CCharacterizing the intersection of QMA and coQMA.
Serge Massar and Miklos Santha Laboratoire d’Information Quantique CP224, Université libre de Bruxelles, B-1050 Brussels, Belgium. CNRS, IRIF, Université Paris Diderot, 75205 Paris, France. Centre for Quantum Technologies & MajuLab, National University of Singapore, Singapore.February 8, 2021
We show that the functional analogueof QMA ∩ coQMA, denoted F( QMA ∩ coQMA ) , equals the complexity class To-tal Functional QMA (TFQMA). To provethis we need to introduce alternative def-initions of QMA ∩ coQMA in terms of asingle quantum verification procedure. Weshow that if TFQMA equals the functionalanalogue of BQP (FBQP), then QMA ∩ coQMA = BQP. We show that if there isa QMA complete problem that (robustly)reduces to a problem in TFQMA, thenQMA ∩ coQMA = QMA. These resultsprovide strong evidence that the inclusionsFBQP ⊆ TFQMA ⊆ FQMA are strict,since otherwise the corresponding inclu-sions in BQP ⊆ QMA ∩ coQMA ⊆ QMAwould become equalities.
Functional NP ( FNP ) is the class of searchproblems defined by polynomial time relations M ( x, y ) (the verifier) where the length of y ispolynomial in the length of x . On an input x the task is to find a witness y (if it exists) suchthat M accepts ( x, y ) .The complexity class Total Functional NP ( TFNP ), introduced in [2], is the subset of
FNP for which it can be shown that for all inputs x ,there exists at least one witness y . It lies be-tween Functional P ( FP ) (the subclass of FNP for which a witness can be found in polynomialtime), and
FNP . TFNP contains many natural and impor-tant problems, including factoring, local searchproblems[3, 4, 5], computational versions ofBrouwer’s fixed point theorem[6], finding Nashequilibria[7, 8]. Although there probably do not
Serge Massar: [email protected] exist complete problems for
TFNP , there aremany syntactically defined subclasses of
TFNP that contain complete problems, and for whichsome of the above natural problems can be shownto be complete. For recent work in this direction,see [9].One of the results of the founding paper [2]concerns the relation between Total Functional NP and other complexity classes. The inclusions FP ⊆ TFNP ⊆ FNP (1) are obvious. But what would be the consequencesif some of these inclusions were not strict, butreplaced by equality?In [2] this question is connected with the inclu-sions P ⊆ NP ∩ coNP ⊆ NP . (2) It was first shown in [2] that Total Functional NP equals the functional analogue of NP ∩ coNP denoted F( NP ∩ coNP ) : TFNP = F( NP ∩ coNP ) ; (3) Furthermore, it is proven in [2] that an inclu-sion in Eq. (1) is strict if and only if the cor-responding inclusion in Eq. (2) is strict. Sincethe inclusions in Eq. (2) are believed to be strict,this provides strong evidence that the inclusionsin Eq. (1) are strict. The proofs of these resultsare very simple, and take only one paragraph, orare even just implicit in [2].The quantum analogue of NP is QMA [10].
QMA has been extensively studied, and containsa rich set of complete problems, see e.g. [11].Functional
QMA , the problem of producing aquantum state that serves as witness for a
QMA problem was first introduced in [12].In a recent work [13] we introduced the com-plexity class Total Functional
QMA ( TFQMA ),the subset of
FQMA for which it can be show a r X i v : . [ qu a n t - ph ] F e b hat for all inputs x , there exists at least one wit-ness | ψ i . TFQMA is an expressive class contain-ing several interesting problems. However in [13]the analog of the complexity results of [2] for
TFQMA where left open. Here we prove theseresults.The analogs of Eqs. (1) and (2) are
FBQP ⊆ TFQMA ⊆ FQMA (4) and
BQP ⊆ QMA ∩ coQMA ⊆ QMA . (5) We first show that
TFQMA equals the func-tional analogue of
QMA ∩ coQMA : TFQMA = F(
QMA ∩ coQMA ) . (6) We then show that if
FBQP = TFQMA , then
BQP = QMA ∩ coQMA .We finally show that if there is a QMA complete problem that reduces (using a slightlystronger notion of reduction than the natural one,which we call robust reduction) to a problem in
TFQMA , then
QMA ∩ coQMA = QMA .Since the inclusions in Eq. (2) are believed tobe strict, these results provide strong evidencethat the inclusions in Eq. (4) are strict.But while the proofs in the classical case are el-ementary, the quantum proofs are more delicate.Showing that
TFQMA = F(
QMA ∩ coQMA ) requires a detailed enquiry into what is the cor-rect definition of F( QMA ∩ coQMA ) . The diffi-culty is that any language L in QMA ∩ coQMA is naturally defined by two quantum verificationprocedures Q and Q . But these two quantumverification procedures do not necessarily com-mute. Therefore it is not clear how to define awitness for QMA ∩ coQMA . Indeed given aninput x and a state | ψ i one cannot test whetherboth Q ( x, | ψ i ) and Q ( x, | ψ i ) accept, since theact of carrying out one of these tests will mod-ify the state | ψ i and render impossible the othertest.We will see that the solution to this conundrumis to append to the state a bit z ∈ { , } , withthe convention that if z = 0 one only tests Q ,and if z = 1 one only tests Q . Thus a witnessfor QMA ∩ coQMA has the form | z i| ψ i .However, while this solution is natural, it isnot clear whether it is the unique way to solvethis problem. In order to address this, we need a deeper understanding of QMA ∩ coQMA . Tothis end we introduce two alternative definitionsof QMA ∩ coQMA . The more important is adefinition in terms of a single 2-outcome quan-tum verification procedure which takes as in-put a single witness. This definition is partic-ularly useful because it allows us to apply to QMA ∩ coQMA the notions of of eigenbasis,spectrum, and eigenspace of a quantum verifica-tion procedure which we introduced in [13] (basedon the earlier work of Marriott and Watrous[14]).These notions are then used to provide a naturaldefinition for F( QMA ∩ coQMA ) which parallelsthe definition for FQMA given in [13].Furthermore, by extending the notion of QMAamplification as developed in [14], we show thatour definition of F( QMA ∩ coQMA ) is indepen-dent of the completeness and soundness bounds.Together these results allow us to define pre-cisely the right hand side of Eq. (6) , in such away that it has the same structure as the lefthand side of Eq. (6) . Proving the equality of thetwo quantities is then rather easy.The paper is structured as follows:Sections 2 to 7 sets the stage, defining quan-tum verification procedures, QMA , coQMA , QMA ∩ coQMA , Functional QMA , Total Func-tional
QMA , and recalling the key notions ofeigenbasis, spectrum and eigenspaces of a quan-tum verification procedure. It closely follows,with appropriate modifications, our recent work[13] on
TFQMA .Section 8 introduces the notion of reductionof quantum verification procedures, including thenotion of robust reduction mentioned above.Section 9 recalls the notion of eigenspace pre-serving map introduced in [13].Section 10 introduces “iterative procedures”,thereby generalising an idea introduced in [14].Iterative procedures modify the acceptance prob-abilities of eigenstates of a quantum verifica-tion procedure without changing the eigenstatesthemselves. We use iterative procedures to showthat the spectrum of a quantum verification pro-cedure can be modified almost arbitrarily.As a first application of iterative procedureswe introduce in Section 11 the notion of “non-destructive” procedure which outputs both a clas-sical bit indicating whether the procedure acceptsor rejects and a quantum state, in such a way thatif the input state is an eigenstate, the output state s also the eigenstate. We show that, essentiallywithout loss of generality, one can take quantumverification procedures to be non-destructive.In Section 12 we present the two additional def-initions of QMA ∩ coQMA mentioned above.We show that the three definitions are equiva-lent. We then introduce in Section 13 functional QMA ∩ coQMA and use the results of Section10 to show that this definition does not dependon the soundness and completeness thresholds.In Section 14 we show that F( QMA ∩ coQMA ) equals the class Total Functional QMA ( TFQMA ).Section 15 contains the proof that if
FBQP = TFQMA , then
BQP = QMA ∩ coQMA .And finally in Section 16 we prove that if thereexists a QMA complete problem that robustlyreduces to a problem in
TFQMA , then
QMA = QMA ∩ coQMA . We denote by H n the Hilbert space of n qubits.For pure states we use the Dirac ket notation | ψ i , whereas for density matrices we just use theGreek letter ρ . We denote by | k i the state of k qubits all in the | i state. We denote by I m theidentity operator acting on m qubits.We denote by poly the set of all non-zero poly-nomials with non-negative integer coefficients.Note that if f ∈ poly , then f maps positive inte-gers to positive integers.We denote by / poly the set of all functionsthat are the inverse of a polynomial in poly : / poly = { g : N → R : ∃ p ∈ poly AND g = 1 /p } . (7) Definition 1. d -Outcome Quantum Verifi-cation Procedure. For any integer d largeror equal to , a d -outcome quantum verificationprocedure is a polynomial time uniform familyof quantum circuits Q = { Q n : n ∈ N } with Q n taking as input ( x, | ψ i ⊗ | k i ) , where x ∈ { , } n is a binary string of length n , | ψ i is a state of m qubits, and both m = m ( n ) and k = k ( n ) be-long to poly . The last k qubits, initialized to thestate | k i , form the ancilla Hilbert space H k , andthe m -qubit states | ψ i form the witness Hilbert space H m . The outcome of the run of Q n isa word w ∈ { , . . . d − } . It is obtained bymeasuring the first d log d e qubits in the computa-tional basis, interpreting the result as an integerin { , ..., d log d e − } and, if this integer is greateror equal to d , replacing it by d − . We denotethis outcome by Q n ( x, | ψ i ) . In most of this article we will consider 2-outcome quantum verification procedures. Forbrevity throughout this work we use the followingterminology which is standard in the literature:
Definition 2. Quantum Verification Proce-dure.
A 2-outcome quantum verification proce-dure is called a quantum verification procedure.
Note that a d -outcome quantum verificationprocedure can of course also take as input a mixedstate ρ , rather than a pure state | ψ i . Mixed statescan be written as convex combinations of purestates. The acceptance (rejection) probability forthe mixed state is the convex combination of theacceptance (rejection) probabilities for the con-stituent pure states.In order to simplify notation, in what fol-lows we will mainly consider the case where Q n takes as input a pure state. In view of theabove remark, the extension to mixed state in-puts is immediate. In some cases the argumentrequires that Q n takes as input a mixed state,in which case, abusing slightly the notation, wewrite Q n ( x, ρ ) for the outcome of the quantumverification procedure on the mixed state ρ . QMA
Definition 3. (a,b)–Quantum VerificationProcedure.
Let a, b : N → (0 , be polynomi-ally time computable functions which satisfy a ( n ) − b ( n ) ≥ /q ( n ) , (8) for some q ∈ poly . We say that a quantum ver-ification procedure Q is an ( a, b ) - quantum verifi-cation procedure (or shortly an ( a, b ) - procedure )if for every x of length n , one of the followingholds: ∃| ψ i , Pr[ Q n ( x, | ψ i ) = 1] ≥ a, (9) ∀| ψ i , Pr[ Q n ( x, | ψ i ) = 1] ≤ b. (10) e call a and b the completeness and soundness probabilities of the quantum verification proce-dure. Note that taking the completeness or sound-ness probabilities equal or require a specificformulation dealing with exact quantum compu-tation, which while theoretically interesting is notimplementable in practice (real quantum com-putation will have some, possibly exponentiallysmall, error probability). One of the results ofthe present work is to show that several com-plexity classes do not depend on the completenessand soundness probabilities used to define them.More precisely, in these results we show that thecompleteness and soundness probabilities can betaken exponentially close to and respectively.These results do not extend to showing that a and b can be taken equal to and . For thesereason we explicitly exclude in Definition 3 andall similar definitions for other kinds of quantumverification procedures the cases when a = 1 and b = 0 , and take them to belong to the open in-terval (0 , . Definition 4. QMA and coQMA.
