Quantum Computation with Supplementary Information

Vol. 1 IPSJ Digital Courier Oct. 2005 Invited Paper Quantum Computation with Supplementary Information

Harumichi Nishimura†

The notion of advised computation was introduced by Karp and Lipton to represent non- uniform complexity in terms of Turing machines. Since then, advised computation has been one of the basic concepts of computational complexity. Recently, the research of advised computation has been originated also in the field of quantum computing. This paper reviews the study of advised quantum computation.

quantum computation are not yet so many, 1. Introduction but the author believes that hereafter the the- Turing machines and Boolean circuits are ory of advised quantum computation should be used mostly as computing models to study furthermore developed from the importance of the uniform and non-uniform complexity, re- non-uniform computation in complexity theory. spectively. We can take two approaches to In this belief, this article surveys the research compare between these two computing models. of quantum computation with advice. One approach is to restrict how to construct This article is organized as follows. In Sec- Boolean circuits, which means that the fam- tion 2, we give uniform complexity classes that ilies of Boolean circuits should satisfy a uni- appear in this article. In Section 3 we introduce form condition. The other is to supplement quantum advice complexity classes and com- Turing machines with non-uniform information, pare them and their basic results with classical called advice. As the latter approach, Karp advice classes. In Section 4, we discuss the am- and Lipton 17) initiated the notion of complex- plitudes of quantum Turing machines from the ity classes with advice in 1980. Since then, the viewpoint of non-uniform quantum complexity study of advised computation has been elabo- classes. Some relation of advised computation rated to understand the power and the limit to one-way communication complexity is seen in of non-uniform computation, and the notion of Section 5. In Section 6 we consider a separation advice complexity classes has appeared in many between the two quantum complexity classes phrases of computational complexity. with polynomial classical advice and with poly- After Shor’s excellent quantum algorithm for nomial quantum advice. In Section 7 we review the factoring problem, a number of complex- quantum complexity classes with short advice. ity notions were imported from classical com- Comparisons among uniform complexity classes plexity theory to quantum computing; non- and quantum advice classes are reviewed in Sec- deterministic computation, finite automata, in- tion 8. Finally, we look at the future work with teractive proof systems, and so on. The no- several open problems. tion of advice is not an exception for this trend. 2. Uniform Complexity Classes The research of advised quantum computation was started just a few years ago. In quantum To classify the power of uniform machines, computation, we have two choices as the sup- a multiplicity of complexity classes for recog- plementary information that is given to quan- nizing sets (that is, computing Boolean func- tum Turing machines; classical advice (binary tions) have been introduced in complexity the- strings) and quantum advice (quantum states). ory. They are often called uniform complexity While classical advice is suitable to character- classes compared to complexity classes with ad- ize polynomial-size quantum circuits, quantum vice, called non-uniform complexity classes.In advice is considered to be a natural quantum what follows, let Σ = {0, 1}. Also, we assume analogue of the non-uniform measure of compu- the familiarity with basics of structural com- tation. The results obtained so far on advised plexity 13) and quantum computing 14),22). First, let us recall several classical uniform † ERATO Quantum Computation and Information complexity classes. Let P (NP, resp.) denote Project, Japan Science and Technology Agency the class of sets recognized by polynomial-time

