arXiv subjects

Vladimir Korepin

Publications and source records attributed to Vladimir Korepin.

At least 37 records · Page 2Linked to original sources

arXiv2025

The Yang-Baxter integrability of the critical Ising chain

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.

hep-th↗

arXiv2025

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

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.

hep-th↗

arXiv2025

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

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.

hep-th↗

arXiv2025

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

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.

hep-th↗

arXiv2025

Near-deterministic quantum search algorithm without phase design

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.

quant-ph↗

arXiv2025

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

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.

hep-th↗

arXiv2024

Geometric representations of braid and Yang-Baxter gates

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.

quant-ph↗

arXiv2024

Unitary tetrahedron quantum gates

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.

quant-ph↗

arXiv2024

Toffoli gates solve the tetrahedron equations

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.

quant-ph↗

arXiv2024

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

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.

hep-th↗

arXiv2024

Yang-Baxter solutions from commuting operators

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.

hep-th↗

arXiv2024

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

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.

hep-ph↗

arXiv2024

Optimal realization of Yang-Baxter gate on quantum computers

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.

quant-ph↗

arXiv2023

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

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.

cond-mat.stat-mech↗

arXiv2023

Can a spin chain relate combinatorics to number theory?

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.

quant-ph↗

arXiv2023

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

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.

hep-ph↗

arXiv2023

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

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.

hep-ph↗

arXiv2022

Quantum multi-programming for Grover's search

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.

quant-ph↗