Showing 1–31 of 31 results for author: Cui, S

The impact of parameter spread of high-temperature superconducting Josephson junctions on the performance of quantum-based voltage standards

Authors: Guanghong Wen, Yi Zhu, Yingxiang Zheng, Shuhe Cui, Ji Wang, Yanyun Ren, Hao Li, Guofeng Zhang, Lixing You

Abstract: Quantum metrology based on Josephson junction array reproduces the most accurate desired voltage by far, therefore being introduced to provide voltage standards worldwide. In this work, we quantitatively analyzed the dependence of the first Shapiro step height of the junction array at 50 GHz on the parameter spread of 10,000 Josephson junctions by numerical simulation with resistively shunted junc… ▽ More Quantum metrology based on Josephson junction array reproduces the most accurate desired voltage by far, therefore being introduced to provide voltage standards worldwide. In this work, we quantitatively analyzed the dependence of the first Shapiro step height of the junction array at 50 GHz on the parameter spread of 10,000 Josephson junctions by numerical simulation with resistively shunted junction model. The results indicate an upper limit spread of the critical current and resistance of the Josephson junctions. Specifically, to keep the maximum first Shapiro step above 0.88 mA, the critical current standard deviation, $σ$, should not exceed 25%, and for it to stay above 0.6 mA, the resistance standard deviation should not exceed 1.5%. △ Less

Submitted 15 June, 2025; originally announced June 2025.

Signless Laplacian State Transfer on Vertex Complemented Coronae

Authors: Ke-Yu Zhu, Gui-Xian Tian, Shu-Yu Cui

Abstract: Given a graph $G$ with vertex set $V(G)=\{v_1,v_2,\ldots,v_{n_1}\}$ and a graph $H$ of order $n_2$, the vertex complemented corona, denoted by $G\tilde{\circ}{H}$, is the graph produced by copying $H$ $n_1$ times, with the $i$-th copy of $H$ corresponding to the vertex $v_i$, and then adding edges between any vertex in $V(G)\setminus\{v_{i}\}$ and any vertex of the $i$-th copy of $H$. The present… ▽ More Given a graph $G$ with vertex set $V(G)=\{v_1,v_2,\ldots,v_{n_1}\}$ and a graph $H$ of order $n_2$, the vertex complemented corona, denoted by $G\tilde{\circ}{H}$, is the graph produced by copying $H$ $n_1$ times, with the $i$-th copy of $H$ corresponding to the vertex $v_i$, and then adding edges between any vertex in $V(G)\setminus\{v_{i}\}$ and any vertex of the $i$-th copy of $H$. The present article deals with quantum state transfer of vertex complemented coronae concerning signless Laplacian matrix. Our research investigates conditions in which signless Laplacian perfect state transfer exists or not on vertex complemented coronae. Additionally, we also provide some mild conditions for the class of graphs under consideration that allow signless Laplacian pretty good state transfer. △ Less

Submitted 20 February, 2025; originally announced February 2025.

Comments: 16 pages

MSC Class: 05C50; 15A18; 81P45

Floquet Codes from Coupled Spin Chains

Authors: Bowen Yan, Penghua Chen, Shawn X. Cui

Abstract: We propose a novel construction of the Floquet 3D toric code and Floquet $X$-cube code through the coupling of spin chains. This approach not only recovers the coupling layer construction on foliated lattices in three dimensions but also avoids the complexity of coupling layers in higher dimensions, offering a more localized and easily generalizable framework. Our method extends the Floquet 3D tor… ▽ More We propose a novel construction of the Floquet 3D toric code and Floquet $X$-cube code through the coupling of spin chains. This approach not only recovers the coupling layer construction on foliated lattices in three dimensions but also avoids the complexity of coupling layers in higher dimensions, offering a more localized and easily generalizable framework. Our method extends the Floquet 3D toric code to a broader class of lattices, aligning with its topological phase properties. Furthermore, we generalize the Floquet $X$-cube model to arbitrary manifolds, provided the lattice is locally cubic, consistent with its Fractonic phases. We also introduce a unified error-correction paradigm for Floquet codes by defining a subgroup, the Steady Stabilizer Group (SSG), of the Instantaneous Stabilizer Group (ISG), emphasizing that not all terms in the ISG contribute to error correction, but only those terms that can be referred to at least twice before being removed from the ISG. We show that correctable Floquet codes naturally require the SSG to form a classical error-correcting code, and we present a simple 2-step Bacon-Shor Floquet code as an example, where SSG forms instantaneous repetition codes. Finally, our construction intrinsically supports the extension to $n$-dimensional Floquet $(n,1)$ toric codes and generalized $n$-dimensional Floquet $X$-cube codes. △ Less

Submitted 23 October, 2024; originally announced October 2024.

Representing arbitrary ground states of toric code by a restricted Boltzmann machine

Authors: Penghua Chen, Bowen Yan, Shawn X. Cui

Abstract: We systematically analyze the representability of toric code ground states by Restricted Boltzmann Machine with only local connections between hidden and visible neurons. This analysis is pivotal for evaluating the model's capability to represent diverse ground states, thus enhancing our understanding of its strengths and weaknesses. Subsequently, we modify the Restricted Boltzmann Machine to acco… ▽ More We systematically analyze the representability of toric code ground states by Restricted Boltzmann Machine with only local connections between hidden and visible neurons. This analysis is pivotal for evaluating the model's capability to represent diverse ground states, thus enhancing our understanding of its strengths and weaknesses. Subsequently, we modify the Restricted Boltzmann Machine to accommodate arbitrary ground states by introducing essential non-local connections efficiently. The new model is not only analytically solvable but also demonstrates efficient and accurate performance when solved using machine learning techniques. Then we generalize our the model from $Z_2$ to $Z_n$ toric code and discuss future directions. △ Less

Submitted 2 January, 2025; v1 submitted 1 July, 2024; originally announced July 2024.

