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arXiv:2402.14191 · Physical Review A

Iterative method to improve the precision of the quantum-phase-estimation algorithm

Junxu Li

Published Mar 7, 2024• 8 citations
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Abstract

Here we revisit the quantum-phase-estimation (QPE) algorithm and devise an iterative method to improve the precision of QPE with propagators over a variety of time spans. For a given propagator and a certain eigenstate as input, QPE with a propagator is introduced to estimate the phase corresponding to an eigenenergy. Due to the periodicity of the complex exponential, we can pinpoint the eigenenergy in a branch of comblike ranges by applying QPE with propagators over longer time spans. Thus, by picking up appropriate time spans, the iterative QPE with corresponding propagators can enable us to pinpoint the eigenenergy more precisely. Moreover, even if there are only a few qubits as ancilla qubits, high precision is still available by the proposed iterative method. Our work provides a feasible and promising means toward precise estimations of an eigenvalue on noisy intermediate-scale quantum devices.

DOI
10.1103/physreva.109.032606

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•Jan 1, 2015•0 citations
Here we revisit the quantum-phase-estimation (QPE) algorithm and devise an iterative method to improve the precision of QPE with propagators over a variety of time spans. For a given propagator and a certain eigenstate as input, QPE with a propagator is introduced to estimate the phase corresponding to an eigenenergy. Due to the periodicity of the complex exponential, we can pinpoint the eigenenergy in a branch of comblike ranges by applying QPE with propagators over longer time spans. Thus, by picking up appropriate time spans, the iterative QPE with corresponding propagators can enable us to pinpoint the eigenenergy more precisely. Moreover, even if there are only a few qubits as ancilla qubits, high precision is still available by the proposed iterative method. Our work provides a feasible and promising means toward precise estimations of an eigenvalue on noisy intermediate-scale quantum devices.
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.
Quantum phase estimation (QPE) is one of the core algorithms for quantum computing. It has been extensively studied and applied in a variety of quantum applications such as the Shor's factoring algorithm, quantum sampling algorithms and the calculation of the eigenvalues of unitary matrices. The QPE algorithm has been combined with Kitaev's algorithm and the inverse quantum Fourier transform (IQFT) which are utilized as a fundamental component of such quantum algorithms. In this paper, we explore the computational challenges of implementing QPE algorithms on noisy intermediate-scale quantum (NISQ) machines using the IBM Q Experience (e.g., the IBMQX4, 5-qubit quantum computing hardware platform). Our experimental results indicate that the accuracy of finding the phase using these QPE algorithms is severely constrained by the NISQ computer's physical characteristics such as coherence time and error rates. To mitigate these physical limitations, we propose implementing a modified solution by reducing the number of controlled rotation gates and phase shift operations, thereby increasing the accuracy of the finding phase in near-term quantum computers.
Quantum phase estimation (QPE) is a crucial subroutine in many quantum algorithms. It is used to estimate the eigenvalues of a unitary operator acting on a target quantum system. Conventional QPE is often expensive to implement because it requires a large number of ancilla qubits and the ability to perform quantum Fourier transform. Recent developments in iterative QPE reduce the implementation cost by repetitive uses of a single ancilla and classical postprocessing. However, both conventional and iterative schemes often require preparation of the target system in an eigenstate of the unitary operator, while it remains ambiguous to achieve QPE of multiple eigenvalues with no need of initial-state preparation. Here we clarify this issue by developing a theoretical framework based on sequential quantum channels for iterative QPE. We find that QPE of multiple eigenvalues can be efficiently realized for arbitrary initial target system state by actively utilizing the measurement backaction of iterative QPE on the target system with a long coherence time. Specifically, we investigate two iterative QPE schemes based on sequential Ramsey interferometry measurements (RIMs) of an ancilla qubit: (a) the repetitive scheme, which conducts repetitive RIMs to achieve the standard quantum limit in estimating the eigenvalues; (b) the adaptive scheme, which adjusts the parameters of each RIM based on prior measurement outcomes to attain the Heisenberg limit. In both schemes, sequential ancilla measurements generate sequential quantum channels on the target system, gradually steering it to the eigenstates of the estimated unitary operator, while the measurement statistics of the ancilla can reveal the embedded information about its eigenvalues with proper postprocessing. We demonstrate the analysis by simulating a central spin model and evaluate the performance and noise resilience of both schemes.
Quantum phase estimation (QPE) is the workhorse behind any quantum algorithm and a promising method for determining ground state energies of strongly correlated quantum systems. Low-cost QPE techniques make use of circuits which only use a single ancilla qubit, requiring classical post-processing to extract eigenvalue details of the system. We investigate choices for phase estimation for a unitary matrix with low-depth noise-free or noisy circuits, varying both the phase estimation circuits themselves as well as the classical post-processing to determine the eigenvalue phases. We work in the scenario when the input state is not an eigenstate of the unitary matrix. We develop a new post-processing technique to extract eigenvalues from phase estimation data based on a classical time-series (or frequency) analysis and contrast this to an analysis via Bayesian methods. We calculate the variance in estimating single eigenvalues via the time-series analysis analytically, finding that it scales to first order in the number of experiments performed, and to first or second order (depending on the experiment design) in the circuit depth. Numerical simulations confirm this scaling for both estimators. We attempt to compensate for the noise with both classical post-processing techniques, finding good results in the presence of depolarizing noise, but smaller improvements in 9-qubit circuit-level simulations of superconducting qubits aimed at resolving the electronic ground state of a H 4 -molecule.
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.
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.