Universal Topological Quantum Computation Research Articles

3-Fermion Topological Quantum Computation

We present a scheme for universal topological quantum computation based on Clifford-complete braiding and fusion of symmetry defects in the 3-fermion anyon theory, supplemented with magic state injection. We formulate a fault-tolerant measurement-based realization of this computational scheme on the lattice using ground states of the Walker-Wang model for the 3-fermion anyon theory with symmetry defects. The Walker-Wang measurement-based topological quantum computation paradigm that we introduce provides a general construction of computational resource states with thermally stable symmetry-protected topological order. We also demonstrate how symmetry defects of the 3-fermion anyon theory can be realized in a two-dimensional subsystem code due to Bombín—making contact with an alternative implementation of the 3-fermion defect-computation scheme via code deformations. Published by the American Physical Society 2024

Simulating the Hadamard gate in the Fibonacci disk code for universal topological quantum computation

Simulating the Hadamard gate in the Fibonacci disk code for universal topological quantum computation

Edge Z3 parafermions in fermionic lattices

Parafermions modes are non-Abelian anyons which were introduced as $\mathbb{Z}_N$ generalizations of $\mathbb{Z}_2$ Majorana states. In particular, $\mathbb{Z}_3$ parafermions can be used to produce Fibonacci anyons, laying a path towards universal topological quantum computation. Due to their fractional nature, much of theoretical work on $\mathbb{Z}_3$ parafermions has relied on bosonization methods or parafermionic quasi-particles. In this work, we introduce a representation of $\mathbb{Z}_3$ parafermions in terms of purely fermionic models operators in the t-J regime. We establish the equivalency of a family of lattice fermionic models written in the $t-J$ model basis with a Kitaev-like chain supporting free $\mathbb{Z}_3$ parafermonic modes at its ends. By using density matrix renormalization group calculations, we are able to characterize the topological phase transition and study the effect of local operators (doping and magnetic fields) on the spatial localization of the parafermionic modes and their stability. Moreover, we discuss the necessary ingredients towards realizing $\mathbb{Z}_3$ parafermions in strongly interacting electronic systems.

Universal topological quantum computation with strongly correlated Majorana edge modes

Majorana-based quantum gates are not complete for performing universal topological quantum computation while Fibonacci-based gates are difficult to be realized electronically and hardly coincide with the conventional quantum circuit models. In reference Hu and Kane (2018 Phys. Rev. Lett. 120 066801), it has been shown that a strongly correlated Majorana edge mode in a chiral topological superconductor can be decomposed into a Fibonacci anyon τ and a thermal operator anyon ɛ in the tricritical Ising model. The deconfinement of τ and ɛ via the interaction between the fermion modes yields the anyon collisions and gives the braiding of either τ or ɛ. With these braidings, the complete members of a set of universal gates, the Pauli gates, the Hadamard gate and extra phase gates for one-qubit as well as controlled-NOT (CNOT) gate for two-qubits, are topologically assembled. Encoding quantum information and reading out the computation results can be carried out through electric signals. With the sparse-dense mixed encodings, we set up the quantum circuit where the CNOT gate turns out to be a probabilistic gate and design the corresponding devices with thin films of the chiral topological superconductor. As an example of the universal topological quantum computing, we show the application to Shor’s integer factorization algorithm.

Anomalous non-Abelian statistics for non-Hermitian generalization of Majorana zero modes

In condensed matter physics, non-Abelian statistics for Majorana zero modes (or Majorana Fermions) is very important, really exotic, and completely robust. The race for searching Majorana zero modes and verifying the corresponding non-Abelian statistics becomes an important frontier in condensed matter physics. In this paper we generalize the Majorana zero modes to non-Hermitian (NH) topological systems that show universal but quite different properties from their Hermitian counterparts. Based on the NH Majorana zero modes, the orthogonal and nonlocal Majorana qubits are well defined. In particular, due to the particle-hole-symmetry breaking, NH Majorana zero modes have anomalous non-Abelian statistics with continuously tunable braiding Berry phase from $\ensuremath{\pi}/8$ to $3\ensuremath{\pi}/8$. This is quite different from the usual non-Abelian statistics with fixed braiding Berry phase $\ensuremath{\pi}/4$. The one-dimensional NH Kitaev model is taken as an example to numerically verify the anomalous non-Abelian statistics for two NH Majorana zero modes. The numerical results are exactly consistent with the theoretical prediction. With the help of braiding these two zero modes, the $\ensuremath{\pi}/8$ gate can be reached and thus universal topological quantum computation becomes possible.