Let a, b be functions as in Definition . The class QMA ( a, b ) is the set of languages L ⊆ { , } ∗ such that there exists an ( a, b ) -procedure Q , wherefor every x, we have x ∈ L if and only if Equa-tion (9) holds (and consequently, x / ∈ L if andonly if Equation (10) holds).We call Q a quantum verification procedurefor L , and for x ∈ L , we say that a | ψ i satisfyingEquation (9) is a witness for x .The class coQMA ( a, b ) is the set of languages L ⊆ { , } ∗ such that there exists an ( a, b ) -quantum verification procedure Q , where for ev-ery x, we have x / ∈ L if and only if Equation (9) holds (and consequently, x ∈ L if and only ifEquation (10) holds). Definition 5. QMA ∩ coQMA. Let a, b and a , b be pairs of functions as in Definition . The class QMA ∩ coQMA ( a, b ; a , b ) is the set of lan-guages L ⊆ { , } ∗ such that L ∈ QMA ( a, b ) and L ∈ coQMA ( a , b ) .More explicitly, QMA ∩ coQMA is the classof languages L ⊆ { , } ∗ such that there existan ( a, b ) -procedure Q = { Q n } , and an ( a , b ) -procedure Q = { Q n } , such that for every x we have x ∈ L if and only if both the following hold: ∃| ψ i , Pr[ Q n ( x, | ψ i ) = 1] ≥ a, (11) ∀| ψ i , Pr[ Q n ( x, | ψ i ) = 1] ≤ b ; (12) and x / ∈ L if and only if both the following hold: ∀| ψ i , Pr[ Q n ( x, | ψ i ) = 1] ≤ b, (13) ∃| ψ i , Pr[ Q n ( x, | ψ i ) = 1] ≥ a . (14) It is of course essential to understand towhat extent the above definitions depend on thebounds ( a, b ) and ( a , b ) . It was shown by Kitaevthat the separation a − b in the definition of QMA could be amplified exponentially by using multi-ple copies of the input state, and multiple copiesof the verification circuit [10], that is by increas-ing both m and k . This was further improvedin [14] (see also [15]) where it was shown thatby running forwards and backwards the originalquantum verification procedure, only one copy ofthe input state was needed to obtain the sameamplification, that is one needs only increase k . Theorem 1. QMA Amplification [10, 14,15].
Let a, b be functions as in Definition .For any polynomial r , we have QMA ( a, b ) ⊆ QMA (1 − − r , − r ) . As a consequence of Theorem 1 the precise val-ues of the bounds ( a, b ) are irrelevant. Tradition-ally they are taken to be / and / . We willdo here the same. Definition 6.
We define the class
QMA as QMA (2 / , / . Definition 7.
We define the class
QMA ∩ coQMA as QMA ∩ coQMA (2 / , /
3; 2 / , / Definition 8. a -Total Quantum VerificationProcedure. Let a : N → [0 , be a polynomiallytime computable function. We say that a quan-tum verification procedure is an a - total quantumverification procedure (or shortly an a - total pro-cedure ) if for every x of length n , the followingholds: ∃| ψ i , Pr[ Q n ( x, | ψ i ) = 1] ≥ a . (15) Note that an a -total procedure is also an ( a, b ) -procedure for all b satisfying the conditions ofDefinition 3. Note that the language associatedto an a -total procedure is L = { , } ∗ . That is he decision problem for total procedures is triv-ial, since for all x ∈ { , } ∗ there exists a wit-ness for x . Therefore for total procedures, theonly interesting questions concern the witnesses.The main topic of the present work is to under-stand the relation between a -total procedures and QMA ∩ coQMA . Recall that a quantum channel is a completelypositive (CP) trace preserving map betweenspaces of operators.
Definition 9. Efficiently implementablequantum channel.
A family Φ of efficiently implementable quan-tum channels is defined by a polynomial time uni-form family of quantum circuits Q = { Q n : n ∈ N } with Q n taking as input ( x, | ψ i ⊗ | k i ) where x ∈ { , } n , where | ψ i ∈ H m with m ∈ poly ,where k ∈ poly , and the output of the channel isobtained by keeping the first m qubits and trac-ing over the remaining m + k − m qubits, where m ∈ poly and m ≤ m + k . We denote the outputof the channel by Φ( x, | ψ i ) : Φ( x, | ψ i ) = Tr m + k − m Q n ( x, | ψ i ⊗ | k i ) . (16) Note that a quantum channel can of course alsoact on a mixed state ρ of m qubits. Definition 10. Efficiently preparablestates.
Let m ∈ poly . A family of densitymatrices { ρ ( x ) : x ∈ { , } n , n ∈ N } is efficientlypreparable if ρ ( x ) acts on H m ( n ) and if thereexists a polynomial time uniform family ofquantum circuits Q = { Q n : n ∈ N } with Q n taking as input ( x, | k i ) with x ∈ { , } n and k ∈ poly with k ≥ m , and where ρ ( x ) is obtainedby tracing out the last k − m qubits of Q n ( x ) . This definition is equivalent to saying that afamily of efficiently preparable states { ρ ( x ) } isthe output of a family of efficiently implementablequantum channels Φ which does not take anyquantum state as input, i.e. if one sets m = 0 in Definition 9: ρ ( x ) = Φ( x, ∅ ) .Note that if we want to use such a ρ (forinstance as input to a quantum algorithm) we would get an equivalent definition if either 1) werequired that the state ρ is only prepared withsome success probability p > / poly ; or 2) we re-quired that the prepared state is pure ρ = | ψ ih ψ | .For 1), the the state would be successfully pre-pared conditional on the measurement of somequbits. However we can also include these qubitsin the prepared state, and any further use of ρ will be conditional on the outcome of the mea-surement of these qubits. For 2) we would nottrace over the last k − m qubits of Q n ( x ) . Anyuser would then receive as input a pure state of k qubits, and would simply not use the last k − m qubits. BQP
Bounded-error quantum polynomial time (
BQP )is the class of decision problems solvable bya quantum computer in polynomial time, withbounded error probability for all instances. Wegive two equivalent definitions of
BQP , followingthe formulation of [13].
Definition 11. BQP.
Let a, b be functions asin Definition . The class BQP ( a, b ) is the setof languages L ⊆ { , } ∗ such that there existsan ( a, b ) -procedure Q = { Q n : n ∈ N } , with Q n taking as input ( x, | k i ) (i.e. there is no wit-ness Hilbert space), where x ∈ { , } n is a binarystring of length n , and where for every x we have x ∈ L ⇔ Pr[ Q n ( x ) = 1] ≥ a , (17) x / ∈ L ⇔ Pr[ Q n ( x ) = 1] ≤ b . (18) By repeating the procedure a polynomial num-ber of times, the thresholds can be made expo-nentially close to and respectively. Thereforethe exact values of the bounds a and b are irrele-vant. Traditionally they are taken to be / and / . We will do here the same. Definition 12.
We define the class
BQP as BQP (2 / , / . We give an alternative definition of
BQP whichis closer to the definition of
QMA . Definition 13. The language class BQP . Let a, b be functions as in Definition . Let BQP ( a, b ) ⊆ QMA ( a, b ) be the set of languages L ⊆ { , } ∗ such that there exists: . an ( a, b ) quantum verification procedure Q = { Q n : n ∈ N } with Q n taking as input ( x, | ψ i ⊗ | k i ) , where x ∈ { , } n is a binarystring of length n , | ψ i is a state of m qubits,with m, k ∈ poly ;2. an efficiently preparable set of density ma-trices { ρ ( x ) } where ρ ( x ) acts on H m ;and where for every x, we have x ∈ L if and onlyif Pr[ Q n ( x, ρ ( x )) = 1] ≥ a, (19) and x / ∈ L if and only if ∀| ψ i , Pr[ Q n ( x, | ψ i ) = 1] ≤ b. (20) Theorem 2. BQP’=BQP[13].
For all a, b functions as in Definition , BQP ( a, b ) = BQP . The witness space has a structure that will becentral in what follows. This structure can bederived using the methods of [14, 15] based onJordan’s lemma, or more easily using the struc-ture of the POVM element corresponding to thequantum verification procedure accepting, see [1].We repeat here the formulation of [13].
Theorem 3. Structure of witness space[14, 1].
Given a quantum verification procedure Q = { Q n } , for all x ∈ { , } n there exists a ba-sis B Q ( x ) = {| ψ i i : 1 ≤ i ≤ m } of the witnessspace H m such that the acceptance probability oflinear combinations of the basis states does notinvolve interferences, that is for all α i such that P i | α i | = 1 , we have Pr[ Q n ( x, P i α i | ψ i i ) = 1]= P i | α i | Pr[ Q n ( x, | ψ i i ) = 1] . (21) Definition 14. Eigenbasis, spectrum andeigenspaces of a quantum verification pro-cedure.
Given a quantum verification procedure Q = { Q n } , for all x ∈ { , } n and for any eigen-basis B Q ( x ) = {| ψ i i} of Q for x ,1. for all | ψ i i ∈ B Q ( x ) , we call p i = Pr[ Q n ( x, | ψ i i ) = 1] (22) the acceptance probability of | ψ i i ; 2. we call the set of acceptance probabilities the spectrum of Q for x : Spect(
Q, x ) = { p ∈ [0 ,
1] : ∃| ψ i i ∈ B Q ( x )such that Pr[ Q n ( x, | ψ i i ) = 1] = p } ;(23)
3. for p ∈ Spect(
Q, x ) , we call H Q ( x, p ) = Span( {| ψ i i ∈ B Q ( x ): Pr[ Q n ( x, | ψ i i ) = 1] = p } )(24) the eigenspace of Q for x with acceptanceprobability p . Theorem 4. Uniqueness of the spectrumand eigenspaces of Q [1, 13]. Given a quan-tum verification procedure Q = { Q n } , for all x ∈ { , } ∗ , the spectrum Spect(
Q, x ) of Q andthe eigenspaces H Q ( x, p ) of Q with acceptanceprobability p ∈ Spect(
Q, x ) are unique and do notdepend on the choice of eigenbasis B Q ( x ) . In this section we are interested in the set of stateson which a quantum verification procedure Q ac-cepts with high probability. We may also be in-terested in the set of states on which Q rejectswith high probability. These sets will allow usto characterise the functional analogs of the com-plexity classes introduced previously.There are two approaches to characterisingthese sets. The first is based on the notion ofwitness introduced in Definition 4. The seconduses the notion of eigenbasis of a quantum verifi-cation procedure, see Theorems 3 and 4. Definition 15. Accepting density matrices.
Let Q = { Q n } be a quantum verification proce-dure and fix a ∈ [0 , .We define the following relations over binarystrings and density matrices: R ≥ aQ ( x, ρ ) = 1 if Pr[ Q n ( x, ρ ) = 1] ≥ a , (25) R ≤ aQ ( x, ρ ) = 1 if Pr[ Q n ( x, ρ ) = 1] ≤ a . (26) o simplify notation, we denote R ≥ aQ ( x ) = n ρ : R ≥ aQ ( x, ρ ) = 1 o , (27) R ≤ aQ ( x ) = n ρ : R ≤ bQ ( x, ρ ) = 1 o . (28) Consider an ( a, b ) -procedure Q . We are in-terested in the functional task of outputting awitness for Q , and in defining the correspond-ing complexity class. This motivates the follow-ing definitions, where the subscript W stands for“Witness”. Definition 16. Witness based definitionof Functional QMA (FQMA W ). Let a, b be functions as in Definition . The class FQMA W ( a ) is the set { R ≥ aQ ( x, ρ ) } where Q isan ( a, b ) -procedure We can also define the functional analog of
BQP , for which we use Definition 13.
