Showing 1–50 of 182 results for author: Korepin, V

Scalene Yang--Baxter triples as a source of hidden symmetries beyond the ordinary Yang--Baxter equation

Authors: Pramod Padmanabhan, Somnath Maity, Vladimir Korepin

Abstract: We study a nearest-neighbor non-Hermitian spin chain obtained from one member of an exact non-braided scalene Yang--Baxter triple. Its local Hamiltonian density violates both the difference-form Reshetikhin condition and its general non-difference counterpart, obstructing its realization by a differentiable homogeneous regular solution of the ordinary Yang--Baxter equation. The transfer matrices c… ▽ More We study a nearest-neighbor non-Hermitian spin chain obtained from one member of an exact non-braided scalene Yang--Baxter triple. Its local Hamiltonian density violates both the difference-form Reshetikhin condition and its general non-difference counterpart, obstructing its realization by a differentiable homogeneous regular solution of the ordinary Yang--Baxter equation. The transfer matrices constructed from the regular member do not commute among themselves at distinct spectral parameters. Nevertheless, the scalene Yang--Baxter relation implies cross-commutativity with another transfer matrix constructed from the third member of the scalene triple. We evaluate the latter for arbitrary chain length and show that, on even periodic chains, it is a finite generating function of a non-obvious staggered nilpotent symmetry. The resulting conserved hierarchy belongs entirely to the algebra generated by this single symmetry and hence does not constitute an extensive family of algebraically independent charges. Nevertheless, this example demonstrates that scalene Yang--Baxter triples can act as an algebraic symmetry-discovery mechanism beyond the ordinary self-commuting transfer-matrix framework. △ Less

Submitted 9 August, 2026; originally announced August 2026.

Comments: 10 pages + Appendix + References

Symmetries of the Generalized Yang--Baxter Equations

Authors: Pramod Padmanabhan, Somnath Maity, Akash Sinha, Vladimir Korepin

Abstract: The generalized Yang-Baxter equations are multi-site versions of the standard Yang-Baxter equation. When spectral parameters are included, such equations are expected to lead to integrable Hamiltonians with local interactions involving multiple degrees of freedom. In this work we characterize both the continuous and discrete symmetries of these equations required to establish an equivalence class… ▽ More The generalized Yang-Baxter equations are multi-site versions of the standard Yang-Baxter equation. When spectral parameters are included, such equations are expected to lead to integrable Hamiltonians with local interactions involving multiple degrees of freedom. In this work we characterize both the continuous and discrete symmetries of these equations required to establish an equivalence class of solutions. We find that the set of such symmetries depend on the number of sites on which the equation is supported. In several cases there are more symmetries than the standard Yang-Baxter equation, thus placing heavy constraints on the number of inequivalent solutions and the associated integrable models. △ Less

Submitted 24 June, 2026; originally announced June 2026.

Comments: 21 pages + References ; Submitted to "A Special Issue in Honour of Jarmo Hietarinta''

Palindromic structure of depth-efficient quantum search algorithms

Authors: Kun Zhang, Hai-Long Shi, Xiao-Hui Wang, Vladimir Korepin

Abstract: Grover's algorithm is optimal in query complexity, but not necessarily in circuit depth. We formulate unstructured quantum search as a circuit-depth optimization problem and identify a critical depth ratio separating query optimality from depth optimality. The resulting depth-efficient search operators exhibit a palindromic structure, in which shallow diffusion-like operators symmetrically replace… ▽ More Grover's algorithm is optimal in query complexity, but not necessarily in circuit depth. We formulate unstructured quantum search as a circuit-depth optimization problem and identify a critical depth ratio separating query optimality from depth optimality. The resulting depth-efficient search operators exhibit a palindromic structure, in which shallow diffusion-like operators symmetrically replace selected Grover diffusion layers while preserving efficient amplitude amplification. This structure yields a simple depth-efficiency criterion and an analytic expression for the minimal expected depth. Applying the framework to $X$-type mixers, local diffusion operators, and nested local diffusion operators, we obtain substantial depth reductions over standard Grover search. In particular, nested local constructions reduce the total circuit depth by about $40\%$ when the oracle and Grover diffusion operators have comparable depth. These results reveal the resource-dependent nature of quantum-search optimality and establish palindromic constructions as a systematic route to depth-efficient quantum search algorithms. △ Less

Submitted 30 May, 2026; originally announced June 2026.

Comments: 7 pages, 5 figures

Hidden Ising models from the generalized Yang-Baxter equation

Authors: Akash Sinha, Somnath Maity, Pramod Padmanabhan, Vladimir Korepin

Abstract: We introduce a one dimensional spin $\frac{1}{2}$ Hamiltonian with multi-site interactions, but still local. The algebra of its Hamiltonian densities resembles that of the transverse field Ising model. Using this fact we show that its spectrum is free-fermionic but with a huge degeneracy for each level. The source of the degeneracy is a set of local conserved quantities that act like a classical b… ▽ More We introduce a one dimensional spin $\frac{1}{2}$ Hamiltonian with multi-site interactions, but still local. The algebra of its Hamiltonian densities resembles that of the transverse field Ising model. Using this fact we show that its spectrum is free-fermionic but with a huge degeneracy for each level. The source of the degeneracy is a set of local conserved quantities that act like a classical background field for the quantum system. The thermodynamics of this system is contrasted with the standard Ising model. At the gapless points in the energy spectrum, we show that this system can be derived from the quantum inverse scattering method adapted to a multi-site generalization of the Yang-Baxter equation as introduced by E. Rowell and Z. Wang. The $R$-matrix is constructed using generators of extraspecial 2-groups. This helps us extract all the conserved charges and lay the framework for a general mechanism to generate such multi-site interaction spin systems that are transverse field Ising models under the hood. A remark on how to obtain P. Fendley's free-fermion in disguise models in this formalism is also included. △ Less

Submitted 28 May, 2026; originally announced May 2026.

Comments: 19 pages + References + Appendices, 7 figures

Asymptotic optimality of Grover-Radhakrishnan-Korepin algorithm

Authors: Kun Zhang, Kang-Yuan Chen, Xiao-Hui Wang, Vladimir Korepin

Abstract: Grover's algorithm is a cornerstone of quantum algorithms and is strictly optimal in oracle-query complexity. While the full search problem admits no further improvement, one may trade accuracy for speed in the partial search problem, where the task is to identify only the block containing the target item. The best known quantum algorithm for the partial search problem is the Grover-Radhakrishnan-… ▽ More Grover's algorithm is a cornerstone of quantum algorithms and is strictly optimal in oracle-query complexity. While the full search problem admits no further improvement, one may trade accuracy for speed in the partial search problem, where the task is to identify only the block containing the target item. The best known quantum algorithm for the partial search problem is the Grover-Radhakrishnan-Korepin (GRK) algorithm, whose optimality has long been conjectured but not proved. In this work, we prove the optimality of GRK in the large-block limit. We formulate partial search as a time-optimal control problem and apply the Pontryagin maximum principle to derive the switching-function dynamics, establish the bang-bang structure of regular extremals, and exclude non-optimal switching patterns. As a result, we show that the optimal regular extremal has the global-local-global form, which yields a control-theoretic proof of the asymptotic optimality of the GRK algorithm in oracle-query complexity. △ Less

Submitted 6 August, 2026; v1 submitted 17 April, 2026; originally announced April 2026.

Comments: Close to published version, 24 pages, 1 figure

Journal ref: J. Phys. A: Math. Theor. 59, 315301 (2026)

Asymptotic bounds on quantum partial search algorithm and its applications to parallel search

Authors: Yan-Bo Jiang, Xiao-Hui Wang, Kun Zhang, Vladimir Korepin

Abstract: Grover's algorithm provides a quadratic speedup over classical algorithms for searching an unstructured database and is known to be strictly optimal in oracle query complexity, with tight bounds on its success probability. Although the standard Grover search cannot be further accelerated in the full-search setting, a trade-off between accuracy and query complexity gives rise to the partial search… ▽ More Grover's algorithm provides a quadratic speedup over classical algorithms for searching an unstructured database and is known to be strictly optimal in oracle query complexity, with tight bounds on its success probability. Although the standard Grover search cannot be further accelerated in the full-search setting, a trade-off between accuracy and query complexity gives rise to the partial search problem. The Grover-Radhakrishnan-Korepin (GRK) algorithm is the standard and most extensively studied protocol for this task. In this work, we provide systematic numerical evidence that the GRK operator sequence gives the highest success probability in all examined cases, supporting it as the optimal ansatz among admissible compositions of global and local Grover operators. Guided by this numerically supported GRK ansatz, we derive an asymptotically tight upper bound on the maximal success probability within the GRK family and establish the corresponding lower bound on the minimal expected number of oracle queries. Furthermore, we investigate parallel quantum search within the partial-search framework. While a direct GRK-based parallelization does not outperform established parallel Grover schemes, we demonstrate that a hybrid strategy combining partial and full search protocols yields a strict, though subleading, improvement over the outer parallel Grover scheme. Our results clarify the fundamental limits of quantum partial search and its role in optimizing parallel quantum search algorithms. △ Less

Submitted 7 July, 2026; v1 submitted 2 March, 2026; originally announced March 2026.

