• Corpus ID: 235368309

Quantum Computers Can Find Quadratic Nonresidues in Deterministic Polynomial Time

@article{Draper2021QuantumCC,
  title={Quantum Computers Can Find Quadratic Nonresidues in Deterministic Polynomial Time},
  author={Thomas G. Draper},
  journal={ArXiv},
  year={2021},
  volume={abs/2106.03991}
}
An integer a is a quadratic nonresidue for a prime p if x2 ≡ a mod p has no solution. Quadratic nonresidues may be found by probabilistic methods in polynomial time. However, without assuming the Generalized Riemann Hypothesis, no deterministic polynomial-time algorithm is known. We present a quantum algorithm which generates a random quadratic nonresidue in deterministic polynomial time. 

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References

SHOWING 1-10 OF 14 REFERENCES
Integer multiplication in time O(n log n)
We present an algorithm that computes the product of two n-bit integers in O(n log n) bit operations, thus confirming a conjecture of Schonhage and Strassen from 1971. Our complexity analysis takes
Quantum Computability
TLDR
It is shown that when quantum Turing machines are restricted to have transition amplitudes which are algebraic numbers, BQP, EQP, and nondeterministic quantum polynomial time (NQP) are all contained in PP, hence in P and PSPACE.
Exact quantum Fourier transforms and discrete logarithm algorithms
We show how the Quantum Fast Fourier Transform (QFFT) can be made exact for arbitrary orders (first showing it for large primes). Most quantum algorithms only need a good approximation of the quantum
An O(M(n) logn) Algorithm for the Jacobi Symbol
The best known algorithm to compute the Jacobi symbol of two n-bit integers runs in time O(M(n)logn), using Schonhage’s fast continued fraction algorithm combined with an identity due to Gauss. We
A fast quantum mechanical algorithm for database search
TLDR
In early 1994, it was demonstrated that a quantum mechanical computer could efficiently solve a well-known problem for which there was no known efficient algorithm using classical computers, i.e. testing whether or not a given integer, N, is prime, in a time which is a finite power of o (logN) .
Quantum complexity theory
TLDR
This dissertation proves that relative to an oracle chosen uniformly at random, the class NP cannot be solved on a quantum Turing machine in time $o(2\sp{n/2}).$ and gives evidence suggesting that quantum Turing Machines cannot efficiently solve all of NP.
Efficient Clifford+T approximation of single-qubit operators
  • P. Selinger
  • Mathematics, Physics
    Quantum Inf. Comput.
  • 2015
TLDR
An efficient randomized algorithm for approximating an arbitrary element of SU(2) by a product of Clifford+T operators, up to any given error threshold e > 0.05, which is within an additive constant of optimal for certain z-rotations.
Evaluating NISQ Devices with Quadratic Nonresidues
TLDR
It is proved quantum computers can find quadratic nonresidues in deterministic polynomial time, whereas the classical version of this problem remains unsolved after hundreds of years.
The Solovay-Kitaev algorithm
TLDR
The algorithm can be used to compile Shor's algorithm into an efficient fault-tolerant form using only Hadamard, controlled-not, and π/8 gates, and is generalized to apply to multi-qubit gates and togates from SU(d).
A course in computational algebraic number theory
  • H. Cohen
  • Computer Science, Mathematics
    Graduate texts in mathematics
  • 1993
TLDR
The first seven chapters guide readers to the heart of current research in computational algebraic number theory, including recent algorithms for computing class groups and units, as well as elliptic curve computations, while the last three chapters survey factoring and primality testing methods.
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