Avishay Tal (Stanford, Simons Institute) Joint with Ran Raz (Princeton University) the Landscape of Complexity Classes PSPACE

Oracle Separation of BQP and PH

Avishay Tal (Stanford, Simons Institute) joint with Ran Raz (Princeton University) The Landscape of Complexity Classes PSPACE

P BQP?

BPP Where does BQP fit in the landscape? BQP: Bounded-error Quantum Polynomial time We know: BPP BQP PSPACE

Open: ⊆ ⊆ BQP NP? NP BQP? BQP PH? PH BQP? ⊆ ⊆ Common Belief: all are false. But can we prove it? ⊆ ⊆ Any separation would imply PSPACE P, and thus seems out of reach. ≠ Oracles

© Kevin Hong for Quanta Magazine Oracles An oracle is a language : 0,1 {0,1}. , , … are the classes of decision∗ problems solvable𝐴𝐴 𝐴𝐴 using devices in𝐴𝐴 P, NP, …→ that could additionally𝐏𝐏 𝐍𝐍𝐏𝐏 ask queries to at a unit cost.

For example: 𝐴𝐴 in a poly-time Turing-machine , that decides whether . 𝐿𝐿 𝐏𝐏 ⟺ ∃ 𝑀𝑀 in a poly-time Turing-machine𝑥𝑥 ∈ 𝐿𝐿 , 𝐴𝐴 making queries to the oracle , 𝐿𝐿 𝐏𝐏 ⟺ that∃ decides whether . 𝑀𝑀 𝐴𝐴 𝑥𝑥 ∈ 𝐿𝐿 Oracle Separations Separating Classical Classes: • oracle : [BGS’75] • oracle : 𝑨𝑨 𝑨𝑨 [BGS’75] ∃ 𝐴𝐴 𝐏𝐏 ≠ 𝐍𝐍𝐏𝐏 • oracle : 𝑨𝑨 𝑨𝑨 [FSS’81, A’83, Y’85] ∃ 𝐴𝐴 𝐍𝐍𝐏𝐏 ≠ 𝐏𝐏𝐇𝐇 𝑨𝑨 𝑨𝑨 Quantum∃ vs.𝐴𝐴 Classical𝐏𝐏𝐇𝐇 ≠ 𝐏𝐏𝐏𝐏𝐏𝐏𝐏𝐏𝐏𝐏Separations:𝐄𝐄 • oracle : [BBBV’97] • oracle : 𝑨𝑨 𝑨𝑨 [BV’93] ∃ 𝐴𝐴 𝐍𝐍𝐏𝐏 ⊈ 𝐁𝐁𝐁𝐁𝐏𝐏 • oracle : 𝑨𝑨 𝑨𝑨 [Watrous’00] ∃ 𝐴𝐴 𝐁𝐁𝐁𝐁𝐏𝐏 ⊈ 𝐁𝐁𝐁𝐁𝐏𝐏 𝑨𝑨 𝑨𝑨 Could∃ it be𝐴𝐴 possible𝐁𝐁𝐁𝐁𝐏𝐏 that⊈ BQP𝐍𝐍𝐍𝐍 PH ? ⊆ Our Main Result: BQP vs. PH Recall: a language in PH iff there exists a constant , and a poly-time computable relation s.t. 𝐿𝐿 𝑘𝑘 … : 𝑅𝑅 , , … , 𝑥𝑥 ∈ 𝐿𝐿 ⟺+ ∃𝑦𝑦1∀+𝑦𝑦2…∃𝑦𝑦+3 𝑄𝑄𝑘𝑘𝑦𝑦𝑘𝑘poly𝑅𝑅 𝑥𝑥 𝑦𝑦1 𝑦𝑦𝑘𝑘 𝑦𝑦1 𝑦𝑦2 𝑦𝑦𝑘𝑘 ≤ 𝑥𝑥 Our Main Result: oracle : 𝑨𝑨 𝑨𝑨 “Even if P were equal to NP, even making that strong assumption, that’s not going∃ to be enough𝐴𝐴 to𝐁𝐁𝐁𝐁 capture𝐏𝐏 quantum⊈ 𝐏𝐏𝐏𝐏 computing.” (Lance Fortnow) The Pseudorandomness Setting

f f

Def’n: a distribution is pseudorandom against a class of functions if : 𝐷𝐷 ~ [ ] 𝒞𝒞 ∀𝑓𝑓 ∈ 𝒞𝒞 𝐄𝐄𝑥𝑥 𝐷𝐷 𝑓𝑓 𝑥𝑥 ≈ 𝐄𝐄𝑥𝑥∼𝑈𝑈 𝑓𝑓 𝑥𝑥 The Pseudorandomness Setting

f f

[Aaronson’10, Fefferman-Shaltiel-Umans-Viola’12]: Find a distribution which is pseudorandom for AC0 but not pseudorandom for poly-log-time quantum algorithms?