Definition 17. Functional BQP (FBQP W ). The class
FBQP W ( a ) is the set of relations { R ≥ aQ ( x, ρ ) } ⊆ FQMA W ( a ) where Q is an ( a, b ) –procedure, and where for each relation R ≥ aQ ( x, ρ ) there exists an efficiently preparable family ofstates { ρ ( x ) } such for all x , ρ ( x ) ∈ R ≥ aQ ( x ) . We can also define the functional class totalfunctional
QMA . Definition 18. Witness based definition oftotal Functional QMA (TFQMA W ). Let a : N → [0 , be a polynomially time com-putable function. The class TFQMA W ( a ) ⊆ FQMA W ( a ) is the set of relations { R ≥ aQ ( x, ρ ) } where Q is an a -total procedure. The above definitions are natural. However theyare not completely satisfactory for several rea-sons.First, we do not know whether
FQMA W ( a ) is independent of the threshold a . We obviouslyhave that FQMA W ( a ) ⊆ FQMA W ( a ) if a ≤ a ,but we do not know if the reverse is true, as FQMA W ( a ) does not transform simply underamplification.Second, while the sets R ≥ aQ ( x, ρ ) are convex,they are not closed under linear combinations ofpure states. That is, if the projectors onto | ψ i and | ψ i belong to R ≥ aQ ( x, ρ ) , then the projector onto the linear combination a | ψ i + b | ψ i does notnecessarily belong to R ≥ aQ ( x, ρ ) .Third, the very useful concept of eigenstateand eigenspace is not captured by the functionalclasses based on witnesses. This is illustrated inthe following example: Example 1.
Consider a function δ : N → [0 , / that decreases faster than / poly( n ) forany polynomial poly( n ) , for instance δ ( n ) =2 − n − . The example consists of a (1 / , / –quantum verification procedure Q example whosespectrum is the set Spect( Q example , x ) = { , − δ ( n ) ,
23 + δ ( n ) } (29) where n = | x | . Let us denote by | ψ / δ i and | ψ / − δ i twoeigenstates with acceptance probability / δ and / − δ respectively. Then the followingstate | ψ i = s δ δ | ψ / δ i + s
11 + δ | ψ / − δ i (30) is a witness (it has acceptance probability / ),hence belongs to R Q ( x, ) . But it has exponen-tially small overlap with the eigenspace H Q ( x, + δ ( n )) .These difficulties are avoided by the followingdefinitions, based on the notion of eigenspace ofa quantum verification procedure. Definition 19. Accepting and rejectingsubspaces.
Let Q = { Q n } be a quantum ver-ification procedure and fix a, b ∈ [0 , .We define the following relations over binarystrings and quantum states: H ≥ aQ ( x, | ψ i ) = 1 if | ψ i ∈ Span( {H Q ( x, p ) : p ≥ a } ) , (31) H ≤ bQ ( x, | ψ i ) = 1 if | ψ i ∈ Span( {H Q ( x, p ) : p ≤ b } ) , (32) where H Q ( x, p ) are the eigenspaces of Q for x .To simplify notation, we denote H ≥ aQ ( x ) = n | ψ i : H ≥ aQ ( x, | ψ i ) = 1 o , (33) H ≤ bQ ( x ) = n | ψ i : H ≤ bQ ( x, | ψ i ) = 1 o , (34) and we will generally express results in terms ofthe subspaces H ≥ aQ ( x ) , H ≤ bQ ( x ) , rather than thecorresponding relations. hese relations provide the basis for an alterna-tive definition of functional classes. These classesare denoted without the subscript W . Definition 20. Functional QMA(FQMA).
Let a, b be functions as inDefinition . The class FQMA ( a, b ) is theset { ( H ≥ aQ ( x, | ψ i ) , H ≤ bQ ( x, | ψ i )) } of pairs ofrelations, where Q is an ( a, b ) -verificationprocedure. This definition is in terms of two relations ( H ≥ aQ and H ≤ bQ ) for reasons which were presented in[13].One can show that this definition of FQMA does not depend on the bounds ( a, b ) : Theorem 5. FQMA is independent of thesoundness and completeness bounds [13].
Let Q be a quantum verification procedure. Let a, b be a pair of functions as in Definition . Forany r ∈ poly and pair of functions a , b as inDefinition satisfying a < − − r and b > − r ,the equality FQMA ( a, b ) = FQMA ( a , b ) holds. Hence we take the traditional values / and / : Definition 21.
We define the class
FQMA as FQMA (2 / , / . We introduced a –total procedures in Definition8. We can now define the corresponding func-tional classes. Definition 22. Totality.
A pair of rela-tions ( H ≥ aQ ( x, | ψ i ) , H ≤ bQ ( x, | ψ i )) in FQMA ( a, b ) is called total if for all inputs x there existsat least one witness | ψ i , i.e. if H ≥ aQ ( x ) is nonempty. Definition 23. Total Functional QMA(TFQMA).
Let a, b be functions as inDefinition . The class TFQMA ( a, b ) isthe set { ( H ≥ aQ ( x, | ψ i ) , H ≤ bQ ( x, | ψ i )) } of pairs oftotal relations, i.e. the set of pairs of relationsin FQMA where Q is an a –total verificationprocedure. Theorem 5 implies that we can take the thresh-olds to have their traditional values:
Definition 24.
We define the class of to-tal relations in
FQMA as TFQMA = TFQMA (2 / , / . A similar definition can be given for the func-tional analog of
BQP . However the definitionof
BQP is that there is a witness that can beefficiently prepared. Therefore functional
BQP needs to be expressed in terms of the existence ofan efficiently preparable witness. The definition
FBQP W thus seems the natural one in this case. QMA
We discuss here how the two definitions of Func-tional
QMA are related.
Theorem 6. Relation between definitionsof Functional QMA[13].
Let a, b be func-tions as in Definition and let Q be an ( a, b ) -procedure. Then,1. We have the inclusion H ≥ aQ ( x ) ⊆ R ≥ aQ ( x ) (35) (where we view H ≥ aQ ( x ) not as a set of purestates, but as the set of density matrices as-sociated to these pure states);2. In the other direction, if R ≥ aQ ( x ) is nonempty, then H ≥ aQ ( x ) is non empty. Definition 25. Reduction.
Let a, b and a , b be pairs of functions as inDefinition .Let L be a language in QMA ( a, b ) , and de-note by Q = { Q n : n ∈ N } the associated ( a, b ) -quantum verification procedure.Let L be a language in QMA ( a , b ) , and de-note by Q = { Q n : n ∈ N } the associated ( a , b ) -quantum verification procedure.A reduction from Q to Q is a pair ( f, Φ) where f : N → N is a polynomial time computablefunction, and Φ is a family of efficiently imple-mentable channel, such that:1. For all x ∈ L , it holds that f ( x ) ∈ L .In other words, for all x such that there ex-ists | ψ i satisfying Pr[ Q x | ( x, | ψ i ) = 1] ≥ a (36) i.e. | ψ i is a witness for Q for x ), it holdsthat there exists | ψ i satisfying Pr[ Q f ( x ) | ( f ( x ) , | ψ i ) = 1] ≥ a , (37) (i.e. | ψ i is a witness for Q for f ( x ) ).2. For all x , for all witnesses | ψ i for Q for f ( x ) , it holds that Φ( x, | ψ i ) is a witness for Q for x .In other words, for all x , for all | ψ i suchthat Pr[ Q f ( x ) | ( f ( x ) , | ψ i ) = 1] ≥ a , (38) it holds that Pr[ Q | x | ( x, Φ( x, | ψ i ) = 1] ≥ a . (39) To illustrate this definition, we recall brieflyKitaev’s construction that k -local-Hamiltonian is QMA complete[10].The k -local Hamiltonian problem, which is apromise problem, is defined as follows. The in-put x is a classical description of a k -local Hamil-tonian acting on n qubits, where we recall thata k -local Hamiltonian H is the sum of polyno-mially many Hermitian matrices (whose norm isbounded by a polynomial) that act on only k qubits. The input also contains two numbers a < b , such that b − a ∈ / poly . The problemis to determine whether the smallest eigenvalueof this Hamiltonian is less than a or greater than b , promised that one of these is the case.Kitaev showed that if Q ∈ QMA , then for ev-ery instance x , there exists an instance f ( x ) of k − local Hamiltonian –which we write H ( f ( x )) –such that:1) if x ∈ L there exists a low energy eigenstate of H ( f ( x )) ;2) if x / ∈ L then H ( f ( x )) does not have low en-ergy eigenstates.Recall that the witnesses in Kitaev’s construc-tion have the form √ L L X j =1 U j ( | ψ i| k i ) | j i (40) where the first register is the witness space, thesecond register is the ancilla space, and the thirdregister is the time counter, and L is the numberof gates in Q .A reduction from Q to k − local Hamiltonianconsists of: 1) the function f ( x ) ; 2) the channel Φ which consists in measuring the time counterof the witness to obtain j and then undo the com-putation conditional on j (i.e. apply U † j ). Theorem 7.
Let a, b , a , b , L , L , Q , Q be functions, languages and procedures as inDefinition .Suppose ( f, Φ) is a reduction from Q to Q .Then there exists an ( a, b ) –procedure Q R suchthat• The language associated to Q R is L ;• If | ψ i is a witness for Q for f ( x ) , then | ψ i is a witness for Q R for x , or expressed dif-ferently R ≥ a Q ( f ( x )) ⊆ R ≥ aQ R ( x ) (41) Proof.
Trivial. The procedure Q R is given by Q R ( x, | ψ i ) = Q ( x, Φ( | ψ i )). We note that it would be interesting to gener-alise slightly the definition of reduction of proce-dures, and only require that the channel Φ suc-ceed with probability greater than / poly . Forinstance in the case of a reduction from Q to k − local Hamiltonian the channel Φ could consistsin measuring the time counter of the witness andrejecting except if the result is j = 0 . Howeveras the above discussion of k − local Hamiltoniansuggests, a chanel Φ that always succeeds seemssufficient. In the present work we will supposethat Φ always succeeds. The case where Φ isprobabilistic would be interesting to study, butis a non–trivial generalisation. For instance it isnot obvious whether Theorem 7 below holds inthis case. Definition 26. QMA Completeness.
Let a, b be functions as in Definition .Let L be a language in QMA ( a, b ) , and de-note by Q = { Q n : n ∈ N } the associated ( a, b ) -quantum verification procedure.The procedure Q is QMA -complete if, for ev-ery a , b functions as in Definition , for everylanguage L in QMA ( a , b ) with Q = { Q n : n ∈ N } the associated ( a , b ) -quantum verifica-tion procedure, there exists a reduction from Q to Q . .2 Robust reductions For many questions concerning
QMA the abovenotion of reduction is sufficient. However Ex-ample 1 shows that it may not be appropriatein some cases. Suppose there exists a reduction ( f, Φ) from a procedure Q example to procedure Q . This reduction tells us how the witness of Q example (the eigenstates which accept with prob-ability / δ ) transform under Φ . But it tells usnothing about how eigenstates with acceptanceprobability / − δ transform under Φ . Butthese eigenstates are operationally indistinguish-able from the genuine witnesses (because theiracceptance probability is so close to / ), exceptpossibly by using the structure of Q example .In Section 16 we will need a slightly strongerform of reduction which we call robust reduction,and which does not suffer from this problem. Definition 27. Robust Reduction.
Let a, b , a , b , L , L , Q , Q be functions, lan-guages and procedures as in Definition .Recall that a − b ≥ /q for some q ∈ poly .Let (cid:15) ∈ / poly be such that (cid:15) ≤ / (2 q ) . (Asa consequence a − (cid:15) lies between a and b , at adistance ≥ / poly from either bound).A robust reduction from Q to Q with param-eter (cid:15) is a reduction from Q to Q via the pair ( f, Φ) where condition 2 of Definition is re-placed by the stronger condition2’. For all x , for all | ψ i such that Pr[ Q f ( x ) | ( f ( x ) , | ψ i ) = 1] ≥ a − (cid:15) , (42) it holds that Pr[ Q | x | ( x, Φ( x, | ψ i ) = 1] ≥ a . (43) Note that a robust reduction with parameter (cid:15) = 0 is a reduction according to Definition . Proposition 1.