Comments: 21 pages, 22 figures

Journal ref: Phys. Rev. B 111, 045101 (2025)

Universal topological quantum computing via double-braiding in SU(2) Witten-Chern-Simons theory

Authors: Adrian L. Kaufmann, Shawn X. Cui

Abstract: We study the problem of universality in the anyon model described by the $SU(2)$ Witten-Chern-Simons theory at level $k$. A classic theorem of Freedman-Larsen-Wang states that for $k \geq 3, \ k \neq 4$, braiding of the anyons of topological charge $1/2$ is universal for topological quantum computing. For the case of one qubit, we prove a stronger result that double-braiding of such anyons alone i… ▽ More We study the problem of universality in the anyon model described by the $SU(2)$ Witten-Chern-Simons theory at level $k$. A classic theorem of Freedman-Larsen-Wang states that for $k \geq 3, \ k \neq 4$, braiding of the anyons of topological charge $1/2$ is universal for topological quantum computing. For the case of one qubit, we prove a stronger result that double-braiding of such anyons alone is already universal. △ Less

Submitted 7 January, 2025; v1 submitted 27 December, 2023; originally announced December 2023.

Journal ref: Quantum Information Processing, vol 24, no 14 (2025)

Quantum circuits for toric code and X-cube fracton model

Authors: Penghua Chen, Bowen Yan, Shawn X. Cui

Abstract: We propose a systematic and efficient quantum circuit composed solely of Clifford gates for simulating the ground state of the surface code model. This approach yields the ground state of the toric code in $\lceil 2L+2+log_{2}(d)+\frac{L}{2d} \rceil$ time steps, where $L$ refers to the system size and $d$ represents the maximum distance to constrain the application of the CNOT gates. Our algorithm… ▽ More We propose a systematic and efficient quantum circuit composed solely of Clifford gates for simulating the ground state of the surface code model. This approach yields the ground state of the toric code in $\lceil 2L+2+log_{2}(d)+\frac{L}{2d} \rceil$ time steps, where $L$ refers to the system size and $d$ represents the maximum distance to constrain the application of the CNOT gates. Our algorithm reformulates the problem into a purely geometric one, facilitating its extension to attain the ground state of certain 3D topological phases, such as the 3D toric model in $3L+8$ steps and the X-cube fracton model in $12L+11$ steps. Furthermore, we introduce a gluing method involving measurements, enabling our technique to attain the ground state of the 2D toric code on an arbitrary planar lattice and paving the way to more intricate 3D topological phases. △ Less

Submitted 29 February, 2024; v1 submitted 4 October, 2022; originally announced October 2022.

Comments: 23 pages, 26 figures

Journal ref: Quantum 8, 1276 (2024)

Ternary Logic Design in Topological Quantum Computing

Authors: Muhammad Ilyas, Shawn Cui, Marek Perkowski

Abstract: A quantum computer can perform exponentially faster than its classical counterpart. It works on the principle of superposition. But due to the decoherence effect, the superposition of a quantum state gets destroyed by the interaction with the environment. It is a real challenge to completely isolate a quantum system to make it free of decoherence. This problem can be circumvented by the use of top… ▽ More A quantum computer can perform exponentially faster than its classical counterpart. It works on the principle of superposition. But due to the decoherence effect, the superposition of a quantum state gets destroyed by the interaction with the environment. It is a real challenge to completely isolate a quantum system to make it free of decoherence. This problem can be circumvented by the use of topological quantum phases of matter. These phases have quasiparticles excitations called anyons. The anyons are charge-flux composites and show exotic fractional statistics. When the order of exchange matters, then the anyons are called non-Abelian anyons. Majorana fermions in topological superconductors and quasiparticles in some quantum Hall states are non-Abelian anyons. Such topological phases of matter have a ground state degeneracy. The fusion of two or more non-Abelian anyons can result in a superposition of several anyons. The topological quantum gates are implemented by braiding and fusion of the non-Abelian anyons. The fault-tolerance is achieved through the topological degrees of freedom of anyons. Such degrees of freedom are non-local, hence inaccessible to the local perturbations. In this paper, the Hilbert space for a topological qubit is discussed. The Ising and Fibonacci anyonic models for binary gates are briefly given. Ternary logic gates are more compact than their binary counterparts and naturally arise in a type of anyonic model called the metaplectic anyons. The mathematical model, for the fusion and braiding matrices of metaplectic anyons, is the quantum deformation of the recoupling theory. We proposed that the existing quantum ternary arithmetic gates can be realized by braiding and topological charge measurement of the metaplectic anyons. △ Less

Submitted 27 September, 2022; v1 submitted 3 April, 2022; originally announced April 2022.

Journal ref: J. Phys. A: Math. Theor. 55 (2022) 305302 (54pp)

Constructing Approximately Diagonal Quantum Gates

Authors: Colton Griffin, Shawn X. Cui

Abstract: We study a method of producing approximately diagonal 1-qubit gates. For each positive integer, the method provides a sequence of gates that are defined iteratively from a fixed diagonal gate and an arbitrary gate. These sequences are conjectured to converge to diagonal gates doubly exponentially fast and are verified for small integers. We systemically study this conjecture and prove several impo… ▽ More We study a method of producing approximately diagonal 1-qubit gates. For each positive integer, the method provides a sequence of gates that are defined iteratively from a fixed diagonal gate and an arbitrary gate. These sequences are conjectured to converge to diagonal gates doubly exponentially fast and are verified for small integers. We systemically study this conjecture and prove several important partial results. Some techniques are developed to pave the way for a final resolution of the conjecture. The sequences provided here have applications in quantum search algorithms, quantum circuit compilation, generation of leakage-free entangled gates in topological quantum computing, etc. △ Less

Submitted 17 November, 2022; v1 submitted 10 September, 2021; originally announced September 2021.

