One of the main topological invariants that characterizes several topologically ordered phases is the many-body Chern number (MBCN). Paradigmatic examples include several fractional quantum Hall phases, which are expected to be realized in different atomic and photonic quantum platforms in the near future. Experimental measurement and numerical computation of this invariant are conventionally based on the linear-response techniques that require having access to a family of states, as a function of an external parameter, which is not suitable for many quantum simulators. Here, we propose an ancilla-free experimental scheme for the measurement of this invariant, without requiring any knowledge of the Hamiltonian. Specifically, we use the statistical correlations of randomized measurements to infer the MBCN of a wave function. Remarkably, our results apply to disklike geometries that are more amenable to current quantum simulator architectures.

}, issn = {0031-9007}, doi = {10.1103/PhysRevLett.126.050501}, author = {Cian, Ze-Pei and Dehghani, Hossein and Elben, Andreas and Vermersch, Benoit and Zhu, Guanyu and Barkeshli, Maissam and Zoller, Peter and Hafezi, Mohammad} } @article {lavasani_measurement-induced_2021, title = {Measurement-induced topological entanglement transitions in symmetric random quantum circuits}, journal = {Nat. Phys.}, volume = {17}, number = {3}, year = {2021}, note = {Place: HEIDELBERGER PLATZ 3, BERLIN, 14197, GERMANY Publisher: NATURE RESEARCH Type: Article}, month = {mar}, pages = {342+}, abstract = {Random quantum circuits, in which an array of qubits is subjected to a series of randomly chosen unitary operations, have provided key insights into the dynamics of many-body quantum entanglement. Recent work has shown that interleaving the unitary operations with single-qubit measurements can drive a transition between high- and low-entanglement phases. Here we study a class of symmetric random quantum circuits with two competing types of measurement in addition to unitary dynamics. We find a rich phase diagram involving robust symmetry-protected topological, trivial and volume law entangled phases, where the transitions are hidden to expectation values of any operator and are only apparent by averaging the entanglement entropy over quantum trajectories. In the absence of unitary dynamics, we find a purely measurement-induced critical point, which maps exactly to two copies of a classical two-dimensional percolation problem. Numerical simulations indicate that this transition is a tricritical point that splits into two critical lines in the presence of arbitrarily sparse unitary dynamics with an intervening volume law entangled phase. Our results show that measurements alone are sufficient to induce criticality and logarithmic entanglement scaling, and arbitrarily sparse unitary dynamics can be sufficient to stabilize volume law entangled phases in the presence of rapid, yet competing, measurements.}, issn = {1745-2473}, doi = {10.1038/s41567-020-01112-z}, author = {Lavasani, Ali and Alavirad, Yahya and Barkeshli, Maissam} } @article {21626, title = {Topological Order and Criticality in (2+1)D Monitored Random Quantum Circuits}, journal = {Phys. Rev. Lett.}, volume = {127}, year = {2021}, month = {Dec}, pages = {235701}, doi = {10.1103/PhysRevLett.127.235701}, url = {https://link.aps.org/doi/10.1103/PhysRevLett.127.235701}, author = {Lavasani, Ali and Alavirad, Yahya and Barkeshli, Maissam} } @article {bulmash_absolute_2020, title = {Absolute anomalies in (2+1){D} symmetry-enriched topological states and exact (3+1){D} constructions}, journal = {Phys. Rev. Res.