Modified group non-membership is in AWPP
aa r X i v : . [ qu a n t - ph ] F e b Modified group non-membership is in AWPP
Tomoyuki Morimae
ASRLD Unit, Gunma University, 1-5-1 Tenjincho,Kiryushi, Gunma, 376-0052, Japan ∗ Harumichi Nishimura
Graduate School of Information Science, Nagoya University,Furocho, Chikusaku, Nagoya, Aichi, 464-8601, Japan † Fran¸cois Le Gall
Department of Computer Science, The University of Tokyo,7-3-1 Hongo, Bunkyoku, Tokyo, 113-8656, Japan ‡ Abstract
It is known that the group non-membership problem is in QMA relative to any group oracle andin SPP ∩ BQP relative to group oracles for solvable groups. We consider a modified version of thegroup non-membership problem where the order of the group is also given as an additional input.We show that the problem is in AWPP relative to any group oracle. To show the result, we usethe idea of the postselected quantum computing. ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] . INTRODUCTION The group non-membership (GNM) is the following problem: • Input: Group elements g , g , ..., g k , and h in some finite group G . • Question: Is h / ∈ H ≡ h g , ..., g k i ?Here, H ≡ h g , ..., g k i is the group generated by g , ..., g k . This problem has long beenstudied for black-box groups [1], which are finite groups whose elements are encoded asstrings of a given length and whose group operations are performed by a group oracle. TheGNM is known to be hard for classical computing: for some group oracle B , GNM is not inBPP B [2, 3]. Furthermore, it was also shown that for some group oracle B , GNM is not inNP B [2, 3], and for some group oracle B , GNM is not in MA B [4].Upper bounds of GNM have also been derived. For example, it was shown that GNM isin coNP B [1], AM B [2, 3], and QMA B [4] for any group oracle B . If we restrict the groupto be solvable, upper bounds can be improved: it was shown that GNM is in BQP B [5] andSPP B [6].In this paper, to deepen our understanding of the upper bounds of GNM, we consider aslightly modified version of GNM, which we call the modified GNM: • Input: Group elements g , g , ..., g k , and h in some finite group G , and |h g , ..., g k i| . • Question: Is h / ∈ H ≡ h g , ..., g k i ?In other words, in the modified GNM, the order |h g , ..., g k i| of the generated group h g , ..., g k i is also given as an additional input.We show that the modified GNM is in AWPP relative to any group oracle. The classAWPP was introduced by Fenner, Fortnow, Kurtz, and Li [7] to understand the structureof counting complexity classes (see also Refs. [8, 9]). AWPP is also famous among quantuminformation scientists, since it is one of the two best upper bounds of BQP [10]. (The otherone is QMA (or QCMA). No direct relation is known between QMA and AWPP. It is atleast known that they share the same upper bound, SBQP, namely, QMA ⊆ SBQP [11]and AWPP ⊆ SBQP.) Therefore, our result implies that if GNM is changed to a bit easierproblem by adding an extra input, its upper bound is improved to the intersection of QMAand AWPP. 2he definition of AWPP is as follows. (Here, we take a simpler definition of AWPP byFenner [8].)
Definition 1.
A language L is in AWPP iff there exist f ∈ FP and g ∈ GapP such that forall w , f ( w ) > w ∈ L then ≤ g ( w ) f ( w ) ≤ w / ∈ L then 0 ≤ g ( w ) f ( w ) ≤ .Here, FP is the class of functions from bit strings to integers that are computable in poly-nomial time by a Turing machine. A GapP function [12] is a function from bit strings tointegers that is equal to the number of accepting paths minus that of rejecting paths of anondeterministic Turing machine which takes the bit strings as input. The FP function f can be replaced with 2 q ( | w | ) for a polynomial q [9, 12], and the error bound ( , ) can bereplaced with (2 − r ( | w | ) , − − r ( | w | ) ) for any polynomial r [7, 9].Our proof is based on the idea of postselected quantum computing. The postselection is afictious ability that one can always obtain a specific measurement result even if its occurringprobability is exponentially small. The class of languages that can be efficiently recognizedby a quantum computer with the postselection is called postBQP, and it is known thatpostBQP = PP [13]. If we consider a restricted version of postBQP where the postselectionprobability is close to an FP function divided by 2 poly , the class was shown to be equal toAWPP [14]. Our proof is based on this relation between postselected quantum comput-ing and AWPP: we first propose a postBQP algorithm that can solve the modified GNM,and then show that the postselection probability satisfies the condition. Then, by usingthe relation between the output probability distribution of quantum computing and GapPfunction [10], we conclude that the modified GNM is in AWPP. Our quantum algorithm isbased on that of Watrous [4]. He showed that if the state P g ∈ H | g i , which is believed to behard to generate with a polynomial-size quantum computing, is given as a witness, GNM isverified efficiently. In our algorithm, the witness is generated by polynomial-size quantumcomputing with postselection. This result itself means GNM ∈ postBQP = PP, which istrivial since it is already known that QMA ⊆ SBQP ⊆ PP. Our contribution is that wepoint out that the postselection probability satisfies a nice condition, and therefore if theGNM is modified as described above, it is in AWPP.3
I. PROOF
Now we show our result that the modified GNM is in AWPP relative to any group oracle.First, let us remember the group oracle and a theorem shown by Babai [2]. A group oracle B can be represented by a family of bijections { B n } with each member having the form B n : { , } n +2 → { , } n +2 and satisfying certain constraints that specify its operation (seeSection 2 in Ref. [4] for the precise definition). We denote the group associated with each B n by G ( B n ). In other words, elements of G ( B n ) form some subset of { , } n and the groupstructure of G ( B n ) is determined by the function B n . The following theorem by Babai [2](see also Ref. [4]) is a basis of our result. Theorem 1.