The following hold for all (cid:15), (cid:15) ≥ :1. A robust reduction with parameter (cid:15) is alsoa robust reduction with parameter (cid:15) for all ≤ (cid:15) ≤ (cid:15) .2. Transitivity of reductions. If there exists arobust reduction from Q to Q with parame-ter (cid:15) , and if there exists a robust reductionfrom Q to Q with parameter (cid:15) , then thereexists a robust reduction from Q to Q withparameter (cid:15) . Proof. Immediate.
We note that k –local Hamiltonian is also QMA complete using the stronger notion of robust re-duction. This follows from the promise that thatthe smallest eigenvalue of the k –local Hamilto-nian is less than a or greater than b . Reductions (robust or not) do not preserve theeigenspace structure of a quantum verificationprocedure. The following definitions and resultsrepeated from [13], define a mapping betweenprocedures that preserves this structure.
Definition 28. Eigenspace preserving mapof quantum verification procedures.
Let Q and Q be two quantum verification procedureswith the same witness space dimensions. We saythat there exists an eigenspace preserving mapfrom Q to Q if there exists a polynomial timecomputable strictly increasing family of functions { f n : n ∈ N } , f n : [0 , → [0 , , such that for all x ∈ { , } ∗ with n = | x | :1. there exists a basis B Q ( x ) = {| ψ i i} of thewitness Hilbert space H m which is a jointeigenbasis of Q and Q for x ;2. for all | ψ i i ∈ B Q ( x ) with p i =Pr[ Q n ( x, | ψ i i ) = 1] the acceptance probabil-ity of | ψ i i for Q n , and p i = Pr[ Q n ( x, | ψ i i ) =1] the acceptance probability of | ψ i i for Q n ,it holds that p i = f n ( p i ) .In what follows we will refer to an eigenspace pre-serving map simply as an e–map . The reason why we require that the functions f n be polynomial time computable is because wewish that the soundness and completeness thresh-olds of Q be mapped onto the soundness and com-pleteness thresholds of Q , where we recall thatthe soundness and completeness thresholds mustbe polynomial time computable, see Definition 3.That is, if Q is an ( a, b ) -procedure that e–mapsto Q via f n , then Q is an a , b -procedure with a ( n ) = f n ( a ( n )) and b ( n ) = f n ( b ( n )) . Proposition 2. [13]
The following hold:1. Conservation of eigenspaces. If Q e-mapsto Q via { f n } , then the eigenspace of Q or x with acceptance probability p equals theeigenspace of Q for x with acceptance prob-ability f n ( p ) : H Q ( x, p ) = H Q ( x, f | x | ( p )) . (44)
2. Conservation of accepting and rejecting sub-spaces. Let Q be an ( a, b ) –procedure and let Q be an ( a , b ) , and suppose that Q e-mapsto Q via { f n } with a ( n ) = f n ( a ( n )) and b ( n ) = f n ( b ( n )) , then H ≥ aQ ( x ) = H ≥ a Q ( x ) H ≤ bQ ( x ) = H ≤ b Q ( x ) (45)
3. Transitivity of reductions. If Q e–maps to Q , and Q e–maps to Q , then Q e–maps to Q .
10 Iterative Procedures
The amplification result for
FQMA , Theorem 5,is based on the method used in [14] in which aquantum verification procedure is run repeatedlybackwards and forwards. Here we generalise thisapproach.
Definition 29. Iterative Procedures.
Let Q = { Q n : n ∈ N } be a quantum verificationprocedure with parameters m (the size of the wit-ness space) and k (the size of the ancilla space).Recall that the Hilbert space H on which Q n acts can be decomposed into H = H wm ⊗ H ak where H wm is the witness Hilbert space comprising m qubits, and H ak is the ancilla Hilbert space com-prising k qubits. The Hilbert space can also bedecomposed into H = H out ⊗ H restm + k − where H out of dimension 2 is the qubit which is measuredat the end of the quantum verification procedure,and H restm + k − is the Hilbert space of the remaining m + k − qubits.Define the following two projectors Π = I m ⊗ | k ih k | (46)Π = V † x (cid:16) | ih | ⊗ I restm + k − (cid:17) V x (47) where V x is the unitary transformation realisedby the quantum verification procedure. That is Π projects onto valid inputs to the quantum ver-ification procedure, and Π projects onto stateson which the quantum verification procedure willaccept with certainty. Let N ∈ poly and let G = { g n : { , ..., N ( n ) } → [0 ,
1] : x ∈ N } (48) be a family of polynomially time computable func-tions.The ( N, G )–iterative procedure obtained from Q is a quantum verification procedure Q it withthe same witness space dimension m as Q , andan ancilla space which we decompose into a firstspace of k qubits (the same dimension as for Q )and an additional working space of dimension k .This second space can be classical. The inputspace of Q it is thus H m ⊗ H k ⊗ H k . We viewthe projectors Π and Π as acting on the space H wm ⊗ H k .On input ( x, | ψ i| k i| k i ) with | ψ i ∈ H m , Q it acts as follows:1. Repeat until N + 1 measurement outcomeshave been registered:(a) measure { Π , − Π } ;(b) measure { Π , − Π } .(Note: the result of the first measurementwill necessarily be since the initial state ofthe first m + k qubits ( | ψ i| k i ) is an eigen-state of Π with eigenvalue ).2. Inspect the sequence of results.(a) If results i and i + 1 differ set z i = 0 ;(b) while if results i and i + 1 are equal set z i = 1 .3. Set s = P Ni =1 z i .4. Generate a random bit c ∈ { , } satisfying Pr[ c = 1] = g | x | ( s ) and Pr[ c = 0] = 1 − g | x | ( s ) .5. Output c . Note that, in view of step 2, the functions g n ( s ) ∈ [0 , can be interpreted as probabilitydistributions over the set { , } . One can ex-tend the above generalised amplification proce-dure and take the g n ( s ) to be probability distri-butions over the alphabet { , ..., d − } , in whichcase the generalised amplification would yield a d -outcome quantum verification procedure.A particularly interesting case of iterative pro-cedure is when the g n are threshold functions: g n ( s ) = 0 if s < s ( n )= 1 if s ≥ s ( n ) (49) or some polynomial time computable function s : N → N with s ( n ) ∈ { , ...N ( n ) } . The QMA amplification described in [14] is based ona threshold iterative procedure.In the following we will denote by f ( k ; N, p ) the probability mass function of the binomialdistribution with parameters N and p . Thus if X ∼ B ( N, p ) , then f ( k ; N, p ) = Pr[ X = k ] = Nk ! p k (1 − p ) N − k . (50) In addition, for any g : { , ..., N } → [0 , , wedenote by P g the function P g : [0 , → [0 , p → P g ( p ) = E [ g ( X )] = N X k =0 f ( k ; N, p ) g ( k ) . (51) Theorem 8. Properties of Iterative Proce-dures.
Let Q = { Q n : n ∈ N } be a quantum ver-ification procedure, let ( N, G ) be as in Definition29, and let Q it be the ( N, G ) -iterative procedureobtained from Q . Then the following hold:1. There exists a basis B Q ( x ) = {| ψ i i} of thewitness Hilbert space H m which is a jointeigenbasis of Q and Q it for x .2. Let | ψ i i be an eigenstate of Q for x (andtherefore also an eigenstate of Q it for x . De-note by p i and p i the acceptance probabilitiesof | ψ i i for Q and Q it respectively.If the initial state of the witness space is | ψ i i ,then the following hold:(a) The variables z i defined in step 2 are − i.i.d. random variables satisfying Pr[ z i = 1] = p i (i.e. equals the accep-tance probability of | ψ i i ).(b) The acceptance probabilities of the iter-ative procedure is given by p i = P g | x | ( p i ) . (52) (c) Whenever at step 1a of Definition 29the outcome of measurement { Π , − Π } is , the state of the witness spaceis the original witness | ψ i i .Proof. Not given. Simple application of the con-struction in [14].
In order to apply these notions to functional
QMA , we need that the ordering of the accep-tance probabilities does not change. In this caseiterative procedures will be e-maps. The follow-ing result give a sufficient condition for this to bethe case.
Theorem 9. Iterative procedures with in-creasing functions g n are eigenspace pre-serving maps. Given a quantum verificationprocedure Q , and ( N, G = { g n } ) as in Definition29 with the functions g n : { , ..., N } → [0 , in-creasing and nonconstant (by this we mean that g ( k +1) ≥ g ( k ) and g ( N ) > g (0) ), then Q e–mapsto the ( N, G ) -generalised amplification procedure Q it obtained from Q .Proof. The condition that g is increasing andnonconstant implies that there is some k ∈{ , ..., N − } such that g ( k + 1) > g ( k ) (witha strict inequality).We need to show that under the conditionsthat the functions g n are polynomial time com-putable, increasing and nonconstant, p i given inEq. (52), viewed as a function of p i , is a poly-nomial time computable strictly increasing func-tion.It is therefore sufficient to show that for anypolynomial time computable strictly increasingfunction g , the function P g ( p ) defined in Eq. (51)is a polynomial time computable strictly increas-ing function of p .The fact that P g ( p ) is polynomial time com-putable follows immediately from the fact thatthe functions g n are polynomial time computable.We now show that P g ( p ) is strictly increasing.First note that for all p ∈ [0 , N X k =0 f ( k ; N, p ) . (53)Taking the derivative of Eq. (53) with respect to p , we have0 = N X k =0 Nk ! p k − (1 − p ) N − k − ( k − N p ) . (54)(Note that as written this expression is well de-fined only for 0 < p <
1. However the limits p → p → p = 0 and p = 1 byreplacing the ill–defined terms by their limit). ence P b Np c k =0 (cid:0) Nk (cid:1) p k − (1 − p ) N − k − ( N p − k )= P Nk = d Np e (cid:0) Nk (cid:1) p k − (1 − p ) N − k − ( k − N p ) , (55)where we note that all terms under summationsigns are positive.Now take the derivative of Eq. (51) with re-spect to p to obtain ∂P g ∂p = N X k =0 Nk ! p k − (1 − p ) N − k − ( k − N p ) g ( k )= − b Np c X k =0 Nk ! p k − (1 − p ) N − k − ( N p − k ) g ( k )+ N X k = d Np e Nk ! p k − (1 − p ) N − k − ( k − N p ) g ( k )(56) ≥ − b Np c X k =0 Nk ! p k − (1 − p ) N − k − ( N p − k ) g ( b N p c )+ N X k = d Np e Nk ! p k − (1 − p ) N − k − ( k − N p ) g ( d N p e )(57)= N X k = d Np e Nk ! p k − (1 − p ) N − k − ( k − N p ) × ( g ( d N p e ) − g ( b N p c )) (58) ≥ k ∈ { , ..., b N p c − } , or k = b N p c , or k ∈ {d N p e , ..., N − } . In the sec-ond case, there is a strict inequality when goingfrom Eq. (58) to Eq. (59). In the first case or thethird case, there is a strict inequality when goingfrom Eq. (56) to Eq. (57). Therefore under thecondition that g is increasing and nonconstant, ∂P g ∂p >
0, i.e. P g is strictly increasing. This result shows that there are many e–maps.We now show that one can almost freely choosethe functions p i = f n ( p i ) that define the e–map. Theorem 10.
Let Q = { Q n } be a quantum ver-ification procedures with witness Hilbert space di-mension m ( n ) . Let u, v, M ∈ poly , and denote (cid:15) ( n ) = 1 /u ( n ) and δ ( n ) = exp( − v ( n )) . Let S and T denote the following sets: S = { S n : n ∈ N } S n = ( p , p , ..., p M , p M +1 ) p i ∈ [0 , , p = 0 , p M +1 = 1 ∀ i : p i − p i − ≥ (cid:15) > ∀ i : p i is polynomial time computableand T = { T n : n ∈ N } T n = ( q , q , ..., q M , q M +1 ) q i ∈ [0 , , q = 0 , q M +1 = 1 ∀ i : q i − q i − ≥ δ > ∀ i : q i is polynomial time computablewhere M = M ( n ) . (Note that the definitions of S and T differ only by Eqs. (60) and (61) whichensure that p i and q i are ranked in increasing or-der, but with different gaps).Then there exists a quantum verification pro-cedure Q such that Q e-maps to Q and the poly-nomial time computable strictly increasing func-tions f n that define the e-map satisfy f n ( p i ) = q i for all p i ∈ S n .Proof. Outline of the proof.
