July 2018 – Fig. Two

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

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

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

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.

Source: Cirac, J. I. and Zoller, P. Quantum Computations with Cold Trapped Ions Phys. Rev. Lett. 74 (20) 4091-4094 (1995)

Within the world of superconducting circuits as quantum computers, there are many different avenues of research. Some choose to focus on noise reduction, some on readout, and others on gate implementation. This paper, from Jerry Chow’s IBM group in 2016, is of the last kind. It’s an exploration of an implementation of the controlled phase gate, a more general implementation of the controlled Z gate, and some of the unique advantages that using a resonator-induced phase (RIP) can have. In full disclosure, I didn’t stumble upon this paper by myself – instead, I was fortunate enough to sit in on a journal club presentation by Jack Qiu at the Oliver Lab at MIT, and he presented on this paper! It sounded interesting enough that I decided to sit down and digest through it more carefully on my own. I do have to warn you – the ending of this paper was rather disappointing, and *spoiler alert*, I don’t think we will be seeing any RIP gates anytime soon.

In general, most superconducting gates are limited in distance. That is, most gates are only “nearest neighbor” gates, meaning that two qubits only interact if they are right next to each other. This seems to make sense – the interaction between two qubits is often mediated by some kind of resonator, or a BUS perhaps, and it would be difficult to construct a large bus to link qubits that are physically far away from each other. This is problematic, as many quantum algorithms require interactions between qubits that may be physically far away from each other. This doesn’t mean that quantum computing is impossible, as you can just put lots of nearest neighbor gates together, acting like one big long distance gate.

However, there is an immediate drawback of this method. Each gate takes some amount of time to execute, and the longer you have to wait, the higher probability that some random error occurred. That is why the lifetime of the gate is slightly less important when compared to the ratio between qubit lifetime and gate implementation. Even if you had a qubit that could live for 5 seconds, an eternity compared to many of the qubits we are using today, if a single gate takes 30 seconds to implement, that qubit would be almost useless!

So, that is where the RIP gate really shines. The RIP gate is able to execute on every single qubit that is coupled to a cavity bus, through a strong coupling in cavity quantum electrodynamics (cQED). By applying a pulsed microwave drive to that shared bus, all of the qubits that are coupled to the bus will begin picking up a phase. This phase that the qubits pick up is only dependent on a few parameters that are related to the microwave drive, chiefly mong them the detuning of the microwave to the resonant frequency of the qubit, and the amplitude of the drive. My mental image of this is that there is an area in parameter space that you can draw a loop around, and the area of that loop determines how much phase you pick up. I believe Shruti Puri wrote an excellent theory paper explaining this concept, although I think I’ll pass on some of the finer details…

So, the bulk of the paper is presenting experimental results. The IBM group fabricated two systems that are very similar to each other, where each has 4 qubits that are coupled to the bus that performs the RIP gate. They then test the fidelity and time it takes to implement each of the gates.

One caveat here – the greatest benefit of the RIP gate is the ability to couple all of the quits to each other. However, that is also a drawback if you /only/ want two qubits to be interacting with each other. Therefore, they need to perform additional operations, a la Hahn echo, to make sure that the other qubits do not interact, or at least refocus by the end.

In fig. 2, they show quite an interesting image in terms of seeing the Ramsey fringes in the excited state population. Over time, the driving decoherence will cause the qubits to oscillate from the ground state up to the excited state, at a rate that is proportional to the frequency of the detuing. Therefore, a shorter detuning frequency will cause more rapid oscillation, while a longer detuning frequency will cause slower “ripples”. I thought this was nicely demonstrated in figure 2e, where they showed the correspondence between experiment and theory, as well as in figure 2a, where they presented experimental data on how these ripple fringes spread out farther and farther at higher detuning frequencies. There is also a presentation of the gate sequence they apply in order to make the RIP gate work for just two or just three qubit interactions.

The coherence times that the RIP gate shows are quite reasonable, ranging between 97% and 98%. That’s not too bad, but it is below the lower coherence limit, so there is still a bit more improvement needed before it can be used for fault tolerant computing. However, the big killer here is the amount of time that it takes for the RIP gate to operate. From Table 2, they show that the gate time is on the order of half a nanosecond, while their qubit lifetimes are on the order of tens of microseconds. Half a microsecond is a really long time, especially when you need to operate at least 8 RIP gates to do a single CZ operation!

This is especially illuminating in light of the final results. The last experiment that the IBM group performs is to create a maximally entangled GHZ state between 4 qubits, or $|0000 \rangle + i |1111 \rangle$ . Interestingly, in the supplemental material, they show two different ways of executing this entanglement – one by using three CZ gates, and one by using a straight sequence of RIP gates. You would expect that they would be using the 4-qubit gate entanglement to create this entaglement sequence, right? But as it turns out, that 4-qubit gate entangement is soooo costly, that it is actually “cheaper” (with regards to time) to just use the three 2-qubit CZ gates!

This was incredibly disappointing to me – as the entire purpose of the RIP gate was to make it more convenient to construct these multi-qubit entangling gates. Even the title hints at that – “demonstration of a resonator-induced phase gate in a multi-qubit circuit QED “. The fact that it is actually really costly to do these multi-qubit gates may be one of the reasons why we haven’t seen much advancement in this direction since this paper. Ah well. Rest in Peace, RIP gate.

Source: Paik, H. et al. Experimental Demonstration of a Resonator-Induced Phase Gate in a Multi-Qubit Circuit QED System Phys. Rev. Lett. 117, 250502 (2016)

Fig. Two

QCJC: Cirac & Zoller 1994

QCJC: Paik 2016