Comments: 25 pages, updated from revisions

From Torus Bundles to Particle-Hole Equivariantization

Authors: Shawn X. Cui, Paul Gustafson, Yang Qiu, Qing Zhang

Abstract: We continue the program of constructing (pre)modular tensor categories from 3-manifolds first initiated by Cho-Gang-Kim using $M$ theory in physics and then mathematically studied by Cui-Qiu-Wang. An important structure involved is a collection of certain $\text{SL}(2, \mathbb{C})$ characters on a given manifold which serve as the simple object types in the corresponding category. Chern-Simons inv… ▽ More We continue the program of constructing (pre)modular tensor categories from 3-manifolds first initiated by Cho-Gang-Kim using $M$ theory in physics and then mathematically studied by Cui-Qiu-Wang. An important structure involved is a collection of certain $\text{SL}(2, \mathbb{C})$ characters on a given manifold which serve as the simple object types in the corresponding category. Chern-Simons invariants and adjoint Reidemeister torsions play a key role in the construction, and they are related to topological twists and quantum dimensions, respectively, of simple objects. The modular $S$-matrix is computed from local operators and follows a trial-and-error procedure. It is currently unknown how to produce data beyond the modular $S$- and $T$-matrices. There are also a number of subtleties in the construction which remain to be solved. In this paper, we consider an infinite family of 3-manifolds, that is, torus bundles over the circle. We show that the modular data produced by such manifolds are realized by the $\mathbb{Z}_2$-equivariantization of certain pointed premodular categories. Here the equivariantization is performed for the $\mathbb{Z}_2$-action sending a simple (invertible) object to its inverse, also called the particle-hole symmetry. It is our hope that this extensive class of examples will shed light on how to improve the program to recover the full data of a premodular category. △ Less

Submitted 27 September, 2022; v1 submitted 3 June, 2021; originally announced June 2021.

Comments: 16 pages, minor changes, to appear in Lett Math Phys

Journal ref: Lett Math Phys 112, 15 (2022)

Ribbon operators in the generalized Kitaev quantum double model based on Hopf algebras

Authors: Penghua Chen, Shawn X. Cui, Bowen Yan

Abstract: Kitaev's quantum double model is a family of exactly solvable lattice models that realize two dimensional topological phases of matter. Originally it is based on finite groups, and is later generalized to semi-simple Hopf algebras. We rigorously define and study ribbon operators in the generalized Kitaev quantum double model. These ribbon operators are important tools to understand quasi-particle… ▽ More Kitaev's quantum double model is a family of exactly solvable lattice models that realize two dimensional topological phases of matter. Originally it is based on finite groups, and is later generalized to semi-simple Hopf algebras. We rigorously define and study ribbon operators in the generalized Kitaev quantum double model. These ribbon operators are important tools to understand quasi-particle excitations. It turns out that there are some subtleties in defining the operators in contrast to what one would naively think. In particular, one has to distinguish two classes of ribbons which we call locally clockwise and locally counterclockwise ribbons. Moreover, this issue already exists in the original model based on finite non-Abelian groups. We show how certain properties would fail even in the original model if we do not distinguish these two classes of ribbons. Perhaps not surprisingly, under the new definitions ribbon operators satisfy all properties that are expected. For instance, they create quasi-particle excitations only at the end of the ribbon, and the types of the quasi-particles correspond to irreducible representations of the Drinfeld double of the input Hopf algebra. However, the proofs of these properties are much more complicated than those in the case of finite groups. This is partly due to the complications in dealing with general Hopf algebras rather than just group algebras. △ Less

Submitted 6 October, 2022; v1 submitted 17 May, 2021; originally announced May 2021.

Comments: 39 pages

Journal ref: J. Phys. A: Math. Theor. 55 (2022) 185201

From Three Dimensional Manifolds to Modular Tensor Categories

Authors: Shawn X. Cui, Yang Qiu, Zhenghan Wang

Abstract: Using M-theory in physics, Cho, Gang, and Kim (JHEP 2020, 115 (2020) ) recently outlined a program that connects two parallel subjects of three dimensional manifolds, namely, geometric topology and quantum topology. They suggest that classical topological invariants such as Chern-Simons invariants of $\text{SL}(2,\mathbb{C})$-flat connections and adjoint Reidemeister torsions of a three manifold c… ▽ More Using M-theory in physics, Cho, Gang, and Kim (JHEP 2020, 115 (2020) ) recently outlined a program that connects two parallel subjects of three dimensional manifolds, namely, geometric topology and quantum topology. They suggest that classical topological invariants such as Chern-Simons invariants of $\text{SL}(2,\mathbb{C})$-flat connections and adjoint Reidemeister torsions of a three manifold can be packaged together to produce a $(2+1)$-topological quantum field theory, which is essentially equivalent to a modular tensor category. It is further conjectured that every modular tensor category can be obtained from a three manifold and a semi-simple Lie group. In this paper, we study this program mathematically, and provide strong support for the feasibility of such a program. The program produces an algorithm to generate the potential modular $T$-matrix and the quantum dimensions of a candidate modular data. The modular $S$-matrix follows from essentially a trial-and-error procedure. We find modular tensor categories that realize candidate modular data constructed from Seifert fibered spaces and torus bundles over the circle that reveal many subtleties in the program. We make a number of improvements to the program based on our computations. Our main result is a mathematical construction of a premodular category from each Seifert fibered space with three singular fibers and a family of torus bundles over the circle with Thurston SOL geometry. The premodular categories from Seifert fibered spaces are related to Temperley-Lieb-Jones categories and the ones from torus bundles over the circle are related to metaplectic categories. We conjecture that a resulting premodular category is modular if and only if the three manifold is a $\mathbb{Z}_2$-homology sphere and condensation of bosons in premodular categories leads to either modular or super-modular categories. △ Less

Submitted 23 August, 2022; v1 submitted 5 January, 2021; originally announced January 2021.

