A stochastic calculus approach to the oracle separation of BQP and PH
AA stochastic calculus approach to the oracle separation of
BQP and PH Xinyu Wu ˚ July 5, 2020
Abstract
After presentations of Raz and Tal’s oracle separation of
BQP and PH result, several people(e.g. Ryan O’Donnell, James Lee, Avishay Tal) suggested that the proof may be simplified bystochastic calculus. In this short note, we describe such a simplification. A recent landmark result of Raz and Tal [RT19] shows there exists an oracle A such that BQP A Ę PH A . Using a correspondence between PH and AC circuits, the question reduces to a lowerbound against AC circuits. Concretely, it suffices to show that there exists a distribution D over t´
1, 1 u N such that1. For any f : t´
1, 1 u N Ñ t
0, 1 u computable by an AC circuit, | E r f p D qs ´ E r f p U N qs | ď polylog p N q? N ,where U N is the uniform distribution on N bits. The notation E r f p D qs means E x „ D r f p x qs .2. There exists a quantum algorithm Q such that | E r Q p D qs ´ E r Q p U N qs | ě Ω ˆ N ˙ .For details, we refer to Raz and Tal’s paper [RT19]. D in Raz and Tal’s work is a truncatedGaussian. In this note, we will describe a construction of D based on Brownian motion, whichsimplifies many details of the analysis. We briefly review some stochastic calculus concepts used in the proof. See for instance [Øks03,Chapter 7] for details.
Definition 1. An N -dimensional standard Brownian motion B : r
8q ˆ R N Ñ R N is a continuous-time stochastic process characterized by the following:(i) B “ B t ` u ´ B t for u ě B s for s ă t . ˚ Computer Science Department, Carnegie Mellon University. [email protected] a r X i v : . [ c s . CC ] J u l iii) B t ` u ´ B t for u ě N -dimensional Gaussian with mean 0 and covari-ance matrix uI N ˆ N .(iv) B t is continuous almost surely.We can describe a large class of stochastic processes, called Itˆo diffusion processes, by thesolutions of stochastic differential equations of the following form: d X t “ b p X t q dt ` σ p X t q d B t . Definition 2.
Let X be an It ˆo diffusion. The infinitesimal generator of f , is defined as A f p x q “ lim t Ñ E x r f p X t qs ´ f p x q t .We use the E x r¨s notation to mean that we let X t evolve with starting point x .If f is twice continuously differentiable with compact support, we have the following expres-sion for A f : A f p x q “ b p x q ¨ ∇ f p x q `
12 tr p σ p x q σ J p x q H p x qq ,where H is the Hessian of f .For example, the infinitesimal generator of a standard 1D Brownian motion is the Laplacianoperator. For a Brownian motion with covariance matrix Σ , the infinitesimal generator would betr p Σ H p x qq .Next we state Dynkin’s formula, which will be the main tool we use in the later proof. Theorem 1 (Dynkin’s formula, [Øks03, Theorem 7.4.1]) . Let X be an Itˆo diffusion, let τ be a stoppingtime with E r τ s ă 8 , and let f : R N Ñ R N be a twice continuously differentiable function with compactsupport. The following holds: E x r f p X τ qs “ f p x q ` E x „ż τ A f p X s q ds , Moreover, if X τ is bounded, Dynkin’s formula with the same expression for A f holds for f which istwice continuously differentiable (without compact support). The main technical part of Raz and Tal’s result [RT19] shows that, for a Boolean function f : t´
1, 1 u N Ñ t´