Symmetry-protected gates of Majorana qubits in a high-$T_c$ higher-order topological superconductor platform

We propose a platform for braiding Majorana non-Abelian anyons based on a heterostructure between a dd-wave high-T_cTc superconductor and a quantum spin-Hall insulator. It has been recently shown that such a setup for a quantum spin-Hall insulator leads to a pair of Majorana zero modes at each corner of the sample, and thus can be regarded as a higher-order topological superconductor. We show that upon applying a Zeeman field in the region, these Majorana modes split in space and can be manipulated for braiding processes by tuning the field and pairing phase. We show that such a setup can achieve full braiding, exchanging, and arbitrary phase gates (including the \pi/8π/8 magic gates) of the Majorana zero modes, all of which are robust and protected by symmetries. As many of the ingredients of our proposed platform have been realized in recent experiments, our results provide a new route toward universal topological quantum computation.

Exact Modular $S$ Matrix for ${\mathbb Z}_{k}$ Parafermion Quantum Hall Islands and Measurement of Non-Abelian Anyons

Using the decomposition of rational conformal filed theory characters for the $\Z_k$ parafermion quantum Hall droplets for general $k=2,3,\dots$, we derive analytically the full modular $S$ matrix for these states, including the $\uu$ parts corresponding to the charged sector of the full conformal field theory and the neutral parafermion contributions corresponding to the diagonal affine coset models. This precise neutral-part parafermion $S$ matrix is derived from the explicit relations between the coset matrix and those for the numerator and denominator of the coset and the latter is expressed in compact form due to the level-rank duality between the affine Lie algebras $\widehat{\frak{su}(k)_2}$ and $\widehat{\frak{su}(2)_k}$. The exact results obtained for the $S$ matrix elements are expected to play an important role for identifying interference patterns of fractional quantum Hall states in Fabry-P\'erot interferometers which can be used to distinguish between Abelian and non-Abelian statistics of quasiparticles localized in the bulk of fractional quantum Hall droplets as well as for nondestructive interference measurement of Fibonacci anyons which can be used for universal topological quantum computation

Chiral topologically ordered insulating phases in arrays of interacting integer quantum Hall islands

We study networks of Coulomb-blockaded integer quantum Hall islands with even fillings $\nu=2k$ ($k$ being an integer), including cases with $2k$ layers each of $\nu=1$ fillings. Allowing only spin-current interactions between the islands (i.e., without any charge transfer), we obtain solvable models leading to a rich set of insulating $SU(2)_k$ topologically ordered phases. The case with $k=1$ is dual to the Kalmeyer-Laughlin phase, $k=2$ to Kitaev's chiral spin liquid and the Moore-Read state, and $k=3$ contains a Fibonacci anyon that may be utilized for universal topological quantum computation. Additionally, we show how the $SU(2)_k$ topological phases may be obtained also in an array of islands with $\nu=2k$ integer quantum Hall states and critical spin chains in a checkerboard pattern. The array and checkerboard constructions gap out the charge mode and additional "flavor" modes by virtue of their geometry. Furthermore, we find that a fine tuning of the system parameter is not needed in the checkerboard configuration and the $\nu=2$ case. We also discuss their bulk excitations, and show that their thermal Hall conductance is universal, reflecting the central charge $c=3k/(k+2)$ of the chiral edge modes.

Parafermions, induced edge states, and domain walls in fractional quantum Hall effect spin transitions

Search for parafermions and Fibonacci anyons, which are excitations obeying non-Abelian statistics, is driven both by the quest for deeper understanding of nature and prospects for universal topological quantum computation. However, physical systems that can host these exotic excitations are rare and hard to realize in experiments. Here we study the domain walls and the edge states formed in spin transitions in the fractional quantum Hall effect. Effective theory approach and exact diagonalization in a disk and torus geometries proves the existence of the counter-propagating edge modes with opposite spin polarizations at the boundary between the two neighboring regions of the two-dimensional electron liquid in spin-polarized and spin-unpolarized phases. By analytical and numerical analysis, we argue that these systems can host parafermions when coupled to an s-wave superconductor and are experimentally feasible. We investigate settings based on $\nu=\frac{2}{3}$, $\nu=\frac{4}{3}$ and $\nu=\frac{5}{3}$ spin transitions and analyze spin-flipping interactions that hybridize counter-propagating modes. Finally, we discuss spin-orbit interactions of composite fermions.