Comments: To appear in Comm. Math. Phys

MSC Class: 18M20; 57K16; 58J28

Nonlinear two-photon Rabi-Hubbard model: superradiance and photon/photon-pair Bose-Einstein condensate

Authors: Shifeng Cui, B. Grémaud, Wenan Guo, G. G. Batrouni

Abstract: We study the ground state phase diagram of a nonlinear two-photon Rabi-Hubbard (RH) model in one dimension using quantum Monte Carlo (QMC) simulations and density matrix renormalization group (DMRG) calculations. Our model includes a nonlinear photon-photon interaction term. Absent this term, the RH model has only one phase, the normal disordered phase, and suffers from spectral collapse at larger… ▽ More We study the ground state phase diagram of a nonlinear two-photon Rabi-Hubbard (RH) model in one dimension using quantum Monte Carlo (QMC) simulations and density matrix renormalization group (DMRG) calculations. Our model includes a nonlinear photon-photon interaction term. Absent this term, the RH model has only one phase, the normal disordered phase, and suffers from spectral collapse at larger values of the photon-qubit interaction or inter-cavity photon hopping. The photon-photon interaction, no matter how small, stabilizes the system which now exhibits {\it two} quantum phase transitions: Normal phase to {\it photon pair} superfluid (PSF) transition and PSF to single particle superfluid (SPSF). The discrete $Z_4$ symmetry of the Hamiltonian spontaneously breaks in two stages: First it breaks partially as the system enters the PSF and then completely breaks when the system finally enters the SPSF phase. We show detailed numerical results supporting this, and map out the ground state phase diagram. △ Less

Submitted 16 June, 2020; originally announced June 2020.

Comments: 9 pages, 11 figures

Journal ref: Phys. Rev. A 102, 033334 (2020)

Kitaev's quantum double model as an error correcting code

Authors: Shawn X. Cui, Dawei Ding, Xizhi Han, Geoffrey Penington, Daniel Ranard, Brandon C. Rayhaun, Zhou Shangnan

Abstract: Kitaev's quantum double models in 2D provide some of the most commonly studied examples of topological quantum order. In particular, the ground space is thought to yield a quantum error-correcting code. We offer an explicit proof that this is the case for arbitrary finite groups. Actually a stronger claim is shown: any two states with zero energy density in some contractible region must have the s… ▽ More Kitaev's quantum double models in 2D provide some of the most commonly studied examples of topological quantum order. In particular, the ground space is thought to yield a quantum error-correcting code. We offer an explicit proof that this is the case for arbitrary finite groups. Actually a stronger claim is shown: any two states with zero energy density in some contractible region must have the same reduced state in that region. Alternatively, the local properties of a gauge-invariant state are fully determined by specifying that its holonomies in the region are trivial. We contrast this result with the fact that local properties of gauge-invariant states are not generally determined by specifying all of their non-Abelian fluxes -- that is, the Wilson loops of lattice gauge theory do not form a complete commuting set of observables. We also note that the methods developed by P. Naaijkens (PhD thesis, 2012) under a different context can be adapted to provide another proof of the error correcting property of Kitaev's model. Finally, we compute the topological entanglement entropy in Kitaev's model, and show, contrary to previous claims in the literature, that it does not depend on whether the "log dim R" term is included in the definition of entanglement entropy. △ Less

Submitted 22 September, 2020; v1 submitted 7 August, 2019; originally announced August 2019.

Comments: added a subsection and two appendices on topological entanglement entropy; fixed some typos; Accepted in Quantum

Journal ref: Quantum 4, 331 (2020)

Two-photon Rabi-Hubbard and Jaynes-Cummings-Hubbard models: photon pair superradiance, Mott insulator and normal phases

Authors: Shifeng Cui, F. Hébert, B. Grémaud, V. G. Rousseau, Wenan Guo, G. G. Batrouni

Abstract: We study the ground state phase diagrams of two-photon Dicke, the one-dimensional Jaynes-Cummings-Hubbard (JCH), and Rabi-Hubbard (RH) models using mean field, perturbation, quantum Monte Carlo (QMC), and density matrix renormalization group (DMRG) methods. We first compare mean field predictions for the phase diagram of the Dicke model with exact QMC results and find excellent agreement. The phas… ▽ More We study the ground state phase diagrams of two-photon Dicke, the one-dimensional Jaynes-Cummings-Hubbard (JCH), and Rabi-Hubbard (RH) models using mean field, perturbation, quantum Monte Carlo (QMC), and density matrix renormalization group (DMRG) methods. We first compare mean field predictions for the phase diagram of the Dicke model with exact QMC results and find excellent agreement. The phase diagram of the JCH model is then shown to exhibit a single Mott insulator lobe with two excitons per site, a superfluid (SF, superradiant) phase and a large region of instability where the Hamiltonian becomes unbounded. Unlike the one-photon model, there are no higher Mott lobes. Also unlike the one-photon case, the SF phases above and below the Mott are surprisingly different: Below the Mott, the SF is that of photon {\it pairs} as opposed to above the Mott where it is SF of simple photons. The mean field phase diagram of the RH model predicts a transition from a normal to a superradiant phase but none is found with QMC. △ Less

Submitted 13 April, 2019; v1 submitted 10 April, 2019; originally announced April 2019.

Comments: 14 pages, 14 figures

Journal ref: Phys. Rev. A 100, 033608 (2019)

The Search For Leakage-free Entangling Fibonacci Braiding Gates

Authors: Shawn X. Cui, Kevin T. Tian, Jennifer F. Vasquez, Zhenghan Wang, Helen M. Wong

Abstract: It is an open question if there are leakage-free entangling Fibonacci braiding gates. We provide evidence to the conjecture for the negative in this paper. We also found a much simpler protocol to generate approximately leakage-free entangling Fibonacci braiding gates than existing algorithms in the literature. It is an open question if there are leakage-free entangling Fibonacci braiding gates. We provide evidence to the conjecture for the negative in this paper. We also found a much simpler protocol to generate approximately leakage-free entangling Fibonacci braiding gates than existing algorithms in the literature. △ Less

Submitted 2 April, 2019; originally announced April 2019.