1, 1 u computable by an AC circuit, and a multivariate Gaussian distribution Z P R N , | E r f p trnc p Z qqs ´ E r f p U N qs| ď O p γ ¨ polylog p n qq ,where γ is a bound on the (pairwise) covariance of the coordinates of Z , trnc truncates Z sothat the resulting random variable is within r´
1, 1 s N , and U N is the uniform distribution over t´
1, 1 u N . The important condition used here is that AC has second level Fourier coefficientsbounded by polylog p n q , and that this holds under any restriction of the function.Another natural way of viewing a multivariate Gaussian distribution is as the result of an N -dimensional Brownian motion stopped at a fixed time. We can also build the truncation intothe stopping time. This allows us to use tools from stochastic calculus to analyze the distribution.We first recall the definition of restrictions of Boolean functions.2 efinition 3. Let f : t´
1, 1 u N Ñ R and let ρ P t´
1, 1, ˚u N . Let free p ρ q be the set of coordinateswith ˚ ’s. We define the restriction of f by ρ as f ρ : t´
1, 1 u N Ñ R , and f ρ p x q is f evaluated at ρ with x replacing the ˚ ’s in ρ . Henceforth, we also identify Boolean functions f : t´
1, 1 u N Ñ R with their multilinear poly-nomial representations (or Fourier expansions) f p x q “ ÿ | S |Ďr N s ˆ f p S q ź i P S x i .We make some observations about Fourier coefficients. First, the Fourier coefficients of f ρ satisfy p f ρ p S q “ S Ę free p ρ q . We also have thatˆ f p S q “ B S f p q , (1)where B S “ ś i P S B i and B i “ BB x i is the usual calculus derivative. Further, because f is multilinear,for any h P R zt u and any standard basis vector e i we have B i f p x q “ f p x ` he i q ´ f p x q h . (2)The following lemma is similar to [CHLT18, Claim A.5], which first appeared in [BB18]and [CHHL19, Claim 3.3]. Lemma 1.
Let f : R N Ñ R be a multilinear polynomial. For any x P r´ {
2, 1 { s N , there exists adistribution R x over restrictions ρ P t´
1, 1, ˚u N , such that for any i , j P r N s , B ij f p x q “ E ρ „ R x “ B ij f ρ p q ‰ . Proof.
We define R x as such: for each coordinate i P r N s we independently set ρ i to be 1 withprobability ` x i , to be ´ ´ x i , and to be ˚ with probability .Using that f is a multilinear polynomial, and that the coordinates are independent, we deducethat for any y P R N , f p x ` y q “ E ρ „ R x “ f ρ p y q ‰ . Then, using Equation (2), B ij f p x q “ f p x ` e i ` e j q ´ f p x ` e i q ´ f p x ` e j q ` f p x q“ E ρ „ R x “ f ρ p e i ` e j q ´ f ρ p e j q ´ f ρ p e i q ` f ρ p q ‰ “ E ρ „ R x “ B ij f ρ p q ‰ .We now show the main result, which is a restatement of [CHLT18, Therorem A.7] and [RT19,Theorem 2.4]. Theorem 2.
Let f : t´
1, 1 u N Ñ t´
1, 1 u be a Boolean function, and let t ą such that for any restric-tion ρ , ÿ S Ďr N s| S |“ | p f ρ p S q| ď t . Let γ ą and let X be an N-dimensional Brownian motion with mean 0 and covariance matrix Σ , in thesense that E rp X t q i s “ for all i P r N s , and Cov pp X t ´ X s q i , p X t ´ X s q j q “ p t ´ s q Σ ij . Further assumethat | Σ ij | ď γ for i ‰ j. Although f ρ ’s domain is t´
1, 1 u N , it only depends on the coordinates in free p ρ q . et ε ą and define the stopping time τ : “ min t ε , first time that X t exits r´ {
2, 1 { s N u . Then, identifying f with its multilinear expansion, we have | E r f p X τ qs ´ E r f p U n qs | ď εγ t . Proof.