407 408 IPSJ Digital Courier Oct. 2005 deterministic (nondeterministic, resp.) Turing ally bounded if there exists a polynomial p machines. Let BPP be the class of sets recog- such that f(n) ≤ p(n) for any n ∈ N). It nized with bounded error (say, with probability is well-known that P/poly exactly character- ≥ 2/3) by polynomial-time probabilistic Turing izes the class of sets recognized by polynomial- machines. Let PP be the class of sets recognized size Boolean circuits 17). According to Defini- with unbounded error (that is, with probability tion 3.1, we can also consider a quantum ad- > 1/2) by polynomial-time probabilistic Tur- vice class BQP/poly. However, it is not known ing machines. Let PSPACE (ESPACE, resp.) whether BQP/poly characterizes the class of be the class of sets recognized by deterministic sets recognized by polynomial-size quantum cir- Turing machines in polynomial space (in space cuits. In fact, it can be easily seen that all sets 2O(n),resp.). in BQP/poly are recognized by polynomial-size The quantum Turing machine, an analogue quantum circuits similar to the classical case of the probabilistic Turing machine, is a math- while the converse fails to show by the follow- ematical model of the quantum computer. Each ing method of the classical case: transition of a quantum Turing machine M is To simulate a polynomial-size circuit by a de- determined by a complex-valued finite transi- terministic Turing machine, the code of the cir- tion function δ as follows: if the inner state cuit is given as advice, and then the set rec- of M is p and the tape head of M scans σ, ognized by the circuit can be also recognized by then the inner state becomes q,andthetape the deterministic Turing machine in polynomial head changes σ to τ and moves to direction time. d with (probability) amplitude δ(p, σ, q, τ, d). The reason why this method fails for Let BQP 7) denote the class of sets recognized BQP/poly is that polynomial-size quantum cir- with bounded error by polynomial-time quan- cuits do not always satisfy the bounded-error tum Turing machines. (In fact, the transition condition. In addition, Definition 3.1 is not amplitudes taken by quantum Turing machines meant for extending advice strings h(n)to are restricted to a subset of complex numbers, quantum states without considering the com- as discussed in Section 4). Since quantum Tur- plexity class of quantum states. Thus, at ing machines can simulate probabilistic Turing present, the definition of the quantum com- machines, BPP is included in BQP 7).More- plexity class with advice is given based on the over, the relationships among the above uni- bounded-error quantum computer 1),24). form complexity classes are given as follows. Definition 3.2 Let f be any function from N Fact 2.1 The following relations hold 3),7),13). to N and let F be any set of functions mapping 1) P ⊆ BPP ⊆ BQP ⊆ PP ⊆ PSPACE ⊆ from N to N. ESPACE. 1. A set A is in BQP/∗f if there exist a 2) P ⊆ NP ⊆ PP. polynomial-time quantum Turing machine M and a function h from N to Σ∗ such that M 3. Quantum Advice Complexity on input (x, h(|x|)) produces A(x)withprob- Classes ∗ ability at least 2/3 for every
x ∈ Σ ,where ∗ ∗ Now we introduce advice complexity classes. |h(n)| = f(n). Let BQP/ F = f∈F BQP/ f. First, we recall a general advice complexity 2. A set A is in BQP/∗Qf if there exist a class by Karp and Lipton 17), which is defined polynomial-time quantum Turing machine M based on the uniform complexity class. and a function h from N to the set of (pure) Definition 3.1 Let C be a class of sets, and quantum states such that M on input (x, h(|x|)) let f be a function from N to N.AsetA is in produces A(x) with probability at least 2/3for ∗ the Karp-Lipton advice class C/f if there exist every x ∈ Σ ,where
h(n)isanf(n)-qubit state. ∗ ∗ ∗ asetB ∈Cand a function h from N to Σ Let BQP/ QF = f∈F BQP/ Qf. such that A = {x |x, h(|x|)∈B} provided Under Definition 3.2, we can exactly char- that
