . r a w = " " > B e l o v s e t a l . , 2016 < / a > ) . < / p > < p > I t i s f u r t h e r c o n j e c t u r e d t h a t e v e n z e r o − e r r o r ( e x a c t ) s e p a r a t i o n o f a n y p a i r i s a c h i e v a b l e w i t h a 2 − s t a t e M C Q F A u s i n g c o m p l e x a m p l i t u d e s ; e v i d e n c e s u p p o r t s t h i s , t h o u g h a c o m p l e t e p r o o f i s p e n d i n g ( < a h r e f = " / p a p e r s / 1602.07967 " t i t l e = " " r e l = " n o f o l l o w " d a t a − t u r b o = " f a l s e " c l a s s = " a s s i s t a n t − l i n k " x − d a t a x − t o o l t i p . r a w = " " > B e l o v s e t a l . , 2016 < / a > ) . A f f i n e f i n i t e a u t o m a t a , i n c o n t r a s t , a c h i e v e e x a c t s e p a r a t i o n o f a n y p a i r w i t h o n l y t w o s t a t e s a n d c a n s e p a r a t e s i n g l e t o n v e r s u s f i n i t e s e t s w i t h in cutpoint-0 mode). The construction employs tensor products of singletons’ 2-state recognizers (<a href="/papers/1602.07967" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">Belovs et al., 2016</a>).</p>
<p>It is further conjectured that even zero-error (exact) separation of any pair is achievable with a 2-state MCQFA using complex amplitudes; evidence supports this, though a complete proof is pending (<a href="/papers/1602.07967" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">Belovs et al., 2016</a>). Affine finite automata, in contrast, achieve exact separation of any pair with only two states and can separate singleton versus finite sets with in c u tp o in t − 0 m o d e ) . T h eco n s t r u c t i o n e m pl oys t e n sor p ro d u c t so f s in g l e t o n s ’2 − s t a t ereco g ni zers ( < ah re f = "/ p a p ers /1602.07967" t i tl e = "" re l = " n o f o ll o w " d a t a − t u r b o = " f a l se " c l a ss = " a ss i s t an t − l ink " x − d a t a x − t oo lt i p . r a w = "" > B e l o v se t a l . , 2016 < / a > ) . < / p >< p > I t i s f u r t h erco nj ec t u re d t ha t e v e n zero − error ( e x a c t ) se p a r a t i o n o f an y p ai r i s a c hi e v ab l e w i t ha 2 − s t a t e MCQF A u s in g co m pl e x am pl i t u d es ; e v i d e n ces u pp or t s t hi s , t h o ug ha co m pl e t e p roo f i s p e n d in g ( < ah re f = "/ p a p ers /1602.07967" t i tl e = "" re l = " n o f o ll o w " d a t a − t u r b o = " f a l se " c l a ss = " a ss i s t an t − l ink " x − d a t a x − t oo lt i p . r a w = "" > B e l o v se t a l . , 2016 < / a > ) . A ff in e f ini t e a u t o ma t a , in co n t r a s t , a c hi e v ee x a c t se p a r a t i o n o f an y p ai r w i t h o n l y tw os t a t es an d c an se p a r a t es in g l e t o n v ers u s f ini t ese t s w i t h s t a t e s , e v e n p r o v i d i n g b o u n d e d − o r z e r o − e r r o r s e t − v e r s u s − s e t s e p a r a t i o n t h r o u g h a n a l o g o u s t e n s o r i n g t e c h n i q u e s . < / p > < h 2 c l a s s = ′ p a p e r − h e a d i n g ′ i d = ′ c l o s u r e − p r o p e r t i e s − a n d − e r r o r − m o d e s ′ > 4. C l o s u r e P r o p e r t i e s a n d E r r o r M o d e s < / h 2 > < p > T h e c l a s s states, even providing bounded- or zero-error set-versus-set separation through analogous tensoring techniques.</p>
<h2 class='paper-heading' id='closure-properties-and-error-modes'>4. Closure Properties and Error Modes</h2>
