QCJC: Cirac & Zoller 1994

Before anyone had even dreamed of quantum computing, trapped ions were used for atomic clocks and setting frequency standards. Their superb isolation from the environment let physicists make the world’s most precise and accurate measurements of energy level structures and frequencies. It also led to interest in using trapped ions as a quantum information platform, especially after the publication of Shor’s factoring algorithm sparked mainstream interest in the field in 1994.

Barely over a week later, PRL received a transcript from Cirac and Zoller detailing how to implement controlled-NOT gates on arbitrary numbers of ions in the same trap. These gates, along with single-qubit rotations around any axis, form a universal set of gates for quantum computation. This means that a sequence of them can approximate any unitary transformation to arbitrary accuracy. Since it was already known how to implement the single-qubit rotation gates, this paper established trapped ions as a candidate platform for universal QC.

In a typical ion trap, the ions are very tightly confined in the Y and Z directions and more loosely trapped in the X direction. This allows them to be trapped in a straight line, where repulsive Coulomb interactions keep them a few microns apart. Just like the internal levels of each ion, the collective movement of the ions in the trap is quantized. There are specific modes of motion that have discrete energies. The lowest-energy motion is the center-of-mass mode, which is when all the ions move together as one, back and forth, in the X direction.

The transition between internal energy states of an ion is driven by applying a laser with the correct frequency, and hence photon energy. Lowering the frequency by the energy gap between the no-phonon (not moving) $|0\rangle$ and one-phonon (center-of-mass mode) $|1\rangle$ states* gives the Hamiltonian

$\hat{H}_{n,q} = \frac{\eta}{\sqrt{N}} \frac{\Omega}{2} [|e_q \rangle_n \langle g| a e^{-i\phi} + |g \rangle_n \langle e_q| a^\dagger e^{i\phi} ]$

where $a$ and $a^\dagger$ are the phonon creation and annihilation operators, the $n$ subscript refers to the nth ion in the chain, and the $q$ subscript denotes the laser polarization and consequently which excited state the ion is pumped into.

Applying this Hamiltonian (by shining laser light on the relevant ion) for a time $t = \frac{k \pi \sqrt{N}}{\Omega \eta}$, also known as the $k\pi$ time, evolves the state according to the unitary operator

$\hat{U}^{k,q}_{n} (\phi) = \exp(\frac{-i k \pi}{2} [|e_q \rangle_n \langle g| a e^{-i\phi} + |g \rangle_n \langle e_q| a^\dagger e^{i\phi} ])$

*Thanks to the detuning, this will leave the states $|g\rangle |0\rangle$ and $|e_q\rangle |1\rangle$ alone while swapping the states $|e_q\rangle |0\rangle$ and $|g\rangle |1\rangle$ for every $\pi$ rotation. When the states swap, they also pick up a phase of $-i$.

If we now consider any two ions that we want to entangle and cool the ion chain to its motional ground state, the four relevant states are $|g \rangle_m |g \rangle_n |0 \rangle$$|g \rangle_m |e_0 \rangle_n |0 \rangle$$|e_0 \rangle_m |g \rangle_n |0 \rangle$, and $|e_0 \rangle_m |e_0 \rangle_n |0 \rangle$. Applying a $\pi$ rotation to the mth ion leads to the changes

$|g\rangle_m |g\rangle_n |0\rangle \rightarrow |g\rangle_m |g\rangle_n |0\rangle$

$|g\rangle_m |e_0\rangle_n |0\rangle \rightarrow |g\rangle_m |e_0\rangle_n |0\rangle$

$|e_0\rangle_m |g\rangle_n |0\rangle \rightarrow -i|g\rangle_m |g\rangle_n |1\rangle$

$|e_0\rangle_m |e_0\rangle_n |0\rangle \rightarrow -i|g\rangle_m |e_0\rangle_n |1\rangle$

The two states with $|e_0\rangle$ changed and picked up phases. The next step in the protocol involves a $2\pi$ rotation applied to the nth ion. This will leave the fourth state alone but apply two swaps and two phases to the third state, ultimately just applying a phase of $-i * -i = -1$. This would leave only one state (the fourth) with a minus sign if not for the fact that a $\pi$ rotation will also affect the second state. To avoid this, we can use polarization that couples the $|g\rangle$ state to $|e_1\rangle$ instead of $|e_0\rangle$. That way, the third state can pick up its phase while the second state remains unaffected due to the fact that $|e_0\rangle$ and $|e_1\rangle$ do not couple through this interaction.

Finally, reapplying a $\pi$ rotation on the mth ion yields the transformations

$|g\rangle_m |g\rangle_n |0\rangle \rightarrow |g\rangle_m |g\rangle_n |0\rangle$

$|g\rangle_m |e_0\rangle_n |0\rangle \rightarrow |g\rangle_m |e_0\rangle_n |0\rangle$

$i|g\rangle_m |g\rangle_n |0\rangle \rightarrow |e_0\rangle_m |g\rangle_n |0\rangle$

$-i|g\rangle_m |e_0\rangle_n |0\rangle \rightarrow -|e_0\rangle_m |e_0\rangle_n |0\rangle$

Thus, there is a phase change only if both ions were initially excited. That this represents a CNOT gate may not be obvious, but if we take the usual definition $|\pm\rangle = \frac{|g\rangle + |e_0\rangle}{\sqrt{2}}$, then the above transformations can be summarized as

$|g\rangle_m |\pm\rangle_n \rightarrow |g\rangle_m |\pm\rangle_n$

$|e_0\rangle_m |\pm\rangle_n \rightarrow |e_0\rangle_m |\mp\rangle_n$

Applying the right X rotations on the nth ion make this transformation what we typically think of when we say CNOT, although it may be more efficient to leave it be.

After going through the implementation of this gate, Cirac and Zoller spend a few paragraphs on a numerical simulation of a trapped ion quantum computer carrying out a Quantum Fourier Transform, the main ingredient in Shor’s algorithm. Only a few months later, the first CNOT gate was implemented experimentally and many of the existing trapped ion research groups turned towards quantum computing, at least in some capacity. Today, we are inching closer to the dream this paper sparked in 1995, that of a universal, trapped-ion quantum computer.

3 thoughts on “QCJC: Cirac & Zoller 1994”

1. Very cool demonstration of the CNOT gate in trapped ions! Reminds me a lot of Rydberg atoms, in the way of using lasers to selectively tune individual ions…

I didn’t quite follow the reasoning in the intermediate step, where you mention that there needs to be a “polarization that couples the |g> state to |e_1> instead of |e_0>”. I don’t understand why you need to care about the 2nd state, as it appears that the nth qubit of the 2nd state is in the excited state, and it seems like this pulse only acts on the ground state? I’m not sure if there is a typo, or if it’s my own lack of understanding!

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2. Jameson O'Reilly says:

The pulse doesn’t ~just~ act on the ground state. It has an effect on the ground state because it drives the transition between |g>|1> and |e_q>|0>, meaning that for every pi pulse |g>|1> will go to |e_q>|0> but also every |e_q>|0> will go to |g>|1>. It’s a Rabi flop, if you leave the laser on the ion will go back and forth between the states. So, we don’t want to apply light that couples to |e_0>|0> since this would take the state to |g>|1> and back, applying an unwanted phase along the way.

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