#quantumcellularautomaton

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Quantum Catalysts for SPT Phase Transitions with Equal FDQCs

Quantum catalysts are exciting and significant extensions of classical chemistry into quantum physics, especially in many-body systems and quantum materials.

What's a quantum catalyst?

Chemical reactions are accelerated by catalysts, which are reusable. Quantum states can operate as “entanglement catalysts” to support transformations between entangled quantum states that local operations and conventional communication cannot. This basic idea works for few-body  quantum mechanics. This perspective has illuminated entanglement theory as well as quantum resource theories and fault-tolerant quantum computer magic states.

Quantum catalysts provide a new and crucial role in many-body quantum physics: they assist things change phases. The time necessary for phase transitions, which increases with particle count, makes transformations in large systems unachievable. A appropriate entangled quantum state can be used as a catalyst to change a system's phase of matter in a time independent of its particle number, speeding up the process for large systems. The basic premise remains: the catalyst is reusable and not consumed.

To be a many-body catalyst, a state must be symmetric and have the same dimensionality as the system it catalyses, meaning it has a fixed number of auxiliary degrees of freedom per site. Potential catalysts for a non-trivial symmetry-protected topological (SPT) phase must be developed from a product state utilising a symmetric finite-depth quantum circuit (FDQC). The catalyst must be as difficult to build as the SPT phase, but since it's reusable, they only need to do it once.

Finite-Depth Quantum Circuits

In many-body quantum physics, finite-depth quantum circuits (FDQCs) are essential to the definition of quantum catalysts. FDQCs are unitary operators that a local Hamiltonian can produce in a finite time. “Finite” means that even in the thermodynamic limit, the time, depth (gate layers), and range of local operations (gate distance) are independent of system size. Thus, FDQCs define a “easy” transformation and are the quantum computer operations most naturally implemented.

Fundamental reasons for FDQCs:

In FDQCs, the equivalence classes of many-body ground states match the topological phase classification of matter. A trivial state can be created from an unentangled state using an FDQC.

When symmetries are imposed on the Hamiltonian or circuit, a state that can be mapped to a product state via an FDQC but not via a symmetric FDQC (where each gate commutes with the symmetry) is said to belong to a non-trivial Symmetry-Protected Topological (SPT) phase of matter.

Quantum Catalysts/SPT Phases

Recent research has focused on creating catalysts that leverage symmetric FDQCs to transition SPT phases, which is impossible without a catalyst. These catalysts are many-body states with diverse physical properties:

Cat-like entanglement states like GHZ-like states are symmetry-breaking states.

Conformal field theories' gapless ground states are critically connected.

With symmetry fractionalisation, topologically ordered states are especially in higher dimensions (d>1).

Spin-glass states are disordered states with SW-SSB characteristics.

If a symmetric quantum cellular automaton (QCA) $\mathcal{U}$ can implement a transformation between SPT phases with a symmetry group G, any state invariant under both G and $\mathcal{U}$ can catalyse the transition. This unites these catalyst kinds under one framework.

The catalyst-quantum oddity link is important. Pure-state catalysts for non-trivial SPT phases in 1D systems require quasi-long-range correlations or long-range entanglement. No short-range entangled state can meet specified symmetry and translational invariance requirements due to the Lieb-Schultz-Mattis (LSM) anomaly. Mixed-state catalysts, which exhibit strong-to-weak spontaneous symmetry breaking (SW-SSB), do not need long-range correlations but could need long-range fidelity correlators. This shows that catalysts can spontaneously violate symmetry like spin-glasses instead of ferromagnets.

Application to State Preparation

Understanding quantum catalysts simplifies quantum transformations, especially when preparing uncommon matter phases for quantum computers.

Effective catalyst creation can be used to efficiently produce various SPT states. Although pure-state catalysts are equally difficult to prepare as SPT stages, they are only done once. Symmetric local channels can build mixed-state catalysts more efficiently in finite time.

Beyond Ancillary Systems: The framework for directly synthesising SPT phases by delicately modifying the target system's Hamiltonian evolution utilising the catalyst's properties, even though catalysts can be created in an ancillary system and used to change a target state.

A unique method for preparing SPT phases using long-range interactions is possible with catalysts. For instance, power-law decaying symmetric Hamiltonians (for particular decay exponents) can yield the catalytic GHZ state in constant time. SPT phases can sometimes be prepared continuously.

SPT phase generation with mixed-state catalysts requires just local symmetric channels/measurements. Specific measurements on a product state can develop a mixed-state catalyst (such as SW-SSB) to catalyse a 1D cluster state.

This seminal work opens up many new avenues for future research, including the use of these concepts in fermionic systems (such as the Kitaev chain, where catalysts are especially attractive because they cannot explicitly break parity symmetry), long-range entangled phases (such as fracton phases), and the study of transformations between mixed states in open quantum systems.