BQP and The Polynomial Hierarchy based on `BQP and The Polynomial Hierarchy' by Scott Aaronson Deepak Sirone J., 17111013 Hemant Kumar, 17111018 Dept. of Computer Science and Engineering Dept. of Computer Science and Engineering Indian Institute of Technology Kanpur Indian Institute of Technology Kanpur Abstract The problem of comparing two complexity classes boils down to either finding a problem which can be solved using the resources of one class but cannot be solved in the other thereby showing they are different or showing that the resources needed by one class can be simulated within the resources of the other class and hence implying a containment. When the relation between the resources provided by two classes such as the classes BQP and PH is not well known, researchers try to separate the classes in the oracle query model as the first step. This paper tries to break the ice about the relationship between BQP and PH, which has been open for a long time by presenting evidence that quantum computers can solve problems outside of the polynomial hierarchy. The first result shows that there exists a relation problem which is solvable in BQP, but not in PH, with respect to an oracle. Thus gives evidence of separation between BQP and PH. The second result shows an oracle relation problem separating BQP from BPPpath and SZK. 1 Introduction The problem of comparing the complexity classes BQP and the polynomial heirarchy has been identified as one of the grand challenges of the field. The paper \BQP and the Polynomial Heirarchy" by Scott Aaronson proves an oracle separation result for BQP and the polynomial heirarchy in the form of two main results: A 1. There exists an oracle A relative to which FBQPA 6⊂ FBPPPH , where FBQP and FBPP are the relation versions of BQP and BPP respectively. 2. There exists an oracle A relative to which BQPA 6⊂ BPPpathA and BQPA 6⊂ SZKA. The problem of comparing BQP and the Polynomial Hierarchy is interesting for the following reasons: 1. The question of what classical resources are needed to simulate quantum computers is still not known. Questions like whether quantum amplitudes can be simulated using approximate counting (which would imply that BQP ⊂ BPPNP are still open). 2. BQP 6⊂ PH would imply that quantum computers can solve problems other than the NP interme- diate problems like graph isomorphism. It might be fruitful to search for quantum algorithms not even in PH. 3. The question of whether BQP could provide exponential speedups even if P = NP is another reason why the question of whether or not BQP 6⊂ PH is interesting. We will give a brief overview of the proofs of the two theorems in proceeding sections. 1 Complexity Classes A brief overview of the various complexity classes presented in this report are as follows: NP A language L 2 NP if there exists a polynomial time Turing machine M such that x 2 L iff 9yM(x; y) = 1 P PH For i ≥ 1, a language L is in Σi if there exists a polynomial time Turing machine M and a polynomial q such that q(jxj) q(jxj) q(jxj) x 2 L iff 9u1 2 f0; 1g 8u2 2 f0; 1g :::Qiui 2 f0; 1g M(x; u1; : : : ui) = 1 where Qi denotes 8 or 9 depending on whether i is even or odd, respectively. The polynomial S P hierarchy is the set PH= Σi i FC The class of functions computable in class C whose output is a bit string rather than a Boolean value. BPP The language L is said to be in class BPP, if there exists a polynomial time probabilistic Turing machine P such that P r[P (x) = L(x)] ≥ 2=3; r where probability is over random strings r chosen by P . [5] BQP A language L is in BQP if and only if there exists a polynomial-time uniform family of quantum circuits fQn : n 2 Ng such that • For all n 2 N, Qn takes n qubits as input and outputs 1 bit. 2 • For all x 2 L, P r[Qjxj(x) = 1] ≥ 3 2 • For all x 62 L, P r[Qjxj(x) = 0] ≥ 3 FBQP FBQP is the class of relations R ⊂ f0; 1g∗ × f0; 1g∗ for which there exists a quantum polynomial time algorithm A, such that given an x 2 f0; 1gn produces an output y such that P r[(x; y) 2 R] = 1 − o(1) where the probability is over A's internal randomness. FBPP FBPP is defined the same way as above except that now A is a classical probabilistic polynomial time algorithm. CA The set of languages solvable by a class C machine when given an oracle access to language A. This means that the C can ask