Comments: Close to published version, 18 pages, 4 figures, 6 tables

Journal ref: Phys. Rev. A 114, 012412 (2026)

Minimal nonintegrable models with three-site interactions

Authors: Wen-Ming Fan, Kun Hao, Yang-Yang Chen, Xiao-Hui Wang, Kun Zhang, Vladimir Korepin

Abstract: We study integrability breaking in translationally invariant spin-$1/2$ chains with genuine three-site interactions. Using a two-qubit composite representation, we prove that the deformed Fredkin spin chain is nonintegrable for any nonzero deformation parameter, although its Hamiltonian decomposes into coupled integrable building blocks. We then extract the minimal injective models, defined as the… ▽ More We study integrability breaking in translationally invariant spin-$1/2$ chains with genuine three-site interactions. Using a two-qubit composite representation, we prove that the deformed Fredkin spin chain is nonintegrable for any nonzero deformation parameter, although its Hamiltonian decomposes into coupled integrable building blocks. We then extract the minimal injective models, defined as the simplest models needed for a rigorous nonintegrability test, from the Hamiltonian density of the deformed Fredkin spin chain. Among the sixteen such models, six satisfy the Reshetikhin condition and are integrable, while two satisfy Hokkyo's criterion and are nonintegrable away from special coefficient values. Returning to one-site translation-invariant spin-$1/2$ chains leaves four genuine three-site models: two integrable models and two generically nonintegrable models. These results give a compact classification of minimal three-site models obtained from the deformed Fredkin spin chain and identify a local mechanism by which coupling individually integrable structures can destroy integrability beyond nearest-neighbor interactions. △ Less

Submitted 20 August, 2026; v1 submitted 8 February, 2026; originally announced February 2026.

Comments: 18 pages, 3 figures, 6 tables

Thermodynamics of the Heisenberg XXX chain with negative spin

Authors: Rong Zhong, Yang-Yang Chen, Kun Hao, Wen-li Yang, Vladimir Korepin

Abstract: We study the thermodynamics of the isotropic Heisenberg XXX spin chain with negative spin, focusing on the case $s=-1$. The model is equivalent to the quantum lattice nonlinear Schrödinger (NLS) model and appears as an effective theory in deep inelastic scattering in high-energy quantum chromodynamics. Owing to its integrability, it admits a consistent Bethe Ansatz description and a well-defined t… ▽ More We study the thermodynamics of the isotropic Heisenberg XXX spin chain with negative spin, focusing on the case $s=-1$. The model is equivalent to the quantum lattice nonlinear Schrödinger (NLS) model and appears as an effective theory in deep inelastic scattering in high-energy quantum chromodynamics. Owing to its integrability, it admits a consistent Bethe Ansatz description and a well-defined thermodynamic limit. Using the thermodynamic Bethe Ansatz, we analyze the ground state, elementary excitations, and finite-temperature properties. In contrast to the conventional positive spin XXX chain, the negative spin model exhibits a distinct vacuum structure and excitation spectrum, leading to modified TBA equations and unconventional low-temperature behavior. Although the integral equations resemble those of the Lieb-Liniger Bose gas, the thermodynamics and scaling properties are qualitatively different and cannot be continuously connected. We derive the free energy, entropy, and specific heat, and identify a quantum phase transition separating different thermodynamic regimes. At zero temperature, the excitation spectrum becomes linear in the continuum limit and can be described by a conformal field theory. The low-temperature regime realizes a Luttinger-liquid like phase with features unique to the negative spin XXX chain. △ Less

Submitted 13 August, 2026; v1 submitted 3 February, 2026; originally announced February 2026.

Comments: 41 pages, 15 figures

Journal ref: JHEP 08 (2026) 101

Noninvertible Kramers-Wannier duality symmetries for the discrete-time quantum Ising chain

Authors: Akash Sinha, Pramod Padmanabhan, Vladimir Korepin

Abstract: Integrable trotterization} provides a method to evolve a continuous time integrable many-body system in discrete time, such that it retains its conserved quantities. Here we explicitly show that the first order trotterization of the critical {\it transverse field Ising model} is integrable. The discrete time conserved quantities are obtained from an inhomogeneous transfer matrix constructed using… ▽ More Integrable trotterization} provides a method to evolve a continuous time integrable many-body system in discrete time, such that it retains its conserved quantities. Here we explicitly show that the first order trotterization of the critical {\it transverse field Ising model} is integrable. The discrete time conserved quantities are obtained from an inhomogeneous transfer matrix constructed using the {\it quantum inverse scattering method}. The inhomogeneity parameter determines the discrete time step. We then focus on the non-invertible {\it Kramers-Wannier} duality-symmetry for the trotterized evolution. We find that the discretization of both space and time leads to a doubling of these duality operators. They account for discrete translations in both space and time. As an interesting application, we find that these operators also provide maps between trotterizations of different orders. This helps us extend our results beyond the trotterization scheme and investigate the Kramers-Wannier duality-symmetry for finite time Floquet evolution of the critical transverse field Ising chain. {Finally, we investigate how these non-invertible operators shape the phase diagram of the discrete-time evolution. This question is particularly interesting in the Floquet setting, which is known to host a richer phase structure than its undriven counterpart. We systematically construct the necessary operators which relate different phases away from criticality for both trotterized and Floquet evolutions. △ Less

Submitted 24 June, 2026; v1 submitted 5 November, 2025; originally announced November 2025.

Comments: 8 pages + Refs + Appendices ; v2 12 pages + Refs + Appendices, Includes discussion of KW duality for Floquet evolution of the TFIM, close to the published version

Journal ref: Phys. Rev. B 113, 235141 (2026)

Clifford Solver for the Tetrahedron Equation and its Variants

Authors: Pramod Padmanabhan, Vivek Kumar Singh, Vladimir Korepin

Abstract: The different forms of the tetrahedron equation appear when all possible ways to label the scattering process of infinitely long straight lines are considered in three dimensional spacetime. This is expected to lead to three dimensional integrability, analogous to the Yang-Baxter equation. Among the three possibilities, we consider two of them and their variants. We show that Clifford algebras sol… ▽ More The different forms of the tetrahedron equation appear when all possible ways to label the scattering process of infinitely long straight lines are considered in three dimensional spacetime. This is expected to lead to three dimensional integrability, analogous to the Yang-Baxter equation. Among the three possibilities, we consider two of them and their variants. We show that Clifford algebras solve both the constant and the spectral parameter dependent versions of all of them. We also present a scheme for canonically solving higher simplex equations using tetrahedron solutions. △ Less

Submitted 27 October, 2025; originally announced October 2025.

Comments: 6 pages, Submitted to "Proceedings of the XIII International Symposium on Quantum Theory and Symmetries (QTS-13)''

Quantum Utility in Simulating the Real-time Dynamics of the Fermi-Hubbard Model using Superconducting Quantum Computers

Authors: Talal Ahmed Chowdhury, Vladimir Korepin, Vincent R. Pascuzzi, Kwangmin Yu

Abstract: The Fermi-Hubbard model is a fundamental model in condensed matter physics that describes strongly correlated electrons. On the other hand, quantum computers are emerging as powerful tools for exploring the complex dynamics of these quantum many-body systems. In this work, we demonstrate the quantum simulation of the one-dimensional Fermi-Hubbard model using IBM's superconducting quantum computers… ▽ More The Fermi-Hubbard model is a fundamental model in condensed matter physics that describes strongly correlated electrons. On the other hand, quantum computers are emerging as powerful tools for exploring the complex dynamics of these quantum many-body systems. In this work, we demonstrate the quantum simulation of the one-dimensional Fermi-Hubbard model using IBM's superconducting quantum computers, employing over 100 qubits. We introduce a first-order Trotterization scheme and extend it to an optimized second-order Trotterization for the time evolution in the Fermi-Hubbard model, specifically tailored for the limited qubit connectivity of quantum architectures, such as IBM's platforms. Notably, both Trotterization approaches are scalable and maintain a constant circuit depth at each Trotter step, regardless of the qubit count, enabling us to precisely investigate the relaxation dynamics in the Fermi-Hubbard model by measuring the expectation value of the Néel observable (staggered magnetization) for time-evolved quantum states. Finally, our successful measurement of expectation values in such large-scale quantum many-body systems, especially at longer time scales with larger entanglement, highlights the quantum utility of superconducting quantum platforms over conventional classical approximation methods. △ Less

Submitted 25 May, 2026; v1 submitted 17 September, 2025; originally announced September 2025.

Comments: 19 pages, 10 figures, matches published version, additional typos corrected

Journal ref: Appl. Phys. Rev. 13, 011434 (2026)

Absence of local conserved charges of the Fredkin spin chain and its truncated versions

Authors: Wen-Ming Fan, Kun Hao, Yang-Yang Chen, Kun Zhang, Xiao-Hui Wang, Vladimir Korepin

Abstract: Conservation laws serve as the hallmark of integrability. The absence of conserved charges typically implies that the model is nonintegrable. The recently proposed Fredkin spin chain exhibits rich structures, and its ground state is analytically known. However, whether the Fredkin spin chain is integrable remains an open question. In this work, through rigorous analytical calculations, we demonstr… ▽ More Conservation laws serve as the hallmark of integrability. The absence of conserved charges typically implies that the model is nonintegrable. The recently proposed Fredkin spin chain exhibits rich structures, and its ground state is analytically known. However, whether the Fredkin spin chain is integrable remains an open question. In this work, through rigorous analytical calculations, we demonstrate that the Fredkin spin chain, under both periodic and open boundary conditions, lacks local conserved charges, thereby confirming its nonintegrable nature. Furthermore, we find that when one or a portion of the Hamiltonian terms are removed (referred to as the truncated Fredkin spin chain), local conserved charges are still absent. Our findings suggest that in models involving three-site interactions, integrable models are generally rare. △ Less

Submitted 23 November, 2025; v1 submitted 5 September, 2025; originally announced September 2025.

