MIP* = RE and Tsirelson's problem THOMAS VIDICK CALIFORNIA INSTITUTE OF TECHNOLOGY Joint work with Zhengfeng Ji Anand Natarajan John Wright Henry Yuen (UTS) (MIT) (UT Austin) (Columbia) Our Results Entangled games are hard(er) to approximate QIP’04 (Waterloo) Cleve-HTW, Consequences and limits of nonlocal strategies QIP’11 (Singapore) Fritz, Tsirelson's problem and Kirchberg's conjecture QIP’13 (Beijing) Ito-V, NEXP in MIP* QIP’17 (Seattle) Ji, Compression of quantum multi-prover interactive proofs QIP’18 (Delft) Natarajan-V, Robust self-testing of many qubit states Slofstra, Tsirelson’s problem and an embedding theorem for groups arising from non-local games QIP’19 (Boulder) Fitzsimons-JVY, Quantum proof systems for iterated exponential time, and beyond QIP’20 (Shenzhen) Natarajan-W, NEEXP in MIP* MIP* = RE BPP, BQP: Problems for which there is a (randomized, quantum) polynomial-time algorithm that always returns the correct answer. MA, QMA: Problems that can be verified in (randomized, quantum) polynomial time, given a polynomial-size proof MIP* = RE RE: Problems for which there is an algorithm that eventually terminates and returns ‘YES’ on positive instances (and doesn’t terminate/returns ‘NO’ on negative instances) MIP*: Problems that can be verified in polynomial time by interacting with quantum provers sharing entanglement Consequences of MIP* = RE • Negative answer to Tsirelson’s problem: the tensor and commuting models for quantum correlations are strictly distinct • Negative answer to Connes’ embedding problem: there exist type 퐼퐼1 von Neumann algebras that are not ‘hyperfinite’ • Verification of quantum systems: asymptotically efficient tests for arbitrarily high-dimensional entanglement • NISQ systems: provable advantage for small noisy quantum systems using deep variational quantum Boltzmann learning Plan for the talk 1) Quantum correlations and Tsirelson’s problem 2) An approach to Tsirelson’s problem via algorithms & complexity 3) Quantum multiprover interactive proofs 4) Open questions Quantum correlations and Tsirelson’s problem Quantum nonlocality TU Delft, Netherlands (2015) Local measurements on distant particles can exhibit unexpected correlations Bell & Tsirelson’s setup 푥 푦 • Correlation: family of distributions 푝 푎, 푏 푥, 푦 | 푥, 푦 ∈ 1, … , 푛 푎, 푏 ∈ 1, … 푘 푎 푏 • Bell’64: some correlations have a quantum model but no classical (LHV) explanation • Tsirelson in the ‘80s introduces two possible representations for quantum correlations: 푥 푦 푝 푎, 푏 푥, 푦 = 휓 푃푎 ⊗ 푄푏 휓 ∶ |휓〉 ∈ ℋ ⊗ ℋ 푥 푦 푝 푎, 푏 푥, 푦 = 휓 푃푎 푄푏 휓 : 휓 ∈ ℋ, 푥 푦 [푃푎 , 푄푏 ] = 0 Tsirelson’s problem Quantum Bell-type inequalities are defined in terms of two (or more) subsystems of a quantum system. The subsystems may be treated either via (local) Hilbert spaces, - tensor factors of the given (global) Hilbert space, or via commuting (local) operator algebras. The latter approach is less restrictive, it just requires that the given operators commute whenever they belong to different subsystems. Are these two approaches equivalent? Tsirelson’s problem 퐶 푛, 푘 = { 휓 푃푥 ⊗ 푄푦 휓 : |휓〉 ∈ ℋ ⊗ ℋ } ⊗ 푎 푏 푎푏푥푦 퐶 푛, 푘 = { 휓 푃푥푄푦 휓 : 푐표푚 푎 푏 푎푏푥푦 퐶푐표푚 푥 푦 |휓〉 ∈ ℋ, [푃푎 , 푄푏 ] = 0} • Both sets are convex 퐶⊗ • 퐶⊗ 푛, 푘 ⊆ 퐶푐표푚 푛, 푘 for all 푛, 푘. 