What is computable and non-computable in the quantum domain: 7 statements and 3 conjectures
arxiv(2024)
摘要
Recent progress in developing computational devices based on quantum effects
and demonstrations of solving various tasks using them has actualized the
question of the origin of the quantum advantage. Although various attempts to
quantify and characterize the nature of quantum computational advantage have
been made, this question in the general context remains open: There is no
universal approach that helps to define a scope of problems that quantum
computers are able to speed up, theoretically and in practice. Here we review
an approach to this question based on the concept of complexity and
reachability of quantum states. On the one hand, the class of quantum states
that is of interest for quantum computing should be complex, i.e.
non-simulatable with classical computers with less than exponential resources.
On the other hand, such quantum states should be reachable on a practical
quantum computer. This means that a unitary corresponding to the transformation
of quantum states from initial to desired can be decomposed in a sequence of
single- and two-qubit gates with of no more than polynomial in the number of
qubits. Our consideration paves the way towards understanding the scope of
problems that can be solved by a quantum computer by formulating a sequence of
statements and conjectures on various sets of quantum states.
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