Nondeterministic testing of SQL
    Propositions on a QC


                Matthew Leifer

     Perimeter Institute for Th...
Outline
1) Introduction

    Logic, complexity and models of Computing


2) Sequential Quantum Logic


3) Testing an SQL p...
1) Introduction


Classical Computational Complexity


    Cook-Levin: SATISFIABILITY is NP complete.


        Given a Bo...
1) Introduction
There is a model of classical computing based directly on Boolean
logic.
1) Introduction
Can we pull off the same trick as in the classical circuit model?


Is there a model of QC in which logic ...
Traditional view of quantum computing

                                       Probabilities
Bit strings, Revesible        ...
2) Sequential Quantum Logic
     • Developed in late 70’s/early 80’s by Stachow and Mittlestaedt.

     • Extended by Isha...
2) SQL: Structure




Not commutative:

Associative:
2) SQL: Hilbert Space Model

Hilbert space:

Elementary propositions:               - projection operators on    .

Notati...
2) SQL: Problems
    SQL works well for situations like this
                                                 {b,¬b}
     ...
3) Testing an SQL Proposition
The algorithm is based on recent QInfo inspired approaches to DMRG.

    Cirac, Latorre, Ric...
3) Testing: History state
Suppose we want to test a simple SQL proposition   .



                              ,



Notat...
3) Testing: History state
        The history state is a kind of PEPS state.


        Starting state:

                  ...
3) Testing: History state

How to prepare the history state:

    i)     Apply            to qubits       and       .

   ...
3) Testing: Renormalization


                                                          a and b.
i) Compute         by app...
3) Testing: Implementing AND


is not a directly implementable. Instead, implement measurement:




    If   , discard 2nd...
3) Testing: Complexity
          n = no. propositions in underlying sequence.
    Let
          m = no. gates in formala.
...
4) Generalizations

 • Testing projectors of arbitrary rank.


 • Testing props on d-dimensional Hilbert space.


        ...
6) Open Questions
Can SQL be modified so that all sequential propositions can be tested?

    Modify definition of sequent...
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Nondeterministic testing of Sequential Quantum Logic Propositions on a Quantum Computer

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Talk given at "Quantum Information, Computation and Logic" workshop at Perimeter Institute in 2005.
Based on the preprint http://arxiv.org/abs/quant-ph/0509193
Talk was recorded and is viewable online at http://pirsa.org/05070102/
I now regard these ideas as flawed, but they may be revisited in future work.

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Nondeterministic testing of Sequential Quantum Logic Propositions on a Quantum Computer