푛2푘2 • 퐶푐표푚(푛, 푘) is closed, but [Slofstra’18] 퐶⊗(푛, 푘) is not! [0,1] Is 퐶⊗ 푛, 푘 = 퐶푐표푚 푛, 푘 for all 푛, 푘 ≥ 2 ? The connection with operator algebras Is 퐶⊗ 푛, 푘 = 퐶푐표푚 푛, 푘 for all 푛, 푘 ≥ 2 ? • [Kirchberg’93, Fritz’11, Junge-NPPSW’11, Ozawa’13]: Tsirelson’s problem ↔ CEP ↔ QWEP •Connes’ 1976 “Embedding Problem” (CEP) : “Every type II1 von Neumann algebra embeds in an ultrapower of the hyperfinite II1 factor ℛ” • Kirchberg’s 1993 QWEP conjecture: ∗ ∗ ? ∗ ∗ 퐶 퐹2 ⊗푚푖푛 퐶 퐹2 = 퐶 퐹2 ⊗푚푎푥 퐶 퐹2 • Multiple reformulations: free entropy (Voiculescu), group theory (Radulescu), etc. An approach to Tsirelson’s problem via algorithms & complexity Computing the quantum bound 퐶 푛, 푘 = { 휓 푃푥 ⊗ 푄푦 휓 : ⊗ 푎 푏 푎푏푥푦 휆 |휓〉 ∈ ℋ ⊗ ℋ } 휔∗ 휆 = sup 휆 ⋅ 푝 푝∈퐶⊗ 푛,푘 = sup ෍ 휆푥푦푎푏 푝(푎, 푏|푥, 푦) 푝∈퐶 푛,푘 ⊗ 푥푦푎푏 퐶 푛, 푘 = { 휓 푃푥푄푦 휓 : 푐표푚 푎 푏 푎푏푥푦 푥 푦 ` |휓〉 ∈ ℋ, [푃푎 , 푄푏 ] = 0} 2 2 푂 [0,1]푛 푘 휔푐표푚 휆 = sup 휆 ⋅ 푝 푝∈퐶푐표푚 푛,푘 Computing the quantum bound • Suppose exhaustive search & NPA 휆 converge to the same value, for all 휆 → Tsirelson’s problem has a positive answer • Suppose exhaustive search & NPA do not converge to the same value,for some 휆 → Tsirelson’s problem has a negative answer • [Fritz-NT’14] Suppose that 휔∗(휆) is uncomputable 2 2 푂 [0,1]푛 푘 → Tsirelson’s problem has a negative answer Quantum multiprover interactive proofs Interactive proofs 푧 ∈ 0,1 푛 푦푒푠/푛표 • BPP/BQP: efficient decision 퐴 Time: randomized/quantum poly(푛) Ex: Primality testing; Connectivity; Linear programming; Factoring 푃 • MA/QMA: efficient verification 휋 |휓⟩ 푧 is “yes” ⇒ ∃ 휋, accepted by 푉 whp 푧 ∈ 0,1 푛 푦푒푠/푛표 푉 푧 is “no” ⇒ ∀ 휋, rejected by 푉 whp Time: randomized/quantum poly(푛) Ex: Graph colorability; Local Hamiltonian Interactive proofs 푃 • IP/QIP: efficient interactive verification 푥 푎 푧 ∈ 0,1 푛 푦푒푠/푛표 Ex: Graph non-colorability, GO, … 푉 [Shamir’90, Jain-JUW’10] IP = QIP = PSPACE Time: randomized/quantum poly(푛) • MIP/QMIP: efficient interactive verification 푃1 푃2 푎 with two provers 푏 푥 푦 푛 푦푒푠/푛표 Ex: Exponential-size graph coloring 푧 ∈ 0,1 푉 [Babai-FL’91,Koabayahi-M’03] MIP = QMIP = NEXP Time: randomized/quantum poly(푛) Interactive proof systems as Bell functionals 푃1 푃2 푎 푏 푥 푦 푧 ∈ 0,1 푛 푉 푦푒푠/푛표 MIP*MIP == QMIP*:QMIP: efficient efficient interactive interactive verification verification withwith two two provers provers sharing entanglement • [Cleve-HTW’04] The class MIP* characterizes the complexity of optimizing over the sets 퐶⊗ 푛, 푘 Interactive proof systems as Bell functionals 푃1 푃2 푎 푏 푥 푦 푧 ∈ 0,1 푛 푉 푦푒푠/푛표 Max acc(푉, 푧) = sup σ푥푦 휋 푥, 푦 σ Prob 푎, 푏 푥, 푦 strategy 푎푏:correct for 푥푦 푥 푦 = sup Σ푥푦푎푏 휋(푥, 푦)1푎푏: correct for 푥푦⟨휓|푃푎 ⊗ 푄푏 |휓⟩ strategy ∗ = 휔 (휆) for 휆푎푏푥푦 = 휋 푥, 푦 1푎푏: correct for 푥푦 Interactive proof systems as Bell functionals 푃1 푃2 푎 푏 푥 푦 푧 ∈ 0,1 푛 푉 푦푒푠/푛표 MIP*: problems that admit efficient interactive verification with two provers sharing entanglement • [Cleve-HTW’04] The class MIP* characterizes the complexity of optimizing over the sets 퐶⊗ 푛, 푘 • What can be said about problems in MIP*? • 휔∗(휆) uncomputable ↔ MIP* contains undecidable languages MIP* ⊇ RE • [Ito-V’12] MIP* contains all problems in MIP = NEXP • Proof shows that error correction-based probabilistically