 an oracle separation between BQP from PH f f

Let be a distribution over 1,1 . Def’n: has advantage distinguishing𝑁𝑁 between 𝐷𝐷 − and with if = ~ . 𝑓𝑓 𝛼𝛼 Main𝐷𝐷 Result:𝑈𝑈 We𝛼𝛼 present𝐄𝐄𝑥𝑥 𝐷𝐷 a 𝑓𝑓distribution𝑥𝑥 − 𝐄𝐄𝑥𝑥∼ 𝑈𝑈 such𝑓𝑓 𝑥𝑥 that: 1. a log(N) time quantum algorithm distinguishing 𝐷𝐷 between and with advantage . ∃ 1 2. Any quasipoly(N)-size constant-depth circuit 𝐷𝐷 𝑈𝑈 Ω log 𝑁𝑁 distinguishes between and with advantage 1 Standard techniques  amplify 𝐷𝐷advantage𝑈𝑈 of quantum alg to 1-1𝑂𝑂�/poly𝑁𝑁(N). Plan: 1. Definition of 2. Quantum algorithm 𝐷𝐷 distinguishing from 3. is pseudorandom for AC0 𝐷𝐷 𝑈𝑈 𝐷𝐷 The Separating Distribution D (Based on Aaronson’s Forrelation distribution) • Let be a power of 2. Let = 1/ log . • The Distribution : draw , … , / i.i.d. from 0, 𝑁𝑁 𝜖𝜖 𝑂𝑂 𝑁𝑁 𝐺𝐺 𝑥𝑥1 𝑥𝑥𝑁𝑁 2 𝒩𝒩 𝜖𝜖 𝑦𝑦…1 = 𝑥𝑥…1 𝑦𝑦2 𝑥𝑥2 / 𝐻𝐻 ⋅ / where is the (𝑦𝑦𝑁𝑁/22) × ( /2)𝑥𝑥Hadamard𝑁𝑁 2 matrix. Output = ( , … . , / , , … , / ). 𝐻𝐻 𝑁𝑁 𝑁𝑁 𝑧𝑧 𝑥𝑥1 𝑥𝑥𝑁𝑁 2 𝑦𝑦1 𝑦𝑦𝑁𝑁 2 The Separating Distribution D (Based on Aaronson’s Forrelation distribution) • Let be a power of 2. Let = 1/ log . • is a multi-variate Gaussian distribution on with zero𝑁𝑁-means and covariance matrix𝜖𝜖 𝑂𝑂 𝑁𝑁 𝑁𝑁 𝐺𝐺 / ℝ / 𝐼𝐼𝑁𝑁 2 𝐻𝐻 where is the ( 𝜖𝜖/2⋅ ) × ( /2) Hadamard matrix with 𝐻𝐻 𝐼𝐼𝑁𝑁 2 1 , , = 𝐻𝐻 1𝑁𝑁 𝑁𝑁 /2 <𝑖𝑖 𝑗𝑗> 𝐻𝐻𝑖𝑖 𝑗𝑗 ⋅ − 𝑁𝑁 Quantum Algorithm Distinguishing from

