Lecture 3 Quantum simulation algorithms Dominic Berry Macquarie

Lecture 3: Quantum simulation algorithms Dominic Berry Macquarie University

Simulation of Hamiltonians 1996 Seth Lloyd

Higher-order simulation n 2007 Berry, Ahokas, Cleve, Sanders

Solving linear systems n 2009 Harrow, Hassidim & Lloyd

Differential equations n n 2010 Berry Discretise the differential equation, then encode as a linear system. Simplest discretisation: Euler method. sets initial condition sets x to be constant

Quantum walks n

Quantum walk on a graph n

Quantum walk on a graph n 1998 Farhi

Quantum walk on a graph n entrance exit 2002 Childs, Farhi, Gutmann

Quantum walk on a graph n entrance exit 2003 Childs, Cleve, Deotto, Farhi, Gutmann, Spielman

2007 NAND tree quantum walk Farhi, Goldstone, Gutmann n In a game tree I alternate making moves with an opponent. n In this example, if I move first then I can always direct the ant to the sugar cube. n What is the complexity of doing this in general? Do we need to query all the leaves? AND OR OR AND AND

2007 NAND tree quantum walk OR OR AND NOT AND NAND NOT NOT Farhi, Goldstone, Gutmann AND

NAND tree quantum walk wave n 2007 Farhi, Goldstone, Gutmann

Simulating quantum walks n wave connected nodes query node

Decomposing the Hamiltonian n 2003 Aharonov, Ta-Shma

2007 Graph colouring n Berry, Ahokas, Cleve, Sanders

2007 Graph colouring n Berry, Ahokas, Cleve, Sanders n Use a string of nodes with equal edge colours, and compress.

General Hamiltonian oracles 2003 Aharonov, Ta-Shma n n More generally, we can perform a colouring on a graph with matrix elements of arbitrary (Hermitian) values. Then we also require an oracle to give us the values of the matrix elements.

Simulating 1 -sparse case 2003 Aharonov, Ta-Shma n Assume we have a 1 -sparse matrix. n How can we simulate evolution under this Hamiltonian? n Two cases: 1. If the element is on the diagonal, then we have a 1 D subspace. 2. If the element is off the diagonal, then we need a 2 D subspace.

Simulating 1 -sparse case n 2003 Aharonov, Ta-Shma

Applications n 2007: Discrete query NAND algorithm – Childs, Cleve, Jordan, Yeung n 2009: Solving linear systems – Harrow, Hassidim, Lloyd n 2009: Implementing sparse unitaries – Jordan, Wocjan n 2010: Solving linear differential equations – Berry n 2013: Algorithm for scattering cross section – Clader, Jacobs, Sprouse

Implementing unitaries n 2009 Jordan, Wocjan

Quantum simulation via walks n Three ingredients: 1. A Szegedy quantum walk 2. Coherent phase estimation 3. Controlled state preparation n The quantum walk has eigenvalues and eigenvectors related to those for Hamiltonian. By using phase estimation, we can estimate the eigenvalue, then implement that actually needed. n

Szegedy Quantum Walk n 2004 Szegedy

Szegedy walk for simulation n 2012 Berry, Childs