Do we need a Logic of Quantum
        Computation?

                Matthew Leifer

     Perimeter Institute for Theoretic...
Outline
1) What is responsible for the power of Quantum Computing?


2) Logic in classical computing: models and complexit...
1) The Power of QC
Which feature of quantum theory is responsible for the exponential
speedup of QC w.r.t. classical compu...
1) The Power of QC
We can prove theorems of the form:

    If a quantum algorithm does not exhibit property X, then it can...
1) The Power of QC
What is responsible for “what is responsible for the power of QC?”?

    Don’t we have to know what is ...
2) Logic in Classical Computing
Classical circuit model:

               a
               b                               ...
2) Logic in Classical Computing
Reversible classical circuit model:




                                d =?



          ...
2) Logic in Classical Computing
Quantum circuit model:




                         d =?



                   €
2) Logic in Classical Computing
   Classical Computational Complexity

        Cook-Levin: SATISFIABILITY is NP complete.
...
2) Logic in Classical Computing
Descriptive Complexity


                             A set of structures T is in NP iff t...
3) The logic of QC
“What is responsible for the power of QC?” is replaced by:

    Can quantum computations be regarded as...
4) BvN Quantum Logic
                          ⇔ basic experimentally testable statements.
Elementary propositions

      ...
4) BvN Quantum Logic
First attempt: Define a model of computing that tests propositions in BvNQL

    BvN Circuit model: N...
Traditional view of quantum computing

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

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




Not commutative:

Associative:
5) SQL: Hilbert Space Model

Hilbert space:

Elementary propositions:               - projection operators on    .

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

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



                              ,



Notat...
6) Testing: History state
        The history state is Matrix Product State.


        Starting state:

                  ...
6) Testing: History state

How to prepare the history state:

    i)     Apply            to qubits       and       .

   ...
6) Testing: Processing


                                                          a and b.
i) Compute         by applying...
6) Testing: Implementing cohAND


is not a directly implementable. Instead, implement measurement:




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

 • Testing projectors of arbitrary rank.


 • Testing props on d-dimensional Hilbert space.


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

    Modify definition of sequent...
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Do we need a logic of quantum computation?

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A longer talk given at the University of Queensland in October 2005 based on the preprint http://arxiv.org/abs/quant-ph/0509193

Note that I now regard some of these ideas as flawed, but I think that the motivation and discussion is still interesting.

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Do we need a logic of quantum computation?