𝑫𝑫 𝑼𝑼 Quantum Algorithm Distinguishing D [Aaronson’10, Aaronson-Ambainis’15]: O(log N)-time quantum algorithm s.t. 1 , Pr accepts input , = + 𝑄𝑄 2 𝐻𝐻𝐻𝐻 𝑦𝑦 𝑄𝑄 𝑥𝑥 𝑦𝑦 𝑁𝑁 , ~ accepts input , = 1/2 1 𝑥𝑥 𝑦𝑦 𝑈𝑈 accepts input , + ( ) 𝐏𝐏𝐏𝐏, ~ 𝑄𝑄 𝑥𝑥 𝑦𝑦 2 𝐏𝐏𝐏𝐏 𝑥𝑥 𝑦𝑦 𝐷𝐷 𝑄𝑄 𝑥𝑥 𝑦𝑦 ≥ Ω 𝜖𝜖 The Quantum Algorithm 1. Prepare the state: 1 1 |0, + |1, [ / ] [ / ] � ⋅ 𝑖𝑖⟩ � ⋅ 𝑖𝑖⟩ 2. Query: 𝑖𝑖∈ 𝑁𝑁 2 𝑁𝑁 𝑖𝑖∈ 𝑁𝑁 2 𝑁𝑁 |0, + |1, [ / ] 𝑥𝑥𝑖𝑖 [ / ] 𝑦𝑦𝑖𝑖 � ⋅ 𝑖𝑖⟩ � ⋅ 𝑖𝑖⟩ 3. Apply the Hadamard𝑖𝑖∈ 𝑁𝑁 2 𝑁𝑁transform to𝑖𝑖∈ 𝑁𝑁first2 half𝑁𝑁 |0, + |1, [ / ] 𝐻𝐻𝐻𝐻 𝑖𝑖 [ / ] 𝑦𝑦𝑖𝑖 � ⋅ 𝑖𝑖⟩ � ⋅ 𝑖𝑖⟩ 4. Measure the first qbit in {|+ , |/ } basis 𝑖𝑖∈ 𝑁𝑁 2 𝑁𝑁1 1 𝑖𝑖∈ 𝑁𝑁 2 𝑁𝑁 Pr accepts , = + 𝑁𝑁 2 2 ⟩ −⟩ 𝑄𝑄 𝑥𝑥 𝑦𝑦 ⋅ � 𝐻𝐻𝐻𝐻 𝑖𝑖𝑦𝑦𝑖𝑖 𝑁𝑁 𝑖𝑖=1 The Main Result: is Pseudorandom for AC0

𝐷𝐷 Bounded Depth Circuits

1 8 19 3 14 26 𝑥𝑥 𝑥𝑥 𝑥𝑥 𝑥𝑥4 𝑥𝑥7 𝑥𝑥57 𝑥𝑥 𝑥𝑥 𝑥𝑥 𝑥𝑥9 𝑥𝑥17 𝑥𝑥8 AC [ , ]: We focus on • 0gates (size of the circuit) AC , 1 𝑠𝑠 𝑑𝑑 • depth 0 polylog 𝑁𝑁 𝑠𝑠 𝑁𝑁 𝑂𝑂 𝑑𝑑 What do we know about AC0? [Furst-Saxe-Sipser’81, Ajtai’83, Yao’85, Håstad’86]: • Parity not in AC , 1 . • requires exp0 polylog/( 𝑁𝑁) size for depth . Parity 𝑁𝑁 𝑂𝑂 1 𝑑𝑑−1 Fourier-analytical proof𝑁𝑁 technique: 𝑑𝑑 • AC0 circuits can be well-approximated (in ) by low-degree polynomials (over ). [Håstad’86, LMN’89] ℓ2 • Parity cannot. ℝ Potential problem with the approach: log time quantum algorithms (BQLogTime) are also well-approximated by low-degree polys. [BBCMW’98] 𝑂𝑂 𝑁𝑁 The Difference between BQLogTime and AC0 Both BQLogtime & AC0 are approximated by low-degree polynomials, but these polynomials are very different!

BQLogtime can have dense low-degree polynomials, e.g. / / 1 , 1 1 + = + 2 2 𝑁𝑁 2 𝑁𝑁 2 , 𝐻𝐻𝐻𝐻 𝑦𝑦 ⋅ � � 𝑦𝑦𝑖𝑖 ⋅ 𝐻𝐻𝑖𝑖 𝑗𝑗 ⋅ 𝑥𝑥𝑗𝑗 𝑁𝑁 𝑁𝑁 𝑖𝑖=1 𝑗𝑗=1 [T’14]: AC0 circuits have sparse low-degree approximations: :: # monomials of degreepolylogpolylog ,…, : 𝑘𝑘 𝑘𝑘 ∀𝑘𝑘𝑘𝑘 � 𝑓𝑓̂ 𝑆𝑆 ≤ 𝑘𝑘 ≤ 𝑁𝑁 𝑁𝑁 𝑆𝑆⊆ 1 𝑁𝑁 𝑆𝑆 =𝑘𝑘 First Attempt: Fourier Analytical Approach Fourier Analytical Approach – First Attempt The Fourier expansion of : 1,1 { 1,1}: 𝑁𝑁 = 𝑓𝑓 − → − { ,…, } 𝑓𝑓 𝑥𝑥 � 𝑓𝑓̂ 𝑆𝑆 ⋅ � 𝑥𝑥𝑖𝑖 The Fourier expansion 𝑆𝑆extends⊆ 1 𝑁𝑁 to non𝑖𝑖∈-𝑆𝑆Boolean inputs!

