arXiv subjects

Chen-Fu Chiang

Publications and source records attributed to Chen-Fu Chiang.

12 recordsLinked to original sources

arXiv2023

Grover Search Inspired Alternating Operator Ansatz of Quantum Approximate Optimization Algorithm for Search Problems

We use the mapping between two computation frameworks , Adiabatic Grover Search (AGS) and Adiabatic Quantum Computing (AQC), to translate the Grover search algorithm into the AQC regime. We then apply Trotterization on the schedule-dependent Hamiltonian of AGS to obtain the values of variational parameters in the Quantum Approximate Optimization Algorithm (QAOA) framework. The goal is to carry the optimal behavior of Grover search algorithm into the QAOA framework without the iterative machine learning processes.

quant-ph↗

arXiv2023

Quantum Walk Inspired Dynamic Adiabatic Local Search

We investigate the irreconcilability issue that raises in translating the search algorithm from the Continuous-Time Quantum Walk (CTQW) framework to the Adiabatic Quantum Computing (AQC) framework. For the AQC formulation to evolve along the same path as the CTQW requires a constant energy gap in the former Hamiltonian throughout the AQC schedule. To resolve the issue, we modify the CTQW-inspired AQC catalyst Hamiltonian with a $Z$ oracle operator. Through simulation we demonstrate that the total running time for the proposed approach remains optimal. Inspired by this solution, we further investigate adaptive scheduling for the catalyst Hamiltonian and its coefficient function in the adiabatic path to improve the adiabatic local search.

quant-ph↗

arXiv2022

Space-efficient Quantization Method for Reversible Markov Chains

In a seminal paper, Szegedy showed how to construct a quantum walk $W(P)$ for any reversible Markov chain $P$ such that its eigenvector with eigenphase $0$ is a quantum sample of the limiting distribution of the random walk and its eigenphase gap is quadratically larger than the spectral gap of $P$. The standard construction of Szegedy's quantum walk requires an ancilla register of Hilbert-space dimension equal to the size of the state space of the Markov chain. We show that it is possible to avoid this doubling of state space for certain Markov chains that employ a symmetric proposal probability and a subsequent accept/reject probability to sample from the Gibbs distribution. For such Markov chains, we give a quantization method which requires an ancilla register of dimension equal to only the number of different energy values, which is often significantly smaller than the size of the state space. To accomplish this, we develop a technique for block encoding Hadamard products of matrices which may be of wider interest.

quant-ph↗

arXiv2018

Optimizing Quantum Walk Search on a Reduced Uniform Complete Multi-Partite Graph

In a recent work by Novo et al. (Sci. Rep. 5, 13304, 2015), the invariant subspace method was applied to the study of continuous-time quantum walk (CTQW). The method helps to reduce a graph into a simpler version that allows more transparent analyses of the quantum walk model. In this work, we adopt the aforementioned method to investigate the optimality of a quantum walk search of a marked element on a complete multi-partite graph. We formulate the eigenbasis that would facilitate the transport between the two lowest energy eigenstates and demonstrate how to set the appropriate coupling factor to preserve the optimality.

quant-ph↗

arXiv2013

Selecting Efficient Phase Estimation With Constant-Precision Phase Shift Operators

We investigate the cost of three phase estimation procedures that require only constant-precision phase shift operators. The cost is in terms of the number of elementary gates, not just the number of measurements. Faster phase estimation requires the minimal number of measurements with a \log * factor of reduction when the required precision n is large. The arbitrary constant-precision approach (ACPA) requires the minimal number of elementary gates with a minimal factor of 14 of reduction in comparison to Kitaev's approach. The reduction factor increases as the precision gets higher in ACPA. Kitaev's approach is with a reduction factor of 14 in comparison to the faster phase estimation in terms of elementary gate counts.

quant-ph↗

arXiv2013

Quantum Phase Estimation with an Arbitrary Number of Qubits

Due to the great difficulty in scalability, quantum computers are limited in the number of qubits during the early stages of the quantum computing regime. In addition to the required qubits for storing the corresponding eigenvector, suppose we have additional k qubits available. Given such a constraint k, we propose an approach for the phase estimation for an eigenphase of exactly n-bit precision. This approach adopts the standard recursive circuit for quantum Fourier transform (QFT) and adopts classical bits to implement such a task. Our algorithm has the complexity of O(n \log k), instead of O(n^2) in the conventional QFT, in terms of the total invocation of rotation gates. We also design a scheme to implement the factorization algorithm by using k available qubits via either the continued fractions approach or the simultaneous diophantine approximation.

