Permutation Symmetry Research Articles

Symmetry-aware reservoir computing.

We demonstrate that matching the symmetry properties of a reservoir computer (RC) to the data being processed dramatically increases its processing power. We apply our method to the parity task, a challenging benchmark problem that highlights inversion and permutation symmetries, and to a chaotic system inference task that presents an inversion symmetry rule. For the parity task, our symmetry-aware RC obtains zero error using an exponentially reduced neural network and training data, greatly speeding up the time to result and outperforming artificial neural networks. When both symmetries are respected, we find that the network size N necessary to obtain zero error for 50 different RC instances scales linearly with the parity-order n. Moreover, some symmetry-aware RC instances perform a zero error classification with only N=1 for n≤7. Furthermore, we show that a symmetry-aware RC only needs a training data set with size on the order of (n+n/2) to obtain such a performance, an exponential reduction in comparison to a regular RC which requires a training data set with size on the order of n2^{n} to contain all 2^{n} possible n-bit-long sequences. For the inference task, we show that a symmetry-aware RC presents a normalized root-mean-square error three orders-of-magnitude smaller than regular RCs. For both tasks, our RC approach respects the symmetries by adjusting only the input and the output layers, and not by problem-based modifications to the neural network. We anticipate that the generalizations of our procedure can be applied in information processing for problems with known symmetries.

Target space entanglement in Matrix Models

We study target space entanglement in gauged multi-matrix models as models of entanglement between groups of D-branes separated by a planar entangling surface, paying close attention to the implementation of gauge invariance. We open with a review of target space entanglement between identical particles, which shares some important features (specifically a gauged permutation symmetry) with our main problem. For our matrix models, we implement a gauge fixing well-adapted to the entangling surface. In this gauge, we map the matrix model problem to that of entanglement of a U(1) gauge theory on a complete or all-to-all lattice. Matrix elements corresponding to open strings stretching across the entangling surface in the target space lead to interesting contributions to the entanglement entropy.

Symmetry enriched phases of quantum circuits

Quantum circuits consisting of random unitary gates and subject to local measurements have been shown to undergo a phase transition, tuned by the rate of measurement, from a state with volume-law entanglement to an area-law state. From a broader perspective, these circuits generate a novel ensemble of quantum many-body states at their output. In this paper, we characterize this ensemble and classify the phases that can be established as steady states. Symmetry plays a nonstandard role in that the physical symmetry imposed on the circuit elements does not on its own dictate the possible phases. Instead, it is extended by dynamical symmetries associated with this ensemble to form an enlarged symmetry. Thus, we predict phases that have no equilibrium counterpart and could not have been supported by the physical circuit symmetry alone. We give the following examples. First, we classify the phases of a circuit operating on qubit chains with Z2 symmetry. One striking prediction, corroborated with numerical simulation, is the existence of distinct volume-law phases in one dimension, which nonetheless support true long-range order. We furthermore argue that owing to the enlarged symmetry, this system can in principle support a topological area-law phase, protected by the combination of the circuit symmetry and a dynamical permutation symmetry. Second, we consider a Gaussian fermionic circuit that only conserves fermion parity. Here the enlarged symmetry gives rise to a U(1) critical phase at moderate measurement rates and a Kosterlitz–Thouless transition to area-law phases. We comment on the interpretation of the different phases in terms of the capacity to encode quantum information. We discuss close analogies to the theory of spin glasses pioneered by Edwards and Anderson as well as crucial differences that stem from the quantum nature of the circuit ensemble.

Circular dichroism metamirror with diversified chiral molecules combinations.

In this paper, we propose a giant circular dichroism (CD) chiral metamirror that differentially absorbs circularly polarized (CP) waves with dual/multi bands. The structure is composed of two variable τresonators with counter split opening directions and different eigenfrequencies. Even more important, these two resonators are aligned vertically, and then integrated into one periodical unit cell, which results in the high-efficiency absorptive CD. The chiral metamirror has a narrow-band absorption of the left-handed circularly polarized (LCP) wave for lower resonance, and a broadband absorption of the right-handed circularly polarized (RCP) wave in the higher-frequency range. Optical resonator designs with diversified chiral molecules combinations are further studied in detail. Through the study of different periodic arrangements, parameters scanning optimization, and power loss distributions, it is proved that the overall permutation symmetry breakdown of total combined chiral molecules is the determining factor for the absorptive CD. Based on the principle in the microwave model, two mid-infrared chiral metamirrors are further realized, which can still show good spin-dependent absorption. This multi-polarization and multifunction response advance novel photonic devices for a variety of applications including CP laser, biomolecules detection, and energy harvester.