<p>The class s t a t es , e v e n p ro v i d in g b o u n d e d − orzero − errorse t − v ers u s − se t se p a r a t i o n t h ro ug hana l o g o u s t e n sor in g t ec hni q u es . < / p >< h 2 c l a ss = ′ p a p er − h e a d in g ′ i d = ′ c l os u re − p ro p er t i es − an d − error − m o d e s ′ > 4. Cl os u re P ro p er t i es an d E rror M o d es < / h 2 >< p > T h ec l a ss , a n d h e n c e t h e n o n d e t e r m i n i s t i c q u a n t u m l a n g u a g e c l a s s , e n j o y s c l o s u r e u n d e r u n i o n , i n t e r s e c t i o n , c o n c a t e n a t i o n , a n d K l e e n e s t a r , b u t n o t u n d e r c o m p l e m e n t ( < a h r e f = " / p a p e r s / 0902.2081 " t i t l e = " " r e l = " n o f o l l o w " d a t a − t u r b o = " f a l s e " c l a s s = " a s s i s t a n t − l i n k " x − d a t a x − t o o l t i p . r a w = " " > 0902.2081 < / a > ) . T h e s e c l o s u r e p r o p e r t i e s f o l l o w f r o m t h e m a t r i x − a n d − t e n s o r c o n s t r u c t i o n s p o s s i b l e i n b o t h G P F A a n d Q F A r e p r e s e n t a t i o n s e m p l o y i n g c u t p o i n t − 0 a c c e p t a n c e . < / p > < p > T w o − s i d e d e r r o r q u a n t u m r e c o g n i t i o n i s c h a r a c t e r i z e d v i a t h e c l a s s , and hence the nondeterministic quantum language class, enjoys closure under union, intersection, concatenation, and Kleene star, but not under complement (<a href="/papers/0902.2081" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">0902.2081</a>). These closure properties follow from the matrix-and-tensor constructions possible in both GPFA and QFA representations employing cutpoint-0 acceptance.</p>
<p>Two-sided error quantum recognition is characterized via the class , an d h e n ce t h e n o n d e t er mini s t i c q u an t u m l an gu a g ec l a ss , e nj oysc l os u re u n d er u ni o n , in t ersec t i o n , co n c a t e na t i o n , an d K l ee n es t a r , b u t n o t u n d erco m pl e m e n t ( < ah re f = "/ p a p ers /0902.2081" t i tl e = "" re l = " n o f o ll o w " d a t a − t u r b o = " f a l se " c l a ss = " a ss i s t an t − l ink " x − d a t a x − t oo lt i p . r a w = "" > 0902.2081 < / a > ) . T h esec l os u re p ro p er t i es f o ll o w f ro m t h e ma t r i x − an d − t e n sorco n s t r u c t i o n s p oss ib l e inb o t h GPF A an d QF A re p rese n t a t i o n se m pl oy in g c u tp o in t − 0 a cce pt an ce . < / p >< p > Tw o − s i d e d error q u an t u m reco g ni t i o ni sc ha r a c t er i ze d v ia t h ec l a ss ( l a n g u a g e s r e a l i z e d b y P F A s w i t h g e n e r a l c u t p o i n t a c c e p t a n c e ) a n d (languages realized by PFAs with general cutpoint acceptance) and ( l an gu a g esre a l i ze d b y PF A s w i t h g e n er a l c u tp o in t a cce pt an ce ) an d . T w o − s i d e d e r r o r l a n g u a g e s c o m p r i s e t h e u n i o n o f p o s i t i v e o n e − s i d e d ( . Two-sided error languages comprise the union of positive one-sided ( . Tw o − s i d e d error l an gu a g esco m p r i se t h e u ni o n o f p os i t i v eo n e − s i d e d ( ) , n e g a t i v e o n e − s i d e d ( ), negative one-sided ( ) , n e g a t i v eo n e − s i d e d ( ) , a n d b o n a f i d e t w o − s i d e d − e r r o r − o n l y c l a s s e s . E x p l i c i t s e p a r a t i o n i s g i v e n b y : ), and bona fide two-sided-error-only classes. Explicit separation is given by: ) , an d b o na f i d e tw o − s i d e d − error − o n l yc l a sses . E x pl i c i t se p a r a t i o ni s g i v e nb y : ( < a h r e f = " / p a p e r s / 0902.2081 " t i t l e = " " r e l = " n o f o l l o w " d a t a − t u r b o = " f a l s e " c l a s s = " a s s i s t a n t − l i n k " x − d a t a x − t o o l t i p . r a w = " " > 0902.2081 < / a > ) . < / p > < h 2 c l a s s = ′ p a p e r − h e a d i n g ′ i d = ′ r e l a t i o n s h i p − t o − c l a s s i c a l − a n d − a f f i n e − a u t o m a t a ′ > 5. R e l a t i o n s h i p t o C l a s s i c a l a n d A f f i n e A u t o m a t a < / h 2 > < p > T a b l e : Q u a n t u m v s . C l a s s i c a l a n d A f f i n e A u t o m a t a < / p > < d i v c l a s s = ′ o v e r f l o w − x − a u t o m a x − w − f u l l m y − 4 ′ > < t a b l e c l a s s = ′ t a b l e b o r d e r − c o l l a p s e w − f u l l ′ s t y l e = ′ t a b l e − l a y o u t : f i x e d ′ > < t h e a d > < t r > < t h > M o d e l < / t h > < t h > M i n i m a l S t a t e s f o r P a i r w i s e S e p a r a t i o n < / t h > < t h > L a n g u a g e s R e c o g n i z e d ( n o n d e t e r m i n i s t i c ) < / t h > < / t r > < / t h e a d > < t b o d y > < t r > < t d > N o n d e t e r m i n i s t i c Q F A ( K W Q F A / M C Q F A ) < / t d > < t d > 2 < / t d > < t d > E x c l u s i v e − s t o c h a s t i c (
