Two-ion logic gates open the way to further advances in quantum computing

Quantum computing is a distant, albeit intensely pursued, goal. To understand its allure, consider how much more information one can process by replacing the classical bit, representing either a 1 or a 0, with a two-state quantum system, such as an ion having two energy levels. This system, called a qubit, might be in either of its two states, representing a 1 or a 0, but it can also be in a superposition of its two states.

Performing an operation between two such qubits can entangle the systems, so that two qubits can be in any superposition of their four states. The information stored thus rises exponentially with the number of qubits. Entanglement also gives the quantum computer unique capabilities. Quantum computers should be much faster than their classical counterparts for certain computational problems, such as factoring large numbers, performing unstructured searches, or simulating quantum behaviors.

One doesn’t need a lot of different basic operations to construct a sophisticated algorithm on a quantum computer: Two operations can suffice, as was pointed out in 1995. ¹ These might be a simple rotation of a qubit (such as flipping the state it’s in) and a two-qubit gate with conditional logic. One possible logic gate is the controlled-NOT (CNOT) gate, which flips the state of one qubit depending on the state of a second qubit. The plot of final state probabilities versus initial states shown in figure 1 approximates a CNOT truth table.

Research groups at the University of Innsbruck, Austria, and at NIST in Boulder, Colorado, recently used different approaches to demonstrate the operation of conditional gates on systems of two trapped ions. ^2,3 The gate built by the Innsbruck team, led by Rainer Blatt, was a CNOT gate. The NIST Boulder group, headed by David Wineland, created a conditional phase gate, which imparts an extra phase factor to certain final states. Either of these two-ion gates could serve as one of the two building blocks for a complicated unitary transformation, and hence they are known as universal gates.

Both groups reported good fidelity for the entangled states that their gates created, with the NIST team attaining a value of 97%. (The fidelity gives the probability that the experimentally realized final state is the intended one.) The basic idea for such experiments was proposed in 1995 by J. Ignacio Cirac (now at the Max Planck Institute for Quantum Optics in Garching, Germany) and Peter Zoller at Innsbruck. ⁴ The eight-year span between the proposal and its implementation reflects the magnitude of the experimental challenge.

The demonstration of a logic gate in a two-ion system gives a boost to those interested in using trapped ions as the basis for a quantum computer. A quantum gate had previously been demonstrated by researchers in the NIST ion group, ⁵ but that gate operated on a single ion: The ion’s two hyperfine states represented one qubit and a pair of vibrational states acted as the other (see Physics Today, March 1996, page 21 ). The NIST team also showed that one can entangle up to four ions ⁶ using an operation that had been proposed by others as a two-qubit universal gate. ⁷

The new work by both the NIST and Innsbruck groups goes beyond the earlier work, notes Emanuel Knill of Los Alamos National Laboratory, because it combines three elements needed to scale up the capabilities of ion traps: the use of multiple ions, the ability to implement any desired algorithm, and the achievement of high accuracy.

Another type of qubit is based on the nuclear spins within single molecules in a liquid sample; the spin states are measured by nuclear magnetic resonance techniques. (What’s actually measured is an ensemble of the qubits because the sample contains many molecules.) High-quality gates have been implemented in such NMR systems, but it’s not clear whether one can scale up the NMR approach to larger numbers of qubits, comments David DiVincenzo of IBM’s T. J. Watson Research Center in Yorktown Heights, New York. By contrast, trapped-ion systems are largely viewed as scalable.

Although NMR and ion-trap qubit systems are the farthest along in development, researchers are also trying to gain precise quantum control over other types of qubits. These include potentially scalable solid-state systems such as charge or flux states in Josephson junctions and double quantum dots (see Physics Today, June 2002, page 14 ). Researchers are also exploring quantum gate implementation in cavity quantum electrodynamics.

A CNOT gate

The CNOT gate demonstrated by the Innsbruck group largely followed the prescription of Cirac and Zoller. In the Innsbruck setup, two calcium-40 ions (⁴⁰Ca⁺) oscillate in a harmonic trap, with two modes of oscillation: the center-of-mass mode, in which the two ions oscillate in phase, and the stretch mode, in which they move in opposition. The basis of the qubit is the internal state of each ion, that is, whether it is in the S_1/2 ground state |S〉 or a metastable D_5/2 state |D〉. The ions can be individually addressed by an optical laser.

The coupling of the ions’ internal states takes place through the external oscillations, which convey information from one ion to another much as a data bus does. The ions add phonons (vibrational quanta) to the bus, or take them away, as the ions interact with laser pulses. The laser interaction stimulates an ion to make a transition between its two internal states, causing the ion to emit or absorb a photon. The photon then transfers momentum to a phonon in an external (vibrational) mode.

To implement the CNOT logic gate, the Innsbruck researchers applied a series of laser pulses. As Blatt explains, the first pulse maps the internal state of the first (control) ion onto an external state by way of the motion in the trap: The initial state of the control ion determines whether or not the bus acquires a phonon. As a sequence of laser pulses then operates on the second ion, that second ion undergoes a transition only if there’s a phonon in the bus. A final laser pulse applied to the control ion returns it to its original state.