  1. 1. Do we need a Logic of Quantum Computation? Matthew Leifer Perimeter Institute for Theoretical Physics quant-ph/0509193 Griffith University 1st November 2005
  2. 2. Outline 1) What is responsible for the power of Quantum Computing? 2) Logic in classical computing: models and complexity 3) The logic of quantum computing 4) Birkhoff-von Neumann Quantum Logic 5) Sequential Quantum Logic 6) Testing an SQL Proposition 7) Open Questions
  3. 3. 1) The Power of QC Which feature of quantum theory is responsible for the exponential speedup of QC w.r.t. classical computing? • Massive Quantum Parallelism • Superposition/Interference • Multi-party entanglement • The entangling power of quantum evolutions • Nonlocality • Contextuality • The dimensionality of Hilbert Space • Quantum Logic
  4. 4. 1) The Power of QC We can prove theorems of the form: If a quantum algorithm does not exhibit property X, then it can be efficiently simulated on a classical computer. We usually cannot prove: All quantum algorithms exhibiting property X cannot be efficiently simulated on a classical computer. Examples: X = a particular kind of multi-party entanglement (Jozsa & Linden 2002, Vidal 2003). The converse is blocked by the Gottesman-Knill theorem.
  5. 5. 1) The Power of QC What is responsible for “what is responsible for the power of QC?”? Don’t we have to know what is the power of QC first? Why don’t we feel the need to ask “what is responsible for the power of classical computing?”? We certainly don’t know what is the power of classical computing. To a large extent classical logic explains the power of classical computing. If we had similar results about some sort of quantum logic for quantum computing then the question might disappear. We also might get some interesting algorithms and complexity results for QC.
  6. 6. 2) Logic in Classical Computing Classical circuit model: a b d € c € € Any algorithm for solving a decision problem can be regarded as a uniform family of Boolean propositions. € d = ¬( a ∧ b) ∧¬c €
  7. 7. 2) Logic in Classical Computing Reversible classical circuit model: d =? €
  8. 8. 2) Logic in Classical Computing Quantum circuit model: d =? €
  9. 9. 2) Logic in Classical Computing 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. Quantum Computational Complexity Kitaev: LOCAL HAMILTONIANS is QMA complete. H1 , H 2 ,K, H r be a set of positive semi-definite operators acting on Let 2 ⊗n (C ) n qubits, out of a t . Each matrix comes with a specification of the total of m qubits, on which it acts and is specified to poly(n ) bits. Let a and b be two real numbers, specified to poly(n ) bits, such that € a − b = 1 poly(n ). Is the smallest eigenvalue of H1 + H 2 +K+ H r € € € smaller than a or larger than b. € € € € € € € €
  10. 10. 2) Logic in Classical Computing Descriptive Complexity A set of structures T is in NP iff there is a Fagin’s Theorem: second order existential formula Φ such that T is the set of all finite structures satisfying Φ. € € Example: 3-COLOURABLE GRAPHS € € € (∃R)(∃Y )(∃B)(∀x )(( R( x) ∨Y ( x ) ∨ B( x)) ∧ (∀y)( E( x, y) → ¬( R( x ) ∧ R( y)) ∧¬(Y ( x ) ∧Y ( y)) ∧¬( B( x ) ∧ B( y))))
  11. 11. 3) The logic of QC “What is responsible for the power of QC?” is replaced by: Can quantum computations be regarded as tests of propositions in a formal logic? Can we prove analogs of Cook-Levin and Fagin’s theorem in such a logic? Aside: Other proposals for a “logic of QC” Extend classical logic • Deutsch, Ekert and Lupacchini (1999) Formalized by dalla Chiara, Guintini and Leoprini (2003) Dynamic Quantum Logic • Baltag and Smets (2004) Categorical Quantum Logic • Abramsky and Coecke (2004)
  12. 12. 4) BvN Quantum Logic ⇔ basic experimentally testable statements. Elementary propositions x and x + dx. The position of the particle is between p and p + dp. The momentum of the particle is between € € € These correspond to closed subspaces of Hilbert Space or projectors. Analogous to the assignment of sets to propositions in classical logic. € € ∧,∨,¬ to assign meaning to heretical propositions. Define connectives The position of the particle is between x and x + dx AND the momentum of the particle is between p and p + dp. Arrive at€ logic describing alternative possible measurements that can be a made at a single time. € € € €
  13. 13. 4) BvN Quantum Logic First attempt: Define a model of computing that tests propositions in BvNQL BvN Circuit model: Nontrivial since there are no truth values. Turns out model can be efficiently simulated on a classical computer. Essentially because size of Hilbert Space required only grows linearly with number of inputs. Surprisingly, BvNSAT is only NP complete. Intuitive reason: BvNQL is about alternative possible measurements whereas computing is about sequences of operations. Need a logic of processes rather than just a logic of properties. In classical logic, AND ⇔ AND THEN, since testing a proposition does not disturb the state of the system. €
  14. 14. 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.
  15. 15. 5) 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 € €
  16. 16. 5) SQL: Structure Not commutative: Associative:
  17. 17. 5) 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
  18. 18. 5) 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 €
  19. 19. 6) Testing an SQL Proposition The algorithm is inspired by 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.
  20. 20. 6) Testing: History state Suppose we want to test a simple SQL proposition . , Notation: , Define: , History state:
  21. 21. 6) Testing: History state The history state is Matrix Product 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 €
  22. 22. 6) 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
  23. 23. 6) Testing: Processing a and b. i) Compute by applying “coherent AND” to qubits € € X ii) Compute by applying to qubit . € d by applying iii) Compute . €
  24. 24. 6) Testing: Implementing cohAND 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.
  25. 25. 5) 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 €
  26. 26. 5) 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.
  27. 27. 7) Open Questions Can SQL be modified so that all sequential propositions can be tested? Modify definition of sequential conjunction. Is SQL the logic of an interesting model of computing? c.f. Aaronson’s QC with post-selection. DMRG: A new paradigm for irreversible quantum computing? Which DMRG schemes are universal for quantum computing? Can they be understood as a kind of Sequential Quantum Logic? Is there a natural quantum logic in which quantum analogs of Cook-Levin and Fagin’s theorem can be proved?
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