For any group oracle B = { B n } , there exists a randomized procedure P actingas follows: On input g , ..., g k ∈ G ( B n ) and ǫ > , the procedure outputs an element of H ≡ h g , ..., g k i in time polynomial in n +log ǫ such that each g ∈ H is output with probabilityin the range ( | H | − ǫ, | H | + ǫ ) . As is explained in Ref. [4], we can simulate the classical randomized procedure P in“quantum way”: Let us assume that a random bit is generated s ( n ) times during P , where s is a polynomial. We first generate the state1 √ N X z ∈{ , } s ( n ) | z i , where N ≡ s ( n ) , | z i is an s ( n )-qubit state, and each z is an s ( n )-bit string representingrandom numbers generated during P . By coupling sufficiently many ancilla qubits andrunning P for each branch controlled by z , we obtain | Ψ i ≡ √ N X z ∈{ , } s ( n ) | η z i ⊗ | φ z i = 1 √ N X g ∈ H √ γ g | g i ⊗ | garbage ( g ) i , where η z is an element of H , φ z is a t ( n )-bit string corresponding to the leftover of theprocedure ( t is a polynomial), and | garbage ( g ) i ≡ √ γ g X z : η z = g | φ z i . γ g is the normalization factor, i.e., the number of z such that η z = g . From Theorem 1, γ g N ∈ (cid:16) | H | − ǫ, | H | + ǫ (cid:17) . (1)Furthermore, due to the normalization of | Ψ i , X g ∈ H γ g N = 1 . Therefore if we write γ g N = 1 | H | + ǫ g , where − ǫ ≤ ǫ g ≤ ǫ , we obtain P g ∈ H ǫ g = 0 . Hence, X g ∈ H (cid:16) γ g N (cid:17) = 1 | H | + 2 | H | X g ∈ H ǫ g + X g ∈ H ǫ g = 1 | H | + X g ∈ H ǫ g (2) ≥ | H | . (3)Let us couple | Ψ i with | + i ≡ ( | i + | i ) / √ h to obtain | Ψ ′ i ≡ √ N X g ∈ H √ γ g (cid:16) | g i ⊗ | i + | gh i ⊗ | i (cid:17) ⊗ | garbage ( g ) i , where the second register is the coupled qubit. Let us apply Hadamard on the secondregister: 12 √ N X g ∈ H √ γ g h ( | g i + | gh i ) | i + ( | g i − | gh i ) | i i | garbage ( g ) i . Let us prepare two copies of them, and add an ancilla qubit | i a :14 N X g,g ′ ∈ H √ γ g √ γ g ′ h ( | g i + | gh i )( | g ′ i + | g ′ h i ) | i| i a + ( | g i + | gh i )( | g ′ i − | g ′ h i ) | i| i a + ( | g i − | gh i )( | g ′ i + | g ′ h i ) | i| i a + ( | g i − | gh i )( | g ′ i − | g ′ h i ) | i| i a i | garbage ( g ) i| garbage ( g ′ ) i . | i :14 N X g,g ′ ∈ H √ γ g √ γ g ′ h ( | g i + | gh i )( | g ′ i + | g ′ h i ) | i| i a + ( | g i + | gh i )( | g ′ i − | g ′ h i ) | i| i a + ( | g i − | gh i )( | g ′ i + | g ′ h i ) | i| i a + ( | g i − | gh i )( | g ′ i − | g ′ h i ) | i| i a i | garbage ( g ) i| garbage ( g ′ ) i . Note that h + ⊗ t ( n ) | garbage ( g ) i = 1 √ γ g X z : η z = g h + ⊗ t ( n ) | φ z i = 1 p γ g t ( n ) γ g = √ γ g √ t ( n ) . Therefore, if we postselect garbage registers onto | + i ⊗ t ( n ) , the (unnormalized) state afterthe postselection is 14 N t ( n ) X g,g ′ ∈ H γ g γ g ′ h ( | g i + | gh i )( | g ′ i + | g ′ h i ) | i| i a + ( | g i + | gh i )( | g ′ i − | g ′ h i ) | i| i a + ( | g i − | gh i )( | g ′ i + | g ′ h i ) | i| i a + ( | g i − | gh i )( | g ′ i − | g ′ h i ) | i| i a i . Let us denote this state by14 N t ( n ) h | h + i| h + i| i| i a + | h + i| h − i| i| i a + | h − i| h + i| i| i a + | h − i| h − i| i| i a i , where | h ± i ≡ X g ∈ H γ g ( | g i ± | gh i ) . P ( p = 1) = 116 N t ( n ) ( h h + | h + i + h h − | h − i ) = ( P g ∈ H γ g ) N t ( n ) (4)= N ( P g ∈ H γ g ) N t ( n ) = N ( P g ∈ H γ g N ) t ( n ) = 12 t ( n ) − s ( n ) (cid:16) | H | + X g ∈ H ǫ g (cid:17) , (5)where we mean p = 1 if the garbage registers are projected on to | + i ⊗ t ( n ) , and we haveused N = 2 s ( n ) , Eq. (2), and the relation h h + | h + i + h h − | h − i = 4 X g ∈ H γ g . Therefore, from Eq. (4), the normalized state after the postselection is14 P g ∈ H γ g h | h + i| h + i| i| i a + | h + i| h − i| i| i a + | h − i| h + i| i| i a + | h − i| h − i| i| i a i . If we project the ancilla qubit onto | i a , the (unnormalized) state after the projection is14 P g ∈ H γ g | h + i| h + i| i , and therefore, P ( o = 0 | p = 1) = (cid:16) h h + | h + i P g ∈ H γ g (cid:17) . Here, we mean o = 0 (resp., o = 1) if the ancilla qubit is projected onto | i a (resp., | i a ). If h / ∈ H , h h + | h + i = 2 X g ∈ H γ g and therefore, P ( o = 0 | p = 1) = 14 ,P ( o = 1 | p = 1) = 1 − P ( o = 0 | p = 1) = 34 . h ∈ H , on the other hand, P ( o = 1 | p = 1) = 1 − P ( o = 0 | p = 1)= 1 − (cid:16) h h + | h + i P g ∈ H γ g (cid:17) = 1 − (cid:16) P g ∈ H γ g − h h − | h − i P g ∈ H γ g (cid:17) = 1 − (cid:16) − h h − | h − i P g ∈ H γ g (cid:17) = 1 − (cid:16) − h h − | h − i P g ∈ H γ g + (cid:16) h h − | h − i P g ∈ H γ g (cid:17) (cid:17) ≤ h h − | h − i P g ∈ H γ g = P g ∈ H ( γ g − γ gh − ) P g ∈ H γ g ≤ ǫ | H | P g ∈ H γ g N ≤ ǫ | H | . (6)Here, we have used Eqs. (1) and (3), and h h − | h − i = X g,g ′ ∈ H γ g γ g ′ ( h g | − h gh | )( | g ′ i − | g ′ h i )= X g,g ′ ∈ H γ g γ g ′ ( δ g,g ′ − δ g,g ′ h − δ gh,g ′ + δ g,g ′ )= X g ∈ H γ g − X g ∈ H γ g γ gh − − X g ′ ∈ H γ g ′ h − γ g ′ + X g ∈ H γ g = X g ∈ H γ g − X g ∈ H γ g γ gh − − X g ∈ H γ gh − γ g + X g ∈ H γ gh − = X g ∈ H ( γ g − γ gh − ) . Now we use the result by Fortnow and Rogers [10]:
Theorem 2.