  1. 1. Nondeterministic testing of SQL Propositions on a QC Matthew Leifer Perimeter Institute for Theoretical Physics QICL Workshop 17th-22nd July 2005
  2. 2. Outline 1) Introduction Logic, complexity and models of Computing 2) Sequential Quantum Logic 3) Testing an SQL proposition: Example 4) Generalizations Higher dimensions, multiple propositions 6) Open questions
  3. 3. 1) Introduction Classical Computational Complexity Cook-Levin: SATISFIABILITY is NP complete. Given a Boolean formula, decide if there are truth value assignments to the elementary propositions that make the formula true.
  4. 4. 1) Introduction There is a model of classical computing based directly on Boolean logic.
  5. 5. 1) Introduction Can we pull off the same trick as in the classical circuit model? Is there a model of QC in which logic of process = logic of properties? • Logic of quantum properties ~ algebra of measurements. Subspace lattice, OM Lattice/Poset, orthoalgebra, etc. • Logic of quantum processes ~ algebra of unitaries? Measurement based models suggest these logics need not be so distinct.
  6. 6. Traditional view of quantum computing Probabilities Bit strings, Revesible Prob. measures on strings, classical logic stochastic operations. Amplitudes Superpositions of strings, unitary operations. von-Neumanized view of quantum computing Probabilities Bit strings, Boolean logic Prob. measures on strings, operations. stochastic operations. Transition from classical to quantum logic Probabilities Subspaces, quantum logic Probabilistic model arising operations. from measures on subspaces.
  7. 7. 2) Sequential Quantum Logic • Developed in late 70’s/early 80’s by Stachow and Mittlestaedt. • Extended by Isham et. al. in mid 90’s as a possible route to QGravity. Up Y Down Up Ψ X Down € Up Y € Down € €
  8. 8. 2) SQL: Structure Not commutative: Associative:
  9. 9. 2) SQL: Hilbert Space Model Hilbert space: Elementary propositions: - projection operators on . Notation: - operator associated to prop. . Negation: “NOT ”. Sequential conjunction: “ AND THEN ”. Note: Usual conjunction can be obtained by including limit propositions
  10. 10. 2) SQL: Problems SQL works well for situations like this {b,¬b} [a] Ψ Ψ {a,¬a} [¬a] Ψ € € {b,¬b} € € € But it does not handle “coarse-grainings” well: € Ψ €
  11. 11. 3) Testing an SQL Proposition The algorithm is based on recent QInfo inspired approaches to DMRG. Cirac, Latorre, Rico Ortega, Verstraete, Vidal, et. al. It has 3 main steps: 1) Prepare a “history” state that encodes the results of the underlying sequence of measurements. 2) Apply rounds of “renormalization” (coherent AND and NOT gates) to get the desired proposition. 3) Measure a qubit to test the proposition. Note: 2) can only be implemented probablistically.
  12. 12. 3) Testing: History state Suppose we want to test a simple SQL proposition . , Notation: , Define: , History state:
  13. 13. 3) Testing: History state The history state is a kind of PEPS state. Starting state: where f a a′ c c′ b b′ Pa Pb Pc 1 = Pa € Pb ⊗ Pc start cm ][b k ][a j ] Ψ €€ [ €€ j €k b m € ⊗ ∑ a c f j,k ,m=0 € € € 1 [x j ]km j x Px = km xx ′ ∑ , j,k,m=0 €
  14. 14. 3) Testing: History state How to prepare the history state: i) Apply to qubits and . ii) Perform a parity measurement on and . , iii) Perform and discard . iv) If outcome occurred then perform on qubit . v) Repeat steps i) - iv) for and . xi ] i [ jk xx' . Note: Equivalent to applying operators ∑ x jk i,j,k
  15. 15. 3) Testing: Renormalization a and b. i) Compute by applying “coherent AND” to qubits € € X ii) Compute by applying to qubit . € d by applying iii) Compute . €
  16. 16. 3) Testing: Implementing AND is not a directly implementable. Instead, implement measurement: If , discard 2nd qubit and proceed. If , abort and restart algorithm from beginning. Note: Algorithm will succeed with exponentially small probability in no. gates.
  17. 17. 3) Testing: Complexity n = no. propositions in underlying sequence. Let m = no. gates in formala. Let Count no. 2-qubit gates: n Preparing entangled states € € O (n ) Preparing history state O( m) Perfoming gates € r = log 2 d . d Generalization to dimensional H.S.: Let € rn Preparing entangled states € O (n ) Preparing history state € € ( ) † ⊗ U T€ O n2 2r Applying U x x €
  18. 18. 4) Generalizations • Testing projectors of arbitrary rank. • Testing props on d-dimensional Hilbert space. d • Need upper bound on needed to get correct probs. for all formulae of length n. • Testing multiple propositions. € € • On disjoint subsets of an underlying sequence of propositions. • By copying qubits in the computational basis.
  19. 19. 6) Open Questions Can SQL be modified so that all sequential propositions can be tested? Modify definition of sequential conjunction. Restrict allowed subspaces. Is SQL the logic of an interesting model of computing? c.f. Aaronson’s QC with post-selection. Renormalization: A new paradigm for irreversible quantum computing? Which DMRG schemes are universal for classical/quantum computing? Is there a natural quantum logic which has SAT problems that are NQP or QMA complete?
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