[Paper Review] Emergent quantum state designs and biunitarity in dual-unitary circuit dynamics

Pieter W. Claeys, Austen Lamacraft|arXiv (Cornell University)|Feb 24, 2022

Quantum many-body systems64 references69 citations

TL;DR

This paper presents a new analytical framework for emergent quantum state designs in dual-unitary quantum circuits, demonstrating that specific measurement bases—derived from complex Hadamard matrices and unitary error bases—generate exact quantum state k-designs after finite time evolution. The construction relies on biunitary symmetries and exact solvability, showing that projected ensembles from bath measurements reproduce Haar-random moments universally, even for non-uniform measurement bases.

ABSTRACT

Recent works have investigated the emergence of a new kind of random matrix behaviour in unitary dynamics following a quantum quench. Starting from a time-evolved state, an ensemble of pure states supported on a small subsystem can be generated by performing projective measurements on the remainder of the system, leading to a projected ensemble. In chaotic quantum systems it was conjectured that such projected ensembles become indistinguishable from the uniform Haar-random ensemble and lead to a quantum state design. Exact results were recently presented by Ho and Choi [Phys. Rev. Lett. 128, 060601 (2022)] for the kicked Ising model at the self-dual point. We provide an alternative construction that can be extended to general chaotic dual-unitary circuits with solvable initial states and measurements, highlighting the role of the underlying dual-unitarity and further showing how dual-unitary circuit models exhibit both exact solvability and random matrix behaviour. Building on results from biunitary connections, we show how complex Hadamard matrices and unitary error bases both lead to solvable measurement schemes.

Motivation & Objective

• To establish a general mechanism for the emergence of exact quantum state k-designs in chaotic quantum systems beyond thermalization.
• To extend the scope of exact results on state designs beyond the kicked Ising model to general dual-unitary circuits with solvable initial states and measurements.
• To clarify the role of dual-unitarity and biunitarity in enabling both exact solvability and random matrix behavior in quantum dynamics.
• To construct solvable measurement schemes using complex Hadamard matrices and unitary error bases that yield exact state designs.
• To demonstrate that non-uniform measurement bases can still lead to Haar-like state ensembles under specific symmetric gate structures.

Proposed method

• Utilizes the transfer matrix formalism to compute moments of projected ensembles in dual-unitary circuits.
• Applies biunitary connections to identify gate structures (e.g., complex Hadamard matrices, unitary error bases) that ensure solvability and symmetry.
• Employs Weinbergter functions and permutation operators to evaluate overlaps in the transfer matrix formalism.
• Derives exact expressions for the k-th moment of the projected ensemble, showing equivalence to the Haar measure.
• Uses the symmetry of dual-unitary gates and boundary contractions to simplify the transfer matrix calculation.
• Demonstrates that the resulting projected ensemble moments match those of the uniform Haar distribution after a time proportional to subsystem size.

Experimental results

Research questions

• RQ1 Can exact quantum state k-designs emerge in dual-unitary circuits beyond the kicked Ising model?
• RQ2 What role do complex Hadamard matrices and unitary error bases play in enabling solvable measurement schemes for state designs?
• RQ3 How does biunitarity extend the class of dual-unitary gates that support exact state design formation?
• RQ4 Can non-uniform measurement bases still lead to Haar-like state ensembles in chaotic systems?
• RQ5 What is the minimal time evolution required for projected ensembles to achieve exact k-designs in these models?

Key findings

• Projected ensembles formed by measuring the bath in a single-site basis using dual-unitary gates constructed from complex Hadamard matrices yield exact quantum state k-designs after a time proportional to the subsystem size.
• For dual-unitary gates built from unitary error bases, measurement in the Bell basis (on neighboring sites) also leads to exact state designs, with all moments matching the Haar distribution.
• The emergence of state designs is guaranteed by the underlying biunitary structure, which ensures the required symmetry and solvability in the transfer matrix formalism.
• The moments of the projected ensemble exactly reproduce those of the Haar measure, even for non-uniform measurement bases, due to the symmetric gate structure.
• The time to achieve exact k-designs scales linearly with the subsystem size, and the result holds in the thermodynamic limit with an infinite bath.
• Numerical validation confirms that the constructed dual-unitary circuits generate state designs with fidelity to the Haar ensemble that is indistinguishable from random matrix predictions.