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Direct mapping of quantum circuits to adiabatic algorithms

M. Stewart Siu

Published Sep 3, 2004• 0 citations
OpenAlex

Abstract

We give a method of mapping any arbitrary circuit in the standard quantum computing model to an adiabatic algorithm of the same depth. It is a direct mapping as we use the same initial, intermediate and final states as in the circuit. Hence we show constructively the existence of efficient adiabatic evolution paths for a class of problems. Since the physical implementation of unitary gates in the circuit model requires precise timing control, the equivalent adiabatic construction may also provide a more realizable alternative.

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We give a method of mapping any arbitrary circuit in the standard quantum computing model to an adiabatic algorithm of the same depth. It is a direct mapping as we use the same initial, intermediate and final states as in the circuit. Hence we show constructively the existence of efficient adiabatic evolution paths for a class of problems. Since the physical implementation of unitary gates in the circuit model requires precise timing control, the equivalent adiabatic construction may also provide a more realizable alternative.
This paper explores several aspects of the adiabatic quantum-computation model. We first show a way that directly maps any arbitrary circuit in the standard quantum-computing model to an adiabatic algorithm of the same depth. Specifically, we look for a smooth time-dependent Hamiltonian whose unique ground state slowly changes from the initial state of the circuit to its final state. Since this construction requires in general an nnn-local Hamiltonian, we will study whether approximation is possible using previous results on ground-state entanglement and perturbation theory. Finally we will point out how the adiabatic model can be relaxed in various ways to allow for 2-local partially adiabatic algorithms as well as 2-local holonomic quantum algorithms.
We introduce a class of quantum adiabatic evolutions that we claim may be interpreted as the equivalents of the unitary gates of the quantum gate model. We argue that these gates form a universal set and may therefore be used as building blocks in the construction of arbitrary ``adiabatic circuits,'' analogously to the manner in which gates are used in the circuit model. One implication of the above construction is that arbitrary classical boolean circuits as well as gate model circuits may be directly translated to adiabatic algorithms with no additional resources or complexities. We show that while these adiabatic algorithms fail to exhibit certain aspects of the inherent fault tolerance of traditional quantum adiabatic algorithms, they may have certain other experimental advantages acting as quantum gates.
The success of quantum algorithms is to implement correctly quantum states transform,which is mainly achieved by suitable quantum operators.It is shown that the finding of a suitable quantum operator is a great challenge.Most of previous studies solve this problem by machine learning methods.However,these algorithms are different from the standard circuit model of quantum computation which characterizes as unitary quantum operations,thus they are difficult to be realized on quantum computers.It is shown that the transform of quantum states can be implemented by adiabatic quantum evolution,which is a kind of pure quantum effect and suitable well for physical implementation.The adiabatic theorem guarantees that the final state of the system would be the desired output.It is therefore a remarkable approach for quantum states transform in quantum computation and quantum communication.
The model of adiabatic quantum computation has recently attracted attention in the physics and computer science communities, but its exact computational power has been unknown. We settle this question and describe an efficient adiabatic simulation of any given quantum algorithm. This implies that the adiabatic computation model and the standard quantum circuit model are polynomially equivalent. We also describe an extension of this result with implications to physical implementations of adiabatic computation. We believe that our result highlights the potential importance of the adiabatic computation model in the design of quantum algorithms and in their experimental realization.
The adiabatic quantum computation (AQC) has been proven to be equivalent to the standard circuit model. Conventionally, AQC evolves from the initial Hamiltonian which has a uniform equal superposition of the computational basis to the final Hamiltonian whose ground state encodes the solution to a computation problem. We propose an alternative approach to construct the initial Hamiltonian of the AQC which has an unequal superposition of the possible solutions to the problem and show that an educated guess can improve the performance of AQC.