checkable proofs used in the proof of NEXP = MIP are sound in the presence of entanglement • [Natarajan-W’19] MIP* contains all problems in NEEXP • Proof leverages entanglement between the provers as a tool to aid verification • [Natarajan-JVYW’20] MIP* ⊇ RE by recursively applying technique from [NW’19] • [Turing’1936] RE contains the halting problem, which is undecidable • MIP* contains undecidable languages Using entanglement to delegate verification [Natarajan-W’19] Probabilistically checkable proofs: p • 푣 verifies that 푝1 and 푝2 simulate the1 interactionand 푝2 prepare between encoded 푉 and certificate 푃1, 푃2 : that they performed the correct 푃1 푃2 1) 푝1 and 푝2 locally generate classical computation. 푣 checks it questions 푋 and 푌 to 푃1, 푃2 efficiently by making small queries 2) 푝 and 푝 locally compute an Self-testing1 : 2푣 verifies correlations Buildsanswer on [Arora 퐴 and- S’98,퐵 to theHarsha’04, question Ben - which certify that each prover has 푧 ∈ 0,1 푛 푉 푦푒푠/푛표 Sassonthey- S’08,++]generated obtained exactly the right information, 푝 푝2 3)in the푝1 rightand 푝 way.2 prove to 푣 that they 1 generated the questions correctly time exp(푛) Buildsand on were [Werner able to-S’88, obtain Mayers valid- Yao’98,answers Natarajan to them-V’18,++] 푧 ∈ 0,1 푛 푣 푦푒푠/푛표 time poly(푛) Recursive compression [Fitzsimons-JVY’18] • Recursive construction yields polynomial-time verifiers for 푛 2푛 languages in NTIME(2푛), NTIME(22 ), NTIME(22 ), … • Main difficulty: identify a class of distributions 풞 for the questions such that sampling from a distribution in 풞 can be tested using a distribution in the same class 풞 • Class 풞 generalizes plane-point distribution from low-degree tests The final step 푉푟푒푐 푛, 푀 푦푒푠/푛표 For any Turing machine 푀, there is a computable 휆 = 휆(푀) such that • 푀 halts → 휔∗ 휆 = 1 1 • 푀 does not halt → 휔∗ 휆 ≤ 2 RE ⊆ MIP∗ Summary 퐶 푛, 푘 = { 휓 푃푥 ⊗ 푄푦 휓 : |휓〉 ∈ ℋ ⊗ ℋ } ⊗ 푎 푏 푎푏푥푦 퐶 푛, 푘 = { 휓 푃푥푄푦 휓 : |휓〉 ∈ ℋ, [푃푥, 푄푦] = 0} 푐표푚 푎 푏 푎푏푥푦 푎 푏 • Tsirelson’s problem: Is 퐶⊗ 푛, 푘 = 퐶푐표푚 푛, 푘 for all 푛, 푘 ≥ 2 ? • We give a negative answer: 퐶⊗(푛, 푘) ≠ 퐶푐표푚(푛, 푘) for some 푛, 푘 • Proof shows that that linear optimization over 퐶⊗ (= computing the quantum value) for specific class of 휆 (coming from interactive proofs) is intractable • Techniques combine proof verification and self-testing. Entanglement used to certify increasingly complex computations in a recursive fashion Open questions Some questions Complexity theory: • What is the complexity of commuting-strategy MIP, MIPco? • Proof requires only two provers. Corollary: MIP(k provers) = MIP(2 provers) • Direct argument? • Can we verify QMA statements using log-length questions and quantum polynomial- time provers (+ access to the witness)? • Can we show uncomputability of 휆 ↦ 휔∗ 휆 for fixed 푛, 푘 ? • Beyond RE: can higher levels of the arithmetical hierarchy be characterized by interactive proof variants? • [Coudron-S’19] characterize zero-gap MIPco • [Mousavi-NY’20] characterize zero-gap MIP* Some questions Operator algebras: • Complexity-theoretic argument implies existence of a correlation that can be realized in the commuting model, but not in the tensor model • Working through the proof gives an explicit example.