Figure 1 shows the probability of each of four possible final, or output, states of the two ions, as measured in the Innsbruck experiment, plotted against the initial, input states. Note that the final state of the second ion has a high probability of being flipped only when the control ion is in state |D〉. If the operation were an ideal CNOT gate, the probabilities seen in the figure would be either 0% or 100%, and the plot would be equivalent to a logical truth table. Experimentally, however, the probabilities fall short: Input/output pairs such as (|SS〉 → |SS〉 and |DS〉 → |DD〉 are found to have probabilities of 70–80% rather than the expected 100%. Blatt and company hope to raise this value to more than 90% by reducing laser noise and by improving control of the laser beams addressing each ion.

The Innsbruck experimenters made one modification to the original Cirac-Zoller proposal. Rather than using one pulse to affect the conditional transition of the second ion, they borrowed from an NMR technique and applied a sequence of four pulses. Although using additional pulses takes a bit longer it reduces losses caused by populations left in unwanted intermediate states. ⁸

A conditional phase gate

At NIST, Dietrich Leibfried and his colleagues operated what they call a conditional phase gate. Specifically, they introduced a geometrical phase by taking the quantum system around a loop in phase space: Any quantum system taken around a closed path will acquire a phase that is determined by the area enclosed by the path. The NIST researchers arranged their experiment so that their system acquired such a geometric phase only when the initial spins of the two ions were antiparallel.

In the NIST experiment, researchers used the two hyperfine states of a beryllium-9 ion (⁹Be⁺) as the two states of a qubit. The two ⁹Be⁺ ions are sufficiently close to one another that two laser beams can each address them both ions simultaneously. The experimenters aim a pair of laser beams at the two ions; the combined effect is to exert on each ion an electric dipole force whose strength depends on the internal state of the ion.

Suppose that the two ions are driven by a dipole force nearly resonant with the stretch mode frequency. If both ions are in the same hyperfine state, they will feel the same force, and the stretch mode will not be excited. To excite the stretch mode, the force must push the ions together or pull them apart. That will happen only if the ions are in different internal states and hence experience different dipole forces. Because the driving force is slightly asynchronous with the stretch mode, the two-ion state is displaced along a circular path in phase space and returns to the starting position once the driving force and stretch mode resynchronize. That resynchronization occurs after a time, which depends on the frequency difference between the dipole driving force and the stretch mode.

Because of this phase-space excursion, a phase of π/2 will be picked up by the wavefunction describing the state in which the ions have opposite spins. This result is logically equivalent to the operation of a CNOT gate. The phase gate can be converted to a CNOT gate by single-bit rotations on the second ion before and after the gate pulse: The first rotation puts the second ion in a superposition of its two states and the final rotation recombines them in different ways depending on the phase that has been acquired.

The conditional phase gate of the NIST Boulder experiment cannot be measured directly, as one can do with the CNOT gate. That’s because the net effect is to alter phases, not populations of ions in the individual states. To assess the result of the gate operation, the experimenters looked for the interference effects caused by the accumulated phases. The technique they adopted is to enclose the gate operation in a π/2-π-π/2 spin-echo sequence familiar from NMR work. This pulse sequence essentially converts the phases into differences in population of the various possible states.

To determine the state populations, Liebfried and company measured the total fluorescence of the two ions (ions fluoresce only if they are in the spin-down state |↓〉. They then compared the total fluorescence to what was expected from a model of their gate.

One such comparison of the measured fluorescence is shown in figure 2. The experimenters prepared both ions with spins down (|↓↓〉). The first laser pulse shown in the inset rotates the initial |↓↓〉 state by π/2, putting it into an equal superposition of the four possible combined-spin states. The displacement pulse then gives those states with oppositely directed spins a phase that depends on the pulse duration t. Sequential pulses of π and π/2 then further rotate the displaced states.

The way that the superposition states recombine during the final rotations is affected by the phases that the states have acquired during the displacement. Varying the displacement time t is then like sweeping the phase and looking at how the interference between the superposition states affects the outcome. In figure 2, for example, we see that at t = 39 µs (one period of the motion), the ions are in the maximally entangled state ( $\left(|\downarrow \downarrow 〉-i|\uparrow \uparrow 〉\right)/\sqrt{2}.$ Because $|\downarrow \downarrow 〉$ and $|\uparrow \uparrow 〉$ are present with equal amplitudes, the fluorescence is cut in half. After another period (at t = 78 µs), the model predicts that only the nonfluorescing $|\uparrow \uparrow 〉$ state will be present, and indeed, virtually no fluorescence is seen. From other measurements of the fluorescence, Leibfried and colleagues determined that the entangled state created after 39 µs was produced with 97% fidelity.

Assembling the building blocks

Zoller said he finds it both personally and scientifically fulfilling to see the physical implementation of the gate he and Cirac proposed. Now, he asserts, experimenters have demonstrated independently all the building blocks for constructing a scalable quantum computer with trapped ions. He acknowledges that success will demand technological improvements that are enormously difficult, but he does not see any fundamental obstacle in the way.

Part of the challenge will be, of course, to scale up to larger numbers of trapped ions. One such proposal is to link many small ion-trap quantum systems, with ions carrying information between traps. ⁹ A team from NIST Boulder has demonstrated the transport and separation of qubits in an array of three traps. ¹⁰ Although ions pick up a motional phase as they travel between traps, these phases factor out of the quantum computation.

For some time, members of Wineland’s group at NIST were virtually the only kids on the block doing studies of entangled trapped ions. As Andrew Steane of the University of Oxford notes, ¹¹ it’s now nice to see successful experiments being done in more than one laboratory. Blatt comments that he and Wineland have a friendly competition: They arranged to have their work appear together in the same issue of Nature.