(<a href="/papers/0902.2081" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">0902.2081</a>).</p>
<h2 class='paper-heading' id='relationship-to-classical-and-affine-automata'>5. Relationship to Classical and Affine Automata</h2>
<p>Table: Quantum vs. Classical and Affine Automata</p>
<div class='overflow-x-auto max-w-full my-4'><table class='table border-collapse w-full' style='table-layout: fixed'><thead><tr>
<th>Model</th>
<th>Minimal States for Pairwise Separation</th>
<th>Languages Recognized (nondeterministic)</th>
</tr>
</thead><tbody><tr>
<td>Nondeterministic QFA (KWQFA/MCQFA)</td>
<td>2</td>
<td>Exclusive-stochastic ( ( < ah re f = "/ p a p ers /0902.2081" t i tl e = "" re l = " n o f o ll o w " d a t a − t u r b o = " f a l se " c l a ss = " a ss i s t an t − l ink " x − d a t a x − t oo lt i p . r a w = "" > 0902.2081 < / a > ) . < / p >< h 2 c l a ss = ′ p a p er − h e a d in g ′ i d = ′ re l a t i o n s hi p − t o − c l a ss i c a l − an d − a ff in e − a u t o ma t a ′ > 5. R e l a t i o n s hi pt o Cl a ss i c a l an d A ff in e A u t o ma t a < / h 2 >< p > T ab l e : Q u an t u m v s . Cl a ss i c a l an d A ff in e A u t o ma t a < / p >< d i v c l a ss = ′ o v er f l o w − x − a u t o ma x − w − f u ll m y − 4 ′ >< t ab l ec l a ss = ′ t ab l e b or d er − co ll a p se w − f u l l ′ s t y l e = ′ t ab l e − l a yo u t : f i x e d ′ >< t h e a d >< t r >< t h > M o d e l < / t h >< t h > M inima lSt a t es f or P ai r w i se S e p a r a t i o n < / t h >< t h > L an gu a g es R eco g ni ze d ( n o n d e t er mini s t i c ) < / t h >< / t r >< / t h e a d >< t b o d y >< t r >< t d > N o n d e t er mini s t i c QF A ( K W QF A / MCQF A ) < / t d >< t d > 2 < / t d >< t d > E x c l u s i v e − s t oc ha s t i c ( ) < / t d > < / t r > < t r > < t d > A f f i n e f i n i t e a u t o m a t a ( A f A ) < / t d > < t d > 2 < / t d > < t d > A l l ( e x a c t p a i r w i s e s e p a r a t i o n , s t r o n g ) < / t d > < / t r > < t r > < t d > N o n d e t e r m i n i s t i c P F A / N F A < / t d > < t d > )</td>
</tr>
<tr>
<td>Affine finite automata (AfA)</td>
<td>2</td>
<td>All (exact pairwise separation, strong)</td>
</tr>
<tr>
<td>Nondeterministic PFA/NFA</td>
<td> ) < / t d >< / t r >< t r >< t d > A ff in e f ini t e a u t o ma t a ( A f A ) < / t d >< t d > 2 < / t d >< t d > A ll ( e x a c tp ai r w i sese p a r a t i o n , s t ro n g ) < / t d >< / t r >< t r >< t d > N o n d e t er mini s t i c PF A / NF A < / t d >< t d > ( p r o b l e m − s p e c i f i c , n o t a l w a y s 2 ) < / t d > < t d > R e g u l a r ( c u t p o i n t − 0 ) < / t d > < / t r > < / t b o d y > < / t a b l e > < / d i v > < p > N o n d e t e r m i n i s t i c Q F A s s t r i c t l y c o n t a i n t h e r e g u l a r l a n g u a g e s , a n d a f f i n e f i n i t e a u t o m a t a ( A f A ) a t t a i n e v e n g r e a t e r s e p a r a t i o n c a p a b i l i t y ( e x a c t z e r o − e r r o r , m i n i m a l − s t a t e ) f o r a r b i t r a r y p a i r s a n d s e t s e p a r a t i o n , w h i c h q u a n t u m m o d e l s c a n n o t a l w a y s g u a r a n t e e f o r f i n i t e s e t s w i t h m i n i m a l − s t a t e r e s o u r c e s ( < a h r e f = " / p a p e r s / 1602.07967 " t i t l e = " " r e l = " n o f o l l o w " d a t a − t u r b o = " f a l s e " c l a s