Quantum Error Mitigation using Symmetry Expansion

Even with the recent rapid developments in quantum hardware, noise remains the biggest challenge for the practical applications of any near-term quantum devices. Full quantum error correction cannot be implemented in these devices due to their limited scale. Therefore instead of relying on engineered code symmetry, symmetry verification was developed which uses the inherent symmetry within the physical problem we try to solve. In this article, we develop a general framework named symmetry expansion which provides a wide spectrum of symmetry-based error mitigation schemes beyond symmetry verification, enabling us to achieve different balances between the estimation bias and the sampling cost of the scheme. We show that certain symmetry expansion schemes can achieve a smaller estimation bias than symmetry verification through cancellation between the biases due to the detectable and undetectable noise components. A practical way to search for such a small-bias scheme is introduced. By numerically simulating the Fermi-Hubbard model for energy estimation, the small-bias symmetry expansion we found can achieve an estimation bias 6 to 9 times below what is achievable by symmetry verification when the average number of circuit errors is between 1 to 2. The corresponding sampling cost for random shot noise reduction is just 2 to 6 times higher than symmetry verification. Beyond symmetries inherent to the physical problem, our formalism is also applicable to engineered symmetries. For example, the recent scheme for exponential error suppression using multiple noisy copies of the quantum device is just a special case of symmetry expansion using the permutation symmetry among the copies.

Measurement-induced purification in large- N hybrid Brownian circuits

Competition between unitary dynamics that scrambles quantum information nonlocally and local measurements that probe and collapse the quantum state can result in a measurement-induced entanglement phase transition. Here we study this phenomenon in an analytically tractable all-to-all Brownian hybrid circuit model composed of qubits. The system is initially entangled with an equal sized reference, and the subsequent hybrid system dynamics either partially preserves or totally destroys this entanglement depending on the measurement rate. Our approach can access a variety of entropic observables which are distinguished by the averaging procedure, and for concreteness we focus on a particular purity quantity for which the averaging is particularly simple. We represent the purity as a path integral coupling four replicas with twisted boundary conditions. Saddle-point analysis reveals a second-order phase transition corresponding to replica permutation symmetry breaking below a critical measurement rate. The transition is mean-field-like and we characterize the critical properties near the transition in terms of a simple Ising field theory in $0+1$ dimensions. In addition to studying the purity of the entire system, we study subsystem purities and relate these results to manifestations of quantum error correction in the model. We also comment on the experimental feasibility for simulating this averaged purity, and corroborate our results with exact diagonalization for modest system sizes.

The Valence-Bond (VB) Model and Its Intimate Relationship to the Symmetric or Permutation Group.

VB and molecular orbital (MO) models are normally distinguished by the fact the first looks at molecules as a collection of atoms held together by chemical bonds while the latter adopts the view that each molecule should be regarded as an independent entity built up of electrons and nuclei and characterized by its molecular structure. Nevertheless, there is a much more fundamental difference between these two models which is only revealed when the symmetries of the many-electron Hamiltonian are fully taken into account: while the VB and MO wave functions exhibit the point-group symmetry, whenever present in the many-electron Hamiltonian, only VB wave functions exhibit the permutation symmetry, which is always present in the many-electron Hamiltonian. Practically all the conflicts among the practitioners of the two models can be traced down to the lack of permutation symmetry in the MO wave functions. Moreover, when examined from the permutation group perspective, it becomes clear that the concepts introduced by Pauling to deal with molecules can be equally applied to the study of the atomic structure. In other words, as strange as it may sound, VB can be extended to the study of atoms and, therefore, is a much more general model than MO.

