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IEEE International Symposium on Multiple-Valued Logic (ISMVL)

Quantum Phase Estimation Using Multivalued Logic

Vamsi Parasa, Marek A. Perkowski

Published May 1, 2011• 12 citations
Publisher

Abstract

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.

DOI
10.1109/ismvl.2011.47

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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.
Quantum phase estimation (QPE) is a central algorithmic primitive that estimates eigenvalues of a Hamiltonian up to precision εεε in Heisenberg-limited time T=Θ(1/ε)T=Θ(1/ε)T=Θ(1/ε). Standard gate-based implementations of QPE require deep controlled time-evolution circuits and are not native to analog hardware. Here, we present a simple adiabatic protocol for QPE that achieves (up to logarithmic factors) the optimal Heisenberg-limited scaling T=O(1εlog⁡(δ−1))T = O\left( \frac{1}ε \log\left(δ^{-1}\right)\right)T=O(ε1​log(δ−1)) in both the precision εεε and failure probability δδδ. By encoding eigenvalues in populations of computational basis states rather than complex phases, our approach is naturally robust against certain dephasing errors. The adiabatic protocol only requires the ability to couple a single ancilla qubit to the system Hamiltonian as well as pairwise couplings within the ancilla register.
Implementations of the quantum phase estimation and quantum counting algorithms with qudits are proposed. The construction of the basic building blocks of the algorithms is described in detail and the fidelity of the quantum counting algorithm with qubits and qudits is simulated numerically. Along with the exponential increase of the size of the target register, the simulations with qudits demonstrate significantly higher probability of finding the number of solutions than qubits and more consistent performance for a different number of solutions.
Quantum algorithms solve certain computational problems faster than the best known classical algorithms. Many algorithms, including Shor's for factoring, Grover's for unstructured search, and the HHL for solving linear systems, rely on quantum phase estimation (QPE) as a fundamental subroutine. The QPE protocol proceeds through the initialization of a control register in a uniform superposition, controlled unitary evolution encoding the eigenphase, and a final inverse quantum Fourier transform followed by measurement to extract the phase. Here, we address a special class of unitary operators that frequently appear in quantum Fourier transform-based protocols, cyclic group representations, and periodically evolving quantum systems. We introduce a QPE algorithm that successfully reduces the circuit complexity from O(n2)\mathcal{O}(n^2)O(n2) to O(n)\mathcal{O}(n)O(n) for a special class of unitary operators and implement it on a four-qubit photonic system. The four-qubit system is realised using a photon pair, with two qubits encoded in its polarization degree of freedom and the remaining two in its path modes. In contrast to previous photonic implementations of QPE based on dual-rail encoding and KLM protocol, where controlled operations are inherently probabilistic and thus reduce the overall success probability of phase estimation, our scheme is fully deterministic. Moreover, it is scalable to higher-dimensional unitaries, provided the underlying structure of the unitaries is preserved.
ABSTRACT Quantum Phase Estimation (QPE) is a cornerstone algorithm in quantum computing, with applications ranging from integer factorization to quantum chemistry simulations. However, the resource demands of standard QPE, which require a large number of coherent qubits and deep circuits, pose significant challenges for current Noisy Intermediate Scale Quantum (NISQ) devices. In this work, we introduce the Adaptive Windowed Quantum Phase Estimation (AWQPE) algorithm, a novel method designed to address the limitations of standard QPE. AWQPE utilizes small, independent blocks of control qubits to estimate multiple phase bits simultaneously within a “window,” thereby significantly reducing the number of coherent qubits required to achieve a desired precision. These independent blocks are amenable to parallelization and, when combined with a robust least‐significant‐bit (LSB) to most‐significant‐bit (MSB) ambiguity resolution mechanism, enhance the algorithm's accuracy while mitigating the risk of error propagation. Our numerical simulations demonstrate AWQPE's accuracy and improved robustness relative to standard QPE under representative noise models, showcasing a favorable balance between resource efficiency and computational speed. This makes AWQPE particularly well‐suited for near‐term quantum platforms.
We propose an analysis of the Quantum Phase Estimation (QPE) algorithm applied to many-electron systems by investigating its free parameters such as the time step, number of phase qubits, initial state preparation, number of measurement shots, and other parameters related to the unitary operators implementation. A deep understanding of these parameters and their impact on QPE probability of success and precision of the results is important to pave the way towards more automation of QPE applied to predictive computational chemistry and material science. We here explicit a constructive method to set the QPE free parameters for ground energy estimation and ground state projection, gathering disseminated results from previous works, refining these results and developing new conditions for achieving target performance. We detail the impact of the QPE `blurring function', related to discretization effects, and propose a method to overcome corresponding pathologies. We finally demonstrate that, using the conditions gathered here, the complexity of the Trotterized version of QPE tends to depend mostly on physical system properties and weakly on the number of phase qubits. Various numerical results illustrate the impact of QPE free parameters on success probability and discretization effects. The impact of Trotterization and other features on the precision of the results are illustrated by first numerical simulations on the H2 molecule, that allows us to derive useful insights.