Comments: Close to published version, 20 pages, 16 tables

Journal ref: Phys. Rev. B 112, 205124 (2025)

Small $x$ behavior in QCD from maximal entanglement and conformal invariance

Authors: Sebastian Grieninger, Kun Hao, Dmitri E. Kharzeev, Vladimir Korepin

Abstract: Recent evidence suggests that, at small Bjorken $x$, QCD evolution drives the proton into a state of maximal entanglement. If the evolution kernel is assumed to be conformally invariant -- as is the case for the Balitsky-Fadin-Kuraev-Lipatov (BFKL) equation -- we can describe it by a conformal field theory. Moreover, the central charge $c$ of the corresponding conformal field theory emerges as the… ▽ More Recent evidence suggests that, at small Bjorken $x$, QCD evolution drives the proton into a state of maximal entanglement. If the evolution kernel is assumed to be conformally invariant -- as is the case for the Balitsky-Fadin-Kuraev-Lipatov (BFKL) equation -- we can describe it by a conformal field theory. Moreover, the central charge $c$ of the corresponding conformal field theory emerges as the key parameter governing the $x$-dependence of both the entanglement entropy and the structure function. Here we apply the exact Bethe Ansatz methods to the quantum spin chain dual to Lipatov's high energy effective action to extract the central charge of the theory, and find that $c=1$. This implies the $\sim x^{-1/3}$ small $x$ behavior for the structure function -- the prediction that can be tested at the forthcoming Electron-Ion Collider. △ Less

Submitted 23 May, 2026; v1 submitted 29 August, 2025; originally announced August 2025.

Comments: v2: added additional clarifications; 5 + 3 pages

Journal ref: Phys. Rev. D 113, 094030 (2026)

Almost local integrable models from supersymmetry algebras

Authors: Somnath Maity, Pramod Padmanabhan, Jarmo Hietarinta, Vladimir Korepin

Abstract: Supersymmetry algebras can be used to obtain algebraic expressions for constant Yang-Baxter solutions, also known as braid group generators. This was done for non-invertible braid operators in \cite{maity2025non}. In this work we extend this construction for the invertible ones. The resulting expressions are then shown to obey relations analogous to those satisfied by quotients of braid groups. Ex… ▽ More Supersymmetry algebras can be used to obtain algebraic expressions for constant Yang-Baxter solutions, also known as braid group generators. This was done for non-invertible braid operators in \cite{maity2025non}. In this work we extend this construction for the invertible ones. The resulting expressions are then shown to obey relations analogous to those satisfied by quotients of braid groups. Examples of the latter include the Iwahori-Hecke algebra and the Birman-Murakami-Wenzl (BMW) algebra. As a result, we can Baxterize the constant Yang-Baxter solutions to yield spectral parameter dependent $R$-matrices. The regularity of these $R$-matrices depends on the representation of SUSY generators. In some cases they are regular in the usual sense and in the remaining they are `almost' regular. In the latter case they are also non-invertible. Nevertheless, we show that they can still help us construct integrable models in all dimensions of the local Hilbert space. These models can be described by Hamiltonian densities that are either local or non-local, depending on the representation chosen for the SUSY generators. We demonstrate this for all constant $4\times 4$ invertible Yang-Baxter solutions. Apart from finding new nearest-neighbor interaction spin $\frac{1}{2}$ systems, we also find their higher spin analogs due to the algebraic [representation independent] approach. △ Less

Submitted 2 June, 2026; v1 submitted 6 August, 2025; originally announced August 2025.

Comments: 40 pages + Appendices + References ; v2 43 pages + References + 1 figure, Section 5 on Higher spin integrable systems expanded, Closer to the published version

Journal ref: J. Phys. A: Math. Theor. 59 205202, 2026

The Yang-Baxter integrability of the critical Ising chain

Authors: Akash Sinha, Tinu Justin, Pramod Padmanabhan, Vladimir Korepin

Abstract: We show that the one dimensional, critical transverse field Ising model is Yang-Baxter integrable. This is done by constructing commuting transfer matrices built out of a $R$-matrix satisfying the Yang-Baxter equation with additive spectral parameters. The $R$-matrix is non-local, as it is expressed in terms of Majorana fermions. It is also non-regular. Nevertheless, we show that the quantum inver… ▽ More We show that the one dimensional, critical transverse field Ising model is Yang-Baxter integrable. This is done by constructing commuting transfer matrices built out of a $R$-matrix satisfying the Yang-Baxter equation with additive spectral parameters. The $R$-matrix is non-local, as it is expressed in terms of Majorana fermions. It is also non-regular. Nevertheless, we show that the quantum inverse scattering method can still be suitably adapted. We then recursively obtain the conserved quantities [in the infinite volume] by the boost operator method. Remarkably, among the conserved charges we also find the Kramers-Wannier duality and other non-invertible symmetries for the periodic transverse field Ising model. △ Less

Submitted 10 October, 2025; v1 submitted 4 June, 2025; originally announced June 2025.

Comments: v3 closer to the published version 29 pages + appendices + references

Journal ref: J. Stat. Mech. (2025) 103102

Non-hermitian integrable systems from constant non-invertible solutions of the Yang-Baxter equation

Authors: Somnath Maity, Pramod Padmanabhan, Vladimir Korepin

Abstract: We construct invertible spectral parameter dependent Yang-Baxter solutions ($R$-matrices) by Baxterizing constant non-invertible Yang-Baxter solutions. The solutions are algebraic (representation independent). They are constructed using supersymmetry (SUSY) algebras. The resulting $R$-matrices are regular leading to local non-hermitian Hamiltonians written in terms of the SUSY generators. As parti… ▽ More We construct invertible spectral parameter dependent Yang-Baxter solutions ($R$-matrices) by Baxterizing constant non-invertible Yang-Baxter solutions. The solutions are algebraic (representation independent). They are constructed using supersymmetry (SUSY) algebras. The resulting $R$-matrices are regular leading to local non-hermitian Hamiltonians written in terms of the SUSY generators. As particular examples we Baxterize the $4\times 4$ constant non-invertible solutions of Hietarinta leading to nearest-neighbor Hamiltonians. On comparing with the literature we find two of the models are new. Apart from being non-hermitian, many of them are also non-diagonalizable with interesting spectrums. With appropriate representations of the SUSY generators we obtain spin chains in all local Hilbert space dimensions. △ Less

Submitted 11 March, 2025; originally announced March 2025.

Comments: 36 pages + references

Journal ref: JHEP05(2025)206

Non-invertible symmetry breaking in a frustration-free spin chain

Authors: Akash Sinha, Vivek Kumar Singh, Pramod Padmanabhan, Kun Hao, Vladimir Korepin

Abstract: A nearest-neighbor, frustration-free spin $\frac{1}{2}$ chain can be constructed {\it via} projectors of various ranks á la Bravyi-Gosset. We show that in the rank 1 case this system is gapped and has two ground states resembling ferromagnetic states. These states spontaneously break the non-invertible symmetry connecting them. The latter is proved using the machinery of algebraic quantum theory.… ▽ More A nearest-neighbor, frustration-free spin $\frac{1}{2}$ chain can be constructed {\it via} projectors of various ranks á la Bravyi-Gosset. We show that in the rank 1 case this system is gapped and has two ground states resembling ferromagnetic states. These states spontaneously break the non-invertible symmetry connecting them. The latter is proved using the machinery of algebraic quantum theory. The non-invertible symmetries of this system do not come from a duality. △ Less

Submitted 17 February, 2025; originally announced February 2025.

Comments: 4 pages, Submitted to the conference proceedings for the 26th DAE-BRNS High Energy Physics (HEP) Symposium 2024, India. A longer version is under preparation

Majorana fermions solve the tetrahedron equations as well as higher simplex equations

Authors: Pramod Padmanabhan, Vladimir Korepin

Abstract: Yang-Baxter equations define quantum integrable models. The tetrahedron and higher simplex equations are multi-dimensional generalizations. Finding the solutions of these equations is a formidable task. In this work we develop a systematic method - constructing higher simplex operators [solutions of corresponding simplex equations] from lower simplex ones. We call it lifting. By starting from solu… ▽ More Yang-Baxter equations define quantum integrable models. The tetrahedron and higher simplex equations are multi-dimensional generalizations. Finding the solutions of these equations is a formidable task. In this work we develop a systematic method - constructing higher simplex operators [solutions of corresponding simplex equations] from lower simplex ones. We call it lifting. By starting from solutions of Yang-Baxter equations we can construct solutions of the tetrahedron equation and simplex equation in any dimension. We then generalize this by starting from a solution of any lower simplex equation and lifting it [construct solution] to another simplex equation in higher dimension. This process introduces several constraints among the different lower simplex operators that are lifted to form the higher simplex operators. We show that braided Yang-Baxter operators [solutions of Yang-Baxter equations independent of spectral parameters] constructed using Majorana fermions satisfy these constraints, thus solving the higher simplex equations. As a consequence these solutions help us understand the action of an higher simplex operator on Majorana fermions. Apart from these we show that solutions constructed using Dirac (complex) fermions and Clifford algebras also satisfy these constraints. Furthermore it is observed that the Clifford solutions give rise to positive Boltzmann weights resulting in the possibility of physical statistical mechanics models in higher dimensions. Finally we also show that anti-Yang-Baxter operators [solutions of Yang-Baxter-like equations with a negative sign on the right hand side] can also be lifted to higher simplex solutions. △ Less

Submitted 13 March, 2025; v1 submitted 26 October, 2024; originally announced October 2024.