Complete set of dimension-eight operators in the standard model effective field theory

We present a complete list of the dimension 8 operator basis in the standard model effective field theory using group theoretic techniques in a systematic and automated way. We adopt a new form of operators in terms of the irreducible representations of the Lorentz group, and identify the Lorentz structures as states in a $SU(N)$ group. In this way, redundancy from equations of motion is absent and that from integration-by-part is treated using the fact that the independent Lorentz basis forms an invariant subspace of the $SU(N)$ group. We also decompose operators into the ones with definite permutation symmetries among flavor indices to deal with subtlety from repeated fields. For the first time, we provide the explicit form of independent flavor-specified operators in a systematic way. Our algorithm can be easily applied to higher dimensional standard model effective field theory and other effective field theories, making these studies more approachable.

R parity from string compactification

The strategy for assigning ${\mathbf{Z}}_{4R}$ parity in the string compactification is presented. For the visible sector, an anti-SU(5) [flipped-SU(5)] grand unification (GUT) model with three families is used to reduce the number of representations compared to the number in the minimal supersymmetric standard models. The SO(32) heterotic string is used to allow a large non-Abelian gauge group $\mathrm{SU}(N)$, $N\ensuremath{\ge}9$, for the hidden sector such that the number of extra U(1) factors is small. A discrete subgroup of the gauge U(1)'s is defined as the ${\mathbf{Z}}_{4R}$ parity. Spontaneous symmetry breaking of anti-SU(5) GUT is achieved by the vacuum expectation values of two index antisymmetric tensor Higgs fields ${\mathbf{10}}_{+1}$ and ${\overline{\mathbf{10}}}_{\ensuremath{-}1}$ that led to our word ``anti-SU(5).'' In the illustrated example, the multiplicity 3 in one twisted sector allows the permutation symmetry ${S}_{3}$ that leads us to select the third family members and one MSSM pair of the Higgs quintets.

Interacting holes in Si and Ge double quantum dots: From a multiband approach to an effective-spin picture

The states of two electrons in tunnel-coupled semiconductor quantum dots can be effectively described in terms of a two-spin Hamiltonian with an isotropic Heisenberg interaction. A similar description needs to be generalized in the case of holes due to their multiband character and spin-orbit coupling, which mixes orbital and spin degrees of freedom and splits $j=3/2$ and $j=1/2$ multiplets. Here we investigate two-hole states in prototypical coupled Si and Ge quantum dots via different theoretical approaches. Multiband $\mathbit{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbit{p}$ and configuration-interaction calculations are combined with entanglement measures in order to thoroughly characterize the two-hole states in terms of band mixing and justify the introduction of an effective spin representation, which we analytically derive a from generalized Hubbard model. We find that, in the weak interdot regime, the ground state and first excited multiplet of the two-hole system display---unlike their electronic counterparts---a high degree of $J$ mixing, even in the limit of purely heavy-hole states. The light-hole component additionally induces $M$ mixing and a weak coupling between spinors characterized by different permutational symmetries.

Flat bands by latent symmetry

The geometry of a lattice may cause energy eigenstates to localize compactly by destructive interference, resulting in flat energy bands. This effect is often induced by symmetry. A ``latent'' symmetry, inspired from graph theory, is a hidden permutation symmetry revealed upon reduction of the original system onto a subsystem; here, an effective two-site dimer. Combined with collective symmetries between possible pathways connecting the dimer sites to special site subsets in the lattice unit cell, latent symmetry provides a novel systematic way to explore and construct flat bands.

Deep Density: Circumventing the Kohn-Sham equations via symmetry preserving neural networks

The recently developed Deep Potential [Phys. Rev. Lett. 120 (2018) 143001 [27]] is a powerful method to represent general inter-atomic potentials using deep neural networks. The success of Deep Potential rests on the proper treatment of locality and symmetry properties of each component of the network. In this paper, we leverage its network structure to effectively represent the mapping from the atomic configuration to the electron density in Kohn-Sham density function theory (KS-DFT). By directly targeting at the self-consistent electron density, we demonstrate that the adapted network architecture, called the Deep Density, can effectively represent the self-consistent electron density as the linear combination of contributions from many local clusters. The network is constructed to satisfy the translation, rotation, and permutation symmetries, and is designed to be transferable to different system sizes. We demonstrate that using a relatively small number of training snapshots, with each snapshot containing a modest amount of data-points, Deep Density achieves excellent performance for one-dimensional insulating and metallic systems, as well as systems with mixed insulating and metallic characters. We also demonstrate its performance for real three-dimensional systems, including small organic molecules, as well as extended systems such as water (up to 512 molecules) and aluminum (up to 256 atoms).

