[Paper Review] Static Quantum Computation

Haiqing Wei, Xin Xue|ArXiv.org|Feb 22, 1997

Quantum Computing Algorithms and Architecture7 references3 citations

TL;DR

This paper introduces a novel model of static quantum computation where a many-body quantum system encodes solutions to computational problems in its ground state. By designing interactions to store binary logic relations, the system can solve any P or NP problem using polynomial-sized quantum circuits, with solutions retrieved by adiabatically relaxing the system—establishing a direct link between computational complexity and quantum dynamics.

ABSTRACT

Tailoring many-body interactions among a proper quantum system endows it with computing ability by means of static quantum computation in the sense that some of the physical degrees of freedom can be used to store binary information and the corresponding binary variables satisfy some given logic relations if and only if the system is in the ground state. Two theorems are proved showing that the universal static quantum computer can encode the solutions for any P and NP problem into its ground state using only polynomial number (in the problem input size) of logic gates. The second step is to read out the solutions by relaxing the system. The time complexity is relevant when one tries to read out the solution by relaxing the system, therefore our model of static quantum computation provides a new connection between the computational complexity and the dynamics of a complex system.

Motivation & Objective

• To propose a new model of quantum computation where computational solutions are encoded in the ground state of a many-body quantum system.
• To demonstrate that universal quantum computation can be achieved using only static interactions, without active gate operations during computation.
• To establish a connection between computational complexity and the dynamics of quantum systems through the time required to relax to the ground state.
• To show that any P or NP problem can be encoded using a polynomial number of quantum logic gates in this static framework.
• To provide a theoretical foundation for using ground-state properties of complex quantum systems as computational resources.

Proposed method

• Design a many-body quantum Hamiltonian such that its ground state encodes the solution to a given computational problem.
• Use only local and two-body interactions among qubits to implement logic relations corresponding to Boolean formulas.
• Construct the system so that the ground state satisfies all constraints of the problem, effectively encoding the solution in the quantum state.
• Employ adiabatic evolution to relax the system from a simple initial Hamiltonian to the final problem Hamiltonian, preparing the ground state.
• Utilize the fact that the ground state of the final Hamiltonian corresponds to a valid solution of the problem, which can be measured upon relaxation.
• Prove that the number of qubits and interactions scales polynomially with the input size of the problem, ensuring efficiency in encoding.

Experimental results

Research questions

• RQ1 Can any computational problem in P or NP be encoded into the ground state of a quantum many-body system using only polynomial resources?
• RQ2 Is it possible to achieve universal quantum computation through static interactions without dynamic gate sequences?
• RQ3 How does the time required to prepare the ground state relate to the computational complexity of the problem?
• RQ4 What is the role of many-body entanglement and interaction design in enabling solution encoding in the ground state?
• RQ5 Can the adiabatic relaxation process serve as a viable mechanism for reading out solutions in this static model?

Key findings

• The paper proves that any problem in P or NP can be encoded into the ground state of a quantum system using only a polynomial number of quantum logic gates.
• The solution is encoded in the ground state of a specially designed many-body Hamiltonian, where logical constraints are enforced via physical interactions.
• The system's ground state corresponds to a valid solution of the problem, and this state can be prepared by adiabatically evolving from a trivial initial Hamiltonian.
• The time required to reach the ground state is directly related to the computational complexity of the problem, linking dynamics to complexity theory.
• The model achieves universality in quantum computation without requiring active gate operations during the computation phase.
• The construction demonstrates that static quantum systems with tailored interactions can serve as universal computational devices, with solution retrieval via relaxation.