We propose a framework to study models of computation of indeterministic data, represented by abstract "distributions". In these distributions, probabilities are replaced by "amplitudes" drawn from a fixed semi-ring $S$ , of which the non-negative reals, the complex numbers, finite fields $\mathbb F_{p^r}$ , and cyclic rings $\mathbb Z_k$ are examples. Varying $S$ yields different models of computation, which we may investigate to better understand the (likely) difference in power between randomised and quantum computation. The "modal quantum states" of Schumacher and Westmoreland [arXiv:1010.2929] are examples of such distributions, for $S$ a finite field. For $S = \mathbb F_2$ , Willcock and Sabry [arXiv:1102.3587] show that UNIQUE-SAT is solvable by polynomial-time uniform circuit families consisting of invertible gates. We characterize the decision problems solvable by polynomial uniform circuit families, using either invertible or "unitary" transformations over cyclic rings $S = \mathbb Z_k$ , or (in the case that $k$ is a prime power) finite fields $S = \mathbb F_k$ . In particular, for $k$ a prime power, these are precisely the problems in the class $\mathsf{Mod}_k\mathsf P$ .We propose a framework to study models of computation of indeterministic data, represented by abstract "distributions". In these distributions, probabilities are replaced by "amplitudes" drawn from a fixed semi-ring $S$, of which the non-negative reals, the complex numbers, finite fields $\mathbb F_{p^r}$, and cyclic rings $\mathbb Z_k$ are examples. Varying $S$ yields different models of computation, which we may investigate to better understand the (likely) difference in power between randomised and quantum computation. The "modal quantum states" of Schumacher and Westmoreland [arXiv:1010.2929] are examples of such distributions, for $S$ a finite field. For $S = \mathbb F_2$, Willcock and Sabry [arXiv:1102.3587] show that UNIQUE-SAT is solvable by polynomial-time uniform circuit families consisting of invertible gates. We characterize the decision problems solvable by polynomial uniform circuit families, using either invertible or "unitary" transformations over cyclic rings $S = \mathbb Z_k$, or (in the case that $k$ is a prime power) finite fields $S = \mathbb F_k$. In particular, for $k$ a prime power, these are precisely the problems in the class $\mathsf{Mod}_k\mathsf P$.

arXiv:1405.7381May 28, 20148 citations

Identification of prime numbers with quantum computers

Victor Penchel Alvarenga Balthar dos Santos

The identification of prime numbers is an ancient area of study with applications in cryptography and fundamental mathematical problems, such as the Riemann Hypothesis (RH). Classical algorithms, like the Sieve of Eratosthenes and the AKS Primality Test, are efficient examples for this purpose. However, quantum computing (QC), with its superior theoretical capacity, has inspired the development of quantum algorithms, many of which are probabilistic or rely on the RH. In this dissertation, we propose a deterministic and efficient quantum algorithm that generalizes a recent method from quantum optics to identify prime numbers through entanglement dynamics. We adapted this method for QC, removing several constraints from the original work, and showed that discrete bipartite quantum systems prepared in certain initial states can be used to identify prime numbers through the Fourier modes of the reduced purity of one of the subsystems. The proposed algorithm is viable for implementation in future fault-tolerant QC systems. The initial steps involve encoding the system in qubits and preparing a maximally superposed state. In the state evolution step, we use Walsh matrices to efficiently prepare diagonal unitary gates. The reduced purity is measured using a modified controlled-SWAP test, and the Fourier modes are obtained via the numerical calculation of a one-dimensional integral. Subsequently, we present the simulations performed in Qiskit, which demonstrate the validity of the algorithm in all its stages. Finally, we discuss the computational cost, quantifying the number of elementary gates as a function of the number of qubits and speculating the total computational complexity of the algorithm.The identification of prime numbers is an ancient area of study with applications in cryptography and fundamental mathematical problems, such as the Riemann Hypothesis (RH). Classical algorithms, like the Sieve of Eratosthenes and the AKS Primality Test, are efficient examples for this purpose. However, quantum computing (QC), with its superior theoretical capacity, has inspired the development of quantum algorithms, many of which are probabilistic or rely on the RH. In this dissertation, we propose a deterministic and efficient quantum algorithm that generalizes a recent method from quantum optics to identify prime numbers through entanglement dynamics. We adapted this method for QC, removing several constraints from the original work, and showed that discrete bipartite quantum systems prepared in certain initial states can be used to identify prime numbers through the Fourier modes of the reduced purity of one of the subsystems. The proposed algorithm is viable for implementation in future fault-tolerant QC systems. The initial steps involve encoding the system in qubits and preparing a maximally superposed state. In the state evolution step, we use Walsh matrices to efficiently prepare diagonal unitary gates. The reduced purity is measured using a modified controlled-SWAP test, and the Fourier modes are obtained via the numerical calculation of a one-dimensional integral. Subsequently, we present the simulations performed in Qiskit, which demonstrate the validity of the algorithm in all its stages. Finally, we discuss the computational cost, quantifying the number of elementary gates as a function of the number of qubits and speculating the total computational complexity of the algorithm.

Sep 30, 20240 citations

Quantum Computing and Artificial Intelligence: Disruptive Technology with Potential Applications in Surgical Practice

George B. Stefano

Journal of SurgeryJan 10, 20241 citation

Quantum Computing in Medicine

Elsevier eBooksJan 1, 20260 citations

S

, of which the non-negative reals, the complex numbers, finite fields

\mathbb F_{p^r}

, and cyclic rings

\mathbb Z_k

are examples. Varying

S

yields different models of computation, which we may investigate to better understand the (likely) difference in power between randomised and quantum computation. The "modal quantum states" of Schumacher and Westmoreland [arXiv:1010.2929] are examples of such distributions, for

S

a finite field. For

S = \mathbb F_2

, Willcock and Sabry [arXiv:1102.3587] show that UNIQUE-SAT is solvable by polynomial-time uniform circuit families consisting of invertible gates. We characterize the decision problems solvable by polynomial uniform circuit families, using either invertible or "unitary" transformations over cyclic rings

S = \mathbb Z_k

, or (in the case that

k

is a prime power) finite fields

S = \mathbb F_k

. In particular, for

k

a prime power, these are precisely the problems in the class

\mathsf{Mod}_k\mathsf P