Comments: 27 pages + References ; v2 36 pages + References : Published version, includes a discussion of positivity of Boltzmann weights and more Majorana solutions

Journal ref: Nuclear Physics B (2025), 0550-3213, 116865

Algebraic classification of Hietarinta's solutions of Yang-Baxter equations~:~invertible $4\times 4$ operators

Authors: Somnath Maity, Vivek Kumar Singh, Pramod Padmanabhan, Vladimir Korepin

Abstract: In order to examine the simulation of integrable quantum systems using quantum computers, it is crucial to first classify Yang-Baxter operators. Hietarinta was among the first to classify constant Yang-Baxter solutions for a two-dimensional local Hilbert space (qubit representation). Including the one produced by the permutation operator, he was able to construct eleven families of invertible solu… ▽ More In order to examine the simulation of integrable quantum systems using quantum computers, it is crucial to first classify Yang-Baxter operators. Hietarinta was among the first to classify constant Yang-Baxter solutions for a two-dimensional local Hilbert space (qubit representation). Including the one produced by the permutation operator, he was able to construct eleven families of invertible solutions. These techniques are effective for 4 by 4 solutions, but they become difficult to use for representations with more dimensions. To get over this limitation, we use algebraic ansätze to generate the constant Yang-Baxter solutions in a representation independent way. We employ four distinct algebraic structures that, depending on the qubit representation, replicate 10 of the 11 Hietarinta families. Among the techniques are partition algebras, Clifford algebras, Temperley-Lieb algebras, and a collection of commuting operators. Using these techniques, we do not obtain the $(2,2)$ Hietarinta class. △ Less

Submitted 2 January, 2025; v1 submitted 9 September, 2024; originally announced September 2024.

Comments: v1 - 26 pages + References ; v2, published version - 29 pages + References

Journal ref: J. High Energ. Phys. 2024, 67 (2024)

Near-deterministic quantum search algorithm without phase design

Authors: Zhen Wang, Kun Zhang, Vladimir Korepin

Abstract: Grover's algorithm solves the unstructured search problem. Grover's algorithm can find the target state with certainty only if searching one out of four. Designing the deterministic search algorithm can avoid any repetition of the algorithm, especially when Grover's algorithm is a subroutine in other algorithms. Grover's algorithm can be deterministic if the phase of the oracle or the diffusion op… ▽ More Grover's algorithm solves the unstructured search problem. Grover's algorithm can find the target state with certainty only if searching one out of four. Designing the deterministic search algorithm can avoid any repetition of the algorithm, especially when Grover's algorithm is a subroutine in other algorithms. Grover's algorithm can be deterministic if the phase of the oracle or the diffusion operator is delicately designed. The precision of the phases could be a problem. A near-deterministic quantum search algorithm without the phase design is proposed. The algorithm has the same oracle and diffusion operators as Grover's algorithm. One additional component is the rescaled diffusion operator. It acts partially on the database. The success probability of Grover's algorithm is improved by the partial diffusion operator in two different ways. The possible cost is one or two more queries to the oracle. The deterministic search algorithm is also designed when searching one out of eight, sixteen, and thirty-two. △ Less

Submitted 13 January, 2025; v1 submitted 15 July, 2024; originally announced July 2024.

Comments: Published version, 9 pages, 3 figures, 2 tables

Journal ref: Adv. Quantum Technol. 2025, 2400557

Unitary tetrahedron quantum gates

Authors: Vivek Kumar Singh, Akash Sinha, Pramod Padmanabhan, Vladimir Korepin

Abstract: Quantum simulations of many-body systems using 2-qubit Yang-Baxter gates offer a benchmark for quantum hardware. This can be extended to the higher dimensional case with $n$-qubit generalisations of Yang-Baxter gates called $n$-simplex operators. Such multi-qubit gates potentially lead to shallower and more efficient quantum circuits as well. Finding them amounts to identifying unitary solutions o… ▽ More Quantum simulations of many-body systems using 2-qubit Yang-Baxter gates offer a benchmark for quantum hardware. This can be extended to the higher dimensional case with $n$-qubit generalisations of Yang-Baxter gates called $n$-simplex operators. Such multi-qubit gates potentially lead to shallower and more efficient quantum circuits as well. Finding them amounts to identifying unitary solutions of the $n$-simplex equations, the building blocks of higher dimensional integrable systems. These are a set of highly non-linear and over determined system of equations making it notoriously hard to solve even when the local Hilbert spaces are spanned by qubits. We systematically overcome this for higher simplex operators constructed using two methods: from Clifford algebras and by lifting Yang-Baxter operators. The $n=3$ or the tetrahedron case is analyzed in detail. For the qubit case our methods produce 13 inequivalent families of unitary tetrahedron operators. 12 of these families are obtained by appending the 5 unitary families of 4 by 4 constant Yang-Baxter operators of Dye-Hietarinta, with a single qubit operator. As applications, universal sets of single, two and three qubit gates are realized using such unitary tetrahedron operators. The ideas presented in this work can be naturally extended to the higher simplex cases. △ Less

Submitted 25 July, 2024; v1 submitted 15 July, 2024; originally announced July 2024.

Comments: 34 pages of main text + 4 pages of appendices + 8 pages of references; v2 Fixed minor typos, updated Acknowledgments

Geometric representations of braid and Yang-Baxter gates

Authors: Kun Zhang, Kun Hao, Kwangmin Yu, Vladimir Korepin, Wen-Li Yang

Abstract: Brick-wall circuits composed of the Yang-Baxter gates are integrable. It becomes an important tool to study the quantum many-body system out of equilibrium. To put the Yang-Baxter gate on quantum computers, it has to be decomposed into the native gates of quantum computers. It is favorable to apply the least number of native two-qubit gates to construct the Yang-Baxter gate. We study the geometric… ▽ More Brick-wall circuits composed of the Yang-Baxter gates are integrable. It becomes an important tool to study the quantum many-body system out of equilibrium. To put the Yang-Baxter gate on quantum computers, it has to be decomposed into the native gates of quantum computers. It is favorable to apply the least number of native two-qubit gates to construct the Yang-Baxter gate. We study the geometric representations of all X-type braid gates and their corresponding Yang-Baxter gates via the Yang-Baxterization. We find that the braid and Yang-Baxter gates can only exist on certain edges and faces of the two-qubit tetrahedron. We identify the parameters by which the braid and Yang-Baxter gates are the Clifford gate, the matchgate, and the dual-unitary gate. The geometric representations provide the optimal decompositions of the braid and Yang-Baxter gates in terms of other two-qubit gates. We also find that the entangling powers of the Yang-Baxter gates are determined by the spectral parameters. Our results provide the necessary conditions to construct the braid and Yang-Baxter gates on quantum computers. △ Less

Submitted 22 October, 2024; v1 submitted 12 June, 2024; originally announced June 2024.

Comments: Published version, 27 pages, 7 figures, 2 tables

Journal ref: J. Phys. A: Math. Theor. 57 445303 (2024)

Toffoli gates solve the tetrahedron equations

Authors: Akash Sinha, Pramod Padmanabhan, Vladimir Korepin

Abstract: The circuit model of quantum computation can be interpreted as a scattering process. In particular, factorised scattering operators result in integrable quantum circuits that provide universal quantum computation and are potentially less noisy. These are realized through Yang-Baxter or 2-simplex operators. A natural question is to extend this construction to higher qubit gates, like the Toffoli ga… ▽ More The circuit model of quantum computation can be interpreted as a scattering process. In particular, factorised scattering operators result in integrable quantum circuits that provide universal quantum computation and are potentially less noisy. These are realized through Yang-Baxter or 2-simplex operators. A natural question is to extend this construction to higher qubit gates, like the Toffoli gates, which also lead to universal quantum computation but with shallower circuits. We show that unitary families of such operators are constructed by the 3-dimensional generalizations of the Yang-Baxter operators known as tetrahedron or 3-simplex operators. The latter satisfy a spectral parameter-dependent tetrahedron equation. This construction goes through for $n$-Toffoli gates realized using $n$-simplex operators. △ Less

Submitted 26 May, 2024; originally announced May 2024.

Comments: 8 pages + References

Solving the Yang-Baxter, tetrahedron and higher simplex equations using Clifford algebras

Authors: Pramod Padmanabhan, Vladimir Korepin

Abstract: Bethe Ansatz was discoverd in 1932. Half a century later its algebraic structure was unearthed: Yang-Baxter equation was discovered, as well as its multidimensional generalizations [tetrahedron equation and $d$-simplex equations]. Here we describe a universal method to solve these equations using Clifford algebras. The Yang-Baxter equation ($d=2$), Zamalodchikov's tetrahedron equation ($d=3$) and… ▽ More Bethe Ansatz was discoverd in 1932. Half a century later its algebraic structure was unearthed: Yang-Baxter equation was discovered, as well as its multidimensional generalizations [tetrahedron equation and $d$-simplex equations]. Here we describe a universal method to solve these equations using Clifford algebras. The Yang-Baxter equation ($d=2$), Zamalodchikov's tetrahedron equation ($d=3$) and the Bazhanov-Stroganov equation ($d=4$) are special cases. Our solutions form a linear space. This helps us to include spectral parameters. Potential applications are discussed. △ Less

Submitted 8 May, 2024; v1 submitted 17 April, 2024; originally announced April 2024.

