AKLT State Research Articles

High-fidelity realization of the AKLT state on a NISQ-era quantum processor

The AKLT state is the ground state of an isotropic quantum Heisenberg spin-1 model. It exhibits an excitation gap and an exponentially decaying correlation function, with fractionalized excitations at its boundaries. So far, the one-dimensional AKLT model has only been experimentally realized with trapped-ions as well as photonic systems. In this work, we successfully prepared the AKLT state on a noisy intermediate-scale quantum (NISQ) era quantum device. In particular, we developed a non-deterministic algorithm on the IBM quantum processor, where the non-unitary operator necessary for the AKLT state preparation is embedded in a unitary operator with an additional ancilla qubit for each pair of auxiliary spin-1/2’s. Such a unitary operator is effectively represented by a parametrized circuit composed of single-qubit and nearest-neighbor CX gates. Compared with the conventional operator decomposition method from Qiskit, our approach results in a much shallower circuit depth with only nearest-neighbor gates, while maintaining a fidelity in excess of 99.99% with the original operator. By simultaneously post-selecting each ancilla qubit such that it belongs to the subspace of spin-up |↑>, an AKLT state can be systematically obtained by evolving from an initial trivial product state of singlets plus ancilla qubits in spin-up on a quantum computer, and it is subsequently recorded by performing measurements on all the other physical qubits. We show how the accuracy of our implementation can be further improved on the IBM quantum processor with readout error mitigation.

AKLT-States as ZX-Diagrams: Diagrammatic Reasoning for Quantum States

From Feynman diagrams to tensor networks, diagrammatic representations of computations in quantum mechanics have catalysed progress in physics. These diagrams represent the underlying mathematical operations and aid physical interpretation, but cannot generally be computed with directly. In this paper we introduce the ZXH-calculus, a graphical language based on the ZX-calculus, that we use to represent and reason about many-body states entirely graphically. As a demonstration, we express the 1D AKLT state, a symmetry protected topological state, in the ZXH-calculus by developing a representation of spins higher than 1/2 within the calculus. By exploiting the simplifying power of the ZXH-calculus rules we show how this representation straightforwardly recovers the AKLT matrix-product state representation, the existence of topologically protected edge states, and the non-vanishing of a string order parameter. Extending beyond these known properties, our diagrammatic approach also allows us to analytically derive that the Berry phase of any finite-length 1D AKLT chain is $\pi$. In addition, we provide an alternative proof that the 2D AKLT state on a hexagonal lattice can be reduced to a graph state, demonstrating that it is a universal quantum computing resource. Lastly, we build 2D higher-order topological phases diagrammatically, which we use to illustrate a symmetry-breaking phase transition. Our results show that the ZXH-calculus is a powerful language for representing and computing with physical states entirely graphically, paving the way to develop more efficient many-body algorithms and giving a novel diagrammatic perspective on quantum phase transitions.

Driven-dissipative control of cold atoms in tilted optical lattices

We present a sequence of driven-dissipative protocols for controlling cold atoms in tilted optical lattices. These experimentally accessible examples are templates that demonstrate how dissipation can be used to manipulate quantum many-body systems. We consider bosonic atoms trapped in a tilted optical lattice, immersed in a superfluid bath, and excited by coherent Raman lasers. With these ingredients, we are able to controllably transport atoms in the lattice and produce self-healing quantum states: a Mott insulator and the topologically ordered spin-1 AKLT state.

Ground states of one-dimensional symmetry-protected topological phases and their utility as resource states for quantum computation

The program of classifying symmetry protected topological (SPT) phases in 1D has been recently completed and has opened the doors to study closely the properties of systems belonging to these phases. It was recently found that being able to constrain the form of ground states of SPT order based on symmetry properties also allows to explore novel resource states for processing of quantum information. In this paper, we generalize the consideration of Else et al. [Phys. Rev. Lett. {\bf 108}, 240505 (2012)] where it was shown that the ground-state form of spin-1 chains protected by $\mathbb{Z}_2 \times \mathbb{Z}_2$ symmetry supports perfect operation of the identity gate, important also for long-distance transmission of quantum information. We develop a formalism to constrain the ground-state form of SPT phases protected by any arbitrary finite symmetry group and use it to examine examples of ground states of SPT phases protected by various finite groups for similar gate protections. We construct a particular Hamiltonian invariant under $A_4$ symmetry transformation which is one of the groups that allows protected identity operation and examine its ground states. We find that there is an extended region where the ground state is the AKLT state, which not only supports the identity gate but also arbitrary single-qubit gates.

Controlled probabilistic quantum key distribution using a ground state

In this paper, we propose two controlled probabilistic quantum key distribution protocols with AKLT states. An AKLT state is a gapped ground state with minimum energy, and owing to the properties of this special state, our proposed protocols incorporate not only the measurement uncertainty in quantum phenomena (entanglement swapping with 1/4 probability) but also additional randomness (Bell measurement on two physical particles with 1/3 probability), in comparison with other PQKD schemes. Therefore, our protocols are more suitable for use by two mutually untrusted communicants, with no authenticated intermediate channel, allowing them to obtain an unpredictable, and therefore secure, key.

Measurement-based quantum computation with an optical two-dimensional Affleck–Kennedy–Lieb–Tasaki state

Measurement-based quantum computation can achieve universal quantum computation via simply performing single-qubit measurements alone on an entangled resource state. Instead of the canonical resource state–cluster state, we pay great attention to another resource state, the Affleck–Kennedy–Lieb–Tasaki (AKLT) state, which is a nondegenerate ground state of a nearest-neighbor two-body interaction Hamiltonian. We propose an optical scheme to generate a two-dimensional AKLT state defined on the honeycomb lattice. In our proposal, three photons from different singlet states are projected onto a spin-32 particle comprising the AKLT state with maximum success probability 38. We also show the controlled-NOT gate can be implemented on this photonic AKLT state via projective measurement using linear optics and photodetection.

