Release the ions!

Jonas Araujo

January 18, 2023

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8 minutes

In my previous article, I wrote that two-level systems are the main resource for quantum computing. They provide quantum bits, or qubits, whose states can be superposed and entangled in quantum algorithms, potentially accelerating many computational tasks. Ion traps use one of the best candidates for qubits: nuclear spins (in fact, a combination of electronic and nuclear spins, also called hyperfine structure).

An ion trap according to stablediffusionweb.com.

Spins arise from intrinsic magnetic dipole moments of subatomic particles — think of a subatomic compass needle. They are vectors purely quantum mechanical in nature, by that I mean they do not behave according to our intuition, which is based on classical physics. For example, measuring the spin component of an electron in any direction can only give you two values that we label 0 and 1, while a classical vector would display a continuum between those values. Similar to electrons, the atomic nucleus can also have an overall spin.

Oversimplified depiction of electron and nuclear spins. The atom’s spin as a whole is a sum of its components’. This sum is done in the framework of the quantum theory of angular momenta. The complete electronic shells are omitted.

The two spin states, also labeled as “up” and “down”, have different energies. You can flip 0 into 1 or vice-versa if the exact energy difference is given or emitted by means of a photon, a particle of light. Controlling the spins means being able to detect their states and manipulate them, but a difficulty arises here. That energy difference is typically 100 000 times smaller than the thermal energy scale at room temperature [1]. It is like whispering to a friend in a nightclub.

Typical values of thermal and spin gap energies. Observe how the thermal energy at room temperature (300 K) is 100 000 times larger than the energy gap between spin states. Observation of quantum effects requires atoms are cooled to thousandths of a K. The constants kB and µN are the Boltzmann constant and the proton’s magnetic moment.

The solution is lowering the thermal energy of the ions, i.e., cooling them down so you can trap them! But one does not simply trap an ion; it takes state-of-the-art engineering. First of all, ions have a net electric charge, right? The trick is using electromagnetic fields to trap them. You resort to a combination of static and oscillating radiofrequency potentials, avoiding Earnshaw’s theorem (which forbids charge confinement by static fields). The spacing between the ions, once they are trapped, is an important quantity. Ideally it should be at least smaller than the wavelength of incident light. This inter-ion spacing is given in terms of the Lamb-Dicke parameter.

Trapped mercury ions. They are illuminated with a probe laser, to which not all mercury isotopes respond. Figure based on the experiments in Ref. [1].

To be fair, fluctuations in the electromagnetic fields that trap the ions can lead to flips of spin states. Nevertheless, these sources of noise can be controlled to the point of not interfering with the spin states in the timescales relevant for calculations or experiments. It also helps that spins are very picky: they care little to frequencies other than that of the transition energy. Sometimes the terms energy and frequency are used interchangeably, that is because the energy of a photon, the carrier of electromagnetic fields, respects energy = h•frequency, where h is the Planck’s constant. Nevertheless, we are missing an invisible but fundamental element of ion traps, the phonons.

The next phonon in the energy scale. A transition between these modes is possible via the emission or absorption of a photon whose energy match the energy gap separating the modes.

As you cool the ions down to make them sit nicely in line, energies are so low that the vibrational modes of the ions as a whole are quantized, or discretized, like the spin. They represent different energy levels of the ions as they vibrate around their equilibrium positions. As long as they are cool enough, the ions do not stray far from the potential well’s center and behave like a harmonic oscillator (much like a series of masses attached by strings). The two lowest-energy vibrational modes are depicted above, just like in a system of coupled springs.

One method used to slow the ions down is called Doppler cooling, in which photons are used to selectively brake the ions. This method works for cooling up to hundreds of micro-Kelvin, beyond that point, interactions between the ions and the phonons are used in what is called sideband cooling. This is in fact how the initial states |00…00> are generated. It is worth noting that, unlike the electronic spin, phonon modes may assume more than 2 levels. Do not worry though! Ion traps are designed so that phonons can occupy only two levels.

Preparing the initial |0…0> state requires sideband cooling. A higher phonon state is excited and undergoes spontaneous emission, releasing photons. As most of the ions prefer lower-energy states, if this process is repeated many times, most of the ions end up in their fundamental state.

The phonon, at first glance, stands at the same level as the ion’s nuclear spins. But do not be fooled: it is far more than that. Because all ions share the same phonon, it serves as a bridge between ions, being therefore the most important qubit in a quantum computer based on trapped ions. This will be clear when we describe the basic quantum operations in an ion trap. But first let us talk about the atomic structure of a typical ion used in experiments.

An ionized mercury atom. In its neutral state, the electronic shells have 2, 8, 18, 32, 18, and 2 electrons. Ionizing the last shell leaves an unpaired electron, whose spin is combined with the nucleus’. The interaction between these spins gives rise to the hyperfine structure.

