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Chinese Physics B

Quantum adiabatic algorithms using unitary interpolation*

Shuo Zhang, Qian-Heng Duan, Tan Li, Xiang-Qun Fu, He-Liang Huang, Xiang Wang, Wan‐Su Bao

Published Dec 5, 2019• 3 citations
Publisher

Abstract

We present two efficient quantum adiabatic algorithms for Bernstein–Vazirani problem and Simon’s problem. We show that the time complexities of the algorithms for Bernstein–Vazirani problem and Simon’s problem are O(1) and O(n), respectively, which are the same complexities as the corresponding algorithms in quantum circuit model. In these two algorithms, the adiabatic Hamiltonians are realized by unitary interpolation instead of standard linear interpolation. Comparing with the adiabatic algorithms using linear interpolation, the energy gaps of our algorithms keep constant. Therefore, the complexities are much easier to analyze using this method.

DOI
10.1088/1674-1056/ab5f02

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We present two efficient quantum adiabatic algorithms for Bernstein–Vazirani problem and Simon’s problem. We show that the time complexities of the algorithms for Bernstein–Vazirani problem and Simon’s problem are O(1) and O(n), respectively, which are the same complexities as the corresponding algorithms in quantum circuit model. In these two algorithms, the adiabatic Hamiltonians are realized by unitary interpolation instead of standard linear interpolation. Comparing with the adiabatic algorithms using linear interpolation, the energy gaps of our algorithms keep constant. Therefore, the complexities are much easier to analyze using this method.
The early work of Wei et al. has shown that it is possible to do the computation in constant time for quantum search in quantum local adiabatic evolution framework if the system Hamiltonian has an extra enlarged factor proportional to the square-root of the size of the problem. Also in our prior work, we have shown that it is possible to apply a linear interpolating path in the system Hamiltonian to achieve an O(1) time complexity for a quantum global adiabatic computation. In this paper, we discuss the issues when implementing these two quantum algorithms on the quantum circuit model, and find that to our surprise, the time slices produced does not equal to the time complexity of the algorithm in each case. This is in contrast to the previous related results, because there these two quantities always coincide. When taking into account of the whole energy-level increase of the system compared with that of the usual quantum adiabatic evolutions and the corresponding time complexity simultaneously, we find that the time slices needed always consist of the two quantities multiplying together. This result is new, and may be hopeful to find application in designing quantum adiabatic algorithm for problems beyond quantum search.
In this paper,we study two different nonlinear interpolating paths in adiabatic evolution algorithms for solving a particular class of quantum search problems where both the initial and final Hamiltonian are one-dimensional projector Hamiltonians on the corresponding ground state.If the overlap between the initial state and final state of the quantum system is not equal to zero,both of these models can provide a constant time speedup over the usual adiabatic algorithms by increasing some another corresponding complexity.But when the initial state has a zero overlap with the solution state in the problem,the second model leads to an infinite time complexity of the algorithm for whatever interpolating functions being applied while the first one can still provide a constant running time.However,inspired by a related reference,a variant of the first model can be constructed which also fails for the problem when the overlap is exactly equal to zero if we want to make up the intrinsic fault of the second model - an increase in energy.Two concrete theorems are given to serve as explanations why neither of these two models can improve the usual adiabatic evolution algorithms for the phenomenon above.These just tell us what should be noted when using certain nonlinear evolution paths in adiabatic quantum algorithms for some special kind of problems.
In this paper, we study two different nonlinear interpolating paths in adiabatic evolution algorithms for solving a particular class of quantum search problems where both the initial and final Hamiltonian are one-dimensional projector Hamiltonians on the corresponding ground state. If the overlap between the initial state and final state of the quantum system is not equal to zero, both of these models can provide a constant time speedup over the usual adiabatic algorithms by increasing some another corresponding "complexity". But when the initial state has a zero overlap with the solution state in the problem, the second model leads to an infinite time complexity of the algorithm for whatever interpolating functions being applied while the first one can still provide a constant running time. However, inspired by a related reference, a variant of the first model can be constructed which also fails for the problem when the overlap is exactly equal to zero if we want to make up the "intrinsic" fault of the second model — an increase in energy. Two concrete theorems are given to serve as explanations why neither of these two models can improve the usual adiabatic evolution algorithms for the phenomenon above. These just tell us what should be noted when using certain nonlinear evolution paths in adiabatic quantum algorithms for some special kind of problems.
We outline an efficient quantum-adiabatic algorithm that solves Simon's problem, in which one has to determine the `period', or xor-mask, of a given black-box function. We show that the proposed algorithm is exponentially faster than its classical counterpart and has the same complexity as the corresponding circuit-based algorithm. Together with other related studies, this result supports a conjecture that the complexity of adiabatic quantum computation is equivalent to the circuit-based computational model in a stronger sense than the well-known, proven polynomial equivalence between the two paradigms. We also discuss the importance of the algorithm and its theoretical and experimental implications for the existence of an adiabatic version of Shor's integer factorization algorithm that would have the same complexity as the original algorithm.