Comments: v2 70 pages, Includes missing solutions, all of which are succinctly captured in Theorems 1 and 2

Journal ref: Nuclear Physics B Volume 1007 , October 2024, 116664

Quantum simulation of entanglement and hadronization in jet production: lessons from the massive Schwinger model

Authors: Adrien Florio, David Frenklakh, Kazuki Ikeda, Dmitri E. Kharzeev, Vladimir Korepin, Shuzhe Shi, Kwangmin Yu

Abstract: The possible link between entanglement and thermalization, and the dynamics of hadronization are addressed by studying the real-time response of the massive Schwinger model coupled to external sources. This setup mimics the production and fragmentation of quark jets, as the Schwinger model and QCD share the properties of confinement and chiral symmetry breaking. By using quantum simulations on cla… ▽ More The possible link between entanglement and thermalization, and the dynamics of hadronization are addressed by studying the real-time response of the massive Schwinger model coupled to external sources. This setup mimics the production and fragmentation of quark jets, as the Schwinger model and QCD share the properties of confinement and chiral symmetry breaking. By using quantum simulations on classical hardware, we study the entanglement between the produced jets, and observe the growth of the corresponding entanglement entropy in time. This growth arises from the increased number of contributing eigenstates of the reduced density matrix with sufficiently large and close eigenvalues. We also investigate the physical nature of these eigenstates, and find that at early times they correspond to fermionic Fock states. We then observe the transition from these fermionic Fock states to meson-like bound states as a function of time. In other words, we observe how hadronization develops in real time. At late times, the local observables at mid-rapidity (such as the fermion density and the electric field) approach approximately constant values, suggesting the onset of equilibrium and approach to thermalization. △ Less

Submitted 29 March, 2024; originally announced April 2024.

Comments: 10 pages, 12 figures

Yang-Baxter solutions from commuting operators

Authors: Pramod Padmanabhan, Kun Hao, Vladimir Korepin

Abstract: We construct $R$-matrices (with a multidimensional spectral parameter) that include additive as well as non-additive parameters. They satisfy the colored Yang-Baxter equation. The solutions depend on a set of commuting operators. They change with the representation of the latter. The associated Yang-Baxter algebra and the spectrum of the transfer matrix are also studied. We construct $R$-matrices (with a multidimensional spectral parameter) that include additive as well as non-additive parameters. They satisfy the colored Yang-Baxter equation. The solutions depend on a set of commuting operators. They change with the representation of the latter. The associated Yang-Baxter algebra and the spectrum of the transfer matrix are also studied. △ Less

Submitted 10 April, 2024; v1 submitted 10 January, 2024; originally announced January 2024.

Comments: 25 pages, v3 shortened version, includes only the new multi-parameter and non-regular R-matrices

Optimal realization of Yang-Baxter gate on quantum computers

Authors: Kun Zhang, Kwangmin Yu, Kun Hao, Vladimir Korepin

Abstract: Quantum computers provide a promising method to study the dynamics of many-body systems beyond classical simulation. On the other hand, the analytical methods developed and results obtained from the integrable systems provide deep insights on the many-body system. Quantum simulation of the integrable system not only provides a valid benchmark for quantum computers but is also the first step in stu… ▽ More Quantum computers provide a promising method to study the dynamics of many-body systems beyond classical simulation. On the other hand, the analytical methods developed and results obtained from the integrable systems provide deep insights on the many-body system. Quantum simulation of the integrable system not only provides a valid benchmark for quantum computers but is also the first step in studying integrable-breaking systems. The building block for the simulation of an integrable system is the Yang-Baxter gate. It is vital to know how to optimally realize the Yang-Baxter gates on quantum computers. Based on the geometric picture of the Yang-Baxter gates, we present the optimal realizations of two types of Yang-Baxter gates with a minimal number of CNOT or $R_{zz}$ gates. We also show how to systematically realize the Yang-Baxter gates via the pulse control. We test and compare the different realizations on IBM quantum computers. We find that the pulse realizations of the Yang-Baxter gates always have a higher gate fidelity compared to the optimal CNOT or $R_{zz}$ realizations. On the basis of the above optimal realizations, we demonstrate the simulation of the Yang-Baxter equation on quantum computers. Our results provide a guideline and standard for further experimental studies based on the Yang-Baxter gate. △ Less

Submitted 7 February, 2024; v1 submitted 31 July, 2023; originally announced July 2023.

Comments: Published version, 14 pages, 11 figures

Journal ref: Adv. Quantum Technol. 2024, 2300345

The case for an EIC Theory Alliance: Theoretical Challenges of the EIC

Authors: Raktim Abir, Igor Akushevich, Tolga Altinoluk, Daniele Paolo Anderle, Fatma P. Aslan, Alessandro Bacchetta, Baha Balantekin, Joao Barata, Marco Battaglieri, Carlos A. Bertulani, Guillaume Beuf, Chiara Bissolotti, Daniël Boer, M. Boglione, Radja Boughezal, Eric Braaten, Nora Brambilla, Vladimir Braun, Duane Byer, Francesco Giovanni Celiberto, Yang-Ting Chien, Ian C. Cloët, Martha Constantinou, Wim Cosyn, Aurore Courtoy , et al. (146 additional authors not shown)

Abstract: We outline the physics opportunities provided by the Electron Ion Collider (EIC). These include the study of the parton structure of the nucleon and nuclei, the onset of gluon saturation, the production of jets and heavy flavor, hadron spectroscopy and tests of fundamental symmetries. We review the present status and future challenges in EIC theory that have to be addressed in order to realize thi… ▽ More We outline the physics opportunities provided by the Electron Ion Collider (EIC). These include the study of the parton structure of the nucleon and nuclei, the onset of gluon saturation, the production of jets and heavy flavor, hadron spectroscopy and tests of fundamental symmetries. We review the present status and future challenges in EIC theory that have to be addressed in order to realize this ambitious and impactful physics program, including how to engage a diverse and inclusive workforce. In order to address these many-fold challenges, we propose a coordinated effort involving theory groups with differing expertise is needed. We discuss the scientific goals and scope of such an EIC Theory Alliance. △ Less

Submitted 23 May, 2023; originally announced May 2023.

Comments: 44 pages, ReVTeX, White Paper on EIC Theory Alliance

Fluctuation of the phase boundary in the six-vertex model with Domain Wall Boundary Conditions: a Monte Carlo study

Authors: Ivar Lyberg, Vladimir Korepin, Jacopo Viti

Abstract: We consider the six-vertex model with Domain Wall Boundary Conditions on a $N\times N$ square lattice. Our main interest is the study of the fluctuations of the extremal lattice path about the arctic curves. We address the problem through Monte Carlo simulations. At $Δ= 0$, the fluctuations of the extremal path along any line parallel to the square diagonal were rigorously proven to follow the Tra… ▽ More We consider the six-vertex model with Domain Wall Boundary Conditions on a $N\times N$ square lattice. Our main interest is the study of the fluctuations of the extremal lattice path about the arctic curves. We address the problem through Monte Carlo simulations. At $Δ= 0$, the fluctuations of the extremal path along any line parallel to the square diagonal were rigorously proven to follow the Tracy-Widom distribution. We provide strong numerical evidence that this is true also for other values of the anisotropy parameter $Δ$ ($0\leq Δ< 1$). We argue that the typical width of the fluctuations of the extremal path about the arctic curves scales as $N^{1/3}$ and provide a numerical estimate for the parameters of the scaling random variable. △ Less

Submitted 19 October, 2023; v1 submitted 26 March, 2023; originally announced March 2023.

Comments: 15 pages, 9 figures, revised version: added the study of the third and fourth cumulant. J. Phys. A: Math. Theor. (2023)

Real-time non-perturbative dynamics of jet production: quantum entanglement and vacuum modification

Authors: Adrien Florio, David Frenklakh, Kazuki Ikeda, Dmitri Kharzeev, Vladimir Korepin, Shuzhe Shi, Kwangmin Yu

Abstract: The production of jets should allow testing the real-time response of the QCD vacuum disturbed by the propagation of high-momentum color charges. Addressing this problem theoretically requires a real-time, non-perturbative method. It is well known that the Schwinger model [QED in $(1+1)$ dimensions] shares many common properties with QCD, including confinement, chiral symmetry breaking, and the ex… ▽ More The production of jets should allow testing the real-time response of the QCD vacuum disturbed by the propagation of high-momentum color charges. Addressing this problem theoretically requires a real-time, non-perturbative method. It is well known that the Schwinger model [QED in $(1+1)$ dimensions] shares many common properties with QCD, including confinement, chiral symmetry breaking, and the existence of vacuum fermion condensate. As a step in developing such an approach, we report here on fully quantum simulations of a massive Schwinger model coupled to external sources representing quark and antiquark jets as produced in $e^+e^-$ annihilation. We study, for the first time, the modification of the vacuum chiral condensate by the propagating jets and the quantum entanglement between the fragmenting jets. Our results indicate strong entanglement between the fragmentation products of the two jets at rapidity separations $Δη\leq 2$, which can potentially exist also in QCD and can be studied in experiments. △ Less

Submitted 13 July, 2023; v1 submitted 27 January, 2023; originally announced January 2023.