Fractional quantum Hall states with gapped boundaries in an extreme lattice limit

We present a detailed microscopic investigation of fractional quantum Hall states with gapped boundaries in a coupled bilayer lattice model featuring holes whose counterpropagating chiral edge states are hybridized and gapped out. We focus on a lattice limit for cold-atom experiments, in which each hole just consists of a single removed site. Although the holes distort the original band structure and lead to ingap remnants of the continuum edge modes, we find that the lowest nearly flat band representing a higher-genus system may naturally form by controlling the local hopping terms that gap out the boundaries. Remarkably, local interactions in this new flat band lead to various Abelian and non-Abelian fractional quantum Hall states with gapped boundaries residing on emergent higher-genus surfaces, which we identify by extracting the nontrivial topological ground-state degeneracies and the fractional statistics of quasiparticles. These results demonstrate the feasibility of realizing novel fractional quantum Hall states with gapped boundaries even in the extreme lattice limit, thus enabling a possible new route towards universal topological quantum computation.

Modular transformations through sequences of topological charge projections

The ground state subspace of a topological phase of matter forms a representation of the mapping class group of the space on which the state is defined. We show that elements of the mapping class group of a surface of genus $g$ can be obtained through a sequence of topological charge projections along at least three mutually intersecting non-contractible cycles. We demonstrate this both through the algebraic theory of anyons and also through an analysis of the topology of the space-time manifold. We combine this result with two observations: (i) that surfaces of genus $g$ can be effectively simulated in planar geometries by using bilayer, or doubled, versions of the topological phase of interest, and inducing the appropriate types of gapped boundaries; and (ii) that the required topological charge projections can be implemented as adiabatic unitary transformations by locally tuning microscopic parameters of the system, such as the energy gap. These observations suggest a possible path towards effectively implementing modular transformations in physical systems. In particular, they also show how the $\text{Ising} \otimes \overline{\text{Ising}}$ state, in the presence of disconnected gapped boundaries, can support universal topological quantum computation.

Fractionalized flux, Majorana fermions and non-Abelian anyons in topological superfluid on optical lattices

Recently, topological excitations obeying non-Abelian statistics attracted intensive attention because of their exotic statistics and possible applications in topological quantum computation. In this paper, based on a topological s-wave superfluid on a Peierls lattice, we find fractionalized flux and non-Abelian anyons, and then give a realistic proposal to realize the universal topological quantum computation, and especially show how to perform the single-qubit gate–phase-shift gate. Finally, we discuss the experimental realization on optical lattices.

Abelian and non-Abelian states inν=2/3bilayer fractional quantum Hall systems

There are several possible theoretically allowed non-Abelian fractional quantum Hall (FQH) states that could potentially be realized in one- and two- component FQH systems at total filling fraction $\nu = n+ 2/3$, for integer $n$. Some of these states even possess quasiparticles with non-Abelian statistics that are powerful enough for universal topological quantum computation, and are thus of particular interest. Here, we initiate a systematic numerical study, using both exact diagonalization and variational Monte Carlo, to investigate the phase diagram of FQH systems at total filling fraction $\nu = n+2/3$, including in particular the possibility of the non-Abelian $Z_4$ parafermion state. In $\nu = 2/3$ bilayers, we determine the phase diagram as a function of interlayer tunneling and repulsion, finding only three competing Abelian states, without the $Z_4$ state. On the other hand, in single-component systems at $\nu = 8/3$, we find that the $Z_4$ parafermion state has significantly higher overlap with the exact ground state than the Laughlin state, together with a larger gap, suggesting that the experimentally observed $\nu = 8/3$ state may be non-Abelian. Our results from the two complementary numerical techniques agree well with each other qualitatively.

