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Simulated Quantum Annealer Danica Bassman Advisor: Max Mintz Senior Project Poster Day 2014 Department of Computer and InformaDon Science – University of Pennsylvania A simulated quantum annealer takes advantage of the adiaba6c theorem to solve problems by transi6oning from a simple to complex quantum state while maintaining minimal energy. We simulate these quantum states, energies, and transi6ons on classical machines by mathema6cal abstrac6ons for proof of concept and to circumvent to the complica6ons of physical implementa6on. Abstract Goals 1. Simulate quantum state on a classical machine 2. Proof of concept of the adiaba3c theorem and its applicaDons to quantum computaDon Quantum CompuDng AdiabaDc Theorem Given a simple quantum system starDng in a minimal energy state, if the system is changed slowly enough, as it moves to a more complex state, it will maintain its minimal energy. Le@: slowly cooled Right: cooled too quickly • n entangled qubits can be abstracted to a 2 n dimensional unit vector in 2 n D complex space • Basis vectors are possible states of collapse • Coordinates are probabiliDes • Classical bits can take on values 1 or 0. • Qubits can take on any value of a unit vector in complex space System Design System Implementa3on SimulaDon Under Quantum Gate Model Qubit Unit Vector Quantum Gate Matrix We mulDply vectors by matrix representaDons of linear transformaDons to simulate quantum gates acDng on qubits. Apply Gate Model AbstracDon Technique to Annealing H B H I H P Simple Hamiltonian with easily found ground quantum state and minimal energy Slow transiDon to more complex Hamiltonians, while maintaining minimal energy Complex Hamiltonian, sDll with minimal energy, whose ground state encodes soluDon Hamiltonian energy funcDon Linear transformaDon on quantum state vector 1. Start at simple H B 2. Define complex H P 3. Define slow moving transiDon funcDon, H, from H B to H P 4. TransiDon slowly over Dme s unDl minimal gap is reached Using Simulated Quantum Annealing to Solve 3SAT Given (x 1 ∨x 2 ∨x 3 )∧…∧(x r ∨x s ∨x t ), find assignments for all x i such that the enDre expression evaluates to true. Under classical compuDng, 3SAT is in NPC. Conclusion and Further Work Current quantum computers use annealing, but their ability to provide exponenDal speedup and computaDonal robustness are not yet certain. SimulaDon of quantum annealing allows us to bypass of the hurdles of physical implementaDon and study annealing’s potenDal uses. Quantum Annealing Physical CNOT Gate Mathema6cal Abstrac6on Simulated Quantum Annealing Algorithm Step Implementa3on H B simple Every qubit in equally weighted superposiDon of 0 and 1 H P complex All clauses that evaluate to true have energy 0; all other clauses have energy 1; Will have minimal energy when the enDre expression is saDsfied TransiDon FuncDon TransiDon slowly from H B to H P by making small changes to H I to a neighboring state SoluDon Stop when minimal gap between current state and H p is reached; Eigenvector for minimal eigenvalue encodes the saDsfying assignments Variable Abstrac3ons Hamiltonian must change slowly to maintain minimal energy (adiabaDc theorem) Quantum state vector is eigenvector for H; represents quantum state for given energy Energy is eigenvalue for H for the quantum state of its corresponding eigenvector