MSC Class: 81P68

Journal ref: Journal of Physics A: Mathematical and Theoretical, vol. 52, no. 45 (2019)

On Generalized Symmetries and Structure of Modular Categories

Authors: Shawn X. Cui, Modjtaba Shokrian Zini, Zhenghan Wang

Abstract: Pursuing a generalization of group symmetries of modular categories to category symmetries in topological phases of matter, we study linear Hopf monads. The main goal is a generalization of extension and gauging group symmetries to category symmetries of modular categories, which include also categorical Hopf algebras as special cases. As an application, we propose an analogue of the classificatio… ▽ More Pursuing a generalization of group symmetries of modular categories to category symmetries in topological phases of matter, we study linear Hopf monads. The main goal is a generalization of extension and gauging group symmetries to category symmetries of modular categories, which include also categorical Hopf algebras as special cases. As an application, we propose an analogue of the classification of finite simple groups to modular categories, where we define simple modular categories as the prime ones without any nontrivial normal algebras. △ Less

Submitted 1 September, 2018; originally announced September 2018.

Journal ref: Science China Mathematics, vol. 62 no. 3 (2019), pp. 417-446

Bit Threads and Holographic Monogamy

Authors: Shawn X. Cui, Patrick Hayden, Temple He, Matthew Headrick, Bogdan Stoica, Michael Walter

Abstract: Bit threads provide an alternative description of holographic entanglement, replacing the Ryu-Takayanagi minimal surface with bulk curves connecting pairs of boundary points. We use bit threads to prove the monogamy of mutual information (MMI) property of holographic entanglement entropies. This is accomplished using the concept of a so-called multicommodity flow, adapted from the network setting,… ▽ More Bit threads provide an alternative description of holographic entanglement, replacing the Ryu-Takayanagi minimal surface with bulk curves connecting pairs of boundary points. We use bit threads to prove the monogamy of mutual information (MMI) property of holographic entanglement entropies. This is accomplished using the concept of a so-called multicommodity flow, adapted from the network setting, and tools from the theory of convex optimization. Based on the bit thread picture, we conjecture a general ansatz for a holographic state, involving only bipartite and perfect-tensor type entanglement, for any decomposition of the boundary into four regions. We also give new proofs of analogous theorems on networks. △ Less

Submitted 28 June, 2019; v1 submitted 15 August, 2018; originally announced August 2018.

Comments: 58 pages, 5 figures; v2: improvements to presentation; references added; essentially matches published version

Journal ref: Commun. Math. Phys. (2019). https://doi.org/10.1007/s00220-019-03510-8

Higher Categories and Topological Quantum Field Theories

Authors: Shawn X. Cui

Abstract: We construct a state-sum type invariant of smooth closed oriented $4$-manifolds out of a $G$-crossed braided spherical fusion category ($G$-BSFC) for $G$ a finite group. The construction can be extended to obtain a $(3+1)$-dimensional topological quantum field theory (TQFT). The invariant of $4$-manifolds generalizes several known invariants in literature such as the Crane-Yetter invariant from a… ▽ More We construct a state-sum type invariant of smooth closed oriented $4$-manifolds out of a $G$-crossed braided spherical fusion category ($G$-BSFC) for $G$ a finite group. The construction can be extended to obtain a $(3+1)$-dimensional topological quantum field theory (TQFT). The invariant of $4$-manifolds generalizes several known invariants in literature such as the Crane-Yetter invariant from a ribbon fusion category and Yetter's invariant from homotopy $2$-types. If the $G$-BSFC is concentrated only at the sector indexed by the trivial group element, a cohomology class in $H^4(G,U(1))$ can be introduced to produce a different invariant, which reduces to the twisted Dijkgraaf-Witten theory in a special case. Although not proven, it is believed that our invariants are strictly different from other known invariants. It remains to be seen if the invariants are sensitive to smooth structures. It is expected that the most general input to the state-sum type construction of $(3+1)$-TQFTs is a spherical fusion $2$-category. We show that a $G$-BSFC corresponds to a monoidal $2$-category with certain extra structure, but that structure does not satisfy all the axioms of a spherical fusion $2$-category given by M. Mackaay. Thus the question of what axioms properly define a spherical fusion $2$-category is open. △ Less

Submitted 3 November, 2019; v1 submitted 24 October, 2016; originally announced October 2016.

Comments: The published version renamed as "Four Dimensional Topological Quantum Field Theories from $G$-crossed Braided Categories" for accuracy

MSC Class: 57R56; 57N13; 18D10;

Journal ref: Quantum Topology, (2019)

Diagonal gates in the Clifford hierarchy

Authors: Shawn X. Cui, Daniel Gottesman, Anirudh Krishna

Abstract: The Clifford hierarchy is a set of gates that appears in the theory of fault-tolerant quantum computation, but its precise structure remains elusive. We give a complete characterization of the diagonal gates in the Clifford hierarchy for prime-dimensional qudits. They turn out to be $p^{m}$-th roots of unity raised to polynomial functions of the basis state to which they are applied, and we determ… ▽ More The Clifford hierarchy is a set of gates that appears in the theory of fault-tolerant quantum computation, but its precise structure remains elusive. We give a complete characterization of the diagonal gates in the Clifford hierarchy for prime-dimensional qudits. They turn out to be $p^{m}$-th roots of unity raised to polynomial functions of the basis state to which they are applied, and we determine which level of the Clifford hierarchy a given gate sits in based on $m$ and the degree of the polynomial. △ Less

Submitted 23 August, 2016; originally announced August 2016.