Quantum phases of collective SU(3) spin systems with bipartite symmetry

We study a bipartite collective spin-$1$ model with exchange interaction between the spins. The bipartite nature of the model manifests itself by the spins being divided into two equal-sized subsystems; within each subsystem the spin-spin interactions are of the same strength, across the subsystems they are also equal, but the two coupling values within and across the subsystem are different. Such a set-up is inspired by recent experiments with ultracold atoms. Using the $\mathrm{SU}(3)$-symmetry of the exchange interaction and the permutation symmetry within the subsystems, we can employ representation theoretic methods to diagonalize the Hamiltonian of the system in the entire parameter space of the two coupling-strengths. These techniques then allow us to explicitly construct and explore the ground-state phase diagram. The phase diagram turns out to be rich containing both gapped and gapless phases. An interesting observation is that one of the five phases features a strong bipartite symmetry breaking, meaning that the two subsystems in the ground states are in different $\mathrm{SU}(3)$ representations.

GPU-Accelerated Neural Network Potential Energy Surfaces for Diffusion Monte Carlo.

Diffusion Monte Carlo (DMC) provides a powerful method for understanding the vibrational landscape of molecules that are not well-described by conventional methods. The most computationally demanding step of these calculations is the evaluation of the potential energy. In this work, a general approach is developed in which a neural network potential energy surface is trained by using data generated from a small-scale DMC calculation. Once trained, the neural network can be evaluated by using highly parallelizable calls to a graphics processing unit (GPU). The power of this approach is demonstrated for DMC simulations on H2O, CH5+, and (H2O)2. The need to include permutation symmetry in the neural network potentials is explored and incorporated into the molecular descriptors of CH5+ and (H2O)2. It is shown that the zero-point energies and wave functions obtained by using the neural network potentials are nearly identical to the results obtained when using the potential energy surfaces that were used to train the neural networks at a substantial savings in the computational requirements of the simulations.

Seemingly asymmetric atom-localized electronic densities following laser-dissociation of homonuclear diatomics.

Recent experiments on laser-dissociation of aligned homonuclear diatomic molecules show an asymmetric forward-backward (spatial) electron-localization along the laser polarization axis. Most theoretical models attribute this asymmetry to interference effects between gerade and ungerade vibronic states. Presumably due to alignment, these models neglect molecular rotations and hence infer an asymmetric (post-dissociation) charge distribution over the two identical nuclei. In this paper, we question the equivalence that is made between spatial electron-localization, observed in experiments, and atomic electron-localization, alluded by these theoretical models. We show that (seeming) agreement between these models and experiments is due to an unfortunate omission of nuclear permutation symmetry, i.e., quantum statistics. Enforcement of the latter requires mandatory inclusion of the molecular rotational degree of freedom, even for perfectly aligned molecules. Unlike previous interpretations, we ascribe spatial electron-localization to the laser creation of a rovibronic wavepacket that involves field-free molecular eigenstates with opposite space-inversion symmetry i.e., even and odd parity. Space-inversion symmetry breaking would then lead to an asymmetric distribution of the (space-fixed) electronic density over the forward and backward hemisphere. However, owing to the simultaneous coexistence of two indistinguishable molecular orientational isomers, our analytical and computational results show that the post-dissociation electronic density along a specified space-fixed axis is equally shared between the two identical nuclei-a result that is in perfect accordance with the principle of the indistinguishability of identical particles.

Connecting minimal chimeras and fully asymmetric chaotic attractors through equivariant pitchfork bifurcations.

Highly symmetric networks can exhibit partly symmetry-broken states, including clusters and chimera states, i.e., states of coexisting synchronized and unsynchronized elements. We address the S_{4} permutation symmetry of four globally coupled Stuart-Landau oscillators and uncover an interconnected web of solutions with different symmetries. Among these are chaotic 2-1-1 minimal chimeras that arise from 2-1-1 periodic solutions in a period-doubling cascade, as well as fully asymmetric chaotic states arising similarly from periodic 1-1-1-1 solutions. A backbone of equivariant pitchfork bifurcations mediate between the two cascades, culminating in equivariant pitchforks of chaotic attractors.