s = " a s s i s t a n t − l i n k " x − d a t a x − t o o l t i p . r a w = " " > B e l o v s e t a l . , 2016 < / a > ) . < / p > < p > I n t h e c o n t e x t o f p u s h d o w n m o d e l s , q u a n t u m p u s h d o w n a u t o m a t a w i t h a g a r b a g e t a p e n o t o n l y s i m u l a t e a l l c l a s s i c a l ( p r o b a b i l i s t i c / n o n d e t e r m i n i s t i c ) p u s h d o w n a u t o m a t a b u t a r e s t r i c t l y m o r e p o w e r f u l : s p e c i f i c p r o b l e m s ( “ P r o b l e m I ” ) a r e d e c i d e d e x a c t l y b y Q P A G b u t n o t b y a n y c l a s s i c a l d e t e r m i n i s t i c o r n o n d e t e r m i n i s t i c P D A . T h e g a r b a g e t a p e ’ s i r r e v e r s i b i l i t y c i r c u m v e n t s t h e r e s t r i c t i o n i n u n i t a r y Q P D A a n d r e a l i z e s f u l l r e v e r s i b l e , q u a n t u m − c o n t r o l c o m p u t a t i o n ( < a h r e f = " / p a p e r s / 1402.3449 " t i t l e = " " r e l = " n o f o l l o w " d a t a − t u r b o = " f a l s e " c l a s s = " a s s i s t a n t − l i n k " x − d a t a x − t o o l t i p . r a w = " " > N a k a n i s h i , 2014 < / a > ) . < / p > < h 2 c l a s s = ′ p a p e r − h e a d i n g ′ i d = ′ c o m p l e x i t y − t h e o r e t i c − c o n s e q u e n c e s − a n d − o p e n − p r o b l e m s ′ > 6. C o m p l e x i t y − T h e o r e t i c C o n s e q u e n c e s a n d O p e n P r o b l e m s < / h 2 > < p > F o r s p a c e − b o u n d e d c o m p u t a t i o n , n o n d e t e r m i n i s t i c q u a n t u m T u r i n g m a c h i n e s o u t p e r f o r m t h e i r c l a s s i c a l a n a l o g u e s f o r a l l s u b l o g a r i t h m i c s p a c e f u n c t i o n s : (problem-specific, not always 2)</td>
<td>Regular (cutpoint-0)</td>
</tr>
</tbody></table></div>
<p>Nondeterministic QFAs strictly contain the regular languages, and affine finite automata (AfA) attain even greater separation capability (exact zero-error, minimal-state) for arbitrary pairs and set separation, which quantum models cannot always guarantee for finite sets with minimal-state resources (<a href="/papers/1602.07967" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">Belovs et al., 2016</a>).</p>
<p>In the context of pushdown models, quantum pushdown automata with a garbage tape not only simulate all classical (probabilistic/nondeterministic) pushdown automata but are strictly more powerful: specific problems (“Problem I”) are decided exactly by QPAG but not by any classical deterministic or nondeterministic PDA. The garbage tape’s irreversibility circumvents the restriction in unitary QPDA and realizes full reversible, quantum-control computation (<a href="/papers/1402.3449" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">Nakanishi, 2014</a>).</p>
<h2 class='paper-heading' id='complexity-theoretic-consequences-and-open-problems'>6. Complexity-Theoretic Consequences and Open Problems</h2>
<p>For space-bounded computation, nondeterministic quantum Turing machines outperform their classical analogues for all sublogarithmic space functions: ( p ro b l e m − s p ec i f i c , n o t a lw a ys 2 ) < / t d >< t d > R e gu l a r ( c u tp o in t − 0 ) < / t d >< / t r >< / t b o d y >< / t ab l e >< / d i v >< p > N o n d e t er mini s t i c QF A ss t r i c tl yco n t ain t h ere gu l a r l an gu a g es , an d a ff in e f ini t e a u t o ma t a ( A f A ) a tt ain e v e n g re a t erse p a r a t i o n c a p abi l i t y ( e x a c t zero − error , minima l − s t a t e ) f or a r bi t r a ry p ai rs an d se t se p a r a t i o n , w hi c h q u an t u mm o d e l sc ann o t a lw a ys gu a r an t ee f or f ini t ese t s w i t hminima l − s t a t ereso u rces ( < ah re f = "/ p a p ers /1602.07967" t i tl e = "" re l = " n o f o ll o w " d a t a − t u r b o = " f a l se " c l a