quant-ph↗

arXiv2011

Quantum Phase Estimation with Arbitrary Constant-precision Phase Shift Operators

While Quantum phase estimation (QPE) is at the core of many quantum algorithms known to date, its physical implementation (algorithms based on quantum Fourier transform (QFT)) is highly constrained by the requirement of high-precision controlled phase shift operators, which remain difficult to realize. In this paper, we introduce an alternative approach to approximately implement QPE with arbitrary constant-precision controlled phase shift operators. The new quantum algorithm bridges the gap between QPE algorithms based on QFT and Kitaev's original approach. For approximating the eigenphase precise to the nth bit, Kitaev's original approach does not require any controlled phase shift operator. In contrast, QPE algorithms based on QFT or approximate QFT require controlled phase shift operators with precision of at least Pi/2n. The new approach fills the gap and requires only arbitrary constant-precision controlled phase shift operators. From a physical implementation viewpoint, the new algorithm outperforms Kitaev's approach.

quant-ph↗

arXiv2011

Hitting Time of Quantum Walks with Perturbation

The hitting time is the required minimum time for a Markov chain-based walk (classical or quantum) to reach a target state in the state space. We investigate the effect of the perturbation on the hitting time of a quantum walk. We obtain an upper bound for the perturbed quantum walk hitting time by applying Szegedy's work and the perturbation bounds with Weyl's perturbation theorem on classical matrix. Based on the definition of quantum hitting time given in MNRS algorithm, we further compute the delayed perturbed hitting time (DPHT) and delayed perturbed quantum hitting time (DPQHT). We show that the upper bound for DPQHT is actually greater than the difference between the square root of the upper bound for a perturbed random walk and the square root of the lower bound for a random walk.

quant-ph↗

arXiv2010

Sensitivity of Quantum Walks with Perturbation

Quantum computers are susceptible to noises from the outside world. We investigate the effect of perturbation on the hitting time of a quantum walk and the stationary distribution prepared by a quantum walk based algorithm. The perturbation comes from quantizing a transition matrix Q with perturbation E (errors). We bound the perturbed quantum walk hitting time from above by applying Szegedy's work and the perturbation bounds with Weyl's perturbation theorem on classical matrix. Based on an efficient quantum sample preparation approach invented in {\em speed-up via quantum sampling} and the perturbation bounds for stationary distribution for classical matrix, we find an upper bound for the total variation distance between the prepared quantum sample and the true quantum sample.

quant-ph↗

arXiv2010

Quantum Algorithm for Preparing Thermal Gibbs States - Detailed Analysis

In a recent work [10], Poulin and one of us presented a quantum algorithm for preparing thermal Gibbs states of interacting quantum systems. This algorithm is based on Grovers's technique for quantum state engineering, and its running time is dominated by the factor D/Z(β), where D and Z(β) denote the dimension of the quantum system and its partition function at inverse temperature β, respectively. We present here a modified algorithm and a more detailed analysis of the errors that arise due to imperfect simulation of Hamiltonian time evolutions and limited performance of phase estimation (finite accuracy and nonzero probability of failure). This modfication together with the tighter analysis allows us to prove a better running time by the effect of these sources of error on the overall complexity. We think that the ideas underlying of our new analysis could also be used to prove a better performance of quantum Metropolis sampling by Temme et al. [12].

physics.comp-ph↗

arXiv2009

Efficient Circuits for Quantum Walks

We present an efficient general method for realizing a quantum walk operator corresponding to an arbitrary sparse classical random walk. Our approach is based on Grover and Rudolph's method for preparing coherent versions of efficiently integrable probability distributions. This method is intended for use in quantum walk algorithms with polynomial speedups, whose complexity is usually measured in terms of how many times we have to apply a step of a quantum walk, compared to the number of necessary classical Markov chain steps. We consider a finer notion of complexity including the number of elementary gates it takes to implement each step of the quantum walk with some desired accuracy. The difference in complexity for various implementation approaches is that our method scales linearly in the sparsity parameter and poly-logarithmically with the inverse of the desired precision. The best previously known general methods either scale quadratically in the sparsity parameter, or polynomially in the inverse precision. Our approach is especially relevant for implementing quantum walks corresponding to classical random walks like those used in the classical algorithms for approximating permanents and sampling from binary contingency tables. In those algorithms, the sparsity parameter grows with the problem size, while maintaining high precision is required.

quant-ph↗

arXiv2009

Quantum Speed-up for Approximating Partition Functions

We achieve a quantum speed-up of fully polynomial randomized approximation schemes (FPRAS) for estimating partition functions that combine simulated annealing with the Monte-Carlo Markov Chain method and use non-adaptive cooling schedules. The improvement in time complexity is twofold: a quadratic reduction with respect to the spectral gap of the underlying Markov chains and a quadratic reduction with respect to the parameter characterizing the desired accuracy of the estimate output by the FPRAS. Both reductions are intimately related and cannot be achieved separately. First, we use Grover's fixed point search, quantum walks and phase estimation to efficiently prepare approximate coherent encodings of stationary distributions of the Markov chains. The speed-up we obtain in this way is due to the quadratic relation between the spectral and phase gaps of classical and quantum walks. Second, we generalize the method of quantum counting, showing how to estimate expected values of quantum observables. Using this method instead of classical sampling, we obtain the speed-up with respect to accuracy.

quant-ph↗