What does it mean to create a topological qubit?

I’ve worked on topological quantum computation, one of Alexei Kitaev’s brilliant innovations, for around 15 years now.& It’s hard to find a more beautiful physics problem, combining spectacular quantum phenomena (non-Abelian anyons) with the promise of transformative technological advances (inherently fault-tolerant quantum computing hardware).& Problems offering that sort of combination originally inspired me to explore quantum matter as a graduate student.&

Non-Abelian anyons are emergent particles born within certain exotic phases of matter.& Their utility for quantum information descends from three deeply related defining features:

• Nucleating a collection of well-separated non-Abelian anyons within a host platform generates a set of quantum states with the same energy (at least to an excellent approximation).& Local measurements give one essentially no information about which of those quantum states the system populates—i.e., any evidence of what the system is doing is hidden from the observer and, crucially, the environment.& In turn, qubits encoded in that space enjoy intrinsic resilience against local environmental perturbations.&
• Swapping the positions of non-Abelian anyons manipulates the state of the qubits.& Swaps can be enacted either by moving anyons around each other as in a shell game, or by performing a sequence of measurements that yields the same effect.& Exquisitely precise qubit operations follow depending only on which pairs the user swaps and in what order.& Properties (1) and (2) together imply that non-Abelian anyons offer a pathway both to fault-tolerant storage and manipulation of quantum information.&
• A pair of non-Abelian anyons brought together can “fuse” into multiple different kinds of particles, for instance a boson or a fermion.& Detecting the outcome of such a fusion process provides a method for reading out the qubit states that are otherwise hidden when all the anyons are mutually well-separated.& Alternatively, non-local measurements (e.g., interferometry) can effectively fuse even well-separated anyons, thus also enabling qubit readout.&

I entered the field back in 2009 during the last year of my postdoc.& Topological quantum computing—once confined largely to the quantum Hall realm—was then in the early stages of a renaissance driven by an explosion of new candidate platforms as well as measurement and manipulation schemes that promised to deliver long-sought control over non-Abelian anyons.& The years that followed were phenomenally exciting, with broadly held palpable enthusiasm for near-term prospects not yet tempered by the practical challenges that would eventually rear their head.&

In 2018, near the height of my optimism, I gave an informal blackboard talk in which I speculated on a new kind of forthcoming NISQ era defined by the birth of a Noisy Individual Semi-topological Qubit.& To less blatantly rip off John Preskill’s famous acronym, I also—jokingly of course—proposed the alternative nomenclature POST-Q (Piece Of S*** Topological Qubit) era to describe the advent of such a device.& The rationale behind those playfully sardonic labels is that the inaugural topological qubit would almost certainly be far from ideal, just as the original transistor appears shockingly crude when compared to modern electronics.& You always have to start somewhere.& But what does it mean to actually create a topological qubit, and how do you tell that you’ve succeeded—especially given likely POST-Q-era performance?

To my knowledge those questions admit no widely accepted answers, despite implications for both quantum science and society.& I would like to propose defining an elementary topological qubit as follows:

A device that leverages non-Abelian anyons to demonstrably encode and manipulate a single qubit in a topologically protected fashion.&

Some of the above words warrant elaboration.& As alluded to above, non-Abelian anyons can passively encode quantum information—a capability that by itself furnishes a quantum memory.& That’s the “encode” part.& The “manipulate” criterion additionally entails exploiting another aspect of what makes non-Abelian anyons special—their behavior under swaps—to enact gate operations.& Both the encoding and manipulation should benefit from intrinsic fault-tolerance, hence the “topologically protected fashion” qualifier.& And very importantly, these features should be “demonstrably” verified.& For instance, creating a device hosting the requisite number of anyons needed to define a qubit does not guarantee the all-important property of topological protection.& Hurdles can still arise, among them: if the anyons are not sufficiently well-separated, then the qubit states will lack the coveted immunity from environmental perturbations; thermal and/or non-equilibrium effects might still induce significant errors (e.g., by exciting the system into other unwanted states); and measurements—for readout and possibly also manipulation—may lack the fidelity required to fruitfully exploit topological protection even if present in the qubit states themselves.&

