We generalize the binary quantum counting algorithm of Lesovik, Suslov, and Blatter [Phys. Rev. A 82, 012316 (2010)] to higher counting bases. The algorithm makes use of qubits, qutrits, and qudits to count numbers in a base-2, base-3, or base-
d representation. In operating the algorithm, the number
n<N=dK is read into a
K-qudit register through its interaction with a stream of
n particles passing in a nearby wire; this step corresponds to a quantum Fourier transformation from the Hilbert space of particles to the Hilbert space of qudit states. An inverse quantum Fourier transformation provides the number
n in the base-
d representation; the inverse transformation is fully quantum at the level of individual qudits, while a simpler semiclassical version can be used on the level of qudit registers. Combining registers of qubits, qutrits, and qudits, where
d is a prime number, with a simpler single-shot measurement allows us to find the powers of 2, 3, and other primes
d in the number
n. We show that the counting task naturally leads to the shift operation and an algorithm based on the quantum Fourier transformation. We discuss possible implementations of the algorithm using quantum spin-
d systems,
d-well systems, and their emulation with spin-1/2 or double-well systems. We establish the analogy between our counting algorithm and the phase estimation algorithm and make use of the latter's performance analysis in stabilizing our scheme. Applications embrace a quantum metrological scheme to measure voltage (an analog to digital converter) and a simple procedure to entangle multiparticle states.
Quantum phase estimation (QPE) is one of the most important quantum algorithms which is used as a subroutine for other important quantum algorithms like Shor's factoring algorithm, simulation of quantum systems, quantum counting and QFT on arbitrary Zp. In this paper we develop the theoretical framework for the multivalued quantum logic version of the QPE algorithm using d valued qudits and show a quantum circuit to implement QPE with a complexity of O(nlogn) single qudit operations. The multivalued QPE algorithm, when compared to the binary quantum logic version, turns out to be more robust and leads to a significant decrease in the number of qudits required along with drastic improvement in the precision and success probability. We derive the requirements to amplify the probability of success to a value very close to 1 (for a given precision), thereby generalizing the previously obtained result in the binary case. Also, we note that the failure probability of QPE algorithm decreases exponentially as d increases.
This papers presents the generalization of the quantum phase estimation (QPE) algorithm to multivalued logic. QPE is a very important subroutine used in many quantum algorithms of practical importance. The QPE using qudits has certain advantages when compared to the version using qubits. The QPE using qudits is more accurate and requires lesser number of qudits (as the value of the radix d increases) to estimate the phase up to a given accuracy with a given success probability. Since the existing quantum computer implementations use only few qubits, it is important to have alternative versions of the QPE which use as few qubits as possible. In this regard, the iterative quantum phase estimation algorithm (IQPE) which uses only single qubit to store the phase information has been developed. The number of iterations involved in the IQPE is directly proportional to the required accuracy in the estimated phase. In this regard, IQPE using qudits requires (logarithmically) lesser number of iterations (and hence is faster) compared to the one using qubits. In this paper, we also develop the multivalued logic version of the IQPE and describe its implementation.
We develop several algorithms for performing quantum phase estimation based on basic measurements and classical post-processing. We present a pedagogical review of quantum phase estimation and simulate the algorithm to numerically determine its scaling in circuit depth and width. We show that the use of purely random measurements requires a number of measurements that is optimal up to constant factors, albeit at the cost of exponential classical post-processing; the method can also be used to improve classical signal processing. We then develop a quantum algorithm for phase estimation that yields an asymptotic improvement in runtime, coming within a factor of log* of the minimum number of measurements required while still requiring only minimal classical post-processing. The corresponding quantum circuit requires asymptotically lower depth and width (number of qubits) than quantum phase estimation.
We develop several algorithms for performing quantum phase estimation based on basic measurements and classical post-processing. We present a pedagogical review of quantum phase estimation and simulate the algorithm to numerically determine its scaling in circuit depth and width. We show that the use of purely random measurements requires a number of measurements that is optimal up to constant factors, albeit at the cost of exponential classical post-processing; the method can also be used to improve classical signal processing. We then develop a quantum algorithm for phase estimation that yields an asymptotic improvement in runtime, coming within a factor of
log∗ of the minimum number of measurements required while still requiring only minimal classical post-processing. The corresponding quantum circuit requires asymptotically lower depth and width (number of qubits) than quantum phase estimation.
In the process of implementing a quantum algorithm as a quantum circuit, the error rate of an actual quantum computer should be considered. In this paper, we confirm through simulation that the one control qubit quantum phase estimation algorithm performs better than the many control qubit quantum phase estimation algorithm on the noisy quantum processor. We use QISKit to simulate a quantum phase estimation algorithm. We performed the simulation considering the error rate of the IBM Q quantum processor. The error rate of the one control qubit quantum phase estimation algorithm is compared with the many control qubit quantum phase estimation algorithm.