We will use The-orem 9. We will show that there exists (
N, G = { ¯ g n } ) as in Definition 29, with the functions ¯ g n : { , ..., N } → [0 ,
1] polynomial time computableand strictly increasing, such that the (
N, G )-generalised amplification procedure Q obtainedfrom Q satisfies the conditions of Theorem 10.The polynomial time computable strictly increas-ing functions f n that define the e-map will begiven by P ¯ g n . Definition of ¯ g n . Choose a function N ∈ poly. We will deter-mine how large N must be at the end of the proof.We denote by∆ i = [ N ( p i − (cid:15) , N ( p i + (cid:15) , ∆ = [0 , N (cid:15) , ∆ M +1 = [1 − N (cid:15) , , Ξ i = ) N ( p i + (cid:15)/ , N ( p i +1 − (cid:15)/ , (62)where i = 1 , ..., M . And we denote by¯∆ i = { n ∈ N : n ∈ ∆ i } , ¯Ξ i = { k ∈ N : n ∈ Ξ i } (63)the integers belonging to these intervals. et X i ∼ B ( N, p i ). For all p i , i = 1 , ..., M ,denote Pr[ X i ∈ ¯∆ i ] = 1 − µ i . (64)Denote µ = max i { µ i } . (65)Note that by taking N sufficiently large, we canensure that µ is exponentially small, and in par-ticular smaller than δ .Let ( λ , λ , ...λ M , λ M +1 ) be parameters we willfix later, except that we fix λ = 0 and λ M +1 = 0.Denote Λ = max i {| λ i |} . (66)Let σ = δ (cid:15) . (67)Consider the continuous, piece wise linearfunctions g n : [0 , N ] → R which on ∆ i is givenby g n ( t ) = ( q i + λ i ) + σ ( t − N p i ) , t ∈ ∆ i , (68)and which are linear outside of the intervals∆ i . Continuity then implies that for t ∈ Ξ i , i = 0 , ..., M , one has the expression: g n ( t ) = ( q i +1 + λ i +1 ) − ( q i + λ i ) − (cid:15)σ/ N ( p i +1 − p i − (cid:15)/ ×× ( t − N ( p i + (cid:15)/ q i + λ i + σ (cid:15) . (69)We denote by ¯ g n : { , ..., N } → [0 ,
1] the func-tions which coincide with the functions g n on theintegers. Strategy for proving the existence of ane–map with the desired properties.
We wish to show that for N large enough, onecan choose the parameters λ i such that:1. Q e-maps to the ( N, G = { g n } ) generalisedamplification procedure obtained from Q ;2. and that for all iq i = N X k =0 f ( k ; N, p i )¯ g n ( k ) . (70)Note that if λ i = 0, then ¯ g n ( N p i ) = q i .Hence if the distributions f ( k ; N, p i ) are stronglypeaked around N p i (which is the case if N is largeenough), one expects that Eqs. (70) can be sat-isfied for small λ i . We show that this is indeedthe case below. Boundary values.
The functions g n satisfy g n (0) = 0 and g n ( N ) = 1. Condition for strict increase.
Consider the slope of the functions g n . Overthe intervals ∆ i the slope is σ which is strictlypositive.Between the intervals ∆ i , the slope is equal to( q i +1 + λ i +1 ) − ( q i + λ i ) − (cid:15)σ/ N ( p i +1 − p i − (cid:15)/ , (71)see Eq. (69). This is strictly positive if ( q i +1 + λ i +1 ) − ( q i + λ i ) − (cid:15)σ/ q i +1 + λ i +1 ) − ( q i + λ i ) − (cid:15)σ/ > δ/ − g n being strictly increasing isΛ < δ/ Computing λ i . Eqs. (70) are a set of M linear equations inthe λ i , with coefficients that can be efficientlycomputed. Hence, since p i and q i are polynomialtime computable, the functions g n are polynomialtime computable.It remains to show that for N large enoughsolving Eqs. (70) for the λ i yields solutions thatsatisfy Eq. (73). Properties of invertible matrices.
Let A ∈ R M × M be an M × M matrix, anddenote by A ij its elements. Denote by k A k = max ij | A ij | (74)their entry–wise 1-norm.Recall that if A, A ∈ R M × M , then the productmatrix AA has norm k AA k ≤ M k A k k A k .From this it follows that if k A k ≤ α , then k A k k ≤ M k − α k .From the identity( I + A + A + ... + A k )( I − A ) = I − A k − (75)it follows that the matrix (1 − A ) is invertible iflim k →∞ A k = 0, in which case( I − A ) − = lim k →∞ ( I + A + A + ... + A k ) . (76) n particular ( I − A ) is invertible if k A k < /M , and one has k ( I − A ) − k ≤ − M k A k . (77)In particular if k A k < / (2 M ), then k ( I − A ) − k < B ∈ R M be an M component vector. De-note by k B k = max i | B i | (78)its entry–wise 1-norm. If A ∈ R M × M and B ∈ R M , then the vector AB has norm bounded by k AB k ≤ M k A k k B k . (79)As a consequence, if one needs to solve thesystem of M linear equations for λ i , with A ∈ R M × M and B ∈ R M ,( I − A ) λ = B , (80)and if k A k < / (2 M ), thenmax i λ i ≤ M max i | B i | (81) System of equations for λ i . Eqs. (70), viewed as equations for the λ i , canbe rewritten in the form (80). We will now boundthe entry–wise 1-norm of the corresponding ma-trix A and vector B .Explicitly, Eqs. (70) take the form q i = X k ∈ ¯∆ i f ( k ; N, p i )( q i + λ i )+ X k ∈ ¯∆ i f ( k ; N, p i ) σ ( k − N p i )+ X k ∈{ ,...,N }\ ¯∆ i f ( k ; N, p i ) g n ( k ) (82)We now consider the contributions of the threeterms in Eq. (82) to | A ij | and | B i | . Left hand side and first term.
The firstterm on the right hand side in Eq. (82) is X k ∈ ∆ i f ( k ; N, p i )( q i + λ i ) = (1 − µ i )( q i + λ i ) . (83)Therefore, the contribution from Eq. (83) to | A ii | is µ i . And the joint contribution of Eq. (83) andthe left hand side of Eq. (82) to | B i | is µ i q i . Second term. σ | X k ∈ ∆ i f ( k ; N, p i )( k − N p i ) | = σ | X k ∈{ ,...,N }\ ∆ i f ( k ; N, p i )( k − N p i ) |≤ µ i σN (84)where we have used Eq. (54). Therefore the con-tribution from Eq. (84) to | B i | is at most µ i σN. Third term.
Note that when λ i = 0 for all i ,then g n ( t ) ∈ [0 , λ ’s are boundedby µ i . Therefore the third term contributes atmost µ i to | B i | .The parts of the third term proportional to λ are given by M +1 X i =0 i = i X k ∈ ¯∆ i f ( k ; N, p i ) λ i (85)+ M X i =0 X k ∈ ¯ S i f ( k ; N, p i ) ×× ( λ i +1 − λ i ) k − N ( p i + (cid:15)/ N ( p i +1 − p i − (cid:15)/
3) (86)+ M X i =0 X k ∈ ¯ S i f ( k ; N, p i ) λ i (87)Note that the coefficient of each λ i in lines(85) and (87) is bounded by µ . Therefore thecontribution from (85) to | A ii | (with i = i ) is atmost µ ; and the contribution from (87) to | A ii | is at most µ .Note that N ( p i +1 − p i − (cid:15)/ ≥ N (cid:15)/ k − N ( p i + (cid:15)/ ≤ N , hence the factor ( k − N ( p i + (cid:15)/ / ( N ( p i +1 − p i − (cid:15)/ /(cid:15) . Hence the coefficient of each λ i and λ i +1 in line (86) is bounded by 3 µ/(cid:15) .Therefore the contribution from (86) to | A ii | isat most 6 µ/(cid:15). For large N , λ i are unique and small. Putting all together, the matrix A in Eq. (80)is bounded by k A k ≤ µ (2 + 6 /(cid:15) ) . (88)And the right hand side of Eq. (80) is boundedby | B i | ≤ µ i ( q i + σN + 1) ≤ µ (2 + σN ) . (89) herefore for large enough N , i.e. µ sufficientlysmall, k A k ≤ / (2 M ). When this is the casethere is a unique solution for the λ ’s, and | λ i | ≤ M (2 + σN ) µ . (90)Since µ decreases exponentially with N , and theother factors increase polynomially with N , bytaking N large enough one can ensure that Eq.(73) is satisfied.
11 Nondestructive procedures
As a first application of Theorem 10, we introducethe notion of nondestructive procedure.
Definition 30. Nondestructive ( a, b ) –Quantum Verification Procedure. Let a, b befunctions as in Definition . An ( a, b ) –quantumverification procedure Q = { Q n : n ∈ N } is nondestructive if it outputs both a classicalbit (the outcome of the quantum verificationprocedure) and a quantum state of m qubits(with m the dimension of the witness Hilbertspace), such that if the input of the procedure isan eigenstate | ψ i i of Q n for x then, conditionalon the classical output bit being (i.e. on theprocedure accepting), the quantum output is theeigenstate | ψ i i . Theorem 11. Properties of nondestructiveprocedures.
Let Q = { Q n : n ∈ N } be a non-destructive procedure as in Definition . Denoteby {| ψ i i} the eigenbasis of Q n for x , and by p i theacceptance probability of | ψ i i . Then the followinghold:1) If the quantum input of Q n is a pure state | ψ in i which we express in the eigenbasis of Q n as | ψ in i = X i α i | ψ i i , (91) then, conditional on the classical output bit being , the quantum output will be the pure state | ψ out i = 1 pP i p i | α i | X i √ p i α i | ψ i i . (92)
2) If one uses the state Eq. (92) as input tothe procedure Q n , the probability of acceptancewill be larger than the probability of acceptanceon the original state Eq. (91) : Pr[ Q n ( x, | ψ out i ) = 1] ≥ Pr[ Q n ( x, | ψ in i ) = 1] . (93) Proof.
The expression Eq. (92) is immediate.We prove Eq. (93). To this end, we introducethe positive operator Q x = P i p i | ψ i ih ψ i | , whichis the POVM element corresponding to the clas-sical output of the procedure being 1 (i.e. accept-ing). Then we can writePr[ Q n ( x, | ψ in i ) = 1] = h ψ in | Q x | ψ in i (94)and | ψ out i = √ Q x | ψ in i q h ψ in | Q x | ψ in i . (95)ConsequentlyPr[ Q n ( x, | ψ out i ) = 1] = h ψ out | Q x | ψ out i = h ψ in | Q x | ψ in ih ψ in | Q x | ψ in i≥ h ψ in | Q x | ψ in ih ψ in | Q x | ψ in ih ψ in | Q x | ψ in i = h ψ in | Q x | ψ in i , (96)where we have used the fact that I m ≥ | ψ in ih ψ in | with I m the identity operator. We now show that without loss of generalitywe can take quantum verification procedures tobe nondestructive.
Theorem 12. Existence of nondestructiveprocedures.
