Learning Haldane Phase on Qudit-Based Quantum Processor
Haldane Stage
Topological Haldane Phase with Qudit Quantum Processor Symmetry Protection
Scientists constructed and investigated the spin-1 Haldane phase on a qudit quantum processor using trapped-ion qutrits, a major quantum computing achievement. This breakthrough allows higher-dimensional quantum phases of matter to be natively realised, which are challenging to explore using typical qubit systems or classical methods due to their complexity and quantum nature.
Symmetry-protected topological (SPT) phases, a new condensed matter physics paradigm, use topological notions for improved metrology, durable quantum information, and innovative materials. Haldane phase, with the spin-1 Heisenberg chain, is a standard SPT phase. In this phase, integer-spin chains are classic SPT states with fascinating condensed matter and quantum information properties, unlike their half-integer spin counterparts.
Trapped-ion qudits could be used to natively study high-dimensional spin systems, according to Alpine Quantum Technologies GmbH and Universität Innsbruck researchers. Spin-1 chains in the Haldane phase are directly engineered using this technology. The researchers say their direct simulation lets them “observe not only the characteristic long-range order and short-range correlations, but also the fractionalisation of fundamental spin-1 particles into effective spin-1/2 degrees of freedom,” a system feature.
Significant study findings and successes include:
The researchers developed a scalable, predictable procedure to prepare the Affleck-Kennedy-Lieb-Tasaki (AKLT) state, a key Haldane phase state. After initialising N qutrits and an ancilla qubit into a product state, the ancilla is attached to each qutrit. This approach requires only 2N entangling gates and removes probabilistic post-selection when used with ancilla measurement feed-forward. Better than earlier qubit-based encodings, which often relied on probabilistically projecting onto a spin-1 subspace, this greatly reduces the allowed measurements for longer chains.
To verify topological features:
Long-Range String Order: Despite short-range correlations and a finite correlation length, the scientists confirmed the AKLT state's concealed antiferromagnetic order, implying a finite energy gap above its ground state. This required measuring a non-local string order parameter. This value was always non-zero, which is essential for SPT states without pairwise correlations and local order.
Spin Fractionalisation and Edge States: Open-boundary chains' symmetry protection induces fascinating quantum number fractionalisation. The researchers found that the physical spin-1 degrees of freedom fractionalise into two unpaired spin-1/2 degrees of freedom at the chain endpoints. This creates a four-fold degenerate ground-state subspace, unlike the unique ground state with closed bounds. Using edge-localized operators to drive Rabi flops showed the presence of these effective qubits. The contrast stayed nearly constant as chain length increased, confirming localisation.
The investigation revealed the Haldane SPT phase's bulk-edge link. The Haldane phase is resilient to global rotations because a global bulk operator is equal to an edge-unitary when constrained to the ground-state manifold.
Quantum Resource Efficiency: The native qudit implementation avoids probabilistic post-selection, d-dimensional spin-qubit encoding and decoding, and a lot of quantum resource overhead. This hardware-efficient technology enables many more quantum modelling applications of non-classical phases of matter.
Sequential coupling via an ancilla qudit can generate matrix product states (MPS) beyond the AKLT state. Because the trapped-ion platform may readily change the ancilla qudit's dimension (d), binding dimensions up to D=7 or more with diverse ion species are feasible. D is the possible bond dimension. Trapped-ion systems are all-to-all connected, hence the coupling order is governed by the application order of unitaries rather than the physical geometry, allowing for arbitrary MPS geometries.
The researchers also investigated the spin-1/2 cluster state, which is similar to the AKLT state. They generated this state experimentally using spin-1/2 trapped-ion qubits and found similar long-range order, short-range correlations, and edge manipulation of an effective qubit to support the bulk-edge correspondence.
This work lays the framework for future research into multidimensional SPT phases to better understand realistic condensed matter systems and materials. Quantum simulations are expected to be necessary for understanding and simulating 2D and 3D models.