the oracle A membership queries without incurring any additional cost in time. AC0 The class of languages computable by circuit families of constant depth and polynomial size, whose gates have unbounded fanin. [5] BPPpath The class of languages L for which there exists a pair of polynomial time Turing machines A and B such that • if x 2 L, then P r[A(x; r)jB(x; r)] ≥ 2 r 3 • if x 62 L, then P r[A(x; r)jB(x; r)] < 1 r 3 • P r[B(x; r)] > 0. [4] r 2 IP A language L is in IP, if there exists a polynomial time probabilistic Turing machine M such that on c input x and random sting r 2 f0; 1gjxj , x 2 L =) 9PP r[M(x; r; P ) = 1] ≥ 2=3; r x 62 L =) 8PP r[M(x; r; P ) = 1] ≤ 1=3; r where P is proof provided by all powerful prover having infinite computation power. [5] SZK Zero-knowledge protocol is a method by which the prover can prove to the verifier that a given statement is true, without conveying any information apart from the fact that the statement is indeed true. In the IP protocol, along with prover and verifier, their exists an expected probabilistic simulator S such that it can predict the conversation between the verifier and the prover. We say the protocol is statistical zero knowledge if the distribution of simulator and the distribution of verifier-prover conversation is statistically close. [3] 2 Preliminaries The paper discusses two problems on boolean functions namely Fourier Fishing and Fourier Check- ing. The boolean functions considered in the following sections will have the form f : f0; 1gn ! {−1; 1g. n ^ The Fourier transform of f in Z2 , denoted as f is 1 X f^ = p (−1)x:zf(x) N x2f0;1gn where N = 2n. 2.1 Fourier Fishing n Given n Boolean functions f1; : : : ; fn : f0; 1g ! {−1; 1g as input. We have to output n strings n ^ z1; : : : ; zn 2 f0; 1g , such that at least 75% of which satisfy jfi(zi)j ≥ 1 and at least 25% of which ^ satisfy jfi(zi)j ≥ 2. Call an n-tuple hf1; : : : ; fni of Boolean functions good if n X X ^ 2 fi(zi) ≥ 0:8Nn; i=1 zi:jf^i(zi)|≥1 n X X ^ 2 fi(zi) ≥ 0:26Nn i=1 zi:jf^i(zi)|≥2 The promise version of Fourier Fishing assumes that we are given n Boolean functions as input which n are promised to be good and we have to output n strings z1; : : : ; zn 2 f0; 1g , such that at least 75% of ^ ^ which satisfy jfi(zi)j ≥ 1 and at least 25% of which satisfy jfi(zi)j ≥ 2. 2.2 Fourier Checking Given two Boolean function f; g : f0; 1gn ! {−1; 1g as input, it is promised that either 1. hf; gi was drawn from a uniform distribution U, which sets each f(x) and g(y) by a fair, independent coin toss. 2. hf; gi was drawn from a forrelated distribution F, defined as the distribution generated by the following process: N A random real vector is chosen v = (vx)x2f0;1gn 2 R , by drawing each entry from a Gaussian distribution N (0; 1). Then f(x) and g(x) are set as follows: v v^ f(x) = x ; g(x) = x jvxj jv^xj n wherev ^ is Fourier transform of v in Z2 . As g is highly correlated to f's Fourier transform, it is called a forrelated distribution. 3 The problem is to accept if hf; gi was drawn from F and reject if drawn from U. The promise version of this problem says that the quantity 0 12 1 X p(f; g) := f(x)(−1)x:yg(y) N 3 @ A x;y2f0;1gn is at least 0.05 or at most 0.01. Accept hf; gi in the first case and reject in the second case. 3 Oracle Separation Between FBQP and FBPP The oracle separation between FBQP and FBPP results from a series of reductions using AC0 circuit n lower bounds. First, we get a circuit lower bound for "-Bias Detection by reducing the 2 -Hamming Weight problem to it and secondly, we reduce "-Bias Detection to Fourier Fishing and thereby derive a circuit lower bound for Fourier Fishing. n 2 -Hamming Weight ! "-Bias Detection ! Fourier Fishing exp Ω n1=(d−1) exp Ω 1="1=(d+2) exp Ω N 1=(2d+8) 3.1 Quantum Algorithm The quantum algorithm for Fourier Fishing FF-ALG with oracle access to hf1; f2; : : : fni is as follows: 1. for i = 1 : : : n (a) Create an n-qubit state j0i 1 P (b) Apply Hadamard gate and create the equal superposition p n jxi N x2f0;1g 1 P (c) Query the f oracle and get the state p n f (x) jxi i N x2f0;1g i (d) Apply Hadamard gate again and measure in the computational basis (e) Output the measurement as zi As shown in the paper, the lemma 7 and lemma 8 claims that the FF-ALG will output the required zi's with probability 1 − 1=exp(n).