Let Q = { Q n } be a quan-tum verification procedures, and let S = { S n } , S n = ( p , p , ..., p M , p M +1 ) , T = { T n } , T n =( q , q , ..., q M , q M +1 ) be as in Theorem . Thenthere exists a nondestructive quantum verifica-tion procedure Q ND such that Q e–maps to Q ND and the polynomial time computable strictly in-creasing functions f n that define the e–map sat-isfy f n ( p i ) = q i for all p i ∈ S n .Proof. Step 1: procedure Q (1) . Use Theorem 10 to construct a procedure Q (1) with the property that the polynomial time com-putable strictly increasing functions f (1) n that de-fine the e–map satisfy f (1) n ( p i ) = √ q i . Step 2: procedure Q ND . Procedure Q ND is obtained as follows: Runthe iterative procedure of Definition 29 with pa-rameter N = 2 on procedure Q (1) , obtaining 2bits z and z . Accept if z = z = 1. Otherwisereject. Step 3: proof that Q ND is a nondestruc-tive procedure with the desired properties. . Q e–maps to Q (1) and Q (1) e–maps to Q ND ,hence Q e–maps to Q ND . Therefore theeigenbasis {| ψ i i} of Q for x is also an eigen-basis Q (1) and of Q ND for x .2. The polynomial time computable strictly in-creasing functions f n that defines the e–mapsatisfies f n ( p ) = (cid:16) f (1) n ( p ) (cid:17) , hence it satisfies f n ( p i ) = q i .3. If Q ND accepts, then the procedure hasended with a measurement of { Π , − Π } with outcome 1. Therefore, by property 2cof Theorem 8, if the original state of the wit-ness space was an eigenstate | ψ i i of Q , thenconditional on Q ND accepting, the state ofthe witness space is | ψ i i .
12 Equivalent definitions of
QMA ∩ coQMA Definition 5 is our starting point for studying
QMA ∩ coQMA . It is however not a very satis-factory for defining functional QMA ∩ coQMA .Indeed, recall that functional QMA is based onthe existence of a quantum verification proce-dure, of witnesses (i.e. of states that are acceptedwith high probability by the quantum verifica-tion procedure and certify that x ∈ L ), and ofthe eigenbasis of the quantum verification proce-dure. However it does not seem possible to in-troduce these notions starting from Definition 5.The difficulty stems from the fact that the twoquantum verification procedures Q and Q do notcommute.To circumvent this we introduce two alterna-tive definitions of QMA ∩ coQMA , the first isbased on a 3-outcome quantum verification pro-cedure, and the second is based on a outcomequantum verification procedure. The later is par-ticularly useful, as it allows us to apply the no-tions of eigenbasis, spectrum, eigenspaces, andamplification in the context of QMA ∩ coQMA .The inspiration for the following definitioncomes from the fact that a quantum verificationprocedure for a language L ∈ QMA has twooutcomes, but these two outcomes play differ-ent roles. Outcome certifies that x ∈ L (upto some uncertainty, since outcomes are proba-bilistic), while outcome does not provide infor- mation whether x ∈ L or x / ∈ L . In the case of QMA ∩ coQMA we need one outcome to certifythat x ∈ L , one outcome to certify that x / ∈ L ,and one outcome that does not provide informa-tion. Definition 31. ( a , b ; a , b ) –Three OutcomeQuantum Verification Procedure. Let a , b and a , b be functions as in Definition . An ( a , b ; a , b ) –Three Outcome Quantum Verifica-tion Procedure is a -outcome procedure Q = { Q n : n ∈ N } whose outcomes are denoted { , L, L } , and such that for every x of length n ,either both of the following hold: ∃| ψ i , Pr[ Q n ( x, | ψ i ) = L ] ≥ a , (97) ∀| ψ i , Pr[ Q n ( x, | ψ i ) = L ] ≤ b ; (98) or both the following hold: ∃| ψ i , Pr[ Q n ( x, | ψ i ) = L ] ≥ a , (99) ∀| ψ i , Pr[ Q n ( x, | ψ i ) = L ] ≤ b . (100) Definition 32. The language class L3.
Let a , b and a , b be functions as in Definition .Let L3 ( a , b ; a , b ) be the set of languages L ⊆{ , } ∗ such that there exists an ( a , b ; a , b ) –three outcome quantum verification procedure Q = { Q n : n ∈ N } , where for every x , we have x ∈ L if and only if Equations (97) and (98) hold(and consequently we have x / ∈ L if and only ifEquations (99) and (100) hold). The following definitions will enable us to pro-vide a definition of
QMA ∩ coQMA based on aquantum verification procedure that only has twooutcomes. The idea behind this definition is thatif outcome occurs with high probability this cer-tifies that x ∈ L , if outcome occurs with highprobability this certifies that x / ∈ L , while if out-comes and occur with approximately the sameprobability, then no information is obtained. Definition 33. ( a , b ; a , b ) –Quantum Ver-ification Procedure. Let a , b and a , b befunctions as in Definition . An ( a , b ; a , b ) –Quantum Verification Procedure is a quantumverification procedure Q = { Q n : n ∈ N } suchthat for every x of length n , either both of thefollowing hold: ∃| ψ i , Pr[ Q n ( x, | ψ i ) = 1] ≥
12 + a , (101) ∀| ψ i , Pr[ Q n ( x, | ψ i ) = 1] ≥ − b r both the following hold: ∃| ψ i , Pr[ Q n ( x, | ψ i ) = 0] ≥
12 + a , (103) ∀| ψ i , Pr[ Q n ( x, | ψ i ) = 0] ≥ − b . (104) Definition 34. The language class L2.
Let a , b and a , b be functions as in Definition .Let L2 ( a , b ; a , b ) be the set of languages theset of languages L ⊆ { , } ∗ such that there existsan ( a , b ; a , b ) –quantum verification procedure Q = { Q n : n ∈ N } , where for every x , we have x ∈ L if and only if Equations (101) and (102) hold (and consequently we have x / ∈ L if and onlyif Equations (103) and (104) hold). We note that the language class L2 is indepen-dent of the bounds used to define it. Theorem 13. L2 ( a , b ; a , b ) is independentof the completeness and soundness proba-bilities. Let a , b and a , b be pairs of functions as inDefinition .For any r ∈ poly , for any ˜ a , ˜ b and ˜ a , ˜ b pairs of functions as in Definition with ˜ a , ˜ a < − − r and ˜ b , ˜ b > − r , it holds that L2 ( a , b ; a , b ) = L2 (˜ a , ˜ b ; ˜ a , ˜ b ) Proof.
The proof is an immediate application ofTheorem 10.
We now show that all these definitions areequivalent.
Theorem 14. QMA ∩ coQMA = L3 = L2.
For all a, b , a, b , pairs of functions as inDefinition , the following equalities hold QMA ∩ coQMA = L3 ( a, b ; a , b ) = L2 ( a, b ; a , b ) . (105) Proof.
Step 1: QMA ∩ coQMA ( a, b ; a , b ) ⊆ L3 ( a, b ; a , b ) . Let a, b and a , b functions as in Definition 3.Let L ∈ QMA ∩ coQMA ( a, b ; a , b ). Then thereexist a, b and a , b functions as in Definition 3,and there exist two quantum verification proce-dures Q = { Q n : n ∈ N } and Q = { Q n : n ∈ N } such that Equations (11) to (14) hold. We nowshow that L ∈ L3 ( a, b ; a , b ).Recall that Q n takes as input ( x, | ψ i ⊗ | k i )where | x | = n , | ψ i is a state of m qubits, and both m = m ( n ) and k = k ( n ) are polynomial functions of n ; and that Q n takes as input ( x, | ψ i ⊗ | k i )where | x | = n , | ψ i is a state of m qubits, andboth m = m ( n ) and k = k ( n ) are polynomialfunctions of n .Define m = max { m, m } + 1, and k =max { k, k } . Define the circuit Q n which on in-put ( x, | ˜ ψ i ⊗ | k i ) with | x | = n and | ˜ ψ i a stateof m qubits, acts as follows:1. Measure the first qubit of | ˜ ψ i in the standardbasis.2. If the result of the first measurement is 1,then carry out the procedure Q n on input( x, | ˜ ψ i ,...,m +1 ⊗ | k i ) where the second reg-ister contains qubits 2 , ..., m + 1 of | ˜ ψ i andthe third register contains the first k ancillaqubits. If the outcome of Q n is 1, then out-put L ; while if the outcome of Q n is 0, thenoutput 0.3. If the result of the first measurement is 0,then carry out the procedure Q n on input( x, | ˜ ψ i ,...,m +1 ⊗ | k i ) where the second reg-ister contains qubits 2 , ..., m + 1 of | ˜ ψ i andthe third register contains the first k ancillaqubits. If the outcome of Q n is 1, then out-put L ; while if the outcome of Q n is 0, thenoutput 0.One easily checks that the family of circuits Q = { Q n : n ∈ N } thus defined satisfy Eqs. (97) and(98) when x ∈ L , and satisfy Eqs. (99) and (100)when x / ∈ L , with a = a, b = b, a = a , b = b . Step 2: L3 ( a, b ; a , b ) ⊆ QMA ∩ coQMA ( a, b ; a , b ) . Let a , b and a , b be pairs of functions asin Definition 3. Let L ∈ L3 ( a , b ; a , b ). Thenthere exists a a three outcome quantum verifica-tion procedure Q = { Q n : n ∈ N } that satisfiesEqs. (97) and (98) when x ∈ L , and satisfies Eqs.(99) and (100) when x / ∈ L . We now show that L ∈ QMA ∩ coQMA ( a , b ; a , b ).Note that Q n takes as input ( x, | ψ i ⊗ | k i )where | x | = n , | ψ i is a state of m qubits, andboth m = m ( n ) and k = k ( n ) belong to poly.Define two quantum verification procedures Q = { Q n : n ∈ N } and Q = { Q n : n ∈ N } as follows: Q n and Q n take as input ( x, | ψ i ⊗ | k i ), with | ψ i a state of m qubits.On input ( x, | ψ i⊗| k i ), run Q n ( x, | ψ i⊗| k i ). n the case of Q n :if Q n ( x, | ψ i ⊗ | k i ) = L output 1,if Q n ( x, | ψ i ⊗ | k i ) = 0 output 0,if Q n ( x, | ψ i ⊗ | k i ) = L output 0.In the case of Q n :if Q n ( x, | ψ i ⊗ | k i ) = L output 0,if Q n ( x, | ψ i ⊗ | k i ) = 0 output 0,if Q n ( x, | ψ i ⊗ | k i ) = L output 1.One easily checks that the family of circuits Q = { Q n : n ∈ N } and Q = { Q n : n ∈ N } thus defined satisfy satisfy Eqs. (11) and (12)when x ∈ L , and satisfy Eqs. (13) and (14) when x / ∈ L , with a = a , b = b and a = a , b = b . Summary of steps 1 and 2.
Since
QMA ∩ coQMA ( a, b ; a , b ) is indepen-dent of the bounds a, b ; a , b , see Definition 7,we have proven the first equality in Eq. (105),including the fact that L3 ( a, b ; a , b ) is indepen-dent of the bounds a, b ; a , b . Step 3: L3 ⊆ L2.
Let L ∈ L3 (3 / , /
4; 3 / , / Q = { Q n : n ∈ N } , n , m , k be defined as in thefirst two paragraphs of Step 2. We show that L ∈ L2 (1 / , /
4; 1 / , / Q = { Q n : n ∈ N } as follows:Run Q n ( x, | ψ i ⊗ | k i ) and• if Q n ( x, | ψ i ) = L then output 1;• if Q n ( x, | ψ i ) = 0, then output a randombit drawn uniformly at random from the set { , } ;• if Q n ( x, | ψ i ) = L then output 0.Let us consider the case x ∈ L . For brevity inwhat follows we omit the arguments of Q . Forall input states | ψ i we havePr[ Q = 1] = Pr[ Q = L ] + 12 Pr[ Q = 0]= Pr[ Q = L ]+ 12 (cid:16) − Pr[ Q = L ] − Pr[ Q = L ] (cid:17) = 12 + 12 Pr[ Q = L ] −
12 Pr[ Q = L ](106)Since x ∈ L , we have that for all | ψ i , Pr[ Q = L ] ≤ /
4. Furthermore we trivially have Pr[ Q = L ] ≥
0. Therefore Eq. (106) implies that for all | ψ i the following holds:Pr[ Q = 1] ≥ −
12 14 = 38 . (107) Since x ∈ L , there exists a | ψ i such thatPr[ Q = L ] ≥ /
4. Therefore Eq. (106) impliesthat for this | ψ i we havePr[ Q = 1] ≥
12 + 12 34 −
12 14 = 34 . (108)Therefore the family of circuits Q = { Q n : n ∈ N } satisfy Eqs. (101) and (102) when x ∈ L with a = 1 / b = 1 / x / ∈ L shows that a =1 / b = 1 / L3 (3 / , /
4; 3 / , / ⊆ L2 (1 / , /
4; 1 / , / . (109) Step 4: L2 ⊆ L3.