|h(n)| = f(n) for all n ∈ N.LetC/F = acterize polynomial-size quantum circuits in f∈F C/f for any set of functions F from N to terms of the quantum advice complexity class N. BQP/∗poly . For the class P and the set poly of poly- Proposition 3.3 AsetA is in BQP/∗poly if nomially bounded functions, we obtain the most representative advice class P/poly. (A In Ref. 1), BQP/∗poly and BQP/∗Qpoly are respec- function f from N to N is called polynomi- tively denoted by BQP/poly and BQP/qpoly. Vol. 1 Quantum Computation with Supplementary Information 409 and only if A can be recognized with probabil- BQPC. According to this line, the uniform con- ity at least 2/3 by a polynomial-size quantum dition of quantum circuit families is defined as circuit family. follows 23): (i) all circuits are constructed using The class P/poly has another characteriza- elementary gates whose matrix representations tion to clarify the relationship between obtain- have polynomial-time computable components; ing non-uniform information from polynomi- and (ii) the construction of each circuit is com- ally long advices and obtaining it from ora- puted in time polynomial in the length of the in- cles. Let TALLY be the collection of subsets put. Using Yao’s simulation of quantum Turing of {0n | n ∈ N}. Then, P/poly = PTALLY . machines by quantum circuits 33),itisshown Similarly, we can see that BQP/∗poly coincides that BQP equals the class of sets recognized with BQPTALLY . with bounded error by uniform quantum circuit Remark 3.4 One might consider that the families 23). On the contrary, to represent the possibility of a gap between two definitions such class BQPC in terms of quantum circuits, we as Definitions 3.1 and 3.2 will occur also for the must consider quantum circuit families that do bounded-error class BPP. However, we can see not satisfy uniform condition (i). Thus, BQPC that BPP/∗poly coincides with BPP/poly us- can be regarded as a kind of non-uniform quan- ing the proof technique to show BPP ⊆ P/poly tum complexity classes. We now bound the (say, see the textbook by Du and Ko 13)). By amount of non-uniform information obtained contrast, it is open whether BQP is included in by the sets in BQPC. Here, log denotes the P/poly (and that inclusion seems not to hold). set of functions f satisfying |f(n)| = O(log n). ∗ Remark 3.5 In Definition 3.1, we take pure Proposition 4.1 BQPC BQP/ log. quantum states as quantum advice, not mixed Remark 4.2 In Ref. 24) a bit worse upper quantum states, the fully general notion of bound was shown. However, almost the same quantum states since we follow the style of the proof leads to Proposition 4.1 by using the 15) classical advice; fixed strings, not randomly se- approximation scheme of Harrow et al. for lected strings, are taken as classical advice. By a quantum state given as advice, instead of 18),22) the broadening of the spirit of advice, we could Solovay-Kitaev theorem used in Ref. 24). ∗ take randomly selected advice or mixed states Proposition 4.1 implies that BQPC/ poly = ∗ ∗ ∗ as advice, which will be shortly mentioned in BQP/ poly and BQPC/ Qpoly=BQP/ Qpoly. ∗ the last section. That is, the complexity classes BQP/ poly and BQP/∗Qpoly are stable for the change of transi- 4. Non-uniformity of Amplitudes tion amplitudes taken by underlying quantum What is the complexity class BQP, which Turing machines, like a quantum analogue of 30) is considered to be the most appropriate class NP, the class NQP . for representing the computational power of By contrast, we can give a lower bound of 19) quantum computers? Roughly speaking, this non-uniformity of BQPC.Ko introduced an- class is the collection of sets that can be recog- other notion of logarithmic advice class Full- / h n nized by polynomial-time bounded-error quan- P log whereanadvicestring(
)canbe m tum Turing machines. However, Adleman, et used to recognize a subset L ∩ m≤n Σ 3) al. showed that, if the transition amplitudes of a set L while it is only available to recog- of quantum Turing machines are any complex nize L ∩ Σn in case of P/ log. The class Full- numbers, quantum Turing machines can rec- P/ log has a nice structure that is closed un- ognize undecidable sets with bounded-error in der polynomial-time Turing reduction, differ- polynomial time. Thus, rigorously, the defi- 7) ent from P/ log. Also, Full-P/ log is charac- nition of BQP is given using quantum Tur- terized as Full-P/ log = PTALLY2 by a rela- ing machines with amplitudes from the set of tivized class 5) like P/poly = PTALLY . Here, polynomial-time computable complex numbers TALLY2 denotes the collection