• Corpus ID: 1869239

The Complexity of Quantum States and Transformations: From Quantum Money to Black Holes

@article{Aaronson2016TheCO,
  title={The Complexity of Quantum States and Transformations: From Quantum Money to Black Holes},
  author={Scott Aaronson},
  journal={Electron. Colloquium Comput. Complex.},
  year={2016},
  volume={TR16}
}
  • S. Aaronson
  • Published 18 July 2016
  • Physics
  • Electron. Colloquium Comput. Complex.
These are lecture notes from a weeklong course in quantum complexity theory taught at the Bellairs Research Institute in Barbados, February 21-25, 2016. The focus is quantum circuit complexity---i.e., the minimum number of gates needed to prepare a given quantum state or apply a given unitary transformation---as a unifying theme tying together several topics of recent interest in the field. Those topics include the power of quantum proofs and advice states; how to construct quantum money… 

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References

SHOWING 1-10 OF 103 REFERENCES

Quantum money from hidden subspaces

The first quantum money scheme that is (1) public-key---meaning that anyone can verify a banknote as genuine, not only the bank that printed it, and (2) cryptographically secure, under a "classical" hardness assumption that has nothing to do with quantum money is proposed.

Limitations of quantum advice and one-way communication

  • S. Aaronson
  • Computer Science
    Proceedings. 19th IEEE Annual Conference on Computational Complexity, 2004.
  • 2004
It is shown in three settings that quantum messages have only limited advantages over classical ones, and the polynomial method is used to give the first correct proof of a direct product theorem for quantum search.

Quantum complexity theory

This paper gives the first formal evidence that quantum Turing machines violate the modern (complexity theoretic) formulation of the Church--Turing thesis, and proves that bits of precision suffice to support a step computation.

A full characterization of quantum advice

It is proved that given any quantum state rho on n qubits, there exists a local Hamiltonian H on poly(n) qubits (e.g., a sum of two-qubit interactions), such that any ground state of H can be used to simulate rho in quantum circuits of fixed polynomial size.

Quantum Copy-Protection and Quantum Money

  • S. Aaronson
  • Computer Science
    2009 24th Annual IEEE Conference on Computational Complexity
  • 2009
There exist quantum oracles relative to which publicly-verifiable quantum money is possible, and any family of functions that cannot be efficiently learned from its input-output behavior can be quantumly copy-protected, the first formal evidence that these tasks are achievable.

Quantum versus Classical Proofs and Advice

  • S. AaronsonG. Kuperberg
  • Mathematics, Computer Science
    Twenty-Second Annual IEEE Conference on Computational Complexity (CCC'07)
  • 2007
This paper shows that any quantum algorithm needs Omega (radic2n-m+1) queries to find an n-qubit "marked state"Psi rang, and gives an explicit QCMA protocol that nearly achieves this lower bound.

Multilinear formulas and skepticism of quantum computing

It is shown that states arising in quantum error-correction require nΩ(log n) additions and tensor products even to approximate, which incidentally yields the first superpolynomial gap between general and multilinear formula size of functions.

Testing Product States, Quantum Merlin-Arthur Games and Tensor Optimization

A test that can distinguish efficiently between product states of n quantum systems and states that are far from product is given, which can be used to construct an efficient test for determining whether a unitary operator is a tensor product, which is a generalization of classical linearity testing.

Classical simulation of commuting quantum computations implies collapse of the polynomial hierarchy

  • M. BremnerR. JozsaD. Shepherd
  • Computer Science, Mathematics
    Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
  • 2010
The class post-IQP of languages decided with bounded error by uniform families of IQP circuits with post-selection is introduced, and it is proved first that post- IQP equals the classical class PP, and that if the output distributions of uniform IQP circuit families could be classically efficiently sampled, then the infinite tower of classical complexity classes known as the polynomial hierarchy would collapse to its third level.

On the power of quantum computation

  • Daniel R. Simon
  • Computer Science
    Proceedings 35th Annual Symposium on Foundations of Computer Science
  • 1994
This work presents here a problem of distinguishing between two fairly natural classes of function, which can provably be solved exponentially faster in the quantum model than in the classical probabilistic one, when the function is given as an oracle drawn equiprobably from the uniform distribution on either class.
...