}, volume = {2}, number = {4}, year = {2020}, note = {Place: ONE PHYSICS ELLIPSE, COLLEGE PK, MD 20740-3844 USA Publisher: AMER PHYSICAL SOC Type: Article}, month = {oct}, abstract = {Certain patterns of symmetry fractionalization in (2+1)-dimensional [(2+1)D] topologically ordered phases of matter can be anomalous, which means that they possess an obstruction to being realized in purely (2+1)D. In this paper we demonstrate how to compute the anomaly for symmetry-enriched topological states of bosons in complete generality. We demonstrate how, given any unitary modular tensor category (UMTC) and symmetry fractionalization class for a global symmetry group G, one can define a (3+1)-dimensional [(3+1)D] topologically invariant path integral in terms of a state sum for a G-symmetry-protected topological (SPT) state. We present an exactly solvable Hamiltonian for the system and demonstrate explicitly a (2+1)D G-symmetric surface termination that hosts deconfined anyon excitations described by the given UMTC and symmetry fractionalization class. We present concrete algorithms that can be used to compute anomaly indicators in general. Our approach applies to general symmetry groups, including anyon-permuting and antiunitary symmetries. In addition to providing a general way to compute the anomaly, our result also shows, by explicit construction, that every symmetry fractionalization class for any UMTC can be realized at the surface of a (3+1)D SPT state. As a by-product, this construction also provides a way of explicitly seeing how the algebraic data that defines symmetry fractionalization in general arises in the context of exactly solvable models. In the case of unitary orientation-preserving symmetries, our results can also be viewed as providing a method to compute the H-4(G, U(1)) obstruction that arises in the theory of G-crossed braided tensor categories, for which no general method has been presented to date.}, doi = {10.1103/PhysRevResearch.2.043033}, author = {Bulmash, Daniel and Barkeshli, Maissam} } @article {16956, title = {Instantaneous braids and Dehn twists in topologically ordered states}, journal = {Phys. Rev. B}, volume = {102}, year = {2020}, month = {Aug}, pages = {075105}, doi = {10.1103/PhysRevB.102.075105}, url = {https://link.aps.org/doi/10.1103/PhysRevB.102.075105}, author = {Zhu, Guanyu and Lavasani, Ali and Barkeshli, Maissam} } @article {zhu_quantum_2020, title = {Quantum origami: {Transversal} gates for quantum computation and measurement of topological order}, journal = {Phys. Rev. Res.}, volume = {2}, number = {1}, year = {2020}, note = {Place: ONE PHYSICS ELLIPSE, COLLEGE PK, MD 20740-3844 USA Publisher: AMER PHYSICAL SOC Type: Article}, month = {mar}, abstract = {In topology, a torus remains invariant under certain nontrivial transformations known as modular transformations. In the context of topologically ordered quantum states of matter supported on a torus geometry in real space, these transformations encode the braiding statistics and fusion rules of emergent anyonic excitations and thus serve as a diagnostic of topological order. Moreover, modular transformations of higher genus surfaces, e.g., a torus with multiple handles, can enhance the computational power of a topological state, in many cases providing a universal fault-tolerant set of gates for quantum computation. However, due to the intrusive nature of modular transformations, which abstractly involve global operations, physical implementations of them in local systems have remained elusive. Here, we show that by engineering an effectively folded manifold corresponding to a multilayer topological system, modular transformations can be applied in a single shot by independent local unitaries, providing a novel class of transversal logical gates for fault-tolerant quantum computation. Specifically, we demonstrate that multilayer topological states with appropriate boundary conditions and twist defects allow modular transformations to be effectively implemented by a finite sequence of local SWAP gates between the layers. We further provide methods to directly measure the modular matrices, and thus the fractional statistics of anyonic excitations, providing a novel way to directly measure topological order. A more general theory of transversal gates and the deep connection to anyon symmetry transformation and symmetry-enriched topological orders are also discussed.