Displacement operator: \(\hat{D}(\alpha)=\exp\left(\alpha\hat{a}^{\dagger}-\alpha^{*}\hat{a}\right)\) where \(\alpha=i\int\varepsilon(t)dt\) acting only on the ground subspace \(\{\left|0\right\rangle \dots\left|n\right\rangle\dots \}\otimes\left|g\right\rangle\)
Nearest neighbours are coupled and by iteration we can couple all levels.
Explicit Algorithm - n to n+1
Given the state \(\left|n\right\rangle\) we want to create the state \(\left|\text{target}\right\rangle=\cos(\theta)\left|n\right\rangle + \sin(\theta)\left|n+1\right\rangle\)
Inspired by the above, consider the non-infinitesimal operation:
Optimize \(F=\left|\left\langle\text{target}\right|\hat{U}_{n}\left|n\right\rangle\right|\) wrt \(\alpha_1\), \(\alpha_2\), and \(\alpha_3\) with some good initial guesses.
Explicit Algorithm - \(\mid 0\rangle\) to \(\mid \psi \rangle\)
Requires \(N\) runs of the previous algorithm (n to n+1) (or just a dictionary lookup)
Optionally a "global" optimization can be run over all parameters (after simplifications this means an optimization over \(2N+1\) parameters)
Apply a final SNAP gate to impart any missing phases.
Explicit Algorithm - \(\mid 0\rangle\) to \(\mid \psi \rangle\)
Fidelity better than \(0.999\).
Final Cost
To create an arbitrary N-dimensional state
To find the pulses
N optimizations over 3 parameters (or N dictionary lookups)
1 optimization over 2N+1 parameters (takes ~5seconds in practice)
To implement the pulses
2N+1 displacement gates
2N SNAP gates
Super Efficient Fock State Preparation
Displace to coherent state \(\mid\alpha\rangle=D(\alpha=\sqrt{n})\mid0\rangle\)
Use \(\mathcal{O}(\sqrt{n})\) rotations to "fold" it into \(\mid n \rangle\)
Super Efficient Fock State Preparation
Universal Control
Universal Control
We want to perform an arbitrary unitary operation on the oscillator state (even when the state is unknown), not just prepare a target state from another given state.
Preparing a 2x2 rotation submatrix
Chaining 2x2 submatrices into an NxN unitary matrix
2-by-2 Rotation Sub-Matrix
As in the case of state preparation for \(\left|n\right\rangle \rightarrow \cos(\theta)\left|n\right\rangle + \sin(\theta)\left|n+1\right\rangle\)
Removing a column requires \(N-1\) 2-by-2 matrices. To ensure high fidelity, at the end of the column removal we perform a global optimization over all \(2N\) displacement parameters.
The cost function is the "leakage" outside of \(\hat{W}_{n-1\times n-1}\)
We repeat the procedure for all columns. Then we do one final global optimization over all the displacement coefficients (~\(N^2\) of them) against the fidelity
where for any \(M\), \(\left\Vert M\right\Vert =\underset{\left\Vert \left|\psi\right\rangle \right\Vert _{2}=1}{\sup}\left\Vert M\left|\psi\right\rangle \right\Vert _{2}\)
People are Already Implementing it on the 4th Floor
Off-topic Topics
Sublinear Preparation of "Sparse" States
The preparation of Fock states by folding can be generalized to the preparation of any "sparse" state where the population is centered around a single Fock state.
Universal Control Including the Spin
We were working on only half the Hilbert space (the \(\mid g \rangle\) subspace).
If we can:
Switch \(\chi\) on and off both for the ground and for the excited state.
Add a \(\hat{\sigma}_z \mid n \rangle\langle n \mid\) term to the Hamiltonian on command.
We can easily extend the protocol to work on the entire Hilbert space.
Efficient Readout Scheme
Ask Chao (MLS, Mar 30, 2015) and Reinier (MLS, next week).
Implementing a binary search for a Fock state using a SNAP-like gate and conditional measurement.
Experimental Implementation
Reinier will be presenting it next week. They are designing even shorter pulses with some in-depth numerical optimizations.
Actual Conclusion
Actual Conclusion
Efficient, Fast and Extendable New Protocol for Universal Control
