Jordan-Wigner Transformation Research Articles

Efficient Construction of Involutory Linear Combinations of Anticommuting Pauli Generators for Large-Scale Iterative Qubit Coupled Cluster Calculations.

We present an efficient method for construction of a fully anticommutative set of Pauli generators (elements of the Pauli group) from a commutative set of operators that are composed exclusively from Pauli x̂ operators (purely X generators) and sorted by an associated numerical measure, such as absolute energy gradients. Our approach uses the Gauss-Jordan elimination applied to a binary matrix that encodes the set of X generators to bring it to the reduced row-echelon form, followed by the construction of an anticommutative system in a standard basis by means of a modified Jordan-Wigner transformation and returning to the original basis. The algorithm complexity is linear in the size of the X set and quadratic in the number of qubits. The resulting anticommutative sets are used to construct the qubit coupled cluster Ansatz with involutory linear combinations of anticommuting Paulis (QCC-ILCAP) proposed in J. Chem. Theory Comput. 2021, 17 (1), 66-78. We applied the iterative qubit coupled cluster method with the QCC-ILCAP Ansatz to calculations of ground-state potential energy curves for symmetric stretching of the water molecule (36 qubits) and dissociation of N2 (56 qubits).

One-dimensional projection of two-dimensional systems using spiral boundary conditions

The spiral boundary conditions can project any two-dimensional or three-dimensional lattice onto a one-dimensional periodic chain with translational symmetry. It enables one to apply efficiently the existing one-dimensional techniques, such as density-matrix renormalization group (DMRG), bosonization, Jordan-Wigner transformation, etc., to studies of lattice systems in higher dimensions. Here, the authors demonstrate the utility of spiral boundary conditions by performing analytical and DMRG calculations for square and honeycomb lattice problems. Thus, the spiral boundary conditions emerge as a principal option in condensed matter physics.

Majorana nanowires, Kitaev chains, and spin models

Motivated by the fact that the idealized Kitaev chain toy model and the experimental semiconductor-superconductor Majorana nanowire can both host the Majorana zero modes, we theoretically investigate the question to what extent the two models are equivalent or similar, using the perspective of the corresponding dual spin models for both. We start with the duality between the Kitaev chain and the transverse-field XY spin model through the Jordan-Wigner transformation with the goal of establishing the connection between the Kitaev chain and the nanowire. By applying the Jordan-Wigner transformation to the nanowire, we find that the corresponding bosonic spin model is a generalized spin cluster model, containing 3-spin and 4-spin terms, with staggered couplings. By projecting out the upper band of the bare semiconductor with higher energy, we obtain an effective low-energy spinless system from the spinful nanowire system deep in the topological regime. Finally, we establish the connection between the Kitaev chain and Majorana nanowire by showing that the spinless Kitaev chain can be viewed as the first-order approximation of the spinful Majorana nanowire deep in the topological regime.

On Araki’s extension of the Jordan–Wigner transformation

In his seminal paper [On the XY-model on two-sided infinite chain, Publ. RIMS Kyoto Univ. 20 (1984) 277–296], Araki introduced an elegant extension of the Jordan–Wigner transformation which establishes a precise connection between quantum spin systems and Fermi lattice gases in one dimension in the so-called infinite system idealization of quantum statistical mechanics. His extension allows in particular for the rigorous study of numerous aspects of the prominent XY chain over the two-sided infinite discrete line without having to resort to a thermodynamic limit procedure at an intermediate or at the final stage. We rigorously review and elaborate this extension from scratch which makes the paper rather self-contained. In the course of the construction, we also present a simple and concrete realization of Araki’s crossed product extension.

DMRG study of the higher-charge Schwinger model and its ’t Hooft anomaly

The charge-q Schwinger model is the (1 + 1)-dimensional quantum electrodynamics (QED) with a charge-q Dirac fermion. It has the ℤq 1-form symmetry and also enjoys the ℤq chiral symmetry in the chiral limit, and there is a mixed ’t Hooft anomaly between those symmetries. We numerically study the charge-q Schwinger model in the lattice Hamiltonian formulation using the density-matrix renormalization group (DMRG). When applying DMRG, we map the Schwinger model to a spin chain with nonlocal interaction via Jordan-Wigner transformation, and we take the open boundary condition instead of the periodic one to make the Hilbert space finite-dimensional. When computing the energy density or chiral condensate, we find that using local operators significantly reduces the boundary effect compared with the computation of corresponding extensive quantities divided by the volume. To discuss the consequence of the ’t Hooft anomaly, we carefully treat the renormalization of the chiral condensates, and then we confirm that Wilson loops generate the discrete chiral transformations in the continuum limit.

Strong-weak duality via Jordan-Wigner transformation: Using fermionic methods for strongly correlated su(2) spin systems.