of all subsets n (that is, complex numbers whose real and imag- of {02 | n ∈ N}. Similarly, we can define inary parts are approximated within 2−n in n another logarithmic quantum advice class Full- time polynomial in ). By contrast, the class BQP/∗ log and see that it equals BQPTALLY2 . of sets recognized by polynomial-time bounded- Then, the proof that BQPC includes undecid- error quantum Turing machines whose ampli- tudes are any complex numbers is often called Ko 19) originally called it Strong-P/poly. 410 IPSJ Digital Courier Oct. 2005 able sets is available for embedding the infor- Σn). Considering Alice’s message as advice, we mation on any set in TALLY2 into the ampli- can regard advised computation as a variant of tudes of quantum Turing machines. Combining the one-way communication complexity model. Proposition 4.1 with this fact, we obtain the fol- This view is often available to show the limit lowing conclusion that figures out the power of of the advised computation that will appear in BQPC. the later sections. Theorem 4.3 The following relations hold. ∗ ∗ ∗ ∗ 6. BQP/ Qpoly =BQP / poly? 1) Full-BQP/ log ⊆ BQPC BQP/ log. TALLY2 TALLY 2) BQP ⊆ BQPC BQP . Is quantum advice more powerful than clas- sical advice? For many ones that have inter- 5. Relation to One-way Communica- est in advised quantum computation, the most tion Complexity natural separation corresponding to this ques- tion would be the separation of BQP/∗poly Since Yao introduced the notion of commu- ∗ 32) / nication complexity in classical as well as from BQP Qpoly. To try to solve this prob- quantum setting 33), it has been intensively lem, Aaronson succeeded to bound from above /∗ studied and applied to various topics in com- BQP Qpoly by the class PP with polynomial puter science 21). As one of such applications, classical advice using a clever algorithm. /∗ ⊆ / 1) we look at the connection between one-way Theorem 6.1 BQP Qpoly PP poly . communication complexity and advised compu- In fact, he gave the following result using tation. a simulation of quantum messages by classical Consider the following cooperative work be- messages in the setting of one-way communica- tween the two parties, Alice and Bob. Alice tion complexity: has a string x and Bob has another string y. Assume that to let Bob compute a Boolean f n× m TheaimforAliceistoletBobcomputethe function from Σ Σ Alice is enough to send l value f(x, y) for a function f by sending a mes- Bob a quantum message of length n.Then, f sage that depends on x only once. We assume Alice can let Bob compute sending a classical O ml l that Bob as well as Alice knows the function message of length ( n log n). f and the time in which Bob computes f(x, y) In this simulation method, Bob is required is unlimited. Hence, if Bob knows x then he to judge whether the original protocol on a can compute f(x, y) correctly. Alice wants to quantum message produces 1 with probabil- > / shorten her message as much as she can (since, ity 1 2, which can be implemented by an for instance, it is expensive for her to send a unbounded-error probabilistic Turing machine. long message). In this setting, the minimal Therefore, the simulation method is also used length of Alice’s message to complete the co- to the proof of Theorem 6.1 by the connec- operative work is called the one-way communi- tion between advice complexity and one-way cation complexity of f. We can also consider communication complexity as follows. Assume L ∈ /∗ poly L L ∩ n the quantum one-way communication complex- BQP Q ,andlet n = Σ . Then, L ity if Alice sends a quantum message to Bob. n is computed by a bounded-error quantum Now assume that Bob is a polynomial-time Turing machine with quantum advice of length p n p Turing machine M, and wants to recognize a ( )where is a polynomial. By incorporating ⊆ 3) set L.IfL is a difficult set, then it is tough the proof technique of BQP PP into the L that Bob computes L(x) for any given string above simulation method, n is computed by x of length n. Intheworstcase,Bobcan- a polynomial-time probabilistic Turing machine O np n p n not compute L(x) for almost all x. Suppose with classical advice of length ( ( )log ( )) > / that Alice completely knows