Editor‘s note: This post was prepared jointly by Ryan Mishmash and Jason Alicea.
Physicists appear to be on the verge of demonstrating proof-of-principle “usefulness” of small quantum computers. Preskill’s notion of quantum supremacy spotlights a particularly enticing goal: use a quantum device to perform some computation—any computation in fact—that falls beyond the reach of the world’s best classical computers. Efforts along these lines are being vigorously pursued along many fronts, from academia to large corporations to startups. IBM’s publicly accessible 16-qubit superconducting device, Google’s pursuit of a 7×7 superconducting qubit array, and the recent synthesis of a 51-qubit quantum simulator using rubidium atoms are a few of many notable highlights. While the number of qubits obtainable within such “conventional” approaches has steadily risen, synthesizing the first “topological qubit” remains an outstanding goal. That ceiling may soon crumble however—vaulting topological qubits into a fascinating new chapter in the quest for scalable quantum hardware.
Why topological quantum computing?
As quantum computing progresses from minimalist quantum supremacy demonstrations to attacking real-world problems, hardware demands will naturally steepen. In, say, a superconducting-qubit architecture, a major source of overhead arises from quantum error correction needed to combat decoherence. Quantum-error-correction schemes such as the popular surface-code approach encode a single fault-tolerant logical qubit in many physical qubits, perhaps thousands. The number of physical qubits required for practical applications can thus rapidly balloon.
The dream of topological quantum computing (introduced by Kitaev) is to construct hardware inherently immune to decoherence, thereby mitigating the need for active error correction. In essence, one seeks physical qubits that by themselves function as good logical qubits. This lofty objective requires stabilizing exotic phases of matter that harbor emergent particles known as “non-Abelian anyons”. Crucially, nucleating non-Abelian anyons generates an exponentially large set of ground states that cannot be distinguished from each other by any local measurement. Topological qubits encode information in those ground states, yielding two key virtues:
(1) Insensitivity to local noise. For reference, consider a conventional qubit encoded in some two-level system, with the 0 and 1 states split by an energy . Local noise sources—e.g., random electric and magnetic fields—cause that splitting to fluctuate stochastically in time, dephasing the qubit. In practice one can engender immunity against certain environmental perturbations. One famous example is the transmon qubit (see “Charge-insensitive qubit design derived from the Cooper pair box” by Koch et al.) used extensively at IBM, Google, and elsewhere. The transmon is a superconducting qubit that cleverly suppresses the effects of charge noise by operating in a regime where Josephson couplings are sizable compared to charging energies. Transmons remain susceptible, however, to other sources of randomness such as flux noise and critical-current noise. By contrast, topological qubits embed quantum information in global properties of the system, building in immunity against all local noise sources. Topological qubits thus realize “perfect” quantum memory.
(2) Perfect gates via braiding. By exploiting the remarkable phenomenon of non-Abelian statistics, topological qubits further enjoy “perfect” quantum gates: Moving non-Abelian anyons around one another reshuffles the system among the ground states—thereby processing the qubits—in exquisitely precise ways that depend only on coarse properties of the exchange.
Disclaimer: Adjectives like “perfect” should come with the qualifier “up to exponentially small corrections”, a point that we revisit below.
The catch is that systems supporting non-Abelian anyons are not easily found in nature. One promising topological-qubit implementation exploits exotic 1D superconductors whose ends host “Majorana modes”—novel zero-energy degrees of freedom that underlie non-Abelian-anyon physics. In 2010, two groups (Lutchyn et al. and Oreg et al.) proposed a laboratory realization that combines semiconducting nanowires, conventional superconductors, and modest magnetic fields.
Since then, the materials-science progress on nanowire-superconductor hybrids has been remarkable. Researchers can now grow extremely clean, versatile devices featuring various manipulation and readout bells and whistles. These fabrication advances paved the way for experiments that have reported increasingly detailed Majorana characteristics: tunneling signatures including recent reports of long-sought quantized response, evolution of Majorana modes with system size, mapping out of the phase diagram as a function of external parameters, etc. Alternate explanations are still being debated though. Perhaps the most likely culprit are conventional localized fermionic levels (“Andreev bound states”) that can imitate Majorana signatures under certain conditions; see in particular Liu et al. Still, the collective experimental effort on this problem over the last 5+ years has provided mounting evidence for the existence of Majorana modes. Revealing their prized quantum-information properties poses a logical next step.
Validating a topological qubit
Ideally one would like to verify both hallmarks of topological qubits noted above—“perfect” insensitivity to local noise and “perfect” gates via braiding. We will focus on the former property, which can be probed in simpler device architectures. Intuitively, noise insensitivity should imply long qubit coherence times. But how do you pinpoint the topological origin of long coherence times, and in any case what exactly qualifies as “long”?
Here is one way to sharply address these questions (for more details, see our work in Aasen et al.). As alluded to in our disclaimer above, logical 0 and 1 topological-qubit states aren’t exactly degenerate. In nanowire devices they’re split by an energy that is exponentially small in the separation distance between Majorana modes divided by the superconducting coherence length . Correspondingly, the qubit states are not quite locally indistinguishable either, and hence not perfectly immune to local noise. Now imagine pulling apart Majorana modes to go from a relatively poor to a perfect topological qubit. During this process two things transpire in tandem: The topological qubit’s oscillation frequency, , vanishes exponentially while the dephasing time becomes exponentially long. That is,
This scaling relation could in fact be used as a practical definition of a topologically protected quantum memory. Importantly, mimicking this property in any non-topological qubit would require some form of divine intervention. For example, even if one fine-tuned conventional 0 and 1 qubit states (e.g., resulting from the Andreev bound states mentioned above) to be exactly degenerate, local noise could still readily produce dephasing.
As discussed in Aasen et al., this topological-qubit scaling relation can be tested experimentally via Ramsey-like protocols in a setup that might look something like the following:
This device contains two adjacent Majorana wires (orange rectangles) with couplings controlled by local gates (“valves” represented by black switches). Incidentally, the design was inspired by a gate-controlled variation of the transmon pioneered in Larsen et al. and de Lange et al. In fact, if only charge noise was present, we wouldn’t stand to gain much in the way of coherence times: both the transmon and topological qubit would yield exponentially long times. But once again, other noise sources can efficiently dephase the transmon, whereas a topological qubit enjoys exponential protection from all sources of local noise. Mathematically, this distinction occurs because the splitting for transmon qubit states is exponentially flat only with respect to variations in a “gate offset” . For the topological qubit, the splitting is exponentially flat with respect to variations in all external parameters (e.g., magnetic field, chemical potential, etc.), so long as Majorana modes still survive. (By “exponentially flat” we mean constant up to exponentially small deviations.) Plotting the energies of the qubit states in the two respective cases versus external parameters, the situation can be summarized as follows:
Outlook: Toward “topological quantum ascendancy”
These qubit-validation experiments constitute a small stepping stone toward building a universal topological quantum computer. Explicitly demonstrating exponentially protected quantum information as discussed above would, nevertheless, go a long way toward establishing practical utility of Majorana-based topological qubits. One might even view this goal as single-qubit-level “topological quantum ascendancy”. Completion of this milestone would further set the stage for implementing “perfect” quantum gates, which requires similar capabilities albeit in more complex devices. Researchers at Microsoft and elsewhere have their sights set on bringing a prototype topological qubit to life in the very near future. It is not unreasonable to anticipate that 2018 will mark the debut of the topological qubit. We could of course be off target. There is, after all, still plenty of time in 2017 to prove us wrong.
What an Outstanding article.
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Agree, impressive indeed, thank you
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