The Jordan-Wigner transformation establishes a duality between su(2) and fermionic algebras. We present qualitative arguments and numerical evidence that when mapping spins to fermions, the transformation makes strong correlation weaker, as demonstrated by the Hartree-Fock approximation to the transformed Hamiltonian. This result can be rationalized in terms of rank reduction of spin shift terms when transformed to fermions. Conversely, the mapping of fermions to qubits makes strong correlation stronger, complicating its solution when one uses qubit-based correlators. The presence of string operators poses challenges to the implementation of quantum chemistry methods on classical computers, but these can be dealt with using established techniques of low computational cost. Our proof of principle results for XXZ and J1-J2 Heisenberg (in 1D and 2D) indicates that the JW transformed fermionic Hamiltonian has reduced complexity in key regions of their phase diagrams and provides a better starting point for addressing challenging spin problems.

Quantum-Compute Algorithm for Exact Laser-Driven Electron Dynamics in Molecules.

In this work, we investigate the capability of known quantum computing algorithms for fault-tolerant quantum computing to simulate the laser-driven electron dynamics of excitation and ionization processes in small molecules such as lithium hydride, which can be benchmarked against the most accurate time-dependent full configuration interaction (TD-FCI) calculations. The conventional TD-FCI wave packet propagation is reproduced using the Jordan-Wigner transformation for wave function and operators and the Trotter product formula for expressing the propagator. In addition, the time-dependent dipole moment, as an example of a time-dependent expectation value, is calculated using the Hadamard test. To include non-Hermitian operators in the ionization dynamics, a similar approach to the quantum imaginary time evolution (QITE) algorithm is employed to translate the propagator, including a complex absorption potential, into quantum gates. The computations are executed on a quantum computer simulator. By construction, all quantum computer algorithms, except for the QITE algorithm used only for ionization but not for excitation dynamics, would scale polynomially on a quantum computer with fully entangled qubits. In contrast, TD-FCI scales exponentially. Hence, quantum computation holds promises for substantial progress in the understanding of electron dynamics of excitation processes in increasingly large molecular systems, as has already been witnessed in electronic structure theory.

Exact weak bosonic zero modes in a spin/fermion chain

We study an exactly solvable one-dimensional spin-$\frac{1}{2}$ model which can support weak zero modes in its ground state manifold. The spin chain has staggered XXZ-type and ZZ-type spin interaction on neighboring bonds and is thus dubbed the (XXZ,Z) chain. The model is equivalent to an interacting fermionic chain by Jordan-Wigner transformation. We study the phase diagram of the system and work out the conditions and properties of its weak zero modes. In the fermion chain representation, these weak zero modes are given by even-order polynomials of Majorana fermion operators and are thus bosonic. The fermionic chain Hamiltonian contains only fermion hopping and interaction terms and may have potential realization in experiments.

Variational solutions to fermion-to-qubit mappings in two spatial dimensions

Through the introduction of auxiliary fermions, or an enlarged spin space, one can map local fermion Hamiltonians onto local spin Hamiltonians, at the expense of introducing a set of additional constraints. We present a variational Monte-Carlo framework to study fermionic systems through higher-dimensional (>1D) Jordan-Wigner transformations. We provide exact solutions to the parity and Gauss-law constraints that are encountered in bosonization procedures. We study thet-Vmodel in 2D and demonstrate how both the ground state and the low-energy excitation spectra can be retrieved in combination with neural network quantum state ansatze.

Probing Operator Spreading via Floquet Engineering in a Superconducting Circuit.

Operator spreading, often characterized by out-of-time-order correlators (OTOCs), is one of the central concepts in quantum many-body physics. However, measuring OTOCs is experimentally challenging due to the requirement of reversing the time evolution of systems. Here we apply Floquet engineering to investigate operator spreading in a superconducting 10-qubit chain. Floquet engineering provides an effective way to tune the coupling strength between nearby qubits, which is used to demonstrate quantum walks with tunable couplings, reversed time evolution, and the measurement of OTOCs. A clear light-cone-like operator propagation is observed in the system with multiple excitations, and has a nearly equal velocity as the single-particle quantum walk. For the butterfly operator that is nonlocal (local) under the Jordan-Wigner transformation, the OTOCs show distinct behaviors with (without) a signature of information scrambling in the near integrable system.

