Quantum proof systems for iterated exponential time, and beyond - PowerPoint PPT Presentation

Quantum proof systems for iterated exponential time, and beyond Henry Yuen University of Toronto Joe Fitzsimons (SUTD) Zhengfeng Ji (UT Sydney) Thomas Vidick (CalTech)

Motivating question What is the computational complexity of nonlocal games?

Nonlocal games • Nonlocal two-player game ! = ($, &) : Player A Player B • Classical verifier interacts with two non-communicating players • Verifier samples questions (, ) ∼ $ sends ( to one player, ) to the other a b • Players respond with answers + and , x y • Players win if & (, ), +, , = 1 . verifier

Example: CHSH game • Questions and answers are bits Player A Player B • Players win if ! ⊕ # = % ∧ ' |;⟩ • Classical value: ( )*+* = 3/4 • Entangled value: ( ∗ )*+* = cos 3 4 ≈ .854 5 • One of the most important classical tests of quantum behavior a b • Loophole-free Bell tests • Device-independent quantum cryptography x y verifier

Nonlocal games • Textbook strategy for the CHSH game is simple ! • Players share an EPR pair 00 + |11⟩ " • Player A measures observables ( or ) , player B measures (( ± ))/ 2 . • This is the unique optimal strategy: • Any quantum strategy that achieves entangled value / ∗ 1232 − 5 must be 6( 5) close to this textbook strategy, up to local isometries. • This is called CHSH game rigidity . • Other nonlocal games • Magic Square game ( / 73 = 8/9 , / ∗ 73 = 1 , 2 EPR pairs) • 3-player GHZ game ( / ;2) = 3/4 , / ∗ ;2) = 1 , GHZ state)

Motivating question What is the complexity of this problem? Given a ! -player nonlocal game " , compute # ∗ " ± & . Games " = (), +) can be described in two ways: • Explicit form: all joint probabilities of ) are given, and + is given as truth table. • Implicit form: the verifier is described as a circuit/randomized Turing Machine.

The complexity of classical games Classical value of a (two-player) game ! : " ! = s%& / 3 4, 5 ⋅ 7(4, 5, 9 4 , : 5 ) ':)→+ 0,2 ,:-→. The supremum is over deterministic strategies for the players. Since the question/answer alphabets are finite, there is a trivial brute force algorithm to compute " ! .

The complexity of classical games The complexity of approximating the classical value of games is extremely important to theoretical computer science. Game Approximation Complexity Reference description Explicit 1/#$%&(() NP-complete Cook-Levin theorem (‘70s) Implicit Ω 1 NEXP-complete MIP=NEXP (late ‘80s) Explicit NP-complete PCP Theorem Ω 1 (‘90s) NEXP: nondeterministic exponential time

The complexity of nonlocal games Entangled value of a (two-player) game ! : 0 ∗ ! = % ⊗ * + ) |#⟩ s34 M P -, / ⋅ ⟨#|& ' 56789:6;9 5 ',+,%,) < ∈ℂ ? ⊗ℂ ? C ',%,+,) NO G ,{I J K } ABCD: E F The supremum is over all dimensions " , bipartite states # , and measurement % = ∑ ) * + ) = , for all -, / . operators for both players. I.e., ∑ % & ' The space of quantum strategies is infinite. There is no a priori upper bound on the dimension needed to win any game optimally. Unclear if there’s a brute force algorithm to estimate 0 ∗ ! .

Why study this question? • Understanding quantum correlations • Is it possible to algorithmically optimize over the set of quantum correlations? • Are there nice mathematical characterizations of this set? • Complexity of quantum proof systems • What is the computational power of the class !"# ∗ ? • Applications to quantum information theory and cryptography • Algorithms for optimizing over quantum correlations are useful in device-independent cryptography. • Connections to deep conjectures in mathematics • The computability of the entangled value of games is implied by positive resolutions to Tsirelson’s problem and Connes’ Embedding Conjecture.

Our main result Theorem : For all ! " , the problem of determining whether a given nonlocal game # satisfies • $ ∗ # = 1 or • $ ∗ # ≤ 1 − Ω + , - . is hard for nondeterministic time 2 , - under polynomial-time reductions. Some fine print: • ! " must be “time-computable.” Hardness holds only for games with at least 15 players (but probably can be reduced to 5) •

Prior work • Kempe-Kobayashi-Matsumoto-Toner-Vidick 2011: • It is NP-hard to approximate ! ∗ # ± % &'()(+) when # is given explicitly. • Nontrivial: does not follow from the NP-hardness of classical games. • Problem : Entanglement could help players convince the verifier to accept, even though it should reject! • Solution : verifier runs a game is immunized against entanglement. • Natarajan-Vidick 2018: • It is QMA-hard under randomized reductions to approximate ! ∗ # ± % - . • Also called “Quantum PCP Theorem for Games.”