Comments: 5 pages + supplementary materials, 10 figures; final version

Journal ref: Phys. Rev. Lett. 131 (2023) 021902

Quantum multi-programming for Grover's search

Authors: Gilchan Park, Kun Zhang, Kwangmin Yu, Vladimir Korepin

Abstract: Quantum multi-programming is a method utilizing contemporary noisy intermediate-scale quantum computers by executing multiple quantum circuits concurrently. Despite early research on it, the research remains on quantum gates or small-size quantum algorithms without correlation. In this paper, we propose a quantum multi-programming (QMP) algorithm for Grover's search. Our algorithm decomposes Grove… ▽ More Quantum multi-programming is a method utilizing contemporary noisy intermediate-scale quantum computers by executing multiple quantum circuits concurrently. Despite early research on it, the research remains on quantum gates or small-size quantum algorithms without correlation. In this paper, we propose a quantum multi-programming (QMP) algorithm for Grover's search. Our algorithm decomposes Grover's algorithm by the partial diffusion operator and executes the decomposed circuits in parallel by QMP. We proved that this new algorithm increases the rotation angle of the Grover operator which, as a result, increases the success probability. The new algorithm is implemented on IBM quantum computers and compared with the canonical Grover's algorithm and other variations of Grover's algorithms. The empirical tests validate that our new algorithm outperforms other variations of Grover's algorithms as well as the canonical Grover's algorithm. △ Less

Submitted 18 December, 2022; v1 submitted 29 July, 2022; originally announced July 2022.

Journal ref: Quantum Inf. Process. 22, 54 (2023)

Can a spin chain relate combinatorics to number theory?

Authors: Kun Hao, Olof Salberger, Vladimir Korepin

Abstract: The Motzkin spin chain is a spin-$1$ frustration-free model introduced by Shor & Movassagh. The ground state is constructed by mapping random walks on the upper half of the square lattice to spin configurations. It has unusually large entanglement entropy [quantum fluctuations]. The ground state of the Motzkin chain can be analytically described by the Motzkin paths. There is no analytical descrip… ▽ More The Motzkin spin chain is a spin-$1$ frustration-free model introduced by Shor & Movassagh. The ground state is constructed by mapping random walks on the upper half of the square lattice to spin configurations. It has unusually large entanglement entropy [quantum fluctuations]. The ground state of the Motzkin chain can be analytically described by the Motzkin paths. There is no analytical description of the excited states. The model is not solvable. We simplify the model by removing one of the local equivalence moves of the Motzkin paths. The system becomes integrable [similar to the XXX spin chain]. We call it free Motzkin chain. From the point of view of quantum integrability, the model is special since its $R$-matrix does not have crossing unitarity. We solve the periodic free Motzkin chain by generalizing the functional Bethe Ansatz method. We construct a $T-Q$ relation with an additional parameter to formulate the energy spectrum. This new parameter is related to the roots of unity and can be described by the Möbius function in number theory. We observe further patterns of number theory. △ Less

Submitted 10 August, 2023; v1 submitted 15 February, 2022; originally announced February 2022.

Comments: 14 pages, no figure

Journal ref: JHEP08(2023)009

Quantum search on noisy intermediate-scale quantum devices

Authors: Kun Zhang, Kwangmin Yu, Vladimir Korepin

Abstract: Quantum search algorithm (also known as Grover's algorithm) lays the foundation for many other quantum algorithms. Although it is very simple, its implementation is limited on noisy intermediate-scale quantum (NISQ) processors. Grover's algorithm was designed without considering the physical resources, such as depth, in the real implementations. Therefore, Grover's algorithm can be improved for NI… ▽ More Quantum search algorithm (also known as Grover's algorithm) lays the foundation for many other quantum algorithms. Although it is very simple, its implementation is limited on noisy intermediate-scale quantum (NISQ) processors. Grover's algorithm was designed without considering the physical resources, such as depth, in the real implementations. Therefore, Grover's algorithm can be improved for NISQ devices. In this paper, we demonstrate how to implement quantum search algorithms better on NISQ devices. We present detailed benchmarks of the five-qubit quantum search algorithm on different quantum processors, including IBMQ, IonQ, and Honeywell quantum devices. We report the highest success probability of the five-qubit search algorithm compared to previous works. Our results show that designing the error-aware quantum search algorithms is possible, which can maximally harness the power of NISQ computers. △ Less

Submitted 27 September, 2022; v1 submitted 31 January, 2022; originally announced February 2022.

Comments: Close to Published version, 12 pages, 6 figures

Journal ref: EPL 140, 18002 (2022)

Local Convertibility in quantum spin systems

Authors: Luigi Amico, Vladimir Korepin, Alioscia Hamma, Salvatore Marco Giampaolo, Fabio Franchini

Abstract: Local Convertibility refers to the possibility of transforming a given state into a target one, just by means of LOCC with respect to a given bipartition of the system and it is possible if and only if all the Renyi-entropies of the initial state are smaller than those of the target state. We apply this concept to adiabatic evolutions and ask whether they can be rendered through LOCC in the sense… ▽ More Local Convertibility refers to the possibility of transforming a given state into a target one, just by means of LOCC with respect to a given bipartition of the system and it is possible if and only if all the Renyi-entropies of the initial state are smaller than those of the target state. We apply this concept to adiabatic evolutions and ask whether they can be rendered through LOCC in the sense above. We argue that a lack of differential local convertibility (dLC) signals a higher computational power of the system's quantum phase, which is also usually connected with the existence of long-range entanglement, topological order, or edge-states. Remarkably, dLC can detect these global properties already by considering small subsystems. Moreover, we connect dLC to spontaneous symmetry breaking by arguing that states with finite order parameters must be the most classical ones and thus be locally convertible. △ Less

Submitted 12 August, 2022; v1 submitted 25 January, 2022; originally announced January 2022.

Comments: Chapter for the Springer Volume "Entanglement in Spin Chains - Theory and Quantum Technology Applications", 33 pages, many figures, typos corrected. arXiv admin note: text overlap with arXiv:1604.06403, arXiv:1306.6685

Report number: RBI-ThPhys-2022-1

Entanglement entropy production in deep inelastic scattering

Authors: Kun Zhang, Kun Hao, Dmitri Kharzeev, Vladimir Korepin

Abstract: Deep inelastic scattering (DIS) samples a part of the wave function of a hadron in the vicinity of the light cone. Lipatov constructed a spin chain which describes the amplitude of DIS in leading logarithmic approximation. Kharzeev and Levin proposed the entanglement entropy as an observable in DIS [Phys. Rev. D 95, 114008 (2017)], and suggested a relation between the entanglement entropy and part… ▽ More Deep inelastic scattering (DIS) samples a part of the wave function of a hadron in the vicinity of the light cone. Lipatov constructed a spin chain which describes the amplitude of DIS in leading logarithmic approximation. Kharzeev and Levin proposed the entanglement entropy as an observable in DIS [Phys. Rev. D 95, 114008 (2017)], and suggested a relation between the entanglement entropy and parton distributions. Here we represent the DIS process as a local quench in the Lipatov's spin chain, and study the time evolution of the produced entanglement entropy. We show that the resulting entanglement entropy depends on time logarithmically, $\mathcal S(t)=1/3 \ln{(t/τ)}$ with $τ= 1/m$ for $1/m \le t\le (mx)^{-1}$, where $m$ is the proton mass and $x$ is the Bjorken $x$. The central charge $c$ of Lipatov's spin chain is determined here to be $c=1$; using the proposed relation between the entanglement entropy and parton distributions, this corresponds to the gluon structure function growing at small $x$ as $xG(x) \sim 1/x^{1/3}$. △ Less

Submitted 4 January, 2022; v1 submitted 10 October, 2021; originally announced October 2021.

Comments: Published version, 8 pages, 1 figure

Journal ref: Phys. Rev. D 105, 014002 (2022)

Implementation of efficient quantum search algorithms on NISQ computers

Authors: Kun Zhang, Pooja Rao, Kwangmin Yu, Hyunkyung Lim, Vladimir Korepin

Abstract: Despite the advent of Grover's algorithm for the unstructured search, its successful implementation on near-term quantum devices is still limited. We apply three strategies to reduce the errors associated with implementing quantum search algorithms. Our improved search algorithms have been implemented on the IBM quantum processors. Using them, we demonstrate three- and four-qubit search algorithm… ▽ More Despite the advent of Grover's algorithm for the unstructured search, its successful implementation on near-term quantum devices is still limited. We apply three strategies to reduce the errors associated with implementing quantum search algorithms. Our improved search algorithms have been implemented on the IBM quantum processors. Using them, we demonstrate three- and four-qubit search algorithm with higher average success probabilities compared to previous works. We present the successful execution of the five-qubit search on the IBM quantum processor for the first time. The results have been benchmarked using degraded ratio, which is the ratio between the experimental and the theoretical success probabilities. The fast decay of the degraded ratio supports our divide-and-conquer strategy. Our proposed strategies are also useful for implementation of quantum search algorithms in the post-NISQ era. △ Less

Submitted 12 July, 2021; v1 submitted 2 February, 2021; originally announced February 2021.

