Hamiltonians and simulating quantum physics
What is a Hamiltonian?
The "Hamiltonian” of a quantum system governs:
Time evolution of a quantum system
The "Hamiltonian” specifies how the state of the system evolves in time, according to Schrödinger’s equation:
Solution:
?
Time evolution of a quantum system
The "Hamiltonian” specifies how the state of the system evolves in time, according to Schrödinger’s equation:
Solution:
Time evolution of a quantum system
The "Hamiltonian” specifies how the state of the system evolves in time, according to Schrödinger’s equation:
Solution:
Is this unitary?
Time evolution of a quantum system
The "Hamiltonian” specifies how the state of the system evolves in time, according to Schrödinger’s equation:
Solution:
Is this unitary?
Yes, when H is Hermitian.
Time evolution of a quantum system
The "Hamiltonian” specifies how the state of the system evolves in time, according to Schrödinger’s equation:
Solution:
Is this unitary?
Yes, when H is Hermitian.
Can we implement
this efficiently?
Time evolution of a quantum system
Simulating the evolution of quantum systems is a critical problem with many important applications
Turns Nitrogen into Ammonia through the help of a catalyst (FeMoco)
1-3% of the world’s energy!
Hamiltonian Simulation, more formally
Hamiltonian Simulation, more formally
Can quantum computers solve this problem efficiently?
Input:
Output:
Local Hamiltonians
Typical Hamiltonians occurring in nature are “local”:
particles interact with nearby ones
Each term captures a “constraint’’ on a few qubits of the
quantum system, e.g. electrical attraction between two particles, nuclear forces..
Local Hamiltonians
Typical Hamiltonians occurring in nature are “local”:
particles interact with nearby ones
Can quantum computers solve Hamiltonian Simulation
efficiently for these kinds of Hamiltonians?
Local Hamiltonians
Can quantum computers solve Hamiltonian Simulation for these kinds of Hamiltonians?
Best classical algorithm runtime: still exponential
Hamiltonian Simulation: the optimistic view
. . .
Hamiltonian Simulation: the cautious view
Understanding nitrogen fixation is not just about
time evolution. It is mostly about computing “energies”.
However, it’s plausible that there are practical speedups for classes of Hamiltonians of practical interest (there are promising heuristics). We are optimistic that we’ll learn more once we have large enough quantum computers.
No provable speedups for computing the ground state energy of a quantum system. This problem is thought to be hard even for quantum computers (in the worst case).
Not as easy!