}, doi = {10.1103/PhysRevResearch.2.013285}, author = {Zhu, Guanyu and Hafezi, Mohammad and Barkeshli, Maissam} } @article { ISI:000527910200017, title = {Reflection and Time Reversal Symmetry Enriched Topological Phases of Matter: Path Integrals, Non-orientable Manifolds, and Anomalies}, journal = {Commun. Math. Phys.}, volume = {374}, number = {2}, year = {2020}, month = {MAR}, pages = {1021-1124}, publisher = {SPRINGER}, type = {Article}, abstract = {We study symmetry-enriched topological (SET) phases in 2+1 space-time dimensions with spatial reflection and/or time-reversal symmetries. We provide a systematic construction of a wide class of reflection and time-reversal SET phases in terms of a topological path integral defined on general space-time manifolds. An important distinguishing feature of different topological phases with reflection and/or time-reversal symmetry is the value of the path integral on non-orientable space-time manifolds. We derive a simple general formula for the path integral on the manifold Sigma(2) x S-1, where Sigma(2) is a two-dimensional non-orientable surface and S-1 is a circle. This also gives an expression for the ground state degeneracy of the SET on the surface Sigma(2) that depends on the reflection symmetry fractionalization class, generalizing the Verlinde formula for ground state degeneracy on orientable surfaces. Consistency of the action of the mapping class group on non-orientable manifolds leads us to a constraint that can detect when a time-reversal or reflection SET phase is anomalous in (2+1)D and, thus, can only exist at the surface of a (3+1)D symmetry protected topological (SPT) state. Given a (2+1)D reflection and/or time-reversal SET phase, we further derive a general formula that determines which (3+1)D reflection and/or time-reversal SPT phase hosts the (2+1)D SET phase as its surface termination. A number of explicit examples are studied in detail.}, issn = {0010-3616}, doi = {10.1007/s00220-019-03475-8}, author = {Barkeshli, Maissam and Bonderson, Parsa and Cheng, Meng and Jian, Chao-Ming and Walker, Kevin} } @article {barkeshli_relative_2020, title = {Relative anomalies in (2+1){D} symmetry enriched topological states}, journal = {SciPost Phys.}, volume = {8}, number = {2}, year = {2020}, note = {Place: C/O J S CAUX, INST PHYSICS, UNIV AMSTERDAM, AMSTERDAM, 1098 XH, NETHERLANDS Publisher: SCIPOST FOUNDATION Type: Article}, month = {feb}, abstract = {Certain patterns of symmetry fractionalization in topologically ordered phases of matter are anomalous, in the sense that they can only occur at the surface of a higher dimensional symmetry-protected topological (SPT) state. An important question is to determine how to compute this anomaly, which means determining which SPT hosts a given symmetry-enriched topological order at its surface. While special cases are known, a general method to compute the anomaly has so far been lacking. In this paper we propose a general method to compute relative anomalies between different symmetry fractionalization classes of a given (2+1)D topological order. This method applies to all types of symmetry actions, including anyon-permuting symmetries and general space-time reflection symmetries. We demonstrate compatibility of the relative anomaly formula with previous results for diagnosing anomalies for Z(2)(T) space-time reflection symmetry (e.g. where time-reversal squares to the identity) and mixed anomalies for U(1) x Z(2)(T) and U (1) (sic) Z(2)(T) symmetries. We also study a number of additional examples, including cases where space-time reflection symmetries are intertwined in non-trivial ways with unitary symmetries, such as Z(4)(T) and mixed anomalies for Z(2) x Z(2)(T) symmetry, and unitary Z(2) x Z(2) symmetry with non-trivial anyon permutations.}, issn = {2542-4653}, doi = {10.21468/SciPostPhys.8.2.028}, author = {Barkeshli, Maissam and Cheng, Meng} } @article {16951, title = {Universal Logical Gates on Topologically Encoded Qubits via Constant-Depth Unitary Circuits}, journal = {Phys. Rev. Lett.