L and n, but she with probability 1 2. does not know which string in Σn is Bob’s in- Unfortunately, Theorem 6.1 means that /∗ put. We can regard this situation as the follow- the separation between BQP Qpoly and /∗ ing cooperative work: Alice’s aim is to let Bob BQP poly is more difficult than a famous compute the value INDEX(Ln,x), where Ln open problem that separates P from the n L ∩ n class PSPACE. In fact, by Theorem 6.1, is the 2 -bit characteristic string of Σ and ∗ ∗ 2n n /  / /  INDEX is the Boolean function from Σ ×Σ BQP poly =BQP Qpoly implies P poly = /  that maps (Ln,x) to the Bin(x)-th bit of Ln. PP poly. It leads to P = PSPACE since other- / (Here, Bin(x) is the lexicographic order of x in wise P = PP by Fact 2.1 and hence P poly = PP/poly. Thus, it cannot be expected to sep- Vol. 1 Quantum Computation with Supplementary Information 411 arate between BQP/∗Qpoly and BQP/∗poly BQP/∗Qlog ⊆ BQP/∗poly can be easily seen (and even P/poly). It is pointed out 1) that by keeping the code of a quantum circuit that the group membership problem by Watrous 28) generates a quantum state of logarithmic length is considered to be a good candidate that sepa- as polynomial classical advice. rates between BQP/∗Qpoly and BQP/∗poly in The proof of Theorem 7.2 is obtained again a relativized world. Nevertheless, a relativized from the connection between advised compu- separation between these classes still remains tation and one-way communication complexity. open. That is, suppose that a polynomial-time quan- tum Turing machine M recognizes any sub- 7. Unrelativized Separation between n set Ln of Σ with the help of quantum advice Classical Advice and Quantum Ad- |φ(Ln) of length m,whereLn(x)=0ifthe vice lexicographical order of x in Σn is larger than In the previous section, we have seen that f(n). This is interpreted as follows in the one- BQP/∗poly and BQP/∗Qpoly are extremely way communication setting: difficult to separate. However, we can show Alice sends the m-qubit state |φ(Ln) to Bob, some unrelativized separation between quan- who can compute any given bit of Ln (in fact, of tum complexity classes with short classical ad- the first f(n) bits in Ln)withprobability≥ 2/3. vice and quantum advice. For instance, we The minimal length of such an m to complete may ask whether BQP/∗ log is different from this cooperative work is known as quantum ran- BQP/∗Q log. This answer is much easily ob- dom access coding 4). tained compared to the separation between Theorem 7.3 (quantum random access cod- polynomial quantum advice and polynomial ing 4))An(n, m, p)-quantum random access classical advice. coding is a function F mapping n-bit strings Theorem 7.1 Let f be a polynomially bounded to m-qubit states satisfying that: for every i ∈ ∗ function. Then, BQP/ Q(O(f(n)logn)) {1,...,n} there is a measurement Oi with out- ∗ 24) BQP/ f(n)n . come 0 or 1 such that the outcome of Oi on The above theorem is obtained using a quan- input F (x) is the ith bit of x with at least p for tum fingerprint 9) (in fact, a quantum finger- all n-bit strings x.Then,m ≥ (1 − H(p))n, print by de Wolf 29)). Consider a “sparse” set where H(p)=−p log p − (1 − p)log(1− p). Ln that cannot be recognized by polynomial- In our case, Alice’s coding is an (f(n),m,2/3)- time quantum Turing machines with any clas- quantum random access coding. Hence, there sical advice of length f(n)n, which is guar- is a set Ln such that M cannot recognize with anteed to exist by a simple counting argu- the help of quantum advice of length at most ment. By contrast, a quantum fingerprint 0.08f(n)(whichis≤ (1−H(2/3))f(n)). On the |φ(Ln) of length O(f(n)logn) enables us to contrary, advice of length f(n) clearly allows a “encode” all the elements of Ln in the fol- deterministic Turing machine to recognize Ln. lowing sense: given |φ(Ln) as supplemen- This argument almost completes the proof of tary information, we can decide if any given Theorem 7.2. input belongs to Ln in quantum polynomial Now we summarize relationships among loga- time. Theorem 7.1 clearly separates be- rithmic and polynomial advice classes of quan- tween BQP/∗ log and BQP/∗Q log as well as tum Turing machines. between BQP/∗polylog and BQP/∗Qpolylog, Corollary 