Lorentz invariance of basis tensor gauge theory

Basis tensor gauge theory (BTGT) is a vierbein analog reformulation of ordinary gauge theories in which the vierbein field describes the Wilson line. After a brief review of the BTGT, we clarify the Lorentz group representation properties associated with the variables used for its quantization. In particular, we show that starting from an SO(1,3) representation satisfying the Lorentz-invariant U(1,3) matrix constraints, BTGT introduces a Lorentz frame choice to pick the Abelian group manifold generated by the Cartan subalgebra of U(1,3) for the convenience of quantization even though the theory is frame independent. This freedom to choose a frame can be viewed as an additional symmetry of BTGT that was not emphasized before. We then show how an [Formula: see text] permutation symmetry and a parity symmetry of frame fields natural in BTGT can be used to construct renormalizable gauge theories that introduce frame-dependent fields but remain frame independent perturbatively without any explicit reference to the usual gauge field.

Semidefinite Programming Relaxations of the Traveling Salesman Problem and Their Integrality Gaps

The traveling salesman problem (TSP) is a fundamental problem in combinatorial optimization. Several semidefinite programming relaxations have been proposed recently that exploit a variety of mathematical structures including, for example, algebraic connectivity, permutation matrices, and association schemes. The main results of this paper are twofold. First, de Klerk and Sotirov [de Klerk E, Sotirov R (2012) Improved semidefinite programming bounds for quadratic assignment problems with suitable symmetry. Math. Programming 133(1):75–91.] present a semidefinite program (SDP) based on permutation matrices and symmetry reduction; they show that it is incomparable to the subtour elimination linear program but generally dominates it on small instances. We provide a family of simplicial TSP instances that shows that the integrality gap of this SDP is unbounded. Second, we show that these simplicial TSP instances imply the unbounded integrality gap of every SDP relaxation of the TSP mentioned in the survey on SDP relaxations of the TSP in section 2 of Sotirov [Sotirov R (2012) SDP relaxations for some combinatorial optimization problems. Anjos MF, Lasserre JB, eds., Handbook on Semidefinite, Conic and Polynomial Optimization (Springer, New York), 795–819.]. In contrast, the subtour linear program performs perfectly on simplicial instances. The simplicial instances thus form a natural litmus test for future SDP relaxations of the TSP.

Single-mode multiphoton polarization states under random Pauli noises

We investigate behaviors of single-mode multiphoton polarization states under lossless random Pauli noises, where ``single-mode'' means that all the photons are in the same spatiotemporal mode and cannot be distinguished by any degree of freedom. We characterize the different decoherence effects of multiphoton and single-photon polarization states via the normalized linear entropy. Our results show that, contrary to single-photon states, multiphoton states cannot always evolve to the maximally mixed states. In particular, decoherence-free states and subspaces are found for the random Pauli noises. We reveal that the distinct behaviors result from the bosonic bunching effect and permutation symmetry, and hence our study may be generalized to other photon degrees of freedom and even some other bosonic systems.

Lorentz- and permutation-invariants of particles

Two theorems of Weyl tell us that the algebra of Lorentz- (and parity-) invariant polynomials in the momenta of n particles are generated by the dot products and that the redundancies which arise when n exceeds the spacetime dimension d are generated by the (d + 1)-minors of the n × n matrix of dot products. We extend the first theorem to include the action of an arbitrary permutation group P ⊂ S n on the particles, to take account of the quantum-field-theoretic fact that particles can be indistinguishable. Doing so provides a convenient set of variables for describing scattering processes involving identical particles, such as pp → jjj, for which we provide an explicit minimal set of Lorentz- and permutation-invariant generators. Additionally, we use the Cohen–Macaulay structure of the Lorentz-invariant algebra to provide a more direct characterisation in terms of a Hironaka decomposition. Among the benefits of this approach is that it can be generalized straightforwardly to when parity is not a symmetry and to cases where a permutation group acts on the particles. In the first non-trivial case, n = d + 1, we give a homogeneous system of parameters that is valid for the action of an arbitrary permutation symmetry and make a conjecture for the full Hironaka decomposition in the case without permutation symmetry. An appendix gives formulæ for the computation of the relevant Hilbert series for d ⩽ 4.