Take the 199-Hg+ isotope of mercury. The electronic shells of a neutral atom have an even number of electrons whose spins are aligned opposite to one another’s. If the atom is ionized (199-Hg+), there will be a single unpaired electron, leaving the electronic cloud with a net spin. Moreover, the atomic nucleus also has a spin, that should be summed to the unpaired electron’s, forming what is called the hyperfine state or structure. When combined, their quantum state can be used as qubits with remarkable coherence times, that is, how long the qubits can stay in a superposition of 1 and 0. This is important because quantum algorithms resort to superposition to implement speedups. The latest result I found claims coherence times longer than a whole hour for ytterbium ions [2]. Compare it with the millisecond-long coherence times of superconducting qubits recently claimed in [3].

As good and stable as trapped ions are, they cannot communicate without the phonon they all share. For that reason, the phonon’s lowest modes are called the “bus” qubit. For example, you can flip the states of ions, but if an operation on ion A is to be done depending on ion B’s state, you will need the phonon qubit to implement it. Ultimately, the phonon qubit enables quantum computation by allowing controlled operations, that is, by providing the necessary entanglement. Now let us illustrate basic quantum operations using trapped ions. A very friendly overview is given in Ref. [4], for those interested in more details.

All spins are connected to the phonon. For this reason, the phonon modes are called the “bus” qubit, since it intermediates the operations between spins.

The first set of operations are rotations. Remember we said above the ions are very picky about the frequencies, they hear? The idea is applying an electromagnetic with that same frequency for a precise amount of time. The simplest, yet very effective, approximation for this process is the Jaynes-Cummings model [5]. The effect of it is rotating the qubit’s state, enabling us to build individual rotations and Hadamard (H) gates. In the figure below, we outline how these gates work in terms of evolution operators.

Individual rotations are obtained by tuning an evolution operator so as to create a rotation. The h with a bar is just the Planck’s constant divided by 2π, while the Ω depends on the electromagnetic field strengths.

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Let us now explore operations that require two qubits. Consider the controlled phase-flip gate — it will be clear soon why we chose to start by it. Here is what it does to the input states:

Effect of the controlled phase-flip gate. It only acts when the first qubit is in state 1, flipping the sign of the second qubit, if the latter is in state 1.

The easiest way of doing it is using an extra level in the ion’s internal quantum state and rotations in another subspace. Before, only 0 and 1 were accessed, so there was a single rotation in this “2-dimensional” space. For the controlled phase-flip, we need a 2π rotation in another subspace, the one spanned by |11> and |20>, as sketched below. In the end, only the state 11 gets an overall (-) sign, the other states remain unchanged, assuming higher energy levels of the phonon are inaccessible.

Operations needed to implement the controlled phase-flip gate. Only the |11> state is affected, gaining an overall minus sign.

Once we have single qubit rotations and controlled phase-flip gates, we can use them to build the ubiquitous controlled-not gates like this:

How to build a controlled-not (C-not) operator with the ones we have — up to a global phase, at least.

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With controlled-not gates and rotations, one can build controlled-phase gates [6], and then assemble circuits for many famous quantum algorithms, such as quantum Fourier transforms. The readout of such algorithms is made by measuring the populations of the hyperfine states. Mismatches between the readout probabilities and expected values can be due to noise introduced or increased in a process called transpilation [7].

An example of transpilation. Limited qubit connectivity requires rewriting the circuit in terms of the available gates. In the case above, the controlled-not has to be replaced by nearest-neighbor gates. This comes at a cost: increasing the circuit depth, which inevitably leads to more errors.

Quantum algorithms often require connectivity between physically distant qubits, like connecting qubit A to the qubit Z. In an ion trap, an operation between two ions should be converted into more basic operations, such as between the first ion and the underlying phonon, and between the phonon and the second ion. This conversion, or transpilation, is the act of translating the required gates into more basic ones that can be implemented in the hardware (see figure above). This almost always leads to deeper circuits, that are more prone to noise and errors, keeping these computers from working as they should, at least until now.

The hope is that technological advances lead to ever-lower error rates so these machines can be used in real-life applications. The latest contender I found is a German startup called eleQtron; their unique MAGIC (Magnetic Gradient Induced Coupling) technology enables, for example, controlled-not operations with 98% fidelity.

At ColibrITD we choose the best hardware available for our customers’ use case. That requires familiarity with the underlying technology and peculiarities of each machine.

References

[1] D. Wineland et. al.. Quantum state manipulation of trapped atomic ions. Proceedings of the Royal Society Lond. A (1998) 454, 411–429.

[2] P. Wang et al.. Single ion qubit with estimated coherence time exceeding one hour. Nature Communications volume 12, Article number: 233 (2021).

[3] Aaron Somoroff et al.. Millisecond coherence in a superconducting qubit. https://arxiv.org/abs/2103.08578

[4] Michael A. Nielsen, and Isaac L. Chuang. Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge University Press. Cambridge, 2010.

[5] E. T. Jaynes and F. W. Cummings. Comparison of quantum and semiclassical radiation theories with application to the beam maser. Proc. IEEE. 51 (1): 89–109 (1963).

[6] Taewan Kim, and Byung-Soo Choi. Efficient decomposition methods for controlled-Rn using a single ancillary qubit. Scientific Reports volume 8, Article number: 5445 (2018).

[7] Scott Johnstun, and Jean-François Van Huele. Understanding and compensating for noise on IBM quantum computers. American Journal of Physics 89, 935 (2021).