Fermionic non-Abelian fractional Chern insulators from dipolar interactions

We study fermions on a triangular lattice model that exhibits topological flatbands characterized by nonzero Chern numbers. Our scheme stems from the well-known Hofstadter model but the next-nearest-neighbor hopping is introduced, which is crucial for tuning the lowest band to be nearly flat. Differing from previous proposals with the necessity of multiparticle interactions, we consider the more realistic long-range dipolar interaction combined with two-body short-range attractions between fermions. We show the realization of the non-Abelian $\nu=1/2$ Moore-Read fractional Chern insulators, and strong evidence for the existence of the more exotic $\nu=3/5$ Read-Rezayi fractional Chern insulators. Our results provide insights for the experimental realization of these exotic states by realistic two-body interactions and thus facilitates the implementation of the universal topological quantum computation.

Exotic topological types of Majorana zero modes and their universal quantum manipulation

From our general index theorem, which characterizes faithfully the topological intrinsic boundary-bulk correspondence of topological superconductors and insulators, we reveal rigorously that four topologically distinct types of Majorana zero modes can emerge at the ends of superconducting wires of various symmetry classes. More intriguingly, we establish three exotic one-dimensional models that have different types of topological charge of Majorana zero modes and disclose exactly the corresponding topological properties, whose distinct topological essences may be tested experimentally. Moreover, we also address their application in universal quantum manipulation, which is promising for realizing universal topological quantum computation.

Universal Topological Quantum Computation from a Superconductor-Abelian Quantum Hall Heterostructure

Non-Abelian anyons promise to reveal spectacular features of quantum mechanics that could ultimately provide the foundation for a decoherence-free quantum computer. A key breakthrough in the pursuit of these exotic particles originated from Read and Green's observation that the Moore-Read quantum Hall state and a (relatively simple) two-dimensional p+ip superconductor both support so-called Ising non-Abelian anyons. Here we establish a similar correspondence between the Z_3 Read-Rezayi quantum Hall state and a novel two-dimensional superconductor in which charge-2e Cooper pairs are built from fractionalized quasiparticles. In particular, both phases harbor Fibonacci anyons that---unlike Ising anyons---allow for universal topological quantum computation solely through braiding. Using a variant of Teo and Kane's construction of non-Abelian phases from weakly coupled chains, we provide a blueprint for such a superconductor using Abelian quantum Hall states interlaced with an array of superconducting islands. Fibonacci anyons appear as neutral deconfined particles that lead to a two-fold ground-state degeneracy on a torus. In contrast to a p+ip superconductor, vortices do not yield additional particle types yet depending on non-universal energetics can serve as a trap for Fibonacci anyons. These results imply that one can, in principle, combine well-understood and widely available phases of matter to realize non-Abelian anyons with universal braid statistics. Numerous future directions are discussed, including speculations on alternative realizations with fewer experimental requirements.

Three- and four-body interactions from two-body interactions in spin models: A route to Abelian and non-Abelian fractional Chern insulators

We describe a method for engineering local $k+1$-body interactions ($k=1,2,3$) from two-body couplings in spin-${1}{2}$ systems. When implemented in certain systems with a flat single-particle band with a unit Chern number, the resulting many-body ground states are fractional Chern insulators which exhibit abelian and non-abelian anyon excitations. The most complex of these, with $k=3$, has Fibonacci anyon excitations; our system is thus capable of universal topological quantum computation. We then demonstrate that an appropriately tuned circuit of qubits could faithfully replicate this model up to small corrections, and further, we describe the process by which one might create and manipulate non-abelian vortices in these circuits, allowing for direct control of the system's quantum information content.

Stability of the k = 3 Read–Rezayi state in chiral two-dimensional systems with tunable interactions

The k = 3 Read–Rezayi (RR) parafermion quantum Hall state hosts non-Abelian excitations which provide a platform for universal topological quantum computation. Although the RR state may be realized at the filling factor ν = 12/5 in GaAs-based two-dimensional electron systems, the corresponding quantum Hall state is weak and at present nearly impossible to study experimentally. Here we argue that the RR state can alternatively be realized in a class of chiral materials with massless and massive Dirac-like band structure. This family of materials encompasses monolayer and bilayer graphene, as well as topological insulators. We show that, compared to GaAs, these systems provide several important advantages in realizing and studying the RR state. Most importantly, the effective interactions can be tuned in situ by varying the external magnetic field, and by designing the dielectric environment of the sample. This tunability enables the realization of RR state with controllable energy gaps in different Landau levels. It also allows one to probe the quantum phase transitions to other compressible and incompressible phases.