Dynamics of spin helices in the one-dimensional XX model

Motivated by cold-atom experiments and a desire to understand far-from-equilibrium quantum transport, we analytically study the dynamics of spin helices in the one-dimensional $XX$ model. We use a Jordan-Wigner transformation to map the spin chain onto a non-interacting Fermi gas with simple equations of motion. The resulting dynamics are nontrivial, however, as the spin-helix initial condition corresponds to a highly nonequilibrium distribution of the fermions. We find a separation of timescales between the in-plane and out-of-plane spin dynamics. We gain insights from analyzing the case of a uniform spin chain and from a semiclassical model. One of our key findings is that the spin correlation functions decay as $t^{-1/2}$ at long time, in contrast to the experimentally observed exponential decay.

A unified framework of transformations based on the Jordan-Wigner transformation.

Quantum simulation of chemical Hamiltonians enables the efficient calculation of chemical properties. Mapping is one of the essential steps in simulating fermionic systems on quantum computers. In this work, a unified framework of transformations mapping fermionic systems to qubit systems is presented and many existing transformations-such as Jordan-Wigner, Bravyi-Kitaev, and parity transformations-are included in this framework. Based on this framework, the Multilayer Segmented Parity (MSP) transformation is proposed. The MSP transformation is a general mapping with an adjustable parameter vector, which can be viewed as a generalization of the above-mentioned mappings. Furthermore, the MSP transformation can adjust flexibly when dealing with different systems. Applying these mappings to the electronic structure Hamiltonians of various molecules, the MSP transformation is found to perform better on a number of Pauli operators and gates needed in the circuit of Hamiltonian simulation. The MSP transformation will reduce the qubit gate requirement for Hamiltonian simulation on noisy intermediate-scale quantum devices, and it will provide a much wider choice of mappings for researchers.

Higher-dimensional Jordan-Wigner transformation and auxiliary Majorana fermions

We discuss a scheme for performing Jordan-Wigner transformation for various lattice fermion systems in two and three dimensions which keeps internal and spatial symmetries manifest. The correspondence between fermionic and bosonic operators is established with the help of auxiliary Majorana fermions. The current construction is applicable to general lattices with even coordination numbers and an arbitrary number of fermion flavors. The approach is demonstrated on the single-orbital square, triangular, and cubic lattices for spin-$1/2$ fermions. We also discuss the relation to some quantum spin liquid models.

Qubit unitary coupled cluster with generalized single and paired double excitations ansatz for variational quantum eigensolver

AbstractVariational quantum eigensolver (VQE) with a unitary coupled cluster (UCC) ansatz has been suggested as a promising method for electronic structure calculations on future quantum computers. However, the complexity of the excitation terms for UCC with the single and double excitations (UCCSD) ansatz is up‐bounded to , where N is the number of electrons and M is the number of spin orbitals. The gate complexity of quantum circuit for the UCCSD ansatz is up‐bounded to using the Jordan–Wigner transformation. These complexities significantly limit the implementation of UCCSD on current Noisy Intermediate‐Scale Quantum (NISQ) devices. Herein, we developed a k‐QUpCCGSD ansatz which is based on the generalized paired double excitation operators and the particle preserving exchange gate. The former reduces the number of the excitation operators, the latter reduces the number of qubit gates for transforming excitation operators to quantum circuit. The gate complexity of the proposed k‐QUpCCGSD ansatz is up‐bounded to , which significantly reduce the complexity of the VQE algorithm on NISQ devices. The performance of the proposed ansatz on VQE is demonstrated by calculating ground‐state dissociation energy curves of the H6, LiH, H2O, and BeH2 molecules with the STO‐3G minimal basis set, and the accuracy is evaluated by comparing to the full configuration interaction (FCI) benchmarks. Moreover, we compare the number of quantum gates, especially the CNOT gates, and accuracies of various ansatzes. The assessments have shown that the accuracy of qubit unitary coupled cluster (QUCC) ansatzes is slightly worse than that of the UCC ones, but the circuit complexity of QUCC is much less than that of UCC. Among the tested QUCC ansatzes, k‐QUpCCGSD achieves higher accuracy with fewer quantum gates than QUCCSD, and k‐QUpCCGSD is a promising ansatz for VQE calculation on NISQ devices.

Quantum computation of nuclear observables involving linear combinations of unitary operators

We present the quantum computation of nuclear observables where the operators of interest are first decomposed in terms of the linear combination of unitaries. Then we utilise the Hadamard test and the linear combination of unitaries (LCU) based methods to compute the expectation values. We apply these methods to calculate the electric quadrupole moment of deuteron. The results are compared for the Jordan-Wigner transformation and Gray code encoding. We discuss the versatility of our approach that can be utilized in general to calculate several observables on a quantum computer.