Comments: 8 pages

Journal ref: Phys. Rev. A 95, 012329 (2017)

Quantum Capacities for Entanglement Networks

Authors: Shawn X Cui, Zhengfeng Ji, Nengkun Yu, Bei Zeng

Abstract: We discuss quantum capacities for two types of entanglement networks: $\mathcal{Q}$ for the quantum repeater network with free classical communication, and $\mathcal{R}$ for the tensor network as the rank of the linear operation represented by the tensor network. We find that $\mathcal{Q}$ always equals $\mathcal{R}$ in the regularized case for the samenetwork graph. However, the relationships bet… ▽ More We discuss quantum capacities for two types of entanglement networks: $\mathcal{Q}$ for the quantum repeater network with free classical communication, and $\mathcal{R}$ for the tensor network as the rank of the linear operation represented by the tensor network. We find that $\mathcal{Q}$ always equals $\mathcal{R}$ in the regularized case for the samenetwork graph. However, the relationships between the corresponding one-shot capacities $\mathcal{Q}_1$ and $\mathcal{R}_1$ are more complicated, and the min-cut upper bound is in general not achievable. We show that the tensor network can be viewed as a stochastic protocol with the quantum repeater network, such that $\mathcal{R}_1$ is a natural upper bound of $\mathcal{Q}_1$. We analyze the possible gap between $\mathcal{R}_1$ and $\mathcal{Q}_1$ for certain networks, and compare them with the one-shot classical capacity of the corresponding classical network. △ Less

Submitted 1 February, 2016; originally announced February 2016.

Improved Quantum Ternary Arithmetics

Authors: Alex Bocharov, Shawn X. Cui, Martin Roetteler, Krysta M. Svore

Abstract: Qutrit (or ternary) structures arise naturally in many quantum systems, particularly in certain non-abelian anyon systems. We present efficient circuits for ternary reversible and quantum arithmetics. Our main result is the derivation of circuits for two families of ternary quantum adders, namely ripple carry adders and carry look-ahead adders. The main difference to the binary case is the more co… ▽ More Qutrit (or ternary) structures arise naturally in many quantum systems, particularly in certain non-abelian anyon systems. We present efficient circuits for ternary reversible and quantum arithmetics. Our main result is the derivation of circuits for two families of ternary quantum adders, namely ripple carry adders and carry look-ahead adders. The main difference to the binary case is the more complicated form of the ternary carry, which leads to higher resource counts for implementations over a universal ternary gate set. Our ternary ripple adder circuit has a circuit depth of $O(n)$ and uses only $1$ ancilla, making it more efficient in both, circuit depth and width than previous constructions. Our ternary carry lookahead circuit has a circuit depth of only $O(\log\,n)$, while using with $O(n)$ ancillas. Our approach works on two levels of abstraction: at the first level, descriptions of arithmetic circuits are given in terms of gates sequences that use various types of non-Clifford reflections. At the second level, we break down these reflections further by deriving them either from the two-qutrit Clifford gates and the non-Clifford gate $C(X): |i,j\rangle \mapsto |i, j + δ_{i,2} \mod 3\rangle$ or from the two-qutrit Clifford gates and the non-Clifford gate $P_9=\mbox{diag}(e^{-2 π\, i/9},1,e^{2 π\, i/9})$. The two choices of elementary gate sets correspond to two possible mappings onto two different prospective quantum computing architectures which we call the metaplectic and the supermetaplectic basis, respectively. Finally, we develop a method to factor diagonal unitaries using multi-variate polynomial over the ternary finite field which allows to characterize classes of gates that can be implemented exactly over the supermetaplectic basis. △ Less

Submitted 9 June, 2016; v1 submitted 11 December, 2015; originally announced December 2015.

Comments: 22 pages, 19 figures, 5 tables, v2 narration improvement after peer review

Journal ref: Quantum Information and Computation, vol. 16, no. 9,10, Rinton Press, July 2016

Quantum Max-flow/Min-cut

Authors: Shawn X. Cui, Michael H. Freedman, Or Sattath, Richard Stong, Greg Minton

Abstract: The classical max-flow min-cut theorem describes transport through certain idealized classical networks. We consider the quantum analog for tensor networks. By associating an integral capacity to each edge and a tensor to each vertex in a flow network, we can also interpret it as a tensor network, and more specifically, as a linear map from the input space to the output space. The quantum max flow… ▽ More The classical max-flow min-cut theorem describes transport through certain idealized classical networks. We consider the quantum analog for tensor networks. By associating an integral capacity to each edge and a tensor to each vertex in a flow network, we can also interpret it as a tensor network, and more specifically, as a linear map from the input space to the output space. The quantum max flow is defined to be the maximal rank of this linear map over all choices of tensors. The quantum min cut is defined to be the minimum product of the capacities of edges over all cuts of the tensor network. We show that unlike the classical case, the quantum max-flow=min-cut conjecture is not true in general. Under certain conditions, e.g., when the capacity on each edge is some power of a fixed integer, the quantum max-flow is proved to equal the quantum min-cut. However, concrete examples are also provided where the equality does not hold. We also found connections of quantum max-flow/min-cut with entropy of entanglement and the quantum satisfiability problem. We speculate that the phenomena revealed may be of interest both in spin systems in condensed matter and in quantum gravity. △ Less

Submitted 30 June, 2016; v1 submitted 19 August, 2015; originally announced August 2015.

Comments: added some motivations; added references on relevant work

MSC Class: 81P45; 94A15; 94A17

Journal ref: J. Math. Phys. 57, 062206 (2016)

Exact analysis of the spectral properties of the anisotropic two-bosons Rabi model

Authors: Shuai Cui, Jun-Peng Cao, Heng Fan, Luigi Amico

Abstract: We introduce the anisotropic two-photon Rabi model in which the rotating and counter rotating terms enters along with two different coupling constants. Eigenvalues and eigenvectors are studied with exact means. We employ a variation of the Braak method based on Bogolubov rotation of the underlying $su(1,1)$ Lie algebra. Accordingly, the spectrum is provided by the analytical properties of a suitab… ▽ More We introduce the anisotropic two-photon Rabi model in which the rotating and counter rotating terms enters along with two different coupling constants. Eigenvalues and eigenvectors are studied with exact means. We employ a variation of the Braak method based on Bogolubov rotation of the underlying $su(1,1)$ Lie algebra. Accordingly, the spectrum is provided by the analytical properties of a suitable meromorphic function. Our formalism applies to the two-modes Rabi model as well, sharing the same algebraic structure of the two-photon model. Through the analysis of the spectrum, we discover that the model displays close analogies to many-body systems undergoing quantum phase transitions. △ Less

Submitted 16 January, 2017; v1 submitted 18 April, 2015; originally announced April 2015.

