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 .