7.4 BQP/∗ log BQP/∗Qlog where polylog is the class of functions f such BQP/∗poly ⊆ BQP/∗Qpoly. that f(n) ≤ p(log n) for some polynomial p. 8. Separating Uniform Complexity On the contrary, we can also show that any Classes from Non-uniform Com- quantum complexity class with quantum advice plexity Classes of length l cannot include a quantum complex- ity class with classical advice whose length is In classical complexity theory, which uniform within a constant factor of l. complexity classes are included in P/poly has Theorem 7.2 For any function f satisfying been investigated at length to unearth the com- f(n) ≤ 2n, P/f(n) BQP/∗Q(0.08f(n)) 24). putational limit of polynomial-size circuits. In Theorem 7.2 implies that BQP/∗Qlog this investigation, it is shown that as a rela- is a proper subset of BQP/∗poly since tivized result there exists an oracle such that 412 IPSJ Digital Courier Oct. 2005 the class NP is not included in P/poly, and be the class of sets recognized by polynomial- as an unrelativized result the class ESPACE time quantum Turing machines with polyno- is not included in P/poly. Similarly, we can mial advice that make queries to the oracle A ask which uniform complexity classes are in- in the truth-table manner. cluded in BQP/∗poly or BQP/∗Qpoly. As a Although the nonadaptive query is restric- relativized result, the following separation is tive, a number of quantum algorithms such as shown. Simon’s algorithm 27) indicate that the quan- Theorem 8.1 There is a set A relative to tum nonadaptive query is still useful compared which NPA BQPA/∗Qpoly 1). to the classical adaptive query. By contrast, By contrast, for the unrelativized case, only a Yamakami 31) showed that there exists a rel- uniform complexity class much higher than NP ativized world that there is a problem that has been shown to separate from advice com- can be solved in deterministic polynomial time, plexity class similar to the classical case. Ear- but cannot be solved by any polynomial-time lier, combining quantum random access coding quantum Turing machine that makes nonadap- with a diagonalization argument, a huge uni- tive queries. This result was extended to the form complexity class EESPACE (the class of case that polynomial-time quantum Turing ma- sets recognized by a deterministic Turing ma- chines have polynomial advice. O(n) chine in space 22 ) was shown to be outside of Theorem 8.4 There exists a set A relative to ∗ 24) A A ∗ 25) BQP/ Qpoly . However, Theorem 6.1, which which P BQPtt/ poly . was proven after the earlier result, implies that 9. Future Work a better uniform complexity class ESPACE is not included in BQP/∗Qpoly. In this article, we have reviewed the study of Corollary 8.2 ESPACE BQP/∗Qpoly. advised quantum computation that is just get- Proof. By Theorem 6.1, BQP/∗Qpoly ⊆ ting started recently. The quantum complex- PP/poly while ESPACE PP/poly can be ity class with classical advice exactly character- shown similar to the proof of ESPACE izes polynomial-size quantum circuits, and en- P/poly 16). This completes the proof. 2 ables us to remove non-uniform information in Finally, we report that the adaptive query transition amplitudes into classical advice. The to an oracle is more powerful than the non- quantum complexity class with quantum ad- adaptive query to the oracle with polynomial vice is another candidate representing a broader advice. The nonadaptive query to an oracle quantum extension of non-uniform computa- A roughly means that a query to A does not tion, but has the computational limit based depend on any query made at previous steps. on comparisons with the quantum complexity In quantum case, the precise definition of the class with classical advice and the uniform com- nonadaptive query is not so simple rather than plexity class. Moreover, the study of quan- the classical case because of quantum inter- tum Turing machines with quantum advice is ference among computation paths. (For in- deeply related to quantum one-way commu- 31) nication complexity. Recently, Bar-Yossef, et stance, see Yamakami for the definition of 6) the parallel query, a pattern of the nonadap- al. showed some problem (but that is not a tive query). Thus, we herewith take a