Comments: Published version, 26 pages, 8 figures

Journal ref: Quantum Inf. Process. 20, 233 (2021)

Shor-Movassagh chain leads to unusual integrable model

Authors: Bin Tong, Olof Salberger, Kun Hao, Vladimir Korepin

Abstract: The ground state of Shor-Movassagh chain can be analytically described by the Motzkin paths. There is no analytical description of the excited states, the model is not solvable. We prove the integrability of the model without interacting part in this paper [free Shor-Movassagh]. The Lax pair for the free Shor-Movassagh open chain is explicitly constructed. We further obtain the boundary $K$-matric… ▽ More The ground state of Shor-Movassagh chain can be analytically described by the Motzkin paths. There is no analytical description of the excited states, the model is not solvable. We prove the integrability of the model without interacting part in this paper [free Shor-Movassagh]. The Lax pair for the free Shor-Movassagh open chain is explicitly constructed. We further obtain the boundary $K$-matrices compatible with the integrability of the model on the open interval. Our construction provides a direct demonstration for the quantum integrability of the model, described by Yang-Baxter algebra. Due to the lack of crossing unitarity, the integrable open chain can not be constructed by the reflection equation (boundary Yang-Baxter equation). △ Less

Submitted 22 September, 2020; originally announced September 2020.

Comments: 12 pages, 1 figure

Entanglement entropy of two disjoint intervals separated by one spin in a chain of free fermion

Authors: L. Brightmore, G. P. Geher, A. R. Its, V. E. Korepin, F. Mezzadri, M. Y. Mo, J. A. Virtanen

Abstract: We calculate the entanglement entropy of a non-contiguous subsystem of a chain of free fermions. The starting point is a formula suggested by Jin and Korepin, \texttt{arXiv:1104.1004}, for the reduced density of states of two disjoint intervals with lattice sites $P=\{1,2,\dots,m\}\cup\{2m+1,2m+2,\dots, 3m\}$, which applies to this model. As a first step in the asymptotic analysis of this system,… ▽ More We calculate the entanglement entropy of a non-contiguous subsystem of a chain of free fermions. The starting point is a formula suggested by Jin and Korepin, \texttt{arXiv:1104.1004}, for the reduced density of states of two disjoint intervals with lattice sites $P=\{1,2,\dots,m\}\cup\{2m+1,2m+2,\dots, 3m\}$, which applies to this model. As a first step in the asymptotic analysis of this system, we consider its simplification to two disjoint intervals separated just by one site, and we rigorously calculate the mutual information between these two blocks and the rest of the chain. In order to compute the entropy we need to study the asymptotic behaviour of an inverse Toeplitz matrix with Fisher-Hartwig symbol using the the Riemann--Hilbert method. △ Less

Submitted 15 June, 2020; v1 submitted 18 December, 2019; originally announced December 2019.

Comments: 25 pages, 6 figures, title changed

MSC Class: 35Q15

Journal ref: J. Phys. A: Math. Theor. 53 345303 (2020)

Bethe Ansatz for XXX chain with negative spin

Authors: Kun Hao, Dmitri Kharzeev, Vladimir Korepin

Abstract: XXX spin chain with spin $s=-1$ appears as an effective theory of Quantum Chromodynamics. It is equivalent to lattice nonlinear Schroediger's equation: interacting chain of harmonic oscillators [bosonic]. In thermodynamic limit each energy level is a scattering state of several elementary excitations [lipatons]. Lipaton is a fermion: it can be represented as a topological excitation [soliton] of o… ▽ More XXX spin chain with spin $s=-1$ appears as an effective theory of Quantum Chromodynamics. It is equivalent to lattice nonlinear Schroediger's equation: interacting chain of harmonic oscillators [bosonic]. In thermodynamic limit each energy level is a scattering state of several elementary excitations [lipatons]. Lipaton is a fermion: it can be represented as a topological excitation [soliton] of original [bosonic] degrees of freedom, described by the group $Z_2$ . We also provide the CFT description (including local quenches) and Yang-Yang thermodynamics of the model. △ Less

Submitted 31 October, 2019; v1 submitted 2 September, 2019; originally announced September 2019.

Comments: 21 pages, 2 figures

Journal ref: Int. J. Mod. Phys. A, Vol. 34, No. 31 (2019) 1950197

Depth optimization of quantum search algorithms beyond Grover's algorithm

Authors: Kun Zhang, Vladimir E. Korepin

Abstract: Grover's quantum search algorithm provides a quadratic speedup over the classical one. The computational complexity is based on the number of queries to the oracle. However, depth is a more modern metric for noisy intermediate-scale quantum computers. We propose a new depth optimization method for quantum search algorithms. We show that Grover's algorithm is not optimal in depth. We propose a quan… ▽ More Grover's quantum search algorithm provides a quadratic speedup over the classical one. The computational complexity is based on the number of queries to the oracle. However, depth is a more modern metric for noisy intermediate-scale quantum computers. We propose a new depth optimization method for quantum search algorithms. We show that Grover's algorithm is not optimal in depth. We propose a quantum search algorithm, which can be divided into several stages. Each stage has a new initialization, which is a rescaling of the database. This decreases errors. The multistage design is natural for parallel running of the quantum search algorithm. △ Less

Submitted 27 March, 2020; v1 submitted 12 August, 2019; originally announced August 2019.

Comments: Published version. 13 pages, 2 figures, 4 tables

Journal ref: Phys. Rev. A 101, 032346 (2020)

Quantum search on Hanoi network

Authors: Pulak Ranjan Giri, Vladimir Korepin

Abstract: Hanoi network has a one-dimensional periodic lattice as its main structure with additional long-range edges, which allow having efficient quantum walk algorithm that can find a target state on the network faster than the exhaustive classical search. In this article, we use regular quantum walks and lackadaisical quantum walks respectively to search for a target state. From the curve fitting of the… ▽ More Hanoi network has a one-dimensional periodic lattice as its main structure with additional long-range edges, which allow having efficient quantum walk algorithm that can find a target state on the network faster than the exhaustive classical search. In this article, we use regular quantum walks and lackadaisical quantum walks respectively to search for a target state. From the curve fitting of the numerical results for Hanoi network of degree three and four we find that their running time for the regular quantum walks followed by amplitude amplification scales as $\mathcal{O}\left(N^{0.79} \sqrt{\log N}\right)$ and $\mathcal{O}\left(N^{0.65} \sqrt{\log N}\right)$ respectively. And for the search by lackadaisical quantum walks the running time scales as $\mathcal{O}\left(N^{0.57}\log N\right)$ and $\mathcal{O}\left(N^{0.50}\log N\right)$ respectively. △ Less

Submitted 27 January, 2020; v1 submitted 19 March, 2019; originally announced March 2019.

Comments: 8 pages, 4 figures, accepted version

Journal ref: International Journal of Quantum InformationVol. 17, No. 07, 1950060 (2019)

Lackadaisical quantum walk for spatial search

Authors: Pulak Ranjan Giri, Vladimir Korepin

Abstract: Lackadaisical quantum walk(LQW) has been an efficient technique in searching a target state from a database which is distributed on a two-dimensional lattice. We numerically study the quantum search algorithm based on the lackadaisical quantum walk on one- and two-dimensions. It is observed that specific values of the self-loop weight at each vertex of the graph is responsible for such speedup of… ▽ More Lackadaisical quantum walk(LQW) has been an efficient technique in searching a target state from a database which is distributed on a two-dimensional lattice. We numerically study the quantum search algorithm based on the lackadaisical quantum walk on one- and two-dimensions. It is observed that specific values of the self-loop weight at each vertex of the graph is responsible for such speedup of the algorithm. Searching for a target state on one-dimensional lattice with periodic boundary conditions is possible using lackadaisical quantum walk, which can find a target state with $\mathcal{O}(1)$ success probability after $\mathcal{O} \left( N \right)$ time steps. In two-dimensions, our numerical simulation upto $M=6$ suggests that lackadaisical quantum walk can search one of the $M$ target states in $\mathcal{O}\left(\sqrt{\frac{N}{M}\log \frac{N}{M}}\right)$ time steps. △ Less

Submitted 3 April, 2019; v1 submitted 14 November, 2018; originally announced November 2018.

Comments: 9 pages, 6 figures

Journal ref: Modern Physics Letters A, Vol. 33, No. 1 (2020) 2050043

Non-Interacting Motzkin Chain - Periodic Boundary Conditions

Authors: Olof Salberger, Pramod Padmanabhan, Vladimir Korepin

Abstract: The Motzkin spin chain is a spin-1 model introduced in \cite{shor} as an example of a system exhibiting a high degree of quantum fluctuations whose ground state can be mapped to Motzkin paths that are generated with local equivalence moves. This model is difficult to solve in general but keeping just the height preserving local equivalence moves we show that the model becomes integrable which when… ▽ More The Motzkin spin chain is a spin-1 model introduced in \cite{shor} as an example of a system exhibiting a high degree of quantum fluctuations whose ground state can be mapped to Motzkin paths that are generated with local equivalence moves. This model is difficult to solve in general but keeping just the height preserving local equivalence moves we show that the model becomes integrable which when projected to certain subspaces of the full Hilbert space is isomorphic to the spin-$\frac{1}{2}$ XXX chain. In fact in the full Hilbert space the system is akin to two non-interacting spin-$\frac{1}{2}$ XXX chains making the spectrum the same as the latter with the change coming in the degeneracy of the states. We then show that including the height-changing local-equivalence move is the same as introducing interactions in the above system. △ Less

Submitted 3 September, 2018; originally announced September 2018.