Goal: 𝑓𝑓 = ′ 1 𝑁𝑁 𝐄𝐄𝑧𝑧 ∼𝐷𝐷 𝑓𝑓 𝑧𝑧′ − 𝐄𝐄𝑢𝑢∼𝑈𝑈 𝑓𝑓 𝑢𝑢 𝑂𝑂� 𝑁𝑁 ′ = (0) 𝐄𝐄𝑧𝑧 ∼𝐷𝐷 𝑓𝑓 𝑧𝑧′ ≈ 𝐄𝐄𝑧𝑧∼𝐺𝐺 𝑓𝑓 𝑧𝑧 𝑢𝑢∼𝑈𝑈 New Goal: 𝐄𝐄 𝑓𝑓 𝑢𝑢0 =𝑓𝑓 1 𝐄𝐄𝑧𝑧∼𝐺𝐺 𝑓𝑓 𝑧𝑧 − 𝑓𝑓 𝑂𝑂� 𝑁𝑁 Fourier Analytical Approach – First Attempt 0 =

𝐄𝐄=𝑧𝑧∼𝐺𝐺 𝑓𝑓 𝑧𝑧 − 𝑓𝑓 (By definition)

�/ 𝑓𝑓̂ 𝑆𝑆 ⋅ 𝐄𝐄𝑧𝑧∼𝐺𝐺 � 𝑧𝑧𝑖𝑖 𝑆𝑆 ≥1 𝑖𝑖∈𝑆𝑆 = 𝑁𝑁 2 (odd moments = 0)

�/ � 𝑓𝑓̂ 𝑆𝑆 ⋅ 𝐄𝐄𝑧𝑧∼𝐺𝐺 � 𝑧𝑧𝑖𝑖 ℓ=1 𝑆𝑆 =2ℓ 𝑖𝑖∈𝑆𝑆 𝑁𝑁 2 ℓ (Isserlis’ Theorem) 𝜖𝜖ℓ ≤ �/ � 𝑓𝑓̂ 𝑆𝑆 ⋅ 𝑂𝑂 ℓ=1 𝑆𝑆 =2ℓ 𝑁𝑁 𝑁𝑁 2 polylog ℓ [T’14] 2ℓ 𝜖𝜖ℓ ≤ � 𝑁𝑁 ⋅ 𝑂𝑂 ℓ=1 𝑁𝑁 Contribution of first O( ) terms: polylog / What about the larger terms? � 𝑁𝑁 𝑁𝑁 𝑁𝑁 Second Attempt: The Random Walk Approach Viewing ~ as a result of a random walk

𝑧𝑧 𝐺𝐺 A Thought Experiment: Instead of sampling ~ at once, we sample vectors ( ), … , ~ 𝑧𝑧 𝐺𝐺 independently,1 𝑡𝑡 and take𝑡𝑡 𝑧𝑧 1𝑧𝑧 𝐺𝐺 = ( ) + + ( ) 1 𝑡𝑡 𝑧𝑧 ⋅ 𝑧𝑧 ⋯ 𝑧𝑧 √𝑡𝑡 Based on the work of [Chattopadhyay, Hatami, Hosseini, Lovett’18]

Picture from http://en.wikipedia.org/wiki/Random_walk Viewing ~ as a result of a random walk Sample vectors ( ), … , ~ Define + 1𝑧𝑧hybrids:𝐺𝐺 1 𝑡𝑡 • =𝑡𝑡0 𝑧𝑧 𝑧𝑧 𝐺𝐺 𝐻𝐻𝑡𝑡 • For 𝑡𝑡= 1, … , 0 𝐻𝐻 𝐻𝐻𝑖𝑖+1 1 ( ) ( ) 𝑖𝑖 =𝑡𝑡 + + 𝐻𝐻𝑖𝑖 1 𝑖𝑖 Observe: ~𝑖𝑖 . 𝐻𝐻 ⋅ 𝑧𝑧 ⋯ 𝑧𝑧 1 Taking yields√𝑡𝑡 a Brownian motion. 𝐻𝐻 𝑡𝑡 0 We take 𝐻𝐻= poly𝐺𝐺 . 𝐻𝐻 𝑡𝑡 → ∞ Claim: for𝑡𝑡 = 0, … ,𝑁𝑁 1, . 𝑖𝑖 𝑡𝑡 − polylog 𝑁𝑁 𝐄𝐄 𝑓𝑓 𝐻𝐻𝑖𝑖+1 − 𝐄𝐄 𝑓𝑓 𝐻𝐻𝑖𝑖 ≤ 𝑡𝑡 𝑁𝑁 Proof by Picture