The preceding discussion raises a natural follow-up question: How do you verify topological protection in practice?  One way forward involves probing qubit lifetimes, and fidelities of gates resulting from anyon swaps, upon varying some global control knob like magnetic field or gate voltage.  As the system moves deeper into the phase of matter hosting non-Abelian anyons, both the lifetime and gate fidelities ought to improve dramatically—reflecting the onset of bona fide topological protection.  First-generation “semi-topological” devices will probably fare modestly at best, though one can at least hope to recover general trends in line with this expectation. 

By the above proposed definition, which I contend is stringent yet reasonable, realization of a topological qubit remains an ongoing effort.& Fortunately the journey to that end offers many significant science and engineering milestones worth celebrating in their own right.& Examples include:

Platform verification.& This most indirect milestone evidences the formation of a non-Abelian phase of matter through (thermal or charge) Hall conductance measurements, detection of some anticipated quantum phase transition, etc.&

Detection of non-Abelian anyons. This step could involve conductance, heat capacity, magnetization, or other types of measurements designed to support the emergence of either individual anyons or a collection of anyons.& Notably, such techniques need not reveal the precise quantum state encoded by the anyons—which presents a subtler challenge.&

Establishing readout capabilities. Here one would demonstrate experimental techniques, interferometry for example, that in principle can address that key challenge of quantum state readout, even if not directly applied yet to a system hosting non-Abelian anyons.&

Fusion protocols.& Readout capabilities open the door to more direct tests of the hallmark behavior predicted for a putative topological qubit.& One fascinating experiment involves protocols that directly test non-Abelian anyon fusion properties.& Successful implementation would solidify readout capabilities applied to an actual candidate topological qubit device.&

Probing qubit lifetimes.& Fusion protocols further pave the way to measuring the qubit coherence times, e.g., and —addressing directly the extent of topological protection of the states generated by non-Abelian anyons.& Behavior clearly conforming to the trends highlighted above could certify the device as a topological quantum memory.& (Personally, I most anxiously await this milestone.)

Fault-tolerant gates from anyon swaps.& Likely the most advanced milestone, successfully implementing anyon swaps, again with appropriate trends in gate fidelity, would establish the final component of an elementary topological qubit.&

Most experiments to date focus on the first two items above, platform verification and anyon detection.& Microsoft’s recent Nature paper, together with the simultaneous announcement of supplementary new results, combine efforts in those areas with experiments aiming to establish interferometric readout capabilities needed for a topological qubit.& Fusion, (idle) qubit lifetime measurements, and anyon swaps have yet to be demonstrated in any candidate topological quantum computing platform, but at least partially feature in Microsoft’s future roadmap.& It will be fascinating to see how that effort evolves, especially given the aggressive timescales predicted by Microsoft for useful topological quantum hardware.& Public reactions so far range from cautious optimism to ardent skepticism; data will hopefully settle the situation one way or another in the near future.& My own take is that while Microsoft’s progress towards qubit readout is a welcome advance that has value regardless of the nature of the system to which those techniques are currently applied, convincing evidence of topological protection may still be far off.&

In the meantime, I maintain the steadfast conviction that topological qubits are most certainly worth pursuing—in a broad range of platforms.& Non-Abelian quantum Hall states seem resurgent candidates, and should not be discounted.& Moreover, the advent of ultra-pure, highly tunable 2D materials provide new settings in which one can envision engineering non-Abelian anyon devices with complementary advantages (and disadvantages) compared to previously explored settings.& Other less obvious contenders may also rise at some point.& The prospect of discovering new emergent phenomena mitigating the need for quantum error correction warrants continued effort with an open mind.