For any uniform family of polynomial-size quantum circuits, there exist g ∈ GapP and a polynomial q such that for any w , the output probability of the quantum circuiton input w is equal to g ( w ) / q ( | w | ) . (Note that this theorem depends on the gate set. In thispaper, we consider the Hadamard and Toffoli gates as a universal gate set.) From this theorem, there exists a GapP function g and a polynomial q such that P ( o = 1 , p = 1) = g ( w )2 q ( n ) , (7)8here w is an input of the modified GNM. In the above, we have shown that if w is a yesinstance of the modified GNM, which means h / ∈ H ,34 = P ( o = 1 | p = 1) ≤ , which means 34 P ( p = 1) = P ( o = 1 , p = 1) ≤ P ( p = 1) . From Eq. (7), it is 34 P ( p = 1) = g ( w )2 q ( n ) ≤ P ( p = 1) . From Eq. (5) and | H | ≤ n , this means12 t ( n ) − s ( n ) (cid:16) | H | + X g ∈ H ǫ g (cid:17)
34 = g ( w )2 q ( n ) ≤ t ( n ) − s ( n ) (cid:16) | H | + X g ∈ H ǫ g (cid:17) ⇔ (cid:16) | H | X g ∈ H ǫ g (cid:17)
34 = g ( w )2 t ( n ) − s ( n ) | H | q ( n ) ≤ (cid:16) | H | X g ∈ H ǫ g (cid:17) ⇒ ≤ g ( w )2 t ( n ) − s ( n ) | H | q ( n ) ≤ (cid:16) | H | ǫ (cid:17) ⇒ ≤ g ( w )2 t ( n ) − s ( n ) | H | q ( n ) ≤ (1 + 2 n ǫ ) ⇔ n ǫ ) ≤ g ( w )2 t ( n ) − s ( n ) | H | q ( n ) (1 + 2 n ǫ ) ≤ . If we take ǫ = 2 − n − in Theorem 1, 23 < n ǫ ) . If we define G ( w ) = g ( w )2 t ( n ) − s ( n ) | H | , F ( w ) = 2 q ( n ) (1 + 2 n ǫ ) , we thus obtain 23 ≤ G ( w ) F ( w ) ≤ . (8)By the definition of the modified GNM, | H | ∈ FP ⊆ GapP. Since GapP functions are closedunder multiplications, G ∈ GapP. Furthermore, since we can assume q ( n ) ≥
12 for all n without loss of generality, we have F ∈ FP for our choice of ǫ .On the other hand, if w is a no instance of the modified GNM, which means h ∈ H , weobtain from Eqs. (7) and (6) that0 ≤ g ( w )2 q ( n ) ≤ ǫ | H | P ( p = 1) ⇔ ≤ g ( w )2 q ( n ) ≤ ǫ | H | t ( n ) − s ( n ) (cid:16) | H | + X g ∈ H ǫ g (cid:17) ⇒ ≤ g ( w )2 t ( n ) − s ( n ) | H | q ( n ) ≤ ǫ | H | (cid:16) ǫ | H | (cid:17) ⇔ ≤ g ( w )2 t ( n ) − s ( n ) | H | q ( n ) (1 + 2 n ǫ ) ≤ ǫ | H | . Since ǫ = 2 − n − , 2 ǫ | H | ≤ − ≤ . We thus obtain 0 ≤ G ( w ) F ( w ) ≤ . (9)Since Eqs. (8) and (9) satisfy the definition of AWPP, we conclude that the modified GNMis in AWPP. 10 cknowledgments TM is supported by the JSPS Grant-in-Aid for Young Scientists (B) No.26730003 and theMEXT JSPS Grant-in-Aid for Scientific Research on Innovative Areas No.15H00850. HNis supported by the JSPS Grant-in-Aid for Scientific Research (A) Nos.23246071, 24240001,26247016, and (C) No.25330012, and the MEXT JSPS Grant-in-Aid for Scientific Researchon Innovative Areas No.24106009. FLG is supported by the JSPS Grant-in-Aid for YoungScientists (B) No. 24700005, the JSPS Grant-in-Aid for Scientific Research (A) No. 24240001,and the MEXT JSPS Grant-in-Aid for Scientific Research on Innovative Areas No. 24106009. [1] L. Babai and E. Szemer´edi, On the complexity of matrix group problems I. Proc. of the 25thAnn. IEEE Sympo. on Found. of Comput. Sci. pp.229-240 (1984).[2] L. Babai, Local expansion of vertex-transitive graphs and random generation in finite groups.In Proc. of 23rd Ann. ACM Sympo. on Theor. of Comput. pp.164-174 (1991).[3] L. Babai, Bounded round interactive proofs in finite groups. SIAM J. Disc. Math. , pp.88-111(1992).[4] J. Watrous, Succinct quantum proofs for properties of finite groups. Proc. of the 41st Ann.IEEE Sympo. on Found. of Comput. Sci. pp.537-546 (2000).[5] J. Watrous, Quantum algorithms for solvable groups. Proc. of the 33rd ACM Sympo. onTheor. of Comput. pp.60-67 (2001).[6] N. V. Vinodchandran, Counting complexity of solvable black-box group problems. SIAM J.Comput. , pp.852-869 (2004).[7] S. Fenner, L. Fortnow, S. Kurtz, and L. Li, An oracle builder’s toolkit. Inf. Comput. , 95-136 (2003); Earlier version in Proc. 8th IEEE Conference on Structure in Complexity Theory,pp.120-131 (1993).[8] S. A. Fenner, PP-lowness and a simple definition of AWPP. Theory Comput. Syst. , 199-212(2003); ECCC Report TR02-036.[9] L. Li, On the counting functions. Ph.D. thesis, University of Chicago (1993).[10] L. Fortnow and J. Rogers, Complexity limitations on quantum computation. J. Comput. Syst.Sci. , 240-252 (1999).
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