Wave function and strange correlator of short-range entangled states.

We demonstrate the following conclusion: If |Ψ⟩ is a one-dimensional (1D) or two-dimensional (2D) nontrivial short-range entangled state and |Ω⟩ is a trivial disordered state defined on the same Hilbert space, then the following quantity (so-called "strange correlator") C(r,r('))=⟨Ω|ϕ(r)ϕ(r('))|Ψ⟩/⟨Ω|Ψ⟩ either saturates to a constant or decays as a power law in the limit |r-r(')|→+∞, even though both |Ω⟩ and |Ψ⟩ are quantum disordered states with short-range correlation; ϕ(r) is some local operator in the Hilbert space. This result is obtained based on both field theory analysis and an explicit computation of C(r,r(')) for four different examples: 1D Haldane phase of spin-1 chain, 2D quantum spin Hall insulator with a strong Rashba spin-orbit coupling, 2D spin-2 Affleck-Kennedy-Lieb-Tasaki state on the square lattice, and the 2D bosonic symmetry-protected topological phase with Z(2) symmetry. This result can be used as a diagnosis for short-range entangled states in 1D and 2D.

Spin quantum Hall effects in featureless nonfractionalized spin-1 magnets

AKLT state (or Haldane phase) in a spin-1 chain represents a large class of gapped topological paramagnets, which hosts symmetry-protected gapless excitations on the boundary. In this work we show how to realize this type of featureless spin-1 states on a generic two-dimensional lattice. These states have a gapped spectrum in the bulk but supports gapless edge states protected by spin rotational symmetry along a certain direction, and are featured by spin quantum Hall effect. Using fermion representation of integer-spins we show a concrete example of such spin-1 topological paramagnets on kagome lattice, and suggest a microscopic spin-1 Hamiltonian which may realize it.

Full counting statistics and the Edgeworth series for matrix product states

We consider the full counting statistics of spin in matrix product states. In particular, we study the approach to a Gaussian distribution for magnetization. We derive the asymptotic corrections to the central limit theorem for magnetization distribution for finite but large blocks in analogy to the Edgeworth series. We also show how central limit theorem like behavior is modified for certain states with topological characteristics such as the AKLT state.

Quantum computational capability of a 2D valence bond solid phase

Quantum phases of naturally-occurring systems exhibit distinctive collective phenomena as manifestation of their many-body correlations, in contrast to our persistent technological challenge to engineer at will such strong correlations artificially. Here we show theoretically that quantum correlations exhibited in the 2D valence bond solid phase of a quantum antiferromagnet, modeled by Affleck, Kennedy, Lieb, and Tasaki (AKLT) as a precursor of spin liquids and topological orders, are sufficiently complex yet structured enough to simulate universal quantum computation when every single spin can be measured individually. This unveils that an intrinsic complexity of naturally-occurring 2D quantum systems—which has been a long-standing challenge for traditional computers—could be tamed as a computationally valuable resource, even if we are limited not to create newly entanglement during computation. Our constructive protocol leverages a novel way to herald the correlations suitable for deterministic quantum computation through a random sampling, and may be extensible to other ground states of various 2D valence bond phases beyond the AKLT state.

Optical spin-1 chain and its use as a quantum-computational wire

Measurement-based quantum computing, a powerful alternative to the standard circuit model, proceeds using only local adaptive measurements on a highly entangled resource state of many spins on a graph or lattice. Along with the canonical cluster state, the valence-bond solid ground state on a chain of spin-1 particles, studied by Affleck, Kennedy, Lieb, and Tasaki (AKLT), is such a resource state. We propose a simulation of this AKLT state using linear optics, wherein we can make use of the high-fidelity projective measurements that are commonplace in quantum-optical experiments, and describe how quantum logic gates can be performed on this chain. In our proposed implementation, the spin-1 particles comprising the AKLT state are encoded on polarization biphotons: three-level systems consisting of pairs of polarized photons in the same spatio-temporal mode. A logical qubit encoded on the photonic AKLT state can be initialized, read out, and have an arbitrary single-qubit unitary applied to it by performing projective measurements on the constituent biphotons. For MBQC, biphoton measurements are required which cannot be deterministically performed using only linear optics and photodetection.

Multiple-spin exchange models on the checkerboard lattice with an exact two-fold degenerate Shastry-Sutherland ground state

Two quantum spin models with bilinear-biquadratic exchange interactions are constructed on the checkerboard lattice. It is proved that, under certain sufficient conditions on the exchange parameters, their ground states consist of two degenerate Shastry-Sutherland singlet configurations. The constructions are studied for arbitrary spin-S. The sufficient conditions for the existence of ferromagnetic ground state are also found exactly. The approximate quantum phase diagrams are presented using the exact results, together with a variational estimate for the N\'eel antiferromagnetic phase. A two-leg spin-1/2 ladder model, based on one of the above constructions, is considered which admits exact solution for a large number of eigenstates. The ladder model is shown to have exact level-crossing between the rung-singlet state and the AKLT state in the singlet ground state. Also introduced is the notion of perpendicularity for quantum spin vectors, which appears in the discussion on one of the two checkerboard models, and is discussed in the Appendix.