[Paper Review] Dual-unitary shadow tomography

Ahmed A. Akhtar, Namit Anand+2 more|arXiv (Cornell University)|Apr 1, 2024

Advanced X-ray and CT Imaging4 citations

TL;DR

This paper introduces dual-unitary classical shadow tomography, demonstrating that random dual-unitary circuits exhibit chiral, massless quasi-particle excitations—left- and right-movers—that propagate ballistically at speed $v=1$, leading to fast thermalization and superior performance in predicting extensive quantum operators. The mean-field description of Pauli weight dynamics $\rho(x,t)$ captures ballistic spreading and efficient sample complexity scaling, outperforming shallow Clifford circuits.

ABSTRACT

We introduce ``dual-unitary shadow tomography'' (DUST), a classical shadow tomography protocol based on dual-unitary brick-wall circuits. To quantify the performance of DUST, we study operator spreading and Pauli weight dynamics in one-dimensional qubit systems, evolved by random two-local dual-unitary gates arranged in a brick-wall structure, ending with a measurement layer. We do this by deriving general constraints on the Pauli weight transfer matrix and specializing to the case of dual-unitarity. Remarkably, we find that operator spreading in these circuits have a rich structure resembling that of relativistic quantum field theories, with massless chiral excitations that can decay or fuse into each other, which we call left- or right-movers. We develop a mean-field description of the Pauli weight in terms of $ρ(x,t)$, which represents the probability of having nontrivial support at site $x$ and depth $t$ starting from a fixed weight distribution. We develop an equation of state for $ρ(x,t)$ and simulate it numerically using Monte Carlo simulations. For the task of predicting operators with (nearly) full support, we show that DUST outperforms brick-wall Clifford shadows of equal depth. This advantage is further pronounced for small system sizes and our results are generally robust to finite-size effects.

Motivation & Objective

• To understand how dual-unitary circuits differ from standard Haar-random or Clifford circuits in their Pauli weight dynamics during quantum state tomography.
• To investigate whether the chiral, relativistic-like excitations in dual-unitary circuits enhance the efficiency of classical shadow tomography for predicting large operators.
• To develop a mean-field description of Pauli weight evolution $\rho(x,t)$ and evaluate its accuracy in capturing thermalization and sample complexity scaling.
• To compare the shadow norm scaling exponent $\beta$ of dual-unitary circuits with Clifford circuits, assessing their relative efficiency in predicting extensive observables.
• To explore the implications of chiral quasi-particles and light-cone dynamics for quantum information scrambling and potential connections to AdS/CFT correspondence.

Proposed method

• Derive general constraints on the Pauli weight transfer matrix and specialize to dual-unitary circuits, leveraging their symmetry to simplify dynamics.
• Introduce a mean-field equation of state for $\rho(x,t)$, representing the probability of nontrivial Pauli support at site $x$ and time $t$.
• Use Monte Carlo simulations to numerically solve the mean-field equation and track the evolution of Pauli weight under dual-unitary evolution.
• Analyze the shadow norm scaling exponent $\beta$ of $n$-local Pauli strings to quantify sample complexity in classical shadow tomography.
• Compare the scaling behavior of dual-unitary circuits with that of shallow brick-wall Clifford circuits, focusing on convergence to the Clifford limit ($\beta=2$).
• Investigate the role of the dual-unitary parameter $\alpha$ in determining interaction strength between chiral excitations and their propagation velocity.

Experimental results

Research questions

• RQ1 How do chiral quasi-particles—left- and right-movers—emerge in the Pauli weight dynamics of dual-unitary circuits?
• RQ2 What is the role of the dual-unitary parameter $\alpha$ in governing the fusion and decay of chiral excitations?
• RQ3 To what extent does the mean-field description of $\rho(x,t)$ accurately capture the ballistic spreading and thermalization in dual-unitary circuits?
• RQ4 How does the shadow norm scaling exponent $\beta$ of dual-unitary circuits compare to that of Clifford circuits in predicting extensive operators?
• RQ5 Can the chiral, relativistic-like structure of Pauli weight dynamics in dual-unitary circuits be linked to deeper geometric or holographic principles, such as those in AdS/CFT?

Key findings

• Dual-unitary circuits exhibit chiral, massless excitations—left- and right-movers—that propagate ballistically at the speed of light ($v=1$), forming the light-cone structure of operator spreading.
• The Pauli weight dynamics $\rho(x,t)$ are well described by a mean-field equation of state, with numerical simulations showing rapid thermalization within the light-cone.
• At the swap point ($\alpha=0$), chiral movers do not interact, confirming the existence of stable, freely propagating excitations.
• Dual-unitary circuits achieve faster convergence to the Clifford limit ($\beta=2$) in shadow norm scaling, indicating lower sample complexity for predicting extensive operators.
• The shadow norm scaling exponent $\beta$ is lower for dual-unitary circuits with larger $\alpha$, suggesting improved efficiency in classical shadow tomography.
• The emergence of chiral dynamics in Pauli weight suggests potential connections to relativistic quantum field theories and possibly the AdS/CFT correspondence, warranting further investigation.