Computation time for adiabatic evolution

In summary, the adiabatic theorem yields an equation for the coefficients, starting in an eigenstate m. When the gap is large and the time derivative of H is small, the sum can be neglected and the coefficient m can be expressed as a_0^m. To determine the first order coefficients, an ansatz is made and the result is a_1^m. The last step relies on perturbation theory and the computation time is determined to be t<<\frac{\langle \psi_{m} | \dot{H}| \psi_{m}\rangle}{(E_{n}-E_{m})^2}. However, it is unclear how this result is obtained and assistance is needed.
  • #1
mupsi
32
1
Hi,

my problem: following the adiabatic theorem we get an equation for the coefficients:

[itex]
\dot{a}_{m}=-a_{m} \langle{\psi_{m}} | \dot{\psi}_{m}\rangle - \sum_{n \neq m} \frac{\langle \psi_{m} | \dot{H}| \psi_{m}\rangle}{E_{n}-E_{m}} exp(\int_ 0^t E_{n}(t')-E_{m}(t') \, dt')
[/itex]

we start in an eigenstate m. When the gap is large and the time derivative of H is small. We can neglect the sum and get for the coefficent m:

[itex]
a_m^0= a_m(0) exp(-i \gamma_m (t))
[/itex]

so far so good. Now we want to determine the 1st order coefficients and make the ansatz:

[itex]
a_m= a_m^0 + a_m^1
[/itex]

and we get:

[itex]
a_m^1= \sum_{n \neq m} \frac{\langle \psi_{m} | \dot{H}| \psi_{m}\rangle}{(E_{n}-E_{m})^2} a_n^0 exp(\int_ 0^t E_{n}(t')-E_{m}(t') \, dt')
[/itex]

the last step is problematic. It's been a while since I used perturbation theory and I don't know how you get this result. The computation time should be:

[itex]
t<<\frac{\langle \psi_{m} | \dot{H}| \psi_{m}\rangle}{(E_{n}-E_{m})^2}
[/itex]

which is clear once you obtain the result above. Can anyone help?
 
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  • #2
mupsi said:
Hi,

my problem: following the adiabatic theorem we get an equation for the coefficients:

[itex]
\dot{a}_{m}=-a_{m} \langle{\psi_{m}} | \dot{\psi}_{m}\rangle - \sum_{n \neq m} \frac{\langle \psi_{m} | \dot{H}| \psi_{m}\rangle}{E_{n}-E_{m}} exp(\int_ 0^t E_{n}(t')-E_{m}(t') \, dt')
[/itex]

we start in an eigenstate m. When the gap is large and the time derivative of H is small. We can neglect the sum and get for the coefficent m:

[itex]
a_m^0= a_m(0) exp(-i \gamma_m (t))
[/itex]

so far so good. Now we want to determine the 1st order coefficients and make the ansatz:

[itex]
a_m= a_m^0 + a_m^1
[/itex]

and we get:

[itex]
a_m^1= \sum_{n \neq m} \frac{\langle \psi_{m} | \dot{H}| \psi_{m}\rangle}{(E_{n}-E_{m})^2} a_n^0 exp(\int_ 0^t E_{n}(t')-E_{m}(t') \, dt')
[/itex]

the last step is problematic. It's been a while since I used perturbation theory and I don't know how you get this result. The computation time should be:

[itex]
t<<\frac{\langle \psi_{m} | \dot{H}| \psi_{m}\rangle}{(E_{n}-E_{m})^2}
[/itex]

which is clear once you obtain the result above. Can anyone help?
not sure how to edit posts... a few remarks: it should be t>>... and gamma is the geometric phase.
 

Related to Computation time for adiabatic evolution

1. What is meant by "computation time" for adiabatic evolution?

Computation time for adiabatic evolution refers to the amount of time it takes for a quantum system to transition from its initial state to its final state. This process involves slowly changing the Hamiltonian of the system, and the computation time is the duration of this gradual change.

2. How is computation time for adiabatic evolution measured?

The computation time for adiabatic evolution is typically measured in terms of the number of steps or operations required to change the Hamiltonian of the system. This can vary depending on the specific algorithm or approach used.

3. What factors affect the computation time for adiabatic evolution?

The computation time for adiabatic evolution can be affected by various factors such as the complexity of the problem being solved, the precision required for the solution, the size of the quantum system, and the resources available for computation.

4. Are there any techniques to reduce computation time for adiabatic evolution?

Yes, there are various techniques that can be used to reduce the computation time for adiabatic evolution. These include improving the efficiency of the algorithm, optimizing the system parameters, and utilizing parallel computing or quantum error correction methods.

5. Is there a limit to how fast adiabatic evolution can be performed?

Yes, there is a theoretical lower bound on the computation time for adiabatic evolution, known as the quantum speed limit. This limit is determined by the energy gap between the ground state and the first excited state of the system and cannot be surpassed without sacrificing accuracy.

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