Especially in the context of many-body systems and quantum materials, quantum catalysts are an intriguing and significant extension of the classical chemical idea into the field of quantum physics.
What are Quantum Catalysts?
Chemistry uses catalysts to speed chemical reactions without being consumed, allowing recurrent use. Quantum states can act as “entanglement catalysts” to support transformations between entangled quantum states that are impossible with local operations and conventional communication. This basic concept has been successfully used to few-body quantum mechanics. In addition to being applied to quantum resource theories and the creation of magic states for fault-tolerant quantum computers, this viewpoint has offered fresh insights into entanglement theory.
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Quantum catalysts provide a new and important role in the context of many-body quantum physics: they help a system transition between different phases of matter. Transformations in big systems are essentially impossible due to the time required for such phase transitions, which typically rises with the number of particles in the system. Nonetheless, a system’s phase of matter can be altered in a time independent of its particle number when a suitable entangled quantum state serves as a catalyst, greatly speeding up the process for huge systems. The fundamental idea is still the same: the catalyst can be used repeatedly and is not consumed throughout the process.
A state must be symmetric (maintaining the symmetry of the entire system) and have the same dimensionality as the system it is catalysing, which means it is defined on a fixed number of auxiliary degrees of freedom per site, in order to qualify as a many-body catalyst. In order to catalyse a non-trivial symmetry-protected topological (SPT) phase, potential catalysts are severely limited by the requirement that they be created from a product state using a symmetric finite-depth quantum circuit (FDQC). Essentially, the catalyst must be as difficult to build as the SPT phase itself, but because the catalyst is reusable, this initial “hard work” only needs to be done once.
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Finite-Depth Quantum Circuits (FDQCs)
A key idea in many-body quantum physics, finite-depth quantum circuits (FDQCs) are inextricably tied to the definition of quantum catalysts. An FDQC is a unitary operator that a local Hamiltonian can produce in a finite amount of time. “Finite” in this context refers to the fact that, even in the thermodynamic limit, the time, depth (the number of gate layers), and range of the local operations the distance a gate acts are all independent of the system size. As a result, FDQCs define what constitutes a “easy” transformation and are the operations that are most naturally implemented in a quantum computer.
FDQCs are fundamental because:
- Classification of Topological Phases: Under FDQCs, the equivalence classes of many-body ground states precisely correlate to the topological phase classification of matter. An FDQC can be used to prepare a trivial state from an unentangled state.
- Symmetric FDQCs: A state that can be mapped to a product state via an FDQC but not via a symmetric FDQC (where each individual gate commutes with the symmetry) is said to belong to a non-trivial Symmetry-Protected Topological (SPT) phase of matter when symmetries are imposed on the Hamiltonian or circuit.
Quantum Catalysts and SPT Phases
Recent work has mostly focused on building catalysts that use symmetric FDQCs to facilitate transitions between different SPT phases of matter, which is not achievable without a catalyst. These catalysts have a variety of physical characteristics and are many-body states:
- Symmetry-breaking states: These include states with cat-like entanglement, such as GHZ-like states.
- Gapless states: Critically correlated ground states of conformal field theories.
- Topologically ordered states: Especially in higher dimensions (d>1), with symmetry fractionalisation.
- Disordered states exhibiting strong-to-weak spontaneous symmetry breaking (SW-SSB) properties are known as spin-glass states.
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Any state invariant under both G and $\mathcal{U}$ can catalyse the transformation via a symmetric FDQC if a transformation between SPT phases with a symmetry group G can be implemented by a symmetric quantum cellular automaton (QCA) $\mathcal{U}$ (also known as the SPT entangler). This unifies all of these different catalyst types under a single framework.
The relationship between these catalysts and quantum oddities is significant. Pure-state catalysts for non-trivial SPT phases in 1D systems, for instance, need to have quasi-long-range correlations (such as power-law fading two-point correlations) or long-range entanglement (LRE). The Lieb-Schultz-Mattis (LSM) anomaly, which asserts that no short-range entangled state can meet specific symmetry and translational invariance requirements, is the cause of this. This LRE restriction is loosened for mixed-state catalysts, which show strong-to-weak spontaneous symmetry breaking (SW-SSB) and do not necessarily require long-range correlations but rather long-range fidelity correlators. This suggests that, instead of a ferromagnet, catalysts can spontaneously violate symmetry in a manner similar to a spin-glass.
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Applications to State Preparation
Gaining knowledge of quantum catalysts makes it easier to accomplish desired quantum transformations, particularly when preparing unusual phases of matter for use in quantum computers.
- Efficiency: A catalyst can be utilised to efficiently prepare a variety of SPT states if it can be created effectively. Although preparing pure-state catalysts can be just as challenging as preparing the SPT stages, the task is only done once. However, symmetric local channels can typically be used to create mixed-state catalysts more effectively, even in finite time.
- Beyond Ancillary Systems: The framework for the direct synthesis of SPT phases by deftly altering the target system’s Hamiltonian evolution using the catalyst’s properties, even though catalysts can be generated in an ancillary system and subsequently utilised to change a target state.
- Long-Range Interactions: Catalysts make it possible to use long-range interactions to prepare SPT phases through novel procedures. For example, power-law decaying symmetric Hamiltonians (for certain decay exponents) can be used to generate the GHZ state, a type of catalyst, in constant time. In certain cases, SPT phases can be prepared in constant time.
- Local Symmetric Channels/Measurements: Only local symmetric measurements are required for SPT phase formation with mixed-state catalysts. To catalyse a 1D cluster state, for instance, particular measurements on a product state can produce a mixed-state catalyst (such one with SW-SSB characteristics).
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The application of these concepts to fermionic systems (such as the Kitaev chain, where catalysts are especially attractive because they cannot explicitly break parity symmetry), their role in long-range entangled phases (such as fracton phases), and the study of transformations between mixed states in open quantum systems are just a few of the many new directions that this seminal work opens up for future investigation.