Let a , b and a , b be functions as inDefinition 3. Let L ∈ L2 ( a , b ; a , b ). Thenthere exists a ( a , b ; a , b )–procedure Q = { Q n : n ∈ N } that satisfies Eqs. (101) and (102)when x ∈ L , and satisfies Eqs. (103) and (104)when x / ∈ L . We now show that L ∈ L3 (cid:18) ( 12 + a , ( 12 + b , ( 12 + a , ( 12 + b (cid:19) . (110)Note that Q n takes as input ( x, | ψ i ⊗ | k i )where | x | = n , | ψ i is a state of m qubits, andboth m = m ( n ) and k = k ( n ) are in poly.Note that since Q is a quantum verificationprocedure, there exists an eigenbasis of Q for x ,denoted B Q ( x ) = {| ψ i i : 1 ≤ i ≤ m } . We de-note by p i = Pr[ Q n ( x, | ψ i i ) = 1] the acceptanceprobability of | ψ i i .Define the polynomial time uniform family ofquantum circuits Q = { Q n : n ∈ N } as follows: Q n takes as input ( x, | ψ i⊗| k i ) where | x | = n , | ψ i is a state of m qubits, and k = k ( n ) is apolynomial function of n .On input ( x, | ψ i ), carry out a slight modifica-tion of the Iterative Procedure of Definition 29(see note below Definition 29) applied to Q asfollows.The parameter N takes the value N = 2.The functions g | x | = g are taken as probabilitydistributions over the alphabet { L, , L } . Theyare independent of | x | , and take the deterministicform: g : { , , } → { L, , L } g (2) = L , g (1) = 0 , g (0) = L (111)Let us bound the probabilities of the outcomes L , 0, and L . We can restrict our discussion to he eigenstates of Q (because of the absence ofinterferences, see Equation (21)).One findsPr[ Q n ( x, | ψ i i ) = L ] = p i Pr[ Q n ( x, | ψ i i ) = 0] = 2 p i (1 − p i )Pr[ Q n ( x, | ψ i i ) = L ] = (1 − p i ) (112)One then easily checks that the family of cir-cuits Q = { Q n : n ∈ N } thus defined satisfyEqs. (97) and (98) when x ∈ L , and satisfy Eqs.(99) and (100) when x / ∈ L , with a = ( + a ) , b = ( + b ) and a = ( + a ) , b = ( + b ) . Summary of Steps 3 and 4.
Since L2 ( a, b ; a , b ) and L3 ( a, b ; a , b ) are in-dependent of the bounds a, b ; a , b , steps 3 and 4show that L2 = L3 . Combining with steps 1 and2, we have the second equality in Eq. (105).
13 Functional
QMA ∩ coQMA . We first generalise Definition 19 as follows:
Definition 35. Accepting and rejectingsubspaces.
Let Q = { Q n } be a quantum ver-ification procedure and fix a, a ∈ [0 , .We define the following binary relations overbinary strings and quantum states: H [0 , − a ] ∪ [ a , Q ( x, | ψ i ) = 1 if | ψ i ∈ Span( H ≥ a Q ( x ) ∪ H ≤ − a Q ( x )) and H [ − a , a ] Q ( x, | ψ i ) = 1 if | ψ i ∈ Span( {H Q ( x, p ) : 1 − a ≤ p ≤ a } ) . and to simplify notation, we denote H [0 , − a ] ∪ [ a , Q ( x ) = ( | ψ i : H [0 , − a ] ∪ [ a , Q ( x, | ψ i ) = 1 ) , H [ − a , a ] Q ( x ) = ( | ψ i : H [ − a , a ] Q ( x, | ψ i ) = 1 ) . (113) We base our definition of functional
QMA ∩ coQMA on ( a , b ; a , b ) –Quantum VerificationProcedures because these have an eigenbasiswhich allows the definition of accepting and re-jecting subspaces. Definition 36. Functional QMA ∩ coQMA( F( QMA ∩ coQMA ) ). Let a , b and a , b be functions as in Definition . The classF( QMA ∩ coQMA )( a , b ; a , b ) is the set { ( H [0 , − a ] ∪ [ a , Q ( x ) , H [ − b , b ] Q ( x )) } of pairsof relations, where Q is an ( a , b ; a , b ) –quantum verification procedure. We already showed in Theorem 13 that thelanguage L2 is independent of the complete-ness and soundness bounds used to define it.We now show, using the same argument, that F( QMA ∩ coQMA ) is also independent of thecompleteness and soundness bounds. Theorem 15. F( QMA ∩ coQMA ) is indepen-dent of the bounds ( a , b ; a , b ) . Let Q bean ( a , b ; a , b ) –quantum verification procedure,with a , b and a , b functions as in Definition .Let ˜ a , ˜ b and ˜ a , ˜ b be functions as inDefinition with ˜ a , ˜ a < − − r and ˜ b , ˜ b > − r , for some r ∈ poly .Then there exists an (˜ a , ˜ b ; ˜ a , ˜ b ) –quantumverification procedure ˜ Q , such that there existsan e-map from Q to ˜ Q and the strictly increas-ing functions { f n } that define the e–map (seeDefinition 28) satisfy f n ( a ) = ˜ a , f n ( b ) = ˜ b , f n ( a ) = ˜ a , f n ( b ) = ˜ b . Consequently for all x H [0 , − ˜ a ] ∪ [ a , Q ( x ) = H [0 , − a ] ∪ [ a , Q ( x )(114) and H [ − ˜ b , b ]˜ Q ( x ) = H [ − b , b ] Q ( x ) . (115) Proof.
The proof is an immediate application ofTheorem 10
As a consequence the precise values of thebounds ( a , b ; a , b ) are irrelevant. We take thetraditional values Definition 37.
We define the class F( QMA ∩ coQMA ) as F( QMA ∩ coQMA )(2 / , /
3; 2 / , / . ince the definition of Functional QMA ∩ coQMA is in terms of Definition 33, what aboutFunctional QMA ∩ coQMA if one uses the orig-inal Definition 5? The proof of Theorem 14 pro-vides a natural mapping from the pair of quantumverification procedures Q and Q used in Defini-tion 5 to the single quantum verification proce-dure Q used in Definition 33. Using this map-ping leads to the following definition. Definition 38. Canonical Definition ofFunctional QMA ∩ coQMA. Let a, b and a , b be pairs of functions as in Definition . Let Q and Q be two quantum verification procedures asin Definition 5. These procedures define a lan-guage L ∈ QMA ∩ coQMA .Let | ψ i i be an eigenbasis of Q for x , and p i thecorresponding acceptance probability; and let | ψ i i be an eigenbasis of Q for x , and p i the corre-sponding acceptance probability.For all x , denote by H + ( x ) = {| i ⊗ | ψ i i : p i > a }∪{| i ⊗ | ψ i i : p i > a } (116) and by H − ( x ) = {| i ⊗ | ψ i i : p i < b }∪{| i ⊗ | ψ i i : p i < b a } (117) Then the pair of relations ( H + ( x ) , H − ( x )) ∈ F( QMA ∩ coQMA )( a, b ; a , b )(118) are the canonical relations associated to the pairof procedures Q and Q with completeness andsoundness probabilities a, b and a , b . From this definition, we see that one of the orig-inal problems we were confronted with, namelythat Q and Q do not commute has been circum-vented by appending to the witnesses a singlequbit whose state, | i or | i , indicates whetherthis is a witness for Q or for Q .
14 Total Functional
QMA equalsFunctional
QMA ∩ coQMA We now prove that Total Functional
QMA equalsFunctional
QMA ∩ coQMA . Theorem 16. F( QMA ∩ coQMA ) = TFQMA
Proof.
Step 1: F( QMA ∩ coQMA ) ⊆ TFQMA
Let Q = { Q n : n ∈ N } be an(2 / , /
3; 2 / , / a = 2 / b = 1 / H [0 , − a ] ∪ [ a , Q ( x ) and H [ − b , b ] Q (( x ). Denote by B Q ( x ) = {| ψ i i the eigenbasis of Q for x and by p i = Pr[ Q n ( x, | ψ i i ) = 1] the acceptanceprobability of | ψ i i .We will show that there exists a a -totalquantum verification procedure Q T = { Q Tn : n ∈ N } such that H ≥ a Q T ( x ) = H [0 , − a ] ∪ [ a , Q ( x ) (119)and H ≤ b Q T ( x ) = H [ − b , b ] Q ( x ) . (120)We define Q T as follows:On input ( x, | ψ i ), run the iterative procedureof Definition 29 on Q n on input ( x, | ψ i ), for pa-rameter N = 2, thereby producing a 2 bit output z , z and functions G = { g | x | } independent of | x | given by: g (0) = g (2) = 1 g (1) = 0 . (121)One easily checks thatPr[ Q Tn ( x, | ψ i i ) = 1] = p i + (1 − p i ) (122)Pr[ Q Tn ( x, | ψ i i ) = 0] = 2 p i (1 − p i ) (123)from which Equations (119) and (120) follow.The completeness and soundness thresholds ofprocedure Q T are a = and b = .They can be changed to 2 / /
3, see Theo-rem 5, which proves the result.
Step 2: TFQMA ⊆ F( QMA ∩ coQMA )Let Q = { Q n : n ∈ N } be an a -total quantumverification procedure. Choose any b such that a, b is a pair of functions as in Definition 3. Thecorresponding accepting and rejecting subspacesare H ≥ aQ ( x ) and H ≤ bQ ( x ).Denote by B Q ( x ) = {| ψ i i the eigenbasis of Q for x and by p i = Pr[ Q n ( x, | ψ i i ) = 1] the accep-tance probability of | ψ i i . e show that there exists an ( a, b ; a, b )–quantum verification procedure Q = { Q n : n ∈ N } such that H [0 , − a ] ∪ [ a , Q ( x ) = H ≥ aQ ( x ) (124)and such that H [ − b , b ] Q ( x ) = H ≤ bQ ( x ) . (125)We define Q ( x, | ψ i ) as follows:Let c be a bit drawn uniformly at random fromthe distribution { , } .If c = 0, acceptIf c = 1, carry out the procedure Q on input( x, | ψ i ). Output Q n ( x, | ψ i ).One easily checks thatPr[ Q n ( x, | ψ i i ) = 1] = 1 + p i H < / Q ( x ) = ∅ ,and consequently H [0 , − a ] ∪ [ a , Q ( x ) = H ≥ a Q ( x )and H [ − b , b ] Q ( x ) = H [ , b ] Q ( x )).
15 If
TFQMA W ⊆ FBQP W then QMA ∩ coQMA = BQP . We show that if the functional
TFQMA class istrivial, in the sense that one can always efficientlyproduce a witness, then
QMA ∩ coQMA = BQP . Theorem 17. If TFQMA W is included inFBQP W then QMA ∩ coQMA equals BQP. Proof.
Step 1: Trivial direction.
It followsfrom Definition 11 that
BQP = coBQP . There-fore BQP ⊆ QMA ∩ coQMA . Step 2 : Preliminaries and idea of theproof.