}, volume = {125}, year = {2020}, month = {Jul}, pages = {050502}, doi = {10.1103/PhysRevLett.125.050502}, url = {https://link.aps.org/doi/10.1103/PhysRevLett.125.050502}, author = {Zhu, Guanyu and Lavasani, Ali and Barkeshli, Maissam} } @article {balram_zn_2020, title = {Z(n) superconductivity of composite bosons and the 7/3 fractional quantum {Hall} effect}, journal = {Phys. Rev. Res.}, volume = {2}, number = {1}, year = {2020}, note = {Place: ONE PHYSICS ELLIPSE, COLLEGE PK, MD 20740-3844 USA Publisher: AMER PHYSICAL SOC Type: Article}, month = {mar}, abstract = {The topological p-wave pairing of composite fermions, believed to be responsible for the 5/2 fractional quantum Hall effect (FQHE), has generated much exciting physics. Motivated by the parton theory of the FQHE, we consider the possibility of a new kind of emergent {\textquotedblleft}superconductivity{\textquotedblright} in the 1/3 FQHE, which involves condensation of clusters of n composite bosons. From a microscopic perspective, the state is described by the n (n) over bar 111 parton wave function P-LLL Phi(n)Phi(n)*Phi(3)(1), where Phi(n) is the wave function of the integer quantum Hall state with n filled Landau levels and P-LLL is the lowest-Landau-level projection operator. It represents a Z(n) superconductor of composite bosons, because the factor Phi(3)(1) similar to Pi(j{\textless}k) (z(j) - z(k))(3), where z(j) = x(j) - iy(j) is the coordinate of the jth electron, binds three vortices to electrons to convert them into composite bosons, which then condense into the Z(n) superconducting state vertical bar Phi(n)vertical bar(2). From a field theoretical perspective, this state can be understood by starting with the usual Laughlin theory and gauging a Z(n) subgroup of the U(1) charge conservation symmetry. We find from detailed quantitative calculations that the 2{\textless}(2)over bar{\textgreater}111 and 3 (3) over bar 111 states are at least as plausible as the Laughlin wave function for the exact Coulomb ground state at filling nu = 7/3, suggesting that this physics is possibly relevant for the 7/3 FQHE. The Z(n) order leads to several observable consequences, including quasiparticles with fractionally quantized charges of magnitude e/(3n) and the existence of multiple neutral collective modes. It is interesting that the FQHE may be a promising venue for the realization of exotic Z(n) superconductivity.}, doi = {10.1103/PhysRevResearch.2.013349}, author = {Balram, Ajit C. and Jain, J. K. and Barkeshli, Maissam} } @article { ISI:000493516700001, title = {Gauging fractons: Immobile non-Abelian quasiparticles, fractals, and position-dependent degeneracies}, journal = {Phys. Rev. B}, volume = {100}, number = {15}, year = {2019}, month = {OCT 29}, pages = {155146}, publisher = {AMER PHYSICAL SOC}, type = {Article}, abstract = {The study of gapped quantum many-body systems in three spatial dimensions has uncovered the existence of quantum states hosting quasiparticles that are confined, not by energetics but by the structure of local operators, to move along lower dimensional submanifolds. These so-called {\textquoteleft}{\textquoteleft}fracton{{\textquoteright}{\textquoteright}} phases are beyond the usual topological quantum field theory description, and thus require new theoretical frameworks to describe them. Here we consider coupling fracton models to topological quantum field theories in (3 + 1) dimensions by starting with two copies of a known fracton model and gauging the Z(2) symmetry that exchanges the two copies. This yields a class of exactly solvable lattice models that we study in detail for the case of the X-cube model and Haah{\textquoteright}s cubic code. The resulting phases host finite-energy non-Abelian immobile quasiparticles with robust degeneracies that depend on their relative positions. The phases also host non-Abelian string excitations with robust degeneracies that depend on the string geometry. Applying the construction to Haah{\textquoteright}s cubic code in particular provides an exactly solvable model with finite energy yet immobile non-Abelian quasiparticles that can only be created at the corners of operators with fractal support.