First, we note that E r f p U N qs “ f p q . Next, let σ “ Σ { . X satisfies the stochastic differen-tial equation d X t “ σ d B t .Note that X τ is always within r´ {
2, 1 { s N . We can apply Theorem 1 E r f p X τ qs ´ f p q “ E »–ż τ ÿ i , j Pr N s Σ ij B ij f p X s q ds fifl .Then, we upper bound τ ď ε , and use that B ii f “ i P r N s because f is multilinear, to get | E r f p X τ qs ´ f p q| ď ε E »– sup s Pr τ s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ÿ i , j Pr N s Σ ij B ij f p X s q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) fifl ď εγ x Pr´ { { s N ÿ i ‰ j (cid:12)(cid:12) B ij f p x q (cid:12)(cid:12) “ εγ sup x Pr´ { { s N ÿ i ‰ j (cid:12)(cid:12)(cid:12)(cid:12) E ρ „ R x “ B ij f ρ p q ‰ (cid:12)(cid:12)(cid:12)(cid:12) (Lemma 1) ď εγ sup x Pr´ { { s N E ρ „ R x »–ÿ i ‰ j (cid:12)(cid:12) B ij f ρ p q (cid:12)(cid:12) fifl ď εγ sup x Pr´ { { s N E ρ „ R x »——– ÿ S Ď free p ρ q| S |“ (cid:12)(cid:12)(cid:12) ˆ f ρ p S q (cid:12)(cid:12)(cid:12) fiffiffifl (Equation (1)) ď εγ t . BQP and PH We now use Theorem 2 to construct D as described in Section 1. The distribution D . Let N “ n , where n is a power of 2, and Σ : “ ˆ I n H n H n I n ˙ ,where H n is the Walsh–Hadamard matrix. Now we define X and τ as in Theorem 2, with ε “ {p N q , and our distribution D will be the distribution defined by X τ . At each time t ,we can also look at X t as a pair of random variables in R n , p x t , y t q such that y t is the Hadamardtransform of x t . 4 C lower bound. Tal showed that [Tal17, Theorem 37] there exists a universal constant c suchthat every function f : t´
1, 1 u N Ñ t´
1, 1 u computable by an AC circuit with at most p ln N q (cid:96) gates and depth d satisfies ÿ S Ďr N s| S |“ k | ˆ f p S q| ď p c ¨ ln (cid:96) N q p d ´ q k .Since AC is closed under restrictions, we can apply Theorem 2 with ε “ {p N q and γ “ ? n ,to deduce that | E r f p X τ qs ´ f p q| ď polylog N ? N . Quantum algorithm.
Finally, we show that a quantum algorithm can distinguish D from theuniform distribution. This is virtually identical to the argument in [RT19, Section 6], but we canagain use some stochastic calculus tools on the stopping time built into the distribution. Usingthe Forrelation query algorithm, there is a quantum algorithm Q with inputs x , y P t´
1, 1 u n which accepts with probability p ` ϕ p x , y qq{
2, where ϕ p x , y q : “ n ÿ i , j Pr n s x i ¨ H ij ¨ y j .We show the following proposition [RT19, Claim 6.3], which implies the existence of a O p log N q -time quantum algorithm distinguishing D from uniform with one query. The quantum algorithmis described in more detail in [Aar10, Section 3.2]. Proposition 1. E p x , y q„ D r ϕ p x , y qs ě ε . Proof.
By the linearity of expectation and optional sampling theorem, E p x , y q„ D r ϕ p x , y qs “ n ÿ i , j Pr n s H ij ¨ E r x i ¨ y j s“ n ÿ i , j Pr n s H ij ¨ E r τ s ¨ H ij “ E r τ s .By Markov’s inequality, E r τ s ě ε Pr r τ ą ε s .If τ ď ε , it must be the case that the path exits r´ {
2, 1 { s N no later than ε . Hence, we can upperbound Pr “ τ ď ε ‰ ď N ¨ Pr “ X t exits “ ´ , ‰ earlier than ε ‰ .Each coordinate of X is a standard 1D Brownian motion since Σ ii “ i . An applicationof Doob’s martingale inequality (e.g. [RY99, Proposition II.1.8]) tells us that, for a standard 1DBrownian motion B t , Pr « sup ď t ď ε { | B t | ě ff ď e ´ { ε “ e ´ N ď N for N ě Pr r τ ď ε s ď , so E r τ s ě ε . 5 Acknowledgments
I would like to thank Ryan O’Donnell and Avishay Tal for helpful discussions and their sugges-tions concerning an early draft. Thanks also to Gregory Rosenthal and anonymous reviewers forhelpful comments.
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