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Equilibrium Value Method For the proof of QIP=PSPACE

Equilibrium Value Method For the proof of QIP=PSPACE. Xiaodi Wu EECS, University of Michigan, Ann Arbor January, 2010. The work was conducted while the author was visiting the Institute for Quantum Computing, University of Waterloo, Ontario, Canada. TexPoint fonts used in EMF.

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Equilibrium Value Method For the proof of QIP=PSPACE

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  1. Equilibrium Value Method For the proof of QIP=PSPACE Xiaodi Wu EECS, University of Michigan, Ann Arbor January, 2010 • The work was conducted while the author was visiting the Institute for Quantum • Computing, University of Waterloo, Ontario, Canada. TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAA

  2. Interactive Proof System : Intuitive Picture Goal : determine whether x is in a language L input x Polynomial Rounds interactions of classical messages input x

  3. Quantum Interactive Proof System Goal : determine whether x is in a language L input x, still classical Polynomial Rounds interactions of quantum messages input x

  4. Previous Result: • IP=PSPACE[Lund, Fortnow, Karloff, and Nisan 1992; Shamir 1992] • Easy Direction : IP PSPACE, since all the messages are of polynomial size. • Hard Direction: PSPACE IP Solved by a method called Arithmetization which constructs a polynomial round interaction protocols for PSPACE-complete problem. Polynomial rounds is necessary !

  5. What about Quantum Case • IP QIP, and thus we know PSPACE QIP • The easier direction becomes hard in quantum case • It is an open problem for almost a decade to show QIP=PSPACE. • However, we do know something non-trivial about the QIP in the very beginning. • QIP=QIP(3)[KW00], 3 rounds are sufficient for quantum case • QIP(3) EXP[KW00], by formulated as a SDP.

  6. 3 rounds Quantum Interactive Proof System all- power , any quantum circuits Qubits 1 2 3 efficient quantum circuits

  7. Notations: linear algebra • : complex Euclidean spaces • : space of operators acting on • : set of positive semidefinite operators (or matrices) acting on Quantum State: One part of the whole state: Given This is called Partial Trace The X part of the state is and Y part of the state is More precisely, we have partial trace be the unique linear mapping such that

  8. Notations: continued Quantum Operations: a operation maps a quantum state to a quantum state. More precisely: auxiliary space • (Complete Positivity) • (Trace Preserving) Quantum Measurement : an irreversible quantum operations defined by a set of positive operators such that . The outcome k occurs with probability

  9. 3 rounds Quantum Interactive Proof System(revisit) all- power , any quantum operation Quantum State on 1 2 3 efficientquantum operations

  10. Roadmap Match from both sides [KW00] General Model for interaction [GW07] Polynomial algorithm for SDP (IPM, Ellipsoid) QIP in EXP [KW00] SDP formulation but with exponential size One Possible Way PSPACE =NC(poly) [Bord77] PSPACE ? How toparallelize? Matrix Multiplicative Weights Update method [AHK05] Parallelizable for someSDP and Equilibrium Value ?

  11. Roadmap(continued) Bad News: old formulation of QIP still open Good News: reformulation becomes solvable SDP reformulation [JJUW09], August 09 1 2 3 Starts with definition, simpler SDP but not that simple. Using MMW to solve SDP involves more complicated steps. Only one classical bit is sent in the second step QIP=QMAM [MW05] Simpler solvable SDP QMAM in NC(poly) too complicated and technical 

  12. Roadmap(continued) Two Key Ingredients A Neater Proof is available[Wu09] August, 09  Quantum Circuit Distinguishability: Given two short quantum circuits, distinguish their distance between two promises. Starts with QIP-Complete problem [RW05] Resulted in a simple Equilibrium Value Problem Solved by Matrix Multiplicative Weight Update Method

  13. Matrix Multiplicative Weight Update Method n agents weights w1 w2 . . . wn Long History (cited from SanjeevArora) Update weights according to performance: wit+1Ã wit (1 +  ¢ performance of i) The answer is Yes by using multiplicative weight update 1$ for correct prediction • N “experts” on TV • Can we perform as good as the best expert ? 0$ for incorrect

  14. MatrixMultiplicative Weight Update Method Density operator Proof Hint: use potential function Tr(W (t)) and matrix inequality. Observation updated in this way some small gap cost of round t any agent’s performance my performance

  15. Equilibrium Value Consider C1, C2 are convex compact sets, function f is a bilinear function on Moreover, there exists an equilibrium point equilibrium value Question: How to compute the equilibrium value ? Pick a random series of points from C1, and then get the maximum over C2 a1 a2 .. .. at b1 b2 .. .. bt a upper bound is obtained easily How about the lower bound ? choose a better series

  16. Equilibrium Value In our settings, C1 is the set of density operators, C2 is the set linear mapping The bilinear function is a1 a2 .. .. at b1 b2 .. .. bt a1 a2 .. .. at b1 b2 .. .. bt MMW helps to generate the series Intuitively thinking : Why this works? • Problem with the upper bound is it can be any large. • MMW helps to make the value less than the “best agent” plus small gap. equilibrium point

  17. Equilibrium Value Get for the round t make own “observation” use MMW to get substitute

  18. Equilibrium Value approximated value use equilibrium point Consider Conclusion: with precision , need rounds!

  19. ConvertQIP-Complete to Equilibrium Value Problem Given any two quantum mixed state circuits Q1, Q2, wants to distinguish between This norm measures the distance between two circuits or channel. It is called diamond norm. Proved to be QIP-Complete when a+b=2, 0<b<1<a<2[RW05] Induced from L1 norm for super-operators: To: better representation of the distinguishing power by using entanglement with auxiliary space. A formulation of equilibrium value simply follows!

  20. The ConvertedProblem two promises! converted from Q0,Q1 1.9 0.1 min equilibrium value two promises with constant gap! constant precision will do the job!

  21. The Conversion : sketch max diamond norm to fidelity fidelity to L1 norm “ – “ sign, min implied then we have a min-max form

  22. Simulation by NC(poly) polynomial constant polynomial time matrix operations matrix operations in NC Finally, polynomial compositions of NC(poly), still NC(poly) ! thus in PSPACE

  23. Conclusions Corollary: QIP=PSPACE A Neater Proof is available[Wu09] August, 09  SDP reformulation [JJUW09], August 09 • Use QIP=QMAM • Use definition to formulate • Solve SDP by MMW • Use QIP-Complete Problem • Formulated as equilibrium value • Solved by MMW

  24. Open Questions • How to make more applications of MMW method? • For quantum, QRG(2), QRG, QMA(2) candidates • In other fields, like algorithmic game theory… • MMW method for Convex Programming, under KKT conditions, or …

  25. Thank You! Q&A

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