}, issn = {2469-9950}, doi = {10.1103/PhysRevB.100.155146}, author = {Bulmash, Daniel and Barkeshli, Maissam} } @article {ISI:000473018000001, title = {Parton construction of particle-hole-conjugate Read-Rezayi parafermion fractional quantum Hall states and beyond}, journal = {Phys. Rev. B}, volume = {99}, number = {24}, year = {2019}, month = {JUN 19}, pages = {241108}, publisher = {AMER PHYSICAL SOC}, type = {Article}, abstract = {The Read-Rezayi (RR) parafermion states form a series of exotic non-Abelian fractional quantum Hall (FQH) states at filling. = k/(k + 2). Computationally, the wave functions of these states are prohibitively expensive to generate for large systems. We introduce a series of parton states, denoted (\$2) over bar (k)1(k+1), and show that they lie in the same universality classes as the particle-hole-conjugate RR ({{\textquoteright}{\textquoteright}}anti-RR{{\textquoteright}{\textquoteright}}) states. Our analytical results imply that a {[}U(1)(k+1) xU(2k)(-1)]/{[}SU(k)(-2) xU(1)(-1)] coset conformal field theory describes the edge excitations of the (2) over bar (k)1(k+1) state, suggesting nontrivial dualities with respect to previously known descriptions. The parton construction allows wave functions in anti-RR phases to be generated for hundreds of particles. We further propose the parton sequence (n) over bar(2) over bar (4), with n = 1, 2, 3, to describe the FQH states observed at nu= 2 + 1/2, 2 + 2/5, and 2 + 3/8.}, issn = {2469-9950}, doi = {10.1103/PhysRevB.99.241108}, author = {Balram, Ajit C. and Barkeshli, Maissam and Rudner, Mark S.} } @article {ISI:000473540500004, title = {Prediction of a Non-Abelian Fractional Quantum Hall State with f-Wave Pairing of Composite Fermions in Wide Quantum Wells}, journal = {Phys. Rev. Lett.}, volume = {123}, number = {1}, year = {2019}, month = {JUL 2}, pages = {016802}, publisher = {AMER PHYSICAL SOC}, type = {Article}, abstract = {We theoretically investigate the nature of the state at the quarter filled lowest Landau level and predict that, as the quantum well width is increased, a transition occurs from the composite fermion Fermi sea into a novel non-Abelian fractional quantum Hall state that is topologically equivalent to f-wave pairing of composite fermions. This state is topologically distinct from the familiar p-wave paired Pfaffian state. We compare our calculated phase diagram with experiments and make predictions for many observable quantities.}, issn = {0031-9007}, doi = {10.1103/PhysRevLett.123.016802}, author = {Faugno, W. N. and Balram, Ajit C. and Barkeshli, Maissam and Jain, J. K.} } @article {ISI:000482956700001, title = {Universal logical gates with constant overhead: instantaneous Dehn twists for hyperbolic quantum codes}, journal = {Quantum}, volume = {3}, year = {2019}, month = {JUL 26}, publisher = {VEREIN FORDERUNG OPEN ACCESS PUBLIZIERENS QUANTENWISSENSCHAF}, type = {Article}, abstract = {A basic question in the theory of fault-tolerant quantum computation is to understand the fundamental resource costs for performing a universal logical set of gates on encoded qubits to arbitrary accuracy. Here we consider qubits encoded with constant space overhead (i.e. finite encoding rate) in the limit of arbitrarily large code distance d through the use of topological codes associated to triangulations of hyperbolic surfaces. We introduce explicit protocols to demonstrate how Dehn twists of the hyperbolic surface can be implemented on the code through constant depth unitary circuits, without increasing the space overhead. The circuit for a given Dehn twist consists of a permutation of physical qubits, followed by a constant depth local unitary circuit, where locality here is defined with respect to a hyperbolic metric that defines the code. Applying our results to the hyperbolic Fibonacci Turaev-Viro code implies the possibility of applying universal logical gate sets on encoded qubits through constant depth unitary circuits and with constant space overhead. Our circuits are inherently protected from errors as they map local operators to local operators while changing the size of their support by at most a constant factor; in the presence of noisy syndrome measurements, our results suggest the possibility of universal fault tolerant quantum computation with constant space overhead and time overhead of O(d/log d). For quantum circuits that allow parallel gate operations, this yields the optimal scaling of space-time overhead known to date.