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arXiv:0805.1387

Adiabatic Quantum Counting by Geometric Phase Estimation

Chi Zhang, Zhaohui Wei, Anargyros Papageorgiou

Published May 9, 2008• 0 citations
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Abstract

We design an adiabatic quantum algorithm for the counting problem, i.e., approximating the proportion, $α$, of the marked items in a given database. As the quantum system undergoes a designed cyclic adiabatic evolution, it acquires a Berry phase $2πα$. By estimating the Berry phase, we can approximate $α$, and solve the problem. For an error bound $ε$, the algorithm can solve the problem with cost of order $(\frac{1}ε)^{3/2}$, which is not as good as the optimal algorithm in the quantum circuit model, but better than the classical random algorithm. Moreover, since the Berry phase is a purely geometric feature, the result may be robust to decoherence and resilient to certain noise.

DOI
10.48550/arxiv.0805.1387

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We design an adiabatic quantum algorithm for the counting problem, i.e., approximating the proportion, ααα, of the marked items in a given database. As the quantum system undergoes a designed cyclic adiabatic evolution, it acquires a Berry phase 2πα2πα2πα. By estimating the Berry phase, we can approximate ααα, and solve the problem. For an error bound εεε, the algorithm can solve the problem with cost of order (1ε)3/2(\frac{1}ε)^{3/2}(ε1​)3/2, which is not as good as the optimal algorithm in the quantum circuit model, but better than the classical random algorithm. Moreover, since the Berry phase is a purely geometric feature, the result may be robust to decoherence and resilient to certain noise.
We outline an algorithm for the quantum counting problem using adiabatic quantum computation (AQC). We show that the mechanism of quantum-adiabatic evolution may be utilized toward estimating the number of solutions to a problem, and not only to find them. Using local adiabatic evolution, a process in which the adiabatic procedure is performed at a variable rate, the problem of counting the number of marked items in an unstructured database is solved quadratically faster than the corresponding classical algorithm. The above algorithm provides further evidence for the potentially powerful capabilities of AQC as a paradigm for more efficient problem solving on a quantum computer, and may be used as the basis for solving more sophisticated problems.
We present an adaptive variational quantum algorithm to estimate the Berry phase accumulated by a nondegenerate ground state under cyclic, adiabatic evolution of a time-dependent Hamiltonian. Our method leverages cyclic adiabatic evolution of the Hamiltonian and employs adaptive variational quantum algorithms for state preparation and evolution, optimizing circuit efficiency while maintaining high accuracy. We benchmark our approach on dimerized Fermi–Hubbard chains with four sites, demonstrating precise Berry phase simulations in both noninteracting and interacting regimes. Our results show that circuit depths reach up to 106 layers for noninteracting systems and increase to 279 layers for interacting systems due to added complexity. In addition, we demonstrate the robustness of our scheme across a wide range of parameters governing adiabatic evolution and variational algorithms. These findings highlight the potential of adaptive variational quantum algorithms for advancing quantum simulations of topological materials and computing geometric phases in strongly correlated systems.
Gate-based quantum computers can in principle simulate the adiabatic dynamics of a large class of Hamiltonians. Here, we consider the cyclic adiabatic evolution of a parameter in the Hamiltonian. We propose a quantum algorithm to estimate the Berry phase and use it to classify the topological order of both single-particle and interacting models, highlighting the differences between the two. This algorithm is extensible to an interacting topological system. Our results evidence the potential of near-term quantum hardware for the topological classification of quantum matter.
Quantum computing has attracted a lot of attention in different research fields, such as mathematics, physics and computer science. Quantum algorithms can solve certain problems significantly faster than classical algorithms. There are many numerical problems, especially those arising from quantum systems, which are notoriously difficult to solve using classical computers, since the computational time required often scales exponentially with the size of the problem. However, quantum computers have the potential to solve these problems efficiently, which is also one of the founding ideas of the field of quantum computing. In this thesis, we explore five computational problems, designing innovative quantum algorithms and studying their computational complexity. First, we design an adiabatic quantum algorithm based on the Berry phases for the counting problem. For the running time, it is not as good as the optimal algorithm in the quantum circuit model, but better than the classical random algorithm. Moreover, since the Berry phase is a purely geometric feature, the result should be robust to decoherence and resilient to certain kinds of noise. Since the counting problem is the foundation of many other numerical problems, such as high-dimensional integration and path integration, our adiabatic algorithms can be directly generalized to these kinds of problems. In addition, we study the quantum PAC learning model, offering an improved lower bound on the query complexity. The lower bound is very close to the best lower bound on query complexity known for the classical PAC learning model. We also study the algorithms and the cost of simulating a system evolving under a given Hamiltonian. We consider high order splitting methods that are particularly applicable in quantum simulation and obtain bounds on the number of exponentials required. Moreover, we derive the optimal order of convergence given the required error bound. We compare our complexity estimates to previously known ones and show the resulting speedup. Furthermore, we consider randomized algorithms for Hamiltonian simulation. The evolution is simulated by a product of exponentials in a random sequence and random evolution times. Hence the final state of the system is approximated by a mixed quantum state. We provide a scheme to bound the error of the final quantum state in a randomized algorithm, and obtain randomized algorithms which have the same efficiency as certain deterministic algorithms but which are simpler to implement. We also apply Hamiltonian simulation in estimating the ground state energy of a multiparticle system, which is also known as the multivariate Sturm-Liouville eigenvalue problem. Since the cost of this problem grows exponentially with the number of particles using deterministic classical algorithms, it suffers from the curse of dimensionality. Quantum computers can vanquish the curse, and we exhibit a quantum algorithm whose total cost are linear in the number of particles.
The Berry phase encodes the geometry of a closed Hamiltonian path, complementing the dynamical phase determined by the accumulated energy. We uncover an adiabatic error-cancellation principle that arises from this geometric character and has no counterpart for energy-based quantities. To extract the Berry phase while canceling the dynamical phase, we combine finite-runtime evolutions generated by ±H\pm H±H along the loop. This construction exactly cancels the leading O(T−1)O(T^{-1})O(T−1) and all higher odd-order nonoscillatory phase errors. Richardson extrapolation further reduces the leading residual error to an oscillatory contribution whose amplitude is set by the endpoints of the loop alone. Beyond this deterministic cancellation, runtime randomization suppresses the remaining oscillatory contribution, reducing the bias after rrr Richardson levels to O(T−(2r+2))O(T^{-(2r+2)})O(T−(2r+2)) for fixed rrr. The same average also suppresses the bias from an imperfectly prepared guiding state, so that the admissible preparation infidelity no longer depends on the target accuracy εB\varepsilon_BεB​. Combining these principles, we obtain a randomized Hadamard-test algorithm for Berry phase estimation that reduces the coherent evolution time to O(εB−1/(2r+2))O(\varepsilon_B^{-1/(2r+2)})O(εB−1/(2r+2)​) while retaining the standard O(εB−2)O(\varepsilon_B^{-2})O(εB−2​) sample complexity. This turns the geometric origin of the cancellation into an algorithmic advantage and makes Berry phase estimation a promising task for early fault-tolerant quantum computers.
We present an adiabatic geometric quantum computation strategy based on the non-degenerate energy eigenstates in (but not limited to) superconducting phase qubit systems. The fidelity of the designed quantum gate was evaluated in the presence of simulated thermal fluctuations in a superconducting phase qubits circuit and was found to be quite robust against random errors. In addition, it was elucidated that the Berry phase in the designed adiabatic evolution may be detected directly via the quantum state tomography developed for superconducting qubits. We also analyze the effects of control parameter fluctuations on the experimental detection of the Berry phase.