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.

1. Nondeterministic testing of SQL
Propositions on a QC
Matthew Leifer
Perimeter Institute for Theoretical Physics
QICL Workshop 17th-22nd July 2005

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. 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. 1) Introduction
There is a model of classical computing based directly on Boolean
logic.

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. 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. 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.
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8. 2) SQL: Structure
Not commutative:
Associative:

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. 2) SQL: Problems
SQL works well for situations like this
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{a,¬a}
[¬a] Ψ
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But it does not handle “coarse-grainings” well:
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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. 3) Testing: History state
Suppose we want to test a simple SQL proposition .
,
Notation: ,
Define: ,
History state:

13. 3) Testing: History state
The history state is a kind of PEPS state.
Starting state:
where
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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
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Note: Equivalent to applying operators ∑ x
jk
i,j,k

15. 3) Testing: Renormalization
a and b.
i) Compute by applying “coherent AND” to qubits
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ii) Compute by applying to qubit .
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d by applying
iii) Compute .
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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. 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
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O (n )
Preparing history state
O( m)
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r = log 2 d .
d
Generalization to dimensional H.S.: Let
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Preparing entangled states
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Preparing history state
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Applying U x x
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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.
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• On disjoint subsets of an underlying sequence of propositions.
• By copying qubits in the computational basis.

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?