ss = " a ss i s t an t − l ink " x − d a t a x − t oo lt i p . r a w = "" > B e l o v se t a l . , 2016 < / a > ) . < / p >< p > I n t h eco n t e x t o f p u s h d o w nm o d e l s , q u an t u m p u s h d o w na u t o ma t a w i t ha g a r ba g e t a p e n o t o n l ys im u l a t e a ll c l a ss i c a l ( p ro babi l i s t i c / n o n d e t er mini s t i c ) p u s h d o w na u t o ma t ab u t a res t r i c tl y m ore p o w er f u l : s p ec i f i c p ro b l e m s ( “ P ro b l e m I ” ) a re d ec i d e d e x a c tl y b y QP A G b u t n o t b y an yc l a ss i c a l d e t er mini s t i cor n o n d e t er mini s t i c P D A . T h e g a r ba g e t a p e ’ s i rre v ers ibi l i t yc i rc u m v e n t s t h eres t r i c t i o nin u ni t a ry QP D A an d re a l i zes f u ll re v ers ib l e , q u an t u m − co n t ro l co m p u t a t i o n ( < ah re f = "/ p a p ers /1402.3449" t i tl e = "" re l = " n o f o ll o w " d a t a − t u r b o = " f a l se " c l a ss = " a ss i s t an t − l ink " x − d a t a x − t oo lt i p . r a w = "" > N akani s hi , 2014 < / a > ) . < / p >< h 2 c l a ss = ′ p a p er − h e a d in g ′ i d = ′ co m pl e x i t y − t h eore t i c − co n se q u e n ces − an d − o p e n − p ro b l e m s ′ > 6. C o m pl e x i t y − T h eore t i c C o n se q u e n ces an d Op e n P ro b l e m s < / h 2 >< p > F ors p a ce − b o u n d e d co m p u t a t i o n , n o n d e t er mini s t i c q u an t u m T u r in g ma c hin eso u tp er f or m t h e i rc l a ss i c a l ana l o gu es f or a ll s u b l o g a r i t hmi cs p a ce f u n c t i o n s : T h i s s t r i c t i n c l u s i o n i s w i t n e s s e d b y l a n g u a g e s l i k e
This strict inclusion is witnessed by languages like T hi ss t r i c t in c l u s i o ni s w i t n esse d b y l an gu a g es l ik e , r e c o g n i z e d b y c o n s t a n t − s p a c e N Q F A b u t n o t b y a n y c l a s s i c a l n o n d e t e r m i n i s t i c a u t o m a t o n w i t h i n , recognized by constant-space NQFA but not by any classical nondeterministic automaton within , reco g ni ze d b yco n s t an t − s p a ce NQF A b u t n o t b y an yc l a ss i c a l n o n d e t er mini s t i c a u t o ma t o n w i t hin 0902.2081 ).
For quantum pushdown automata, QPAGs act as a natural, reversible, unitary model strictly above classical context-free language recognition (CFL) in the one-sided/zero-error setting, yet remain subsumed by PSPACE. A key open question is the full closure properties and comparative power of QPAG versus unitary quantum pushdown models without the garbage tape: current evidence and conjecture suggest incomparability, with the garbage tape being essential for full reversibility in arbitrary stack manipulation (Nakanishi, 2014 ).
7. Summary and Outlook
Nondeterministic quantum recognition, across finite and stack-automata regimes, yields a rich variety of structural, language-theoretic, and complexity-theoretic phenomena. The one-sided (cutpoint-0) quantum paradigm admits all exclusive-stochastic languages, strictly extending regular languages and outperforming classical nondeterministic/probabilistic automata in both separation and expressive capacity. Minimal-state quantum separation for word pairs is optimal, and affine automata attain even more robust separability. The design of garbage tapes in stack-based quantum automata restores reversibility and produces models with strictly stronger recognition power, even solving promise problems beyond classical reach. Open directions include the pursuit of zero-error minimal-state quantum separation for all pairs, further delineation of closure properties, and rigorous classification of the boundaries of quantum and affine nondeterministic language recognition capabilities (0902.2081 , Belovs et al., 2016 , Nakanishi, 2014 ).
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