We will show that under the condi-tion
TFQMA W ( a ) ⊆ FBQP W ( a ), it holds that QMA ∩ coQMA ⊆ BQP .Let L ∈ QMA ∩ coQMA . Then there exists an(2 / , /
3; 2 / , / Q = { Q n : n ∈ N } such that for every x of length n , either both Eqs. (101) and (102), orboth Eqs. (103) and (104) hold.Denote by B Q ( x ) = {| ψ i i} the eigenbasis of Q for x and by p i = Pr[ Q n ( x, | ψ i i ) = 1] theacceptance probability of | ψ i i . Note that the definition of (2 / , /
3; 2 / , / x ∈ L then 13 ≤ p i ≤ , (127)if x / ∈ L then 0 ≤ p i ≤ . (128)The idea of the proof is (in Step 3) to constructfrom Q a total procedure Q T , as in the proof ofTheorem 16. Our hypothesis implies that Q T hasan efficiently preparable witness ρ .We then construct (in Step 4) another proce-dure Q P such that Q e–maps to Q P . We showthat that the language defined by Q P is the samelanguage as that defined by Q . Furthermore,the procedures Q , Q T , and Q P all have thesame eigenbasis. This will allow us to show thatfor all x , ρ is an efficiently preparable witnessfor Q P . Therefore Q P satisfies the conditions ofDefinition 13 and L ∈ BQP . Step 3: -total procedure Q T . Procedure Q allows us to define a -total procedure Q T = { Q Tn : n ∈ N } exactly as in Step 1 of the proofof Theorem 16, following the construction in theparagraph surrounding Eqs. (121), (122), (123).Note that Q T has the same eigenbasis B Q ( x )as Q , but that the acceptance probability ofeigenstate | ψ i i is β ( p i ) = p i + (1 − p i ) , (129)see Eq. (122).The hypothesis TFQMA W ( a ) ⊆ FBQP W ( a )implies that there exists an efficiently preparablefamily of density matrices { ρ ( x ) } such that forall x Pr[ Q Tn ( x, ρ ( x )) = 1] ≥ β (cid:18) (cid:19) = 1318 . (130)We derive some properties of this ρ . We de-note ρ ij = h ψ i | ρ | ψ j i the matrix elements of ρ inthe eigenbasis of Q . Because there is no inter-ferences between eigenbasis states, we havePr[ Q n ( x, ρ ( x )) = X i p i ρ ii , Pr[ Q Tn ( x, ρ ( x )) = X i β ( p i ) ρ ii . (131) ence, for x ∈ L , we have β (cid:18) (cid:19) ≤ Pr[ Q Tn ( x, ρ ( x )) = 1]= X i : ≤ p i < β ( p i ) ρ ii + X i : ≤ p i ≤ β ( p i ) ρ ii ≤ X i : ≤ p i < β (cid:18) (cid:19) ρ ii + X i : ≤ p i ≤ β (1) ρ ii = β (cid:18) (cid:19) + (cid:18) − β (cid:18) (cid:19)(cid:19) X i : ≤ p i ≤ ρ ii where we have used Eq. (127), β (1) = 1, and thefact that over the interval [1 / , / β ( p ) has itsmaximum at 3 /
4: max p ∈ [1 / , / β ( p ) = β (3 / X i : ≤ p i ≤ ρ ii ≥ β (cid:16) (cid:17) − β (cid:16) (cid:17) − β (cid:16) (cid:17) = 727 . (132) Step 4: Procedure Q P . Theorem 10 impliesthat there exists a quantum verification proce-dure Q P such that Q e–maps to Q P and suchthat the polynomial time computable strictly in-creasing functions { f n } that define the reductionsatisfy f n (3 /
4) = 67 , (133) f n (2 /
3) = 17 . (134)Note that Q P has the same eigenbasis B Q ( x )as Q (and thus as Q T ), but that the acceptanceprobability of eigenstate | ψ i i is f n ( p i ).We bound the success probability of Q Pn ( x, ρ ( x )) when x ∈ L and ρ ( x ) is the ef-ficiently preparable state that satisfies Eq.(130):Pr[ Q Pn ( x, ρ ( x )) = 1] = X i f n ( p i ) ρ ii ≥ X i : ≤ p i ≤ f n ( p i ) ρ ii ≥ X i : ≤ p i ≤ ρ ii ≥
727 67 = 29 (135)where we have used Eq. (132), Eq. (133), andthe fact that f n is strictly increasing. Let us bound the success probability of Q Pn ( x, ρ ) when x / ∈ L and ρ is an arbitrary state:Pr[ Q Pn ( x, ρ ) = 1] = X i f n ( p i ) ρ ii = X i :0 ≤ p i ≤ f n ( p i ) ρ ii ≤ X i :0 ≤ p i ≤ f n ( 23 ) ρ ii = 17 (136)where we have used Eq. (128), Eq. (134), andthe fact that f n is strictly increasing.Therefore Q P is a (2 / , / L as Q . Furthermore,when x ∈ L there exists an efficiently prepara-ble witness. Thus Q P satisfies the conditions ofDefinition 13, and L ∈ BQP ( , ) = BQP .
16 If there exists a
QMA completeproblem that robustly reduces to aproblem in
TFQMA , then
QMA = QMA ∩ coQMA . Theorem 18.
Let a : N → [0 , be a poly-nomially time computable function. Let a, b befunctions as in Definition . Suppose that thereexists 1) an ( a, b ) -quantum verification procedure Q = { Q n : n ∈ N } for a QMA -complete lan-guage L , and 2) an a –total quantum verifica-tion procedure Q ( T ) = { Q ( T ) n : n ∈ N } , and 3) (cid:15) ∈ / poly such that there exists a robust re-duction from Q to Q ( T ) with parameter (cid:15) , then QMA = QMA ∩ coQMA .Proof. Step 0: idea of the proof.
We denote by ( f, Φ) the pair of a polynomialtime computable function and an efficiently im-plementable channel that define the robust re-duction of Q to Q ( T ) , and denote by (cid:15) the pa-rameter of the robust reduction, see Definition27.We denote by L the language associated to Q .We denote by ¯ L the complement of L . We willshow that under the hypothesis of Theorem 18,¯ L ∈ QMA . To this end we will construct a proce-dure Q CO which rejects on all states when x ∈ L ,but which accepts on some states when x ∈ ¯ L .The basic idea behind the construction of Q CO is similar to the proof in the classical case given n [2]. Namely, given x and given an input state | ψ i , we first check whether | ψ i is a witness for Q ( T ) ( f ( x )). If this check is successful, we checkif Q ( x, Φ( | ψ i )) rejects in which case we accept,while if Q ( x, Φ( | ψ i )) accepts we reject.That Q CO thus constructed should have thedesired properties follows from the following rea-soning. If x ∈ L then either | ψ i is not a witnessfor Q ( T ) ( f ( x )) and Q CO rejects, or | ψ i is a wit-ness for Q ( T ) ( f ( x )), in which case Q ( x, Φ( | ψ i ))will accept, and therefore Q CO again rejects. Onthe other hand, if x ∈ ¯ L then the second step(testing whether one has a witness for Q ) will al-ways fail, and in this case Q CO accepts. Because Q ( T ) is a total procedure, one can always find astate on which the first step accepts. Therefore,for x ∈ ¯ L there exists at least one a state onwhich Q CO accepts.However there are several complications in thequantum case. We have prepared for these com-plications by our earlier theorems and definitions.First it is not obvious that we can use the inputstate twice. We solve this by replacing Q ( T ) by anondestructive version of Q ( T ) , which we denote˜ Q ( T ) (Theorem 12).Second, we need to be able to amplify thecompleteness and soundness probabilities with-out distorting the structure of the witness states.This is achieved through e-maps (Theorems 10and 12).Thirdly, we need to deal with the eigenstates of Q ( T ) which have acceptance probability equal to a − δ , with δ exponentially small (see Example 1).This is addressed by requiring in the statement ofthe Theorem that the reduction from Q to Q ( T ) is an robust reduction (Definition 27). This en-sures that eigenstates with acceptance probabil-ity equal to a − δ (if they exist) can be treated inthe same way as the real witnesses (i.e. the eigen-states that have acceptance probability greater orequal to a ). Step 1: preliminary definitions.
Procedure Q n has as input a classical bit stringof length n , a witness state of m ( n ) qubits, andancilla state of k ( n ) qubits. We denote η = 2 − m − . (137)Using Theorem 12 we construct a (1 − η )–totalnondestructive procedure ˜ Q ( T ) such that Q ( T ) e–maps to ˜ Q ( T ) , and such that the polynomialtime computable strictly increasing functions ˜ f ( T ) n that define the e–map satisfy ˜ f ( T ) n ( a ) =1 − η , ˜ f ( T ) n ( a − (cid:15) ) = η .Using Theorem 10 we construct a (1 − η, η )–procedure ˜ Q such that Q e–maps to ˜ Q , and suchthat the polynomial time computable strictly in-creasing functions ˜ f n that define the e–map sat-isfy ˜ f n ( a ) = 1 − η , ˜ f n ( b ) = η . Step 2: procedure Q CO . We define procedure Q CO as follows:1. Denote by ( x, | ψ i ) the input to Q CO .2. Run ˜ Q ( T ) on input ( f ( x ) , | ψ i ).3. If ˜ Q ( T ) ( f ( x ) , | ψ i ) = 0 (i.e. ˜ Q ( T ) rejects),output 0 (i.e. Q CO rejects).4. If ˜ Q ( T ) ( f ( x ) , | ψ i ) = 1, then apply Φ to thequantum output of ˜ Q ( T ) . Denote by | ˜ ψ i thestate so obtained.5. Run ˜ Q on input ( x, | ˜ ψ i ).6. If ˜ Q n ( x, | ˜ ψ i ) = 1 (i.e. ˜ Q accepts), output 0(i.e. Q CO rejects).7. If ˜ Q n ( x, | ˜ ψ i ) = 0 (i.e. ˜ Q rejects), output 1(i.e. Q CO accepts).We now show that Q CO is a ((1 − η ) , / Q CO is ¯ L , which proves the result. Step 3: Existence of a witness when x ∈ ¯ L . Take any x ∈ ¯ L . Let | ψ i i be an eigenstate of Q ( T ) with acceptance probability p i ≥ a . Suchan eigenstate exists, since Q ( T ) is an a –total pro-cedure. We will show that Q CO accepts on input( x, | ψ i i ) with probability ≥ (1 − η ) .To this end note that on this input, step 4 ofprocedure Q CO succeeds with probability ≥ − η ,and that the quantum output at step 4 is | ψ i i , i.e.it is not affected running ˜ Q ( T ) in step 2. This iswhere the nondestructiveness of ˜ Q ( T ) is used.However x ∈ ¯ L . Therefore in step 7, ˜ Q outputs0 with probability at least 1 − η . Hence the overallprobability that Q CO accepts on input ( x, | ψ i i ) isat least (1 − η ) . Step 4: soundness probability of Q CO oneigenstates of Q ( T ) . Take any x ∈ L . Let | ψ i i be an eigenstateof Q ( T ) with acceptance probability p i . Con-sider the acceptance probability of Q CO on input( x, | ψ i i ). f p i < a − (cid:15) , then Q CO rejects with probabilityat least 1 − η at step 3.If p i ≥ a − (cid:15) , then Q CO either rejects at step 3,or passes to step 5. In the latter case, the input of˜ Q is ( x, Φ( x, | ψ i i )), where Φ( x, | ψ i i ) is a witnessfor ˜ Q for x (see definition of robust reductions).That is ˜ Q will accept on input ( x, Φ( x, | ψ i i )) withprobability at least 1 − η . Hence the probabilitythat Q CO rejects at step 6 is ≥ − η .Thus on all eigenstates | ψ i i of Q ( T ) , Q CO re-jects with probability at least 1 − η .Note that this is the step where we use thatthe reduction from Q to Q ( T ) is robust with pa-rameter (cid:15) = 1 / poly. If we set (cid:15) to zero, then instep 4 we have no control over how the reductionacts on eigenstates of Q ( T ) which have acceptanceprobability equal to a − δ with δ exponentiallysmall. Step 5: soundness probability of Q CO onarbitrary states. Let us now consider the probability that Q CO rejects on an arbitrary input state ρ . The accep-tance probability of Q CO can be written asPr[ Q COn ( x, ρ ) = 1] = Tr( M ρ ) (138)where M is the POVM element correspondingto Q CO accepting. Note that M is a positivematrix of size 2 m × m .We have established at Step 4 that for all i ≤ h ψ i | M | ψ i i ≤ η . (139)It then follows that for all i, j |h ψ i | M | ψ j i| ≤ η . 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