}, issn = {2521-327X}, author = {Lavasani, Ali and Zhu, Guanyu and Barkeshli, Maissam} } @article { ISI:000448933900006, title = {Fractional Quantum Hall Effect at nu=2+6/13: The Parton Paradigm for the Second Landau Level}, journal = {PHYSICAL REVIEW LETTERS}, volume = {121}, number = {18}, year = {2018}, month = {NOV 1}, pages = {186601}, issn = {0031-9007}, doi = {10.1103/PhysRevLett.121.186601}, author = {Balram, Ajit C. and Mukherjee, Sutirtha and Park, Kwon and Barkeshli, Maissam and Rudner, Mark S. and Jain, J. K.} } @article { ISI:000434628400002, title = {Higgs mechanism in higher-rank symmetric U(1) gauge theories}, journal = {PHYSICAL REVIEW B}, volume = {97}, number = {23}, year = {2018}, month = {JUN 8}, pages = {235112}, issn = {2469-9950}, doi = {10.1103/PhysRevB.97.235112}, author = {Bulmash, Daniel and Barkeshli, Maissam} } @article { ISI:000450258700003, title = {Low overhead Clifford gates from joint measurements in surface, color, and hyperbolic codes}, journal = {PHYSICAL REVIEW A}, volume = {98}, number = {5}, year = {2018}, month = {NOV 15}, pages = {052319}, issn = {2469-9926}, doi = {10.1103/PhysRevA.98.052319}, author = {Lavasani, Ali and Barkeshli, Maissam} } @article { ISI:000439057800004, title = {Parton construction of a wave function in the anti-Pfaffian phase}, journal = {PHYSICAL REVIEW B}, volume = {98}, number = {3}, year = {2018}, month = {JUL 18}, pages = {035127}, issn = {2469-9950}, doi = {10.1103/PhysRevB.98.035127}, author = {Balram, Ajit C. and Barkeshli, Maissam and Rudner, Mark S.} } @article { ISI:000444775000003, title = {Time-reversal and spatial-reflection symmetry localization anomalies in (2+1)-dimensional topological phases of matter}, journal = {PHYSICAL REVIEW B}, volume = {98}, number = {11}, year = {2018}, month = {SEP 17}, pages = {115129}, issn = {2469-9950}, doi = {10.1103/PhysRevB.98.115129}, author = {Barkeshli, Maissam and Cheng, Meng} } @article { ISI:000437838000015, title = {Topological Exciton Fermi Surfaces in Two-Component Fractional Quantized Hall Insulators}, journal = {PHYSICAL REVIEW LETTERS}, volume = {121}, number = {2}, year = {2018}, month = {JUL 9}, pages = {026603}, issn = {0031-9007}, doi = {10.1103/PhysRevLett.121.026603}, author = {Barkeshli, Maissam and Nayak, Chetan and Papic, Zlatko and Young, Andrea and Zaletel, Michael} } @article { ISI:000382008100013, title = {Charge 2e/3 Superconductivity and Topological Degeneracies without Localized Zero Modes in Bilayer Fractional Quantum Hall States}, journal = {PHYSICAL REVIEW LETTERS}, volume = {117}, number = {9}, year = {2016}, month = {AUG 24}, issn = {0031-9007}, doi = {10.1103/PhysRevLett.117.096803}, author = {Barkeshli, Maissam} } @article { ISI:000357261300002, title = {Abelian and non-Abelian states in nu=2/3 bilayer fractional quantum Hall systems}, journal = {PHYSICAL REVIEW B}, volume = {92}, number = {3}, year = {2015}, month = {JUL 2}, issn = {1098-0121}, doi = {10.1103/PhysRevB.92.035103}, author = {Peterson, Michael R. and Wu, Yang-Le and Cheng, Meng and Barkeshli, Maissam and Wang, Zhenghan and S. Das Sarma} } @article { ISI:000365773300003, title = {Chirality-protected Majorana zero modes at the gapless edge of Abelian quantum Hall states}, journal = {PHYSICAL REVIEW B}, volume = {92}, number = {19}, year = {2015}, month = {NOV 30}, issn = {1098-0121}, doi = {10.1103/PhysRevB.92.195152}, author = {Cano, Jennifer and Cheng, Meng and Barkeshli, Maissam and Clarke, David J. and Nayak, Chetan} }