Comments: 10 pages, 7 figures; typos corrected. To be published in JPA

Efficient Topological Compilation for Weakly-Integral Anyon Model

Authors: Alex Bocharov, Shawn X. Cui, Vadym Kliuchnikov, Zhenghan Wang

Abstract: A class of anyonic models for universal quantum computation based on weakly-integral anyons has been recently proposed. While universal set of gates cannot be obtained in this context by anyon braiding alone, designing a certain type of sector charge measurement provides universality. In this paper we develop a compilation algorithm to approximate arbitrary $n$-qutrit unitaries with asymptotically… ▽ More A class of anyonic models for universal quantum computation based on weakly-integral anyons has been recently proposed. While universal set of gates cannot be obtained in this context by anyon braiding alone, designing a certain type of sector charge measurement provides universality. In this paper we develop a compilation algorithm to approximate arbitrary $n$-qutrit unitaries with asymptotically efficient circuits over the metaplectic anyon model. One flavor of our algorithm produces efficient circuits with upper complexity bound asymptotically in $O(3^{2\,n} \, \log{1/\varepsilon})$ and entanglement cost that is exponential in $n$. Another flavor of the algorithm produces efficient circuits with upper complexity bound in $O(n\,3^{2\,n} \, \log{1/\varepsilon})$ and no additional entanglement cost. △ Less

Submitted 9 June, 2016; v1 submitted 13 April, 2015; originally announced April 2015.

Comments: 15 pages, 5 figures

ACM Class: D.3.4

Journal ref: Phys. Rev. A 93, 012313 (2016)

Generalized Graph States Based on Hadamard Matrices

Authors: Shawn X Cui, Nengkun Yu, Bei Zeng

Abstract: Graph states are widely used in quantum information theory, including entanglement theory, quantum error correction, and one-way quantum computing. Graph states have a nice structure related to a certain graph, which is given by either a stabilizer group or an encoding circuit, both can be directly given by the graph. To generalize graph states, whose stabilizer groups are abelian subgroups of the… ▽ More Graph states are widely used in quantum information theory, including entanglement theory, quantum error correction, and one-way quantum computing. Graph states have a nice structure related to a certain graph, which is given by either a stabilizer group or an encoding circuit, both can be directly given by the graph. To generalize graph states, whose stabilizer groups are abelian subgroups of the Pauli group, one approach taken is to study non-abelian stabilizers. In this work, we propose to generalize graph states based on the encoding circuit, which is completely determined by the graph and a Hadamard matrix. We study the entanglement structures of these generalized graph states, and show that they are all maximally mixed locally. We also explore the relationship between the equivalence of Hadamard matrices and local equivalence of the corresponding generalized graph states. This leads to a natural generalization of the Pauli $(X,Z)$ pairs, which characterizes the local symmetries of these generalized graph states. Our approach is also naturally generalized to construct graph quantum codes which are beyond stabilizer codes. △ Less

Submitted 25 February, 2015; originally announced February 2015.

Comments: 16 pages, 11 figures

Journal ref: J. Math. Phys. 56, 072201 (2015)

Deterministic coupling of delta-doped NV centers to a nanobeam photonic crystal cavity

Authors: Jonathan C. Lee, David O. Bracher, Shanying Cui, Kenichi Ohno, Claire A. McLellan, Xingyu Zhang, Paolo Andrich, Benjamin Aleman, Kasey J. Russell, Andrew P. Magyar, Igor Aharonovich, Ania Bleszynski Jayich, David Awschalom, Evelyn L. Hu

Abstract: The negatively-charged nitrogen vacancy center (NV) in diamond has generated significant interest as a platform for quantum information processing and sensing in the solid state. For most applications, high quality optical cavities are required to enhance the NV zero-phonon line (ZPL) emission. An outstanding challenge in maximizing the degree of NV-cavity coupling is the deterministic placement o… ▽ More The negatively-charged nitrogen vacancy center (NV) in diamond has generated significant interest as a platform for quantum information processing and sensing in the solid state. For most applications, high quality optical cavities are required to enhance the NV zero-phonon line (ZPL) emission. An outstanding challenge in maximizing the degree of NV-cavity coupling is the deterministic placement of NVs within the cavity. Here, we report photonic crystal nanobeam cavities coupled to NVs incorporated by a delta-doping technique that allows nanometer-scale vertical positioning of the emitters. We demonstrate cavities with Q up to ~24,000 and mode volume V ~ $0.47(λ/n)^{3}$ as well as resonant enhancement of the ZPL of an NV ensemble with Purcell factor of ~20. Our fabrication technique provides a first step towards deterministic NV-cavity coupling using spatial control of the emitters. △ Less

Submitted 3 November, 2014; originally announced November 2014.

Comments: 13 pages, 3 figures

Universal Quantum Computation with Metaplectic Anyons

Authors: Shawn X. Cui, Zhenghan Wang

Abstract: We show that braidings of the metaplectic anyons $X_ε$ in $SO(3)_2=SU(2)_4$ with their total charge equal to the metaplectic mode $Y$ supplemented with measurements of the total charge of two metaplectic anyons are universal for quantum computation. We conjecture that similar universal computing models can be constructed for all metaplectic anyon systems $SO(p)_2$ for any odd prime $p\geq 5$. In o… ▽ More We show that braidings of the metaplectic anyons $X_ε$ in $SO(3)_2=SU(2)_4$ with their total charge equal to the metaplectic mode $Y$ supplemented with measurements of the total charge of two metaplectic anyons are universal for quantum computation. We conjecture that similar universal computing models can be constructed for all metaplectic anyon systems $SO(p)_2$ for any odd prime $p\geq 5$. In order to prove universality, we find new conceptually appealing universal gate sets for qutrits and qupits. △ Less

Submitted 15 March, 2015; v1 submitted 30 May, 2014; originally announced May 2014.

