Fundamental charges for dual-unitary circuits
1Department of Physics and Astronomy, University College London, United Kingdom
2Department of Computer Science, University College London, United Kingdom
3London Centre for Nanotechnology, University College London, United Kingdom
| Published: | 2025-01-30, volume 9, page 1615 |
| Editor: | Abhinav Deshpande |
| Eprint: | arXiv:2312.14148v3 |
| Doi: | https://doi.org/10.22331/q-2025-01-30-1615 |
| Citation: | Quantum 9, 1615 (2025). |
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Abstract
Dual-unitary quantum circuits have recently attracted attention as an analytically tractable model of many-body quantum dynamics. Consisting of a 1+1D lattice of 2-qudit gates arranged in a 'brickwork' pattern, these models are defined by the constraint that each gate must remain unitary under swapping the roles of space and time. This dual-unitarity restricts the dynamics of local operators in these circuits: the support of any such operator must grow at the effective speed of light of the system, along one or both of the edges of a causal light cone set by the geometry of the circuit. Using this property, it is shown here that for 1+1D dual-unitary circuits the set of width-$w$ conserved densities (constructed from operators supported over $w$ consecutive sites) is in one-to-one correspondence with the set of width-$w$ solitons – operators which, up to a multiplicative phase, are simply spatially translated at the effective speed of light by the dual-unitary dynamics. A number of ways to construct these many-body solitons (explicitly in the case where the local Hilbert space dimension $d=2$) are then demonstrated: firstly, via a simple construction involving products of smaller, constituent solitons; and secondly, via a construction which cannot be understood as simply in terms of products of smaller solitons, but which does have a neat interpretation in terms of products of fermions under a Jordan-Wigner transformation. This provides partial progress towards a characterisation of the microscopic structure of complex many-body solitons (in dual-unitary circuits on qubits), whilst also establishing a link between fermionic models and dual-unitary circuits, advancing our understanding of what kinds of physics can be explored in this framework.
Featured image: A cartoon of a soliton in a dual-unitary circuit. These non-interacting quasiparticles ballistically propagate along the edge of a causal light cone defined by the circuit geometry. They can be thought of as fundamental charges for these circuits – any conserved quantity of the circuit (up to some weak locality-based assumptions) must be generated by them.
Popular summary
We can, however, make progress without quantum computers; some many-body quantum systems have sufficient structure that they can be exactly solved essentially by hand. Typically, this relies on the existence of an extensive number of simple conserved quantities, a property known as integrability. This is a fine-tuned property, however; generic systems harbour only a few simple conserved quantities (or none at all) and behave chaotically. Examples of solvable chaotic quantum systems are few and far between.
Recently, a number of 'toy’ models have been identified in the framework of brickwork quantum circuits (quantum circuits being the standard mathematical description of computations in digital quantum computers) that are both highly chaotic yet, to varying degrees, analytically solvable. Dual-unitary circuits are one such example; in these circuits the inclusion of a special space-time symmetry – the circuit must remain a valid quantum circuit when the roles of space and time are swapped – allows for remarkable simplifications of calculations of almost all dynamical quantities one could consider. Calculating these quantities, however, reveals the model to be generically chaotic – maximally so, in fact.
In this paper, we ask what kind of conserved quantities can be added in to dual-unitary circuits. It is well established that at certain fine-tuned points dual-unitary circuits can host emergent, non-interacting quasiparticles called solitons. These solitons generate an extensive number of conserved quantities and a strong form of integrability, breaking the sense in which the circuit is chaotic. We prove a converse result, showing that every conserved quantity (up to some reasonable technical assumptions) in a dual-unitary circuit must always be constructed from solitons. This presents a sharp dichotomy realised by dual-unitary circuits: either they have solitons, which generate strong integrability and a breakdown of chaos, or they harbour no simple conserved quantities at all and exhibit maximal quantum chaos.
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