simple function) that quantum one-way communica- definition for the nonadaptive query, called a tion complexity is exponentially smaller than truth-table query, as follows. classical one. As long as the author knows, any connection between this problem and the Definition 8.3 We say that a quantum Tur- ∗ ∗ ing machine M queries to an oracle A in the BQP/ Qpoly =BQP / poly problem has not truth-table manner if (i) M produces a super- been shown. It would be interesting to exam- ine which problem of one-way communication position x αx|x of lists of query words in the first register without querying A, (ii) quan- complexity has a connection to an oracle that /∗ /∗ tumly receives the answers A(x1),...,A(xm) separates BQP poly from BQP Qpoly. We x x1,...,xm left a number of open problems other than the for each list =( ) from the oracle in ∗ ∗ the second register, and (iii) completes the com- BQP/ poly =BQP / Qpoly problem. A ∗ putation without querying A.LetBQPtt/ poly ( 1 ) Do we give a uniform complexity class It is also shown that the exponential-time Merlin- lower than the class NP that is out- 10) ∗ Arthur class MAEXP is outside of P/poly . side of BQP/ Qpoly in a relativized Vol. 1 Quantum Computation with Supplementary Information 413

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of Computing, ACM, New York, pp.302–309 28) Watrous, J.: Succinct quantum proofs for (1980). An extended version appeared as: Tur- properties of finite groups, Proc. 41st Annual ing machines that take advice, L’Enseignement IEEE Symposium on Foundations of Computer Mathematique, Vol.28, pp.191–209 (1982). Science, IEEE, New York, pp.537–546 (2000). 18) Kitaev, A.: Quantum computations: algo- 29) de Wolf, R.: Quantum Computing and Com- rithms and error correction, Russian Math.Sur- munication Complexity, PhD dissertation, Uni- veys, Vol.52, pp.1191–1249 (1997). versity of Amsterdam (2001). 19) Ko, K.: On helping by robust oracle machines, 30) Yamakami, T. and Yao, A.C.: NQPC =co- Theoret. Comput. Sci., Vol.52, pp.15–36 (1987). C=P, Inform. Process. Lett., Vol.71, pp.63–69 20) K¨obler, J. and Watanabe, O.: New collapse (1999). consequences of NP having small circuits, 31) Yamakami, T.: Analysis of quantum func- SIAM J. Comput., Vol.28, pp.311–324 (1998). tions, International Journal of Foundations of 21)Kushilevitz,E.andNisan,N.:Communica- Computer Science, Vol.14, pp.815–852 (2003). tion Complexity, Cambridge University Press 32) Yao, A.C.: Some questions related to dis- (1997). tributed computing, Proc. 11th ACM Sympo- 22) Nielsen, M.A. and Chuang, I.L.: Quantum sium on Theory of Computing,ACM,New Computation and Quantum Information,Cam- York, pp.209–213 (1979). bridge University Press (2000). 33) Yao, A.C.: Quantum circuit complexity, Proc. 23) Nishimura, H. and Ozawa, M.: Computational 34th Annual IEEE Symposium on Founda- complexity of uniform quantum circuit families tions of Computer Science, IEEE, New York, and quantum Turing machines, Theoret. Com- pp.352–361 (1993). put. Sci., Vol.276, pp.147–181 (2002). (Received January 31, 2005) 24) Nishimura, H. and Yamakami, T.: Polynomial (Accepted July 4, 2005) time quantum computation with advice, In- (Released October 5, 2005) form. Process. Lett., Vol.90, pp.195–204 (2004). 25) Nishimura, H. and Yamakami, T.: An algo- Harumichi Nishimura is rithmic argument for nonadaptive query com- plexity lower bounds on advised quantum a researcher of ERATO Quan- computation, Proc. 29th International Sympo- tum Computation and Infor- sium on Mathematical Foundations of Com- mation Project, Japan Science puter Science, Lecture Notes in Computer Sci- and Technology. He received ence, pp.827–838 (2004). B.E. degree from Department 26) Raz, R.: Quantum information and the PCP of Mathematics, Nagoya Univer- theorem, To appear in Proc. 46th Annual IEEE sity in 1993, and M.S. and Ph.D. degrees from Symposium on Foundations of Computer Sci- Human Informatics, Nagoya University in 1997 ence (2005). and 2001, respectively. His research interests http://arxiv.org/quant-ph/0504075 are mainly quantum computing, computational 27) Simon, D.: On the power of quantum compu- complexity, and cryptography. tation, SIAM J.Comput., Vol.26, pp.1474–1483 (1997).