Prior work • Zhengfeng Ji 2017: • Approximating ! ∗ # ± % &'() * when # is given explicitly is hard for NEXP. • Recall the complexity of deciding whether a classical game # has value 1 is NP-complete. • An exponential separation between the complexities of classical and nonlocal games!

Prior work • Zhengfeng Ji 2017: • Approximating ! ∗ # ± % &'() * when # is given explicitly is hard for NEXP. • Proved using a protocol compression technique • Compresses a nonlocal game into an equivalent one that is exponentially smaller, but gap also shrinks. • However, can only compress once. • The starting point of our result.

Prior work • William Slofstra 2016: • It is undecidable to determine whether ! ∗ # = 1 for general games # . • Proved via an intricate, but beautiful, method to embed the word problem for groups into a two-player nonlocal game. • We give a different proof of this result.

Prior work Complexity lower bounds for nonlocal games Accuracy Lower bound Reference Ω(1) QMA-hard Natarajan-Vidick ‘18 1/&'()(*) NEXP-hard Ji ‘17 = 1 vs. < 1 undecidable Slofstra ‘16

Prior work Complexity lower bounds for nonlocal games Accuracy Lower bound Reference Ω(1) QMA-hard Natarajan-Vidick ‘18 1/&'()(*) NEXP-hard Ji ‘17 ./012[4 , - ] -hard +/,(-) This result = 1 vs. < 1 undecidable Slofstra ‘16

Compressing quantum protocols

Ji’s Compression Theorem Generalization of nonlocal games where verifier exchanges quantum messages over " -rounds with the players. There exists a poly time transformation where • Input : ! -player, " -round quantum interactive protocol # where verifier runs in poly()) time. • Output : (! + 7) -player nonlocal game - with • Has message length O log ) and size 0123 ) # has gap : • If 4 ∗ # = 1 , then 4 ∗ - = 1 - has gap :/0123()) . • If 4 ∗ # ≤ 1 − : , then 4 ∗ - ≤ 1 − ; <=>?(@) .

Protocol circuits [Kempe, et al. ‘08] Every quantum interactive protocol can be transformed into an equivalent 1-round protocol. Protocol circuit for ! -player, " -round quantum interactive protocol ( ! = 2 , " = 1 ) ) ' ’s private registers ) ' Messages to/from ) ' |0⟩ & & Verifier’s private registers ' ( Messages to/from ) ( ) ( ) ( ’s private registers

Protocol circuits [Kempe, et al. ‘08] Every quantum interactive protocol can be transformed into an equivalent 1-round protocol. Protocol circuit for ! -player, " -round quantum interactive protocol ( ! = 2 , " = 1 ) & ' ’s private registers & ' Messages to/from & ' |0⟩ Verifier’s private registers Messages to/from & ( & ( & ( ’s private registers

Protocol history states • Let ! be protocol circuit of size " = poly()) . • The output of Ji’s Compression Theorem is a game + that checks if the players share a history state of the circuit ! . 3 1 , = / 4 ⊗ 6 0 6 078 ⋯ 6 8 |, 2 ⟩ " + 1 012 Phase Unitaries Locality > 8 < 6 8 , … , 6 A 2 qubit gates 8 |, 2 ⟩ < < 8 = > 6 AB8 unbounded 8 > = Unbounded > = 6 AB= < 6 ABC , … , 6 3 2 qubit gates =

Checking protocol history states • Idea of history state comes from Kitaev’s proof that the local Hamiltonians problem is QMA -complete. • Quantum computations can be encoded into history states. • History states can be checked via local measurements. • Main idea of compression: run a “games” version of history state test. • Introduce extra “trusted” player ! " to simulate the verifier of # . • New verifier $′ asks players ! & , ! ( , ! " to measure their local state and check they share a history state of # . • Furthermore, check if the shared state encodes an accepting history of # . • This game has O(log .) -length questions.

Checking protocol history states • Idea of history state comes from Kitaev’s proof that the local Issue 1 : how can we trust the players’ measurements? Hamiltonians problem is QMA -complete. Issue 2 : protocol circuit # involves non-local “gates” • Quantum computations can be encoded into history states. (i.e. players’ actions) • History states can be checked via local measurements. • Main idea of compression: run a “games” version of history state test. • Introduce extra “trusted” player ! " to simulate the verifier of # . • New verifier $′ asks players ! & , ! ( , ! " to measure their local state and check they share a history state of # . • Furthermore, check if the shared state encodes an accepting history of # . • This game has O(log .) -length questions.