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arXiv:quant-ph/0509193

Nondeterministic testing of Sequential Quantum Logic propositions on a quantum computer

Matthew Leifer

Published Sep 28, 2005• 0 citations
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Abstract

In the past few years it has been shown that universal quantum computation can be obtained by projective measurements alone, with no need for unitary gates. This suggests that the underlying logic of quantum computing may be an algebra of sequences of quantum measurements rather than an algebra of products of unitary operators. Such a Sequential Quantum Logic (SQL) was developed in the late 70's and has more recently been applied to the consistent histories framework of quantum mechanics as a possible route to the theory of quantum gravity. In this letter, I give a method for deciding the truth of a proposition in SQL with nonzero probability of success on a quantum computer.

DOI
10.48550/arxiv.quant-ph/0509193

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In the past few years it has been shown that universal quantum computation can be obtained by projective measurements alone, with no need for unitary gates. This suggests that the underlying logic of quantum computing may be an algebra of sequences of quantum measurements rather than an algebra of products of unitary operators. Such a Sequential Quantum Logic (SQL) was developed in the late 70's and has more recently been applied to the consistent histories framework of quantum mechanics as a possible route to the theory of quantum gravity. In this letter, I give a method for deciding the truth of a proposition in SQL with nonzero probability of success on a quantum computer.
According to quantum mechanics, statements about the future made by sentient beings like us are, in general, neither true nor false; they must satisfy a many-valued logic. I propose that the truth value of such a statement should be identified with the probability that the event it describes will occur. After reviewing the history of related ideas in logic, I argue that it gives an understanding of probability which is particularly satisfactory for use in quantum mechanics. I construct a lattice of future-tense propositions, with truth values in the interval [0, 1], and derive logical properties of these truth values given by the usual quantum-mechanical formula for the probability of a history.
Quantum logic has been introduced by Birkhoff and von Neumann as an attempt to base the logical primitives, the propositions and the relations and operations among them, on quantum theoretical entities, and thus on the related empirical evidence of the quantum world. We give a brief outline of quantum logic, and some of its algebraic properties, such as nondistributivity, whereby emphasis is given to concrete experimental setups related to quantum logical entities. A probability theory based on quantum logic is fundamentally and sometimes even spectacularly different from probabilities based on classical Boolean logic. We give a brief outline of its nonclassical aspects; in particular violations of Boole-Bell type consistency constraints on joint probabilities, as well as the Kochen-Specker theorem, demonstrating in a constructive, finite way the scarcity and even nonexistence of two-valued states interpretable as classical truth assignments. A more complete introduction of the author can be found in the book "Quantum Logic" (Springer, 1998)
An introduction is given to an algebraic formulation and generalisation of the consistent histories approach to quantum theory. The main technical tool in this theory is an orthoalgebra of history propositions that serves as a generalised temporal analogue of the lattice of propositions of standard quantum logic. Particular emphasis is placed on those cases in which the history propositions can be represented by projection operators in a Hilbert space, and on the associated concept of a `history group'.
A probabilistic propositional logic, endowed with an epistemic component for asserting (non-)compatibility of diagonizable and bounded observables, is presented and illustrated for reasoning about the random results of projective measurements made on a given quantum state. Simultaneous measurements are assumed to imply that the underlying observables are compatible. A sound and weakly complete axiomatization is provided relying on the decidable first-order theory of real closed ordered fields. The proposed logic is proved to be a conservative extension of classical propositional logic.
We consider sequential hypothesis testing among multiple samples of one amongMMMpure quantum states in an equidistant ensemble, i.e., those with identical pair-wise inner products. Each measurement in the sequence is a binary projective measurement that collapses the sample of the state measured at that instant into a linear span of a collection of states from the ensemble or its orthogonal complement. The algorithm adaptively decides if additional samples are needed or sufficient observation has been gathered. We show that our sequential measurement algorithm outperforms the sequential testing (ST) receiver, whose error-probability exponent is known to achieve the quantum Chernoff bound asymptotically in the limit of large (and fixed) number of samples. Even though our algorithm does not attain the quantum limit of minimum error probability (the Helstrom limit), it paves the way for future research on more advanced sequential quantum hypothesis tests, e.g., those that go beyond binary projective measurements on each sample.
In quantum logic, i.e., within the structure of the Hilbert lattice imposed on all closed linear subspaces of a Hilbert space, the assignment of truth values to quantum propositions (i.e., experimentally verifiable propositions relating to a quantum system) is unambiguously determined by the state of the system. So, if only pure states of the system are considered, can a probability measure mapping the probability space for truth values to the unit interval be assigned to quantum propositions? In other words, is a probability concept contingent or emergent in the logic of quantum propositions? Until this question is answered, the cause of probabilities in quantum theory cannot be completely understood. In the present paper it is shown that the interaction of the quantum system with its environment causes the irreducible randomness in the relation between quantum propositions and truth values.