Entanglement entropy in quantum spin chains with broken parity number symmetry

Consider a generic quantum spin chain that can be mapped to free quadratic fermions via Jordan-Wigner (JW) transformation. In the presence of arbitrary boundary magnetic fields, this Hamiltonian is no longer a quadratic Hamiltonian after JW transformation. Using ancillary sites and enlarging the Hamiltonian we first introduce a bigger quadratic Hamiltonian. Then we diagonalize this enlarged Hamiltonian in its most generic form and show that all the states are degenerate because of the presence of a zero mode. The eigenstates of the original spin chain with boundary magnetic fields can be derived after appropriate projection. We study in depth the properties of the eigenstates of the enlarged Hamiltonian. In particular we find: 1) the eigenstates in configuration bases, 2) calculate all the correlation functions, 3) find the reduced density matrices, 4) calculate the entanglement entropy. We show that the generic eigenstate of the enlarged Hamiltonian (including the eigenstates of the original spin chain) breaks the parity number symmetry and consequently one needs to take care of some technicalities regarding the calculation of the reduced density matrix and entanglement entropy. Interestingly we show that the entanglement structure of these eigenstates is quite universal and independent of the Hamiltonian. We support our results by applying them to a couple of examples.

Characterizing quantum criticality and steered coherence in the XY -Gamma chain

In this paper, we show that an effective spin Hamiltonian with various types of couplings can be engineered using quantum simulators in atomic-molecular-optical laboratories, dubbed the $XY$-Gamma model. We analytically solve the one-dimensional short-range interacting case with the Jordan-Wigner transformation and establish the phase diagram. In the gapless phase, an incommensurate spiral order is manifested by the vector-chiral correlations. Between distinct gapped phases, a logarithmic scaling behavior of local measures, including spin correlations and the steered quantum coherence, is identified for the quantum critical points, yielding a compelling value of the correlation-length critical exponent. We derive explicit scaling forms of the excitation gap near the quantum critical points. The extracted critical exponents reveal the quantum phase transition on the boundary of Tomonaga-Luttinger liquid belongs to Lifshitz universality class. Our results may provide useful insights into the underlying mechanism in quantum criticality for state-of-the-art experiments of quantum simulation.

The transverse field XY model on the diamond chain

We consider the s=1/2 transverse field XY model on the frustrated diamond chain, considering anisotropic exchange parameters between nearest neighbor spins. To this end, we employ three different methodologies: mean-field approximations, and state-of-the-art exact diagonalizations (ED), and density matrix renormalization group (DMRG) simulations. Within a mean-field theory, the Hamiltonian is fermionized by introducing the Jordan–Wigner transformation, and the interacting (many-body) terms are approximated to single-particle ones by a Hartree–Fock approach. We analyze the behavior of the induced and spontaneous magnetization as functions of the external field, investigating the magnetic properties at the ground state, and at finite temperatures. Interestingly, the mean-field results are in reasonable agreement with the ED and DMRG ones, in particular for the distorted chain, or at an intermediate/large spin anisotropy parameter. As our key results, we present phase diagrams anisotropy × magnetic field at zero temperature, discussing the emergence of phases and its quantum critical points. Finally, our analysis at finite temperature provides a range of parameters in which an unusual behavior of the induced magnetization occurs — with it increasing as a function of temperature. This work presents a global picture of the XY model on the diamond chain, which may be useful to understand features of magnetism in more complex geometries.

The spectral gap of a fractional quantum Hall system on a thin torus

We study a fractional quantum Hall system with maximal filling ν = 1/3 in the thin torus limit. The corresponding Hamiltonian is a truncated version of Haldane’s pseudopotential, which upon a Jordan–Wigner transformation is equivalent to a one-dimensional quantum spin chain with periodic boundary conditions. Our main result is a lower bound on the spectral gap of this Hamiltonian, which is uniform in the system size and total particle number. The gap is also uniform with respect to small values of the coupling constant in the model. The proof adapts the strategy of individually estimating the gap in invariant subspaces used for the bosonic ν = 1/2 model to the present fermionic case.

Hidden continuous quantum phase transition without gap closing in non-Hermitian transverse Ising model

Continuous phase transition in quantum matters is a significant issue in condensed matter physics. In general, the continuous quantum phase transitions in many-body systems occur with gap closing. On the other hand, non-Hermitian systems could display quite different properties as their Hermitian counterparts. In this paper, we show that a hidden, continuous quantum phase transition occurs without gap closing in non-Hermitian transverse Ising model. By using a projected Jordan–Wigner transformation, the one-dimensional (1D) non-Hermitian transverse Ising model with ferromagnetic order is mapped on to 1D non-Hermitian Kitaev model with topological superconducting order and becomes exactly solvable. A hidden, continuous quantum phase transition is really normal–abnormal transition for fermionic correlation in the 1D non-Hermitian Kitaev model. In addition, similar hidden, continuous quantum phase transition is discovered in two-dimensional non-Hermitian transverse Ising model and thus becomes a universal feature in certain non-Hermitian many-body systems.