[CHHL’18]: i-th step first step, using closure under restrictions. i-th step ≈

First Step: Simple Fourier Analysis first step Only second level matters. Base Case

1 𝐄𝐄= 𝑓𝑓 𝐻𝐻1 − 𝐄𝐄 𝑓𝑓 𝐻𝐻0 0 / 𝐄𝐄𝑧𝑧∼𝐺𝐺 𝑓𝑓 𝑧𝑧 − 𝑓𝑓 1 = 𝑁𝑁 2 √𝑡𝑡

�/ � 𝑓𝑓̂ 𝑆𝑆 ⋅ 𝐄𝐄𝑧𝑧∼𝐺𝐺 � 𝑧𝑧𝑖𝑖 ℓ=1 𝑆𝑆 =2ℓ 𝑖𝑖∈𝑆𝑆 √𝑡𝑡 𝑁𝑁 2 ℓ 𝜖𝜖ℓ ≤ �/ � 𝑓𝑓̂ 𝑆𝑆 ⋅ 𝑂𝑂 ℓ=1 𝑆𝑆 =2ℓ 𝑡𝑡 𝑁𝑁 𝑁𝑁 2 polylog ℓ 2ℓ 𝜖𝜖ℓ ≤ polylog� 𝑁𝑁 ⋅ 𝑂𝑂1 ℓ=1 + 𝑡𝑡 𝑁𝑁 (for large enough) 𝑁𝑁 ≤ 𝑜𝑜 𝑡𝑡 𝑡𝑡 𝑁𝑁 𝑡𝑡 𝑁𝑁 General Case: Reduction to Base Case

Lemma [CHHL’18]: for any fixed 0.5, 0.5 the fnc + 𝑁𝑁 𝑣𝑣 ∈ − can be written as 2 0 where is a random restriction𝑔𝑔 𝑧𝑧 of ≝(whose𝑓𝑓 𝑣𝑣 marginals𝑧𝑧 − 𝑓𝑓 𝑣𝑣depend on ). 𝐄𝐄𝜌𝜌 𝑓𝑓𝜌𝜌 ⋅ 𝑧𝑧 − 𝑓𝑓𝜌𝜌 𝑓𝑓𝜌𝜌 Analysis of step i+1: 𝑓𝑓 𝑣𝑣 Conditioned on 0.5, 0.5 (happens whp): 𝑁𝑁 𝐻𝐻𝑖𝑖 ∈ − + ( ) 𝐄𝐄 𝑓𝑓 𝐻𝐻𝑖𝑖+1 − 𝐄𝐄 𝑓𝑓 𝐻𝐻𝑖𝑖 1 𝑖𝑖+1 polylog ≤ 𝐄𝐄 𝑓𝑓 𝐻𝐻𝑖𝑖 √𝑡𝑡( ⋅ 𝑧𝑧 ) −0𝑓𝑓 𝐻𝐻𝑖𝑖 2 𝑖𝑖+1 𝑁𝑁 ≤ 𝐄𝐄 𝑓𝑓𝜌𝜌 √𝑡𝑡 ⋅ 𝑧𝑧 − 𝑓𝑓𝜌𝜌 ≤ 𝑡𝑡 𝑁𝑁 Recap 1. Defined a distribution based on a MVG . 2. is not pseudorandom for log(N)-time quantum algorithms. [Aaronson𝐷𝐷’10, Aaronson-Ambainis𝐺𝐺 ’15] 3. 𝐷𝐷 is pseudorandom for AC0 (our contribution) 1 𝐷𝐷 . - Thought𝐄𝐄𝑧𝑧∼𝐺𝐺 Experiment:𝑓𝑓 𝑧𝑧 − 𝐄𝐄𝑢𝑢 View∼𝑈𝑈 𝑓𝑓 ~𝑢𝑢 as ≤a result𝑂𝑂� of a random walk making tiny steps. 𝑁𝑁 - AC0 circuits are approximated𝑧𝑧 by𝐺𝐺 sparse low-degree polynomials𝑡𝑡 [T’14]  first step has advantage 1 - [Chattopadhyay, Hatami, Hosseini, Lovett ’18]: 𝑂𝑂� 𝑡𝑡 𝑁𝑁  -th step has advantage 1 𝑖𝑖 𝑂𝑂� 𝑡𝑡 𝑁𝑁 Thank You!