Comments: To appear in Journal of Mathematical Physics

MSC Class: 57R56; 81P68

Journal ref: J. Math. Phys. 56, 032202 (2015)

Universal quantum computation with weakly integral anyons

Authors: Shawn X. Cui, Seung-Moon Hong, Zhenghan Wang

Abstract: Harnessing non-abelian statistics of anyons to perform quantum computational tasks is getting closer to reality. While the existence of universal anyons by braiding alone such as the Fibonacci anyon is theoretically a possibility, accessible anyons with current technology all belong to a class that is called weakly integral---anyons whose squared quantum dimensions are integers. We analyze the com… ▽ More Harnessing non-abelian statistics of anyons to perform quantum computational tasks is getting closer to reality. While the existence of universal anyons by braiding alone such as the Fibonacci anyon is theoretically a possibility, accessible anyons with current technology all belong to a class that is called weakly integral---anyons whose squared quantum dimensions are integers. We analyze the computational power of the first non-abelian anyon system with only integral quantum dimensions---$D(S_3)$, the quantum double of $S_3$. Since all anyons in $D(S_3)$ have finite images of braid group representations, they cannot be universal for quantum computation by braiding alone. Based on our knowledge of the images of the braid group representations, we set up three qutrit computational models. Supplementing braidings with some measurements and ancillary states, we find a universal gate set for each model. △ Less

Submitted 20 February, 2014; v1 submitted 28 January, 2014; originally announced January 2014.

Comments: add 2 references and corrected many minor typos

Journal ref: Quantum Information Processing(2015) 14: 2687-2727

Anisotropic Rabi model

Authors: Qiong-Tao Xie, Shuai Cui, Jun-Peng Cao, Luigi Amico, Heng Fan

Abstract: We define the anisotropic Rabi model as the generalization of the spin-boson Rabi model: The Hamiltonian system breaks the parity symmetry; the rotating and counter-rotating interactions are governed by two different coupling constants; a further parameter introduces a phase factor in the counter-rotating terms. The exact energy spectrum and eigenstates of the generalized model is worked out. The… ▽ More We define the anisotropic Rabi model as the generalization of the spin-boson Rabi model: The Hamiltonian system breaks the parity symmetry; the rotating and counter-rotating interactions are governed by two different coupling constants; a further parameter introduces a phase factor in the counter-rotating terms. The exact energy spectrum and eigenstates of the generalized model is worked out. The solution is obtained as an elaboration of a recent proposed method for the isotropic limit of the model. In this way, we provide a long sought solution of a cascade of models with immediate relevance in different physical fields, including i) quantum optics: two-level atom in single mode cross electric and magnetic fields; ii) solid state physics: electrons in semiconductors with Rashba and Dresselhaus spin-orbit coupling; iii) mesoscopic physics: Josephson junctions flux-qubit quantum circuits. △ Less

Submitted 20 May, 2014; v1 submitted 22 January, 2014; originally announced January 2014.

Comments: 5 pages+ 6 pages supplementary, 7 figures, accepted by Phys. Rev. X

Journal ref: Phys. Rev. X 4, 021046 (2014)

Thermalization from a general canonical principle

Authors: Shuai Cui, Jun-Peng Cao, Hui Jing, Heng Fan, Wu-Ming Liu

Abstract: We investigate the time evolution of a generic and finite isolated quantum many-body system starting from a pure quantum state. We find the kinematical general canonical principle proposed by Popescu-Short-Winter for statistical mechanics can be built in a more solid ground by studying the thermalization, i.e. comparing the density matrices themselves rather than the measures of distances. In part… ▽ More We investigate the time evolution of a generic and finite isolated quantum many-body system starting from a pure quantum state. We find the kinematical general canonical principle proposed by Popescu-Short-Winter for statistical mechanics can be built in a more solid ground by studying the thermalization, i.e. comparing the density matrices themselves rather than the measures of distances. In particular, this allows us to explicitly identify that, from any instantaneous pure state after thermalization, the state of subsystem is like from a microcanonical ensemble or a generalized Gibbs ensemble, but neither a canonical nor a thermal ones due to finite-size effect. Our results are expected to bring the task of characterizing the state after thermalization to completion. In addition, thermalization of coupled systems with different temperatures corresponding to mixed initial states is studied. △ Less

Submitted 16 January, 2012; v1 submitted 20 October, 2011; originally announced October 2011.

Comments: 9 pages, 15 figures

Probing the quantum ground state of a spin-1 Bose-Einstein condensate with cavity transmission spectra

Authors: J. M. Zhang, S. Cui, H. Jing, D. L. Zhou, W. M. Liu

Abstract: We propose to probe the quantum ground state of a spin-1 Bose-Einstein condensate with the transmission spectra of an optical cavity. By choosing a circularly polarized cavity mode with an appropriate frequency, we can realize coupling between the cavity mode and the magnetization of the condensate. The cavity transmission spectra then contain information of the magnetization statistics of the c… ▽ More We propose to probe the quantum ground state of a spin-1 Bose-Einstein condensate with the transmission spectra of an optical cavity. By choosing a circularly polarized cavity mode with an appropriate frequency, we can realize coupling between the cavity mode and the magnetization of the condensate. The cavity transmission spectra then contain information of the magnetization statistics of the condensate and thus can be used to distinguish the ferromagnetic and antiferromagnetic quantum ground states. This technique may also be useful for continuous observation of the spin dynamics of a spinor Bose-Einstein condensate. △ Less

Submitted 7 July, 2009; originally announced July 2009.

Comments: 6 pages, 1 figure

Journal ref: Phys.Rev.A 80, 043623 (2009)