Comments: 18 pages, 5 figures

Renyi entropy of highly entangled spin chains

Authors: Fumihiko Sugino, Vladimir Korepin

Abstract: Entanglement is one of the most intriguing features of quantum theory and a main resource in quantum information science. Ground states of quantum many-body systems with local interactions typically obey an "area law" meaning the entanglement entropy proportional to the boundary length. It is exceptional when the system is gapless, and the area law had been believed to be violated by at most a log… ▽ More Entanglement is one of the most intriguing features of quantum theory and a main resource in quantum information science. Ground states of quantum many-body systems with local interactions typically obey an "area law" meaning the entanglement entropy proportional to the boundary length. It is exceptional when the system is gapless, and the area law had been believed to be violated by at most a logarithm for over two decades. Recent discovery of Motzkin and Fredkin spin chain models is striking, since these models provide significant violation of the entanglement beyond the belief, growing as a square root of the volume in spite of local interactions. Although importance of intensive study of the models is undoubted to reveal novel features of quantum entanglement, it is still far from their complete understanding. In this article, we first analytically compute the Renyi entropy of the Motzkin and Fredkin models by careful treatment of asymptotic analysis. The Renyi entropy is an important quantity, since the whole spectrum of an entangled subsystem is reconstructed once the Renyi entropy is known as a function of its parameter. We find non-analytic behavior of the Renyi entropy with respect to the parameter, which is a novel phase transition never seen in any other spin chain studied so far. Interestingly, similar behavior is seen in the Renyi entropy of Rokhsar-Kivelson states in two-dimensions. △ Less

Submitted 19 September, 2018; v1 submitted 11 June, 2018; originally announced June 2018.

Comments: 14+22 pages, 8 figures; (v2) references added, (v3) version to be published in International Journal of Modern Physics B

Journal ref: Int. J. Mod. Phys. B 32 No.28, 1850306 (2018)

Quantum Phase Transitions and Localization in Semigroup Fredkin Spin Chain

Authors: Pramod Padmanabhan, Fumihiko Sugino, Vladimir Korepin

Abstract: We construct an extended quantum spin chain model by introducing new degrees of freedom to the Fredkin spin chain. The new degrees of freedom called arrow indices are partly associated to the symmetric inverse semigroup $\cS^3_1$. Ground states of the model fall into three different phases, and quantum phase transition takes place at each phase boundary. One of the phases exhibits logarithmic viol… ▽ More We construct an extended quantum spin chain model by introducing new degrees of freedom to the Fredkin spin chain. The new degrees of freedom called arrow indices are partly associated to the symmetric inverse semigroup $\cS^3_1$. Ground states of the model fall into three different phases, and quantum phase transition takes place at each phase boundary. One of the phases exhibits logarithmic violation of the area law of entanglement entropy and quantum criticality, whereas the other two obey the area law. As an interesting feature arising by the extension, there are excited states due to disconnections with respect to the arrow indices. We show that these states are localized without disorder. △ Less

Submitted 30 March, 2018; originally announced April 2018.

Comments: 31 pages, 15 figures. arXiv admin note: text overlap with arXiv:1710.10426

Journal ref: Quantum Inf Process (2019) 18: 69

Phase separation in the six-vertex model with a variety of boundary conditions

Authors: Ivar Lyberg, Vladimir Korepin, G. A. P. Ribeiro, Jacopo Viti

Abstract: We present numerical results for the six-vertex model with a variety of boundary conditions. Adapting an algorithm proposed by Allison and Reshetikhin for domain wall boundary conditions, we examine some modifications of these boundary conditions. To be precise, we discuss partial domain wall boundary conditions, reflecting ends and half turn boundary conditions (domain wall boundary conditions wi… ▽ More We present numerical results for the six-vertex model with a variety of boundary conditions. Adapting an algorithm proposed by Allison and Reshetikhin for domain wall boundary conditions, we examine some modifications of these boundary conditions. To be precise, we discuss partial domain wall boundary conditions, reflecting ends and half turn boundary conditions (domain wall boundary conditions with half turn symmetry). △ Less

Submitted 9 May, 2018; v1 submitted 21 November, 2017; originally announced November 2017.

Comments: 12+2 pages, 11 figures, invited contribution for Journal of Mathematical Physics special issue "To the memory of Ludwig Faddeev"

Journal ref: Journal of Mathematical Physics 59, 053301 (2018)

Quantum partial search for uneven distribution of multiple target items

Authors: Kun Zhang, Vladimir Korepin

Abstract: Quantum partial search algorithm is approximate search. It aims to find a target block (which has the target items). It runs a little faster than full Grover search. In this paper, we consider quantum partial search algorithm for multiple target items unevenly distributed in database (target blocks have different number of target items). The algorithm we describe can locate one of the target block… ▽ More Quantum partial search algorithm is approximate search. It aims to find a target block (which has the target items). It runs a little faster than full Grover search. In this paper, we consider quantum partial search algorithm for multiple target items unevenly distributed in database (target blocks have different number of target items). The algorithm we describe can locate one of the target blocks. Efficiency of the algorithm is measured by number of queries to the oracle. We optimize the algorithm in order to improve efficiency. By perturbation method, we find that the algorithm runs the fastest when target items are evenly distributed in database. △ Less

Submitted 24 October, 2017; v1 submitted 24 September, 2017; originally announced September 2017.

Comments: 12 pages, 1 figure

Journal ref: Quantum Inf. Process. (2018) 17: 143

Haldane Topological Orders in Motzkin Spin Chains

Authors: L. Barbiero, L. Dell'Anna, A. Trombettoni, V. E. Korepin

Abstract: Motzkin spin chains are frustration-free models whose ground-state is a combination of Motzkin paths. The weight of such path contributions can be controlled by a deformation parameter t. As a function of the latter these models, beside the formation of domain wall structures, exhibit a Berezinskii-Kosterlitz-Thouless phase transition for t=1 and gapped Haldane topological orders with constant dec… ▽ More Motzkin spin chains are frustration-free models whose ground-state is a combination of Motzkin paths. The weight of such path contributions can be controlled by a deformation parameter t. As a function of the latter these models, beside the formation of domain wall structures, exhibit a Berezinskii-Kosterlitz-Thouless phase transition for t=1 and gapped Haldane topological orders with constant decay of the string order parameters for t < 1. By means of numerical calculations we show that the topological properties of the Haldane phases depend on the spin value. This allows to classify different kinds of hidden antiferromagnetic Haldane gapped regimes associated to nontrivial features like symmetry-protected topological order. Our results from one side allow to clarify the physical properties of Motzkin frustration-free chains and from the other suggest them as a new interesting and paradigmatic class of local spin Hamiltonians. △ Less

Submitted 8 November, 2017; v1 submitted 20 January, 2017; originally announced January 2017.

Comments: 5 pages, 3 figures

Journal ref: Phys. Rev. B 96, 180404 (2017)

The density profile of the six vertex model with domain wall boundary conditions

Authors: Ivar Lyberg, Vladimir Korepin, Jacopo Viti

Abstract: We study numerically the density profile in the six-vertex model with domain wall boundary conditions. Using a Monte Carlo algorithm originally proposed by Allison and Reshetikhin we numerically evaluate the inhomogeneous density profiles in the disordered and antiferromagnetic regimes where frozen corners appear. At the free fermion point we present an exact finite-size formula for the density on… ▽ More We study numerically the density profile in the six-vertex model with domain wall boundary conditions. Using a Monte Carlo algorithm originally proposed by Allison and Reshetikhin we numerically evaluate the inhomogeneous density profiles in the disordered and antiferromagnetic regimes where frozen corners appear. At the free fermion point we present an exact finite-size formula for the density on the horizontal edges that relies on the imaginary time transfer matrix approach. In all cases where exact analytic forms for the density and the arctic curves are known the numerical method shows perfect agreement with them. This also suggests the possibility of its use for accurate quantitative purposes. △ Less

Submitted 8 May, 2017; v1 submitted 20 December, 2016; originally announced December 2016.

Comments: 13 pages, 7 pictures, Published version

Deformed Fredkin Spin Chain with Extensive Entanglement

Authors: Olof Salberger, Takuma Udagawa, Zhao Zhang, Hosho Katsura, Israel Klich, Vladimir Korepin

Abstract: We introduce a new spin chain which is a deformation of the Fredkin spin chain and has a phase transition between bounded and extensive entanglement entropy scaling. In this chain, spins have a local interaction of three nearest neighbors. The Hamiltonian is frustration-free and its ground state can be described analytically as a weighted superposition of Dyck paths. In the purely spin $1/2$ case,… ▽ More We introduce a new spin chain which is a deformation of the Fredkin spin chain and has a phase transition between bounded and extensive entanglement entropy scaling. In this chain, spins have a local interaction of three nearest neighbors. The Hamiltonian is frustration-free and its ground state can be described analytically as a weighted superposition of Dyck paths. In the purely spin $1/2$ case, the entanglement entropy obeys an area law: it is bounded from above by a constant, when the size of the block $n$ increases (and $t>1$). When a local color degree of freedom is introduced the entanglement entropy increases linearly with the size of the block (and $t>1$). The entanglement entropy of half of the chain is tightly bounded by ${ n}\log s$ where $n$ is the size of the block, and $s$ is the number of colors. Our chain fosters a new example for a significant boost to entropy and for the existence of the associated critical rainbow phase where the entanglement entropy scales with volume that has recently been discovered in Zhang et al. (arXiv:1606.07795) △ Less

Submitted 15 November, 2016; originally announced November 2016.

Journal ref: J. Stat. Mech. (2017) 063103