hiiiiii i need help
how do i discretely apply a time evolution operator to a wavefunction in steps of Δt? i'm trying to write something down as an optimization problem with discrete timesteps
@er1n How is your wavefunction represented? If it's just stored as a vector, then time evolution is just matrix multiplication. The matrix exponential of a Hamiltonian times (-i) times your timestep Δt divided by Planck's constant will give the time-evolution operator for Δt. If Δt is very small, then this will Taylor-approximate to the identity matrix minus iHΔt / hbar.
@bstacey okay this is what i thought, but i wanted to make sure. the system is a josephson junction qubit where we're modelling only 3 states, so i think the 'wavefunction' is just sth in ℂ³ right?
@er1n If it's a 3-level system, then it'd be called a "qutrit" rather than a "qubit", but yes, a wavefunction for such a system would be a unit vector in ℂ³. A density operator, which you'd use to model imprecise preparations, thermal noise, etc., would be a 3-by-3 self-adjoint matrix of complex numbers, with trace equal to 1 and nonnegative eigenvalues.
@bstacey okay yeah it's obvious i'm an engineer and not a physicist :) i was saying qubit bc the full model has at least seven dominant levels they consider, but for optimization we truncate to 3, and we're actually trying to minimize leakage to |3〉
thank you for your help! the free evolution hamiltonian for this qutrit is actually diagonalizable which should make the matrix exponentiation much nicer
@bstacey oh they actually do it for us, for the _driven_ operator
@bstacey also i should've just said "transmon which we're attempting to use as a qubit, truncated to a 3-level system" which would've been more accurate
@er1n At least now you know the word "qutrit" and can quote it to impress your friends :-)
@bstacey haha, the worst part is i was already corrected on that in a meeting too, and also my pronunciation of qubit
@er1n It's pronounced like "cubit" because it was literally coined as a joke. Ben Schumacher and Bill Wootters were stumped trying to do an old-school information theory calculation in quantum physics. "Maybe we just need a whole new notion of